IEEE Transactions on Antennas and Propagation [volume 58 number 10]

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OCTOBER 2010

VOLUME 58

NUMBER 10

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Design of Frequency Reconfigurable Antennas Using the Theory of Network Characteristic Modes ... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ..... K. A. Obeidat, B. D. Raines, R. G. Rojas, and B. T. Strojny Lower Bounds on the Q of Electrically Small Dipole Antennas . ......... ..... .... ......... A. D. Yaghjian and H. R. Stuart Analysis and Design of a Differentially-Fed Frequency Agile Microstrip Patch Antenna ..... S. V. Hum and H. Y. Xiong Coplanar Capacitively Coupled Probe Fed Microstrip Antennas for Wideband Applications .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... V. G. Kasabegoudar and K. J. Vinoy Theory of Electromagnetic Time-Reversal Mirrors ...... ......... ......... ........ .... J. de Rosny, G. Lerosey, and M. Fink Microstrip-Fed Wideband Circularly Polarized Printed Antenna ......... . ...... X. L. Bao, M. J. Ammann, and P. McEvoy Wideband and High-Gain Composite Cavity-Backed Crossed Triangular Bowtie Dipoles for Circularly Polarized Radiation ....... ......... ........ ......... ......... ........ ......... ......... ........ ...... S.-W. Qu, C. H. Chan, and Q. Xue Regime of a Ferrite-Loaded Open Low-Profile Leaky Wave Electric Monopole Loop Antenna Using the Waveguide ..... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... T. Kodera and C. Caloz Ultrawideband Hemispherical Helical Antennas . ........ ......... ......... ........ ..... H. W. Alsawaha and A. Safaai-Jazi Transmitter and Receiver Isolation by Concentric Antenna Structure .... ........ ...... W.-G. Lim, H.-L. Lee, and J.-W. Yu Arrays True-Time-Delay Beamforming With a Rotman-Lens for Ultrawideband Antenna Systems .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... A. Lambrecht, S. Beer, and T. Zwick Frequency Agile Switched Beam Antenna Array System ........ ......... . ....... ......... . J. R. De Luis and F. De Flaviis A Measurement System for the Complex Far-Field of Physically Large Antenna Arrays Under Noisy Conditions Utilizing the Equivalent Electric Current Method ........ ........ ......... ......... ........ ... T. Lindgren, J. Ekman, and S. Backén Electromagnetics Scalar and Tensor Holographic Artificial Impedance Surfaces ... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... .... B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper Reflections From Multiple Surfaces Without Edges ..... ......... ......... ........ ......... ......... ........ .... W. B. Gordon On the Convergence of the Eigencurrent Expansion Method Applied to Linear Embedding via Green’s Operators (LEGO) ......... ......... ........ ......... ......... ........ ......... ......... .. V. Lancellotti, B. P. de Hon, and A. G. Tijhuis Unified Time- and Frequency-Domain Approach for Accurate Modeling of Electromagnetic Radiation Processes in Ultrawideband Antennas ....... ......... ......... ........ ..... ..... ......... ........ ......... ..... D. Caratelli and A. Yarovoy Enhanced A-EFIE With Perturbation Method .... ........ ......... ......... ........ ......... ..... Z.-G. Qian and W. C. Chew

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(Contents Continued on p. 3105)

(Contents Continued from Front Cover) Numerical Exponentially Converging Nystrom Methods in Scattering From Infinite Curved Smooth Strips—Part 1: TM-Case ... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... J. L. Tsalamengas Exponentially Converging Nystrom Methods in Scattering From Infinite Curved Smooth Strips—Part 2: TE-Case .... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... J. L. Tsalamengas Frequency-Dependent FDTD Simulation of the Interaction of Microwaves With Rocket-Plume ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... K. Kinefuchi, I. Funaki, and T. Abe Scattering and Imaging Two-Dimensional Microwave Imaging Based on Hybrid Scatterer Representation and Differential Evolution . ......... .. .. ........ ......... ......... ........ ......... ......... ...... A. Semnani, I. T. Rekanos, M. Kamyab, and T. G. Papadopoulos Scattering by an Infinite Dielectric Cylinder Having an Elliptic Metal Core: Asymptotic Solutions .... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ...... G. P. Zouros and J. A. Roumeliotis On the Absorption Mechanism of Ultra Thin Absorbers ......... ......... ........ ......... A. Kazemzadeh and A. Karlsson Scattering and Radiation from/by 1-D Periodic Metallizations Residing in Layered Media ..... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... . D. Vande Ginste, H. Rogier, and D. De Zutter Wireless Analysis of Antenna Coupling in Near-Field Communication Systems .. ....... .. ..... Y.-S. Chen, S.-Y. Chen, and H.-J. Li Uncoupled Matching for Active and Passive Impedances of Coupled Arrays in MIMO Systems ....... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ..... M. A. Jensen and B. K. Lau Elevation Plane Beam Scanning of a Novel Parasitic Array Radiator Antenna for 1900 MHz Mobile Handheld Terminals ....... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... . M. R. Islam and M. Ali Depth and Rate of Fading on Fixed Wireless Channels Between 200 MHz and 2 GHz in Suburban Macrocell Environments .. ......... ........ ......... ......... ........ ......... ......... . K. N. Sivertsen, A. Liou, and D. G. Michelson Internal Broadband Antennas for Digital Television Receiver in Mobile Terminals ..... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ....... J. Holopainen, O. Kivekäs, C. Icheln, and P. Vainikainen

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COMMUNICATIONS

Design of Yagi-Uda Antenna Using Biogeography Based Optimization . ........ ... U. Singh, H. Kumar, and T. S. Kamal A Wideband Circularly Polarized H-Shaped Patch Antenna ..... ......... ........ ...... .... ......... ........ ..... K. L. Chung Polarization Reconfigurable U-Slot Patch Antenna ...... ......... ....... P.-Y. Qin, A. R. Weily, Y. J. Guo, and C.-H. Liang High-Efficiency On-Chip Dielectric Resonator Antenna for mm-Wave Transceivers .... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ....... M.-R. Nezhad-Ahmadi, M. Fakharzadeh, B. Biglarbegian, and S. Safavi-Naeini Investigation Into the Effects of the Reflection Phase Characteristics of Highly-Reflective Superstrates on Resonant Cavity Antennas ....... ......... ........ ......... ......... ........ ......... ......... ........ ......... ...... A. Foroozesh and L. Shafai Optimization of Uniaxial Multilayer Cylinders Used for Invisible Cloak Realization ... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... B. Ivsic, T. Komljenovic, and Z. Sipus A Hybrid Optimization Algorithm and Its Application for Conformal Array Pattern Synthesis . ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ...... W. T. Li, X. W. Shi, Y. Q. Hei, S. F. Liu, and J. Zhu Enhanced Total-Field/Scattered-Field Technique for Isotropic-Dispersion FDTD Scheme ...... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ....... H. Kim, I.-S. Koh, and J.-G. Yook Multilevel Fast Multipole Acceleration in the Nyström Discretization of Surface Electromagnetic Integral Equations for Composite Objects ..... ........ ......... ......... ........ . ........ ......... ........ ......... ..... M. S. Tong and W. C. Chew Frequency Selective Surface Using Nested Split Ring Slot Elements as a Lens With Mechanically Reconfigurable Beam Steering Capability ..... ........ ......... ......... ........ ......... ......... ........ ......... ........ M. Euler and V. F. Fusco Combined Dynamic Channel Simulator for High Capacity Broadband Fixed Wireless Access Systems ........ ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ..... M. Cheffena, L. E. Bråten, T. Ekman, and T. Tjelta Planar Printed Strip Monopole With a Closely-Coupled Parasitic Shorted Strip for Eight-Band LTE/GSM/UMTS Mobile Phone .. ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ...... F.-H. Chu and K.-L. Wong Bandwidth Enhancement of the Small-Size Internal Laptop Computer Antenna Using a Parasitic Open Slot for Penta-Band WWAN Operation ...... ........ ......... ......... ........ ......... ... K.-L. Wong, W.-J. Chen, L.-C. Chou, and M.-R. Hsu

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CORRECTIONS

Corrections to “On the use of Nonsingular Kernels in Certain Integral Equations for Thin-Wire Circular-Loop Antennas” ...... ......... ........ ......... ......... ........ ......... ... G. Fikioris, P. J. Papakanellos, and H. T. Anastassiu Corrections to “Comparison of Interpolating Functions and Interpolating Points in Full-Wave Multilevel Green’s Function Interpolation Method” . ........ ......... ......... ........ ....... ... ......... ........ ......... ......... .. Y. Shi and C. H. Chan

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Digital Object Identifier 10.1109/TAP.2010.2083535

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Design of Frequency Reconfigurable Antennas Using the Theory of Network Characteristic Modes Khaled A. Obeidat, Member, IEEE, Bryan D. Raines, Member, IEEE, Roberto G. Rojas, Fellow, IEEE, and Brandan T. Strojny, Student Member, IEEE

Abstract—This paper demonstrates a design procedure for frequency tunable reconfigurable antennas based on the application of reactive loads. Unlike other design procedures, antennas of arbitrary geometry can be tuned utilizing the proposed design framework. The design technique utilizes the theory of network characteristic modes to systematically compute reactive load values required to resonate any antenna at many frequency points in a wide frequency range. For simplicity, a 1.2 m dipole antenna is used to demonstrate the design procedure by tuning it at four loading ports along the antenna body. Both simulations and measurements demonstrate wide frequency tunability characteristics of the dipole input impedance (tunability range wider than 1:4) while preserving the radiation pattern and polarization at seven different frequency states. Lastly, a loaded PIFA is briefly examined as a more complex application of the procedure. Index Terms—Antennas, characteristic modes, electrically small antennas (ESAs), loading, reconfigurable antennas.

I. INTRODUCTION

M

ULTIFUNCTION electrically small antennas have been gaining attention in the last ten years due to the increased demand to cover large frequency bands with a single small antenna while maintaining efficient behavior. As a result, a single reconfigurable antenna with narrow instantaneous bandwidth tunable over a wide frequency range can become part of the new generation of wireless products, especially with recent advances on software defined radios (SDRs). This work proposes a general design methodology applicable to electrically small to mid size antennas to effectively control the antenna current distribution at different frequency points over a wide frequency range using discrete reactive loading. Furthermore, this design methodology allows any type of full wave simulators to calculate the required loads for any type of antenna geometry. As a specific example, the proposed method will be applied to a linear wire dipole loaded with various discrete loads as shown in Fig. 1. A more advanced example, a loaded PIFA with discrete loads, will also be examined following the dipole Manuscript received May 19, 2009; revised January 29, 2010; accepted April 08, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. K. A. Obeidat was with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, Columbus, OH 43212 USA. He is now with the Department of Electrical and Computer Engineering, Brigham Young University Provo, UT 84602 USA (e-mail: [email protected]). B. D. Raines, R. J. Rojas, and B. T. Strojny are with the ElectroScience Laboratory, Department of Electrical and Computer Engineering, Columbus, OH 43212 USA (e-mail: [email protected]; [email protected]; strojnyb@ece. osu.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055787

Fig. 1. Center-fed dipole antenna loaded with four reactive lumped loads.

discussion. While discrete passive loading [1] using the theory of characteristic modes have been used in the past to alter the radar cross section (RCS) of a scatterer, its use to design reconfigurable frequency antenna is believed to be novel. Although several related works [2], [3] were performed in the past to calculate the required passive reactance loads, their analytical framework depended on transmission line theory, which is not generally applicable when applied to electrically small and arbitrarily-shaped antennas. In other related work, [4], [5], calculation of the required loads relies on empirical parametric studies or computer optimization rather than a systematic design technique. The proposed design methodology has the advantage of being systematic. In this paper, an example is presented where the goal is to excite one mode of operation (of the original unloaded antenna) at all the frequency points of interest, including frequencies where higher order modes exist. In previous work [3], the antenna geometry has been modified to eliminate the excitation of higher order modes; however, no systematic design methodology was introduced. II. DESIGN METHODOLOGY If the current distribution on the body of an antenna is controlled over a wide frequency range using tunable loads at discrete frequency points (states), both the antenna input impedance and its radiation pattern are said to be widely tunable over that frequency range. The theory of network characteristic modes (NCM) [1] can systematically compute the load values which are required to force the antenna to resonate different current shapes at different desired states. However, prior to calculating the load values, the locations of those loads along the antenna body should be carefully chosen. Furthermore, identifying the required current is very critical to obtain acat the antenna feed, and to ceptable reflection coefficient produce the desired radiation pattern at various states. At this time, there is no systematic method of determining the number and location of the ports to achieve optimum control of the current distribution over a general antenna. For the two antennas considered here, the port locations were determined heuristically by studying the distribution of the desired current. More ports are placed in locations where the desired current

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OBEIDAT et al.: DESIGN OF FREQUENCY RECONFIGURABLE ANTENNAS USING THE THEORY OF NCM

magnitude is large and less ports are placed when the magnitude is small. It is also clear that the larger the number of ports, the better control one has of the currents over larger frequency bands. Although the number and location of the ports can be determined by some optimization algorithm, the long term goal is to develop a more robust and systematic technique for an arbitrary antenna geometry and desired current distribution. The discussion of such an algorithm is beyond the scope of the work presented here. The theory of NCM for an antenna with N-ports can be represented by an N-port Z matrix. The N eigenmodes are computed using the following generalized eigenvalue problem at some radial frequency [1] (1) where and are, respectively, the real and the open circuit impedance imaginary parts of the N-port is the th eigencurrent, and is the matrix of the antenna, corresponding th eigenvalue. The total current is therefore a weighted summation of all these modes [1] (2) is the N-port open circuit voltage column vector of where , and indicates the N-port network characterized by a Hermitian transpose. Each modal current expansion coefficient in (2) has two important components. The denominator essentially depends on the frequency-dependent eigenvalue , which is a function solely of the antenna geometry and materials. The numerator is determined by an inner product between a frequency-depenand the applied excitation and dent eigencurrent therefore determines the feeding network needed to excite the desired current modes. The required lossless reactance values at each port are given by [1] (3) where the subscript denotes the th port and is the required is in resoequiphase current distribution. An eigencurrent nance when its corresponding eigenvalue equals zero. The are used load values described by the diagonal matrix to be zero. Consequently, to enforce the quantity the N-port current becomes the dominant eigencurrent at the various desired frequency points. III. APPLICATION: FREQUENCY RECONFIGURABLE DIPOLE ANTENNA The proposed design technique is applicable to any antenna geometry; however, for simplicity, a planar copper dipole antenna of length 1.2 m and 6 mm width mounted on 3 cm foam is considered first. To feed this antenna, the dipole was connected to a ZFSCJ-2-1-S (mini-circuits) Balun through two 0.33 m coaxial cables at each port of the balun. Seven arbitrary frequency states were chosen to cover the frequency band

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, where m is the radius down to 70 MHz ( of the smallest circumscribing sphere about the dipole) and up , with a minimum operational tunable to 300 MHz range of 4:1. The radiation pattern must substantially resemble dipole. We have chosen to use the the pattern shape of a method of moments (MoM) codes ESP5.4 [6] and Agilent Momentum [7]. They will be used to extract the network Z parameters of the multiport dipole antenna, as well as input impedance and radiation pattern. A. Port Setup The first characteristic eigencurrent of the unloaded dipole [8] was chosen as the desired current. This current mode should resonate at the chosen frequencies states in order to produce the desired pattern shape at each state (spherical mode). The first eigencurrent of the unloaded dipole has its first resonance frequency at 115 MHz. Note that, for a center-fed dipole, Mode 2 does not contribute to the total current because it is an odd mode [8]. Five ports were selected, with four loading ports symmetrically distributed along the dipole and one feeding port at the center of the antenna as illustrated in Fig. 1. The locations of the load ports were chosen such that two ports are close to the center of the dipole where the current distribution is high in magnitude, and the remaining 2 ports are closer to the dipole ends. The port locations were chosen keeping in mind that the actual antenna current distribution is largely influenced (or controlled) by a lumped load in its immediate vicinity. Thus, the two ports near the dipole center provide some amount of control over the input impedance, while the remaining two loads ensure that the current distribution shape near the ends resembles the desired current distribution. The number of ports was determined empirically with four ports yielding an acceptable tradeoff between complexity and achievable tunable bandwidth. As previously mentioned, the tunable bandwidth can be increased by increasing the number of ports. Given the port locations, the 5-port Z network parameters can be computed using (1). B. Load Computation The desired current distribution, which in this case corresponds to the first eigenmode of the 5-port dipole at 115 MHz, is computed to be , where indicates transpose. was applied to (3) to extract the reactance values required to force the current to be in resonance at seven frequencies distributed over 50 to 350 MHz as tabulated in Table I. Reactance values for each state are shown in Fig. 2 as seven heavy dots. Positive reactance values in Fig. 2 were approximated by passive inductors at the corresponding frequency points, while negative reactance values were approximated by passive capacitors at the corresponding frequency points. Reactive element values for each state are tabulated in Table I. State 2 represents the unloaded dipole antenna. States 0 and 1 represent the antenna with inductive loading, while States 3-6 represent the antenna with capacitive loading. For this particular example, capacitive loading increases the resonant frequency, while inductive loading decreases the resonant frequency. This result is intuitive in that capacitive loading effectively shrinks

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Fig. 2. Required series reactances at port 2 of the seven states as specified in Table I.

Fig. 3. Eigenvalue spectrum of the 5-port 1.2 m dipole for both the unloaded case and for the capacitive loading case.

TABLE I LOAD ELEMENT VALUES OF THE LOADED 4 PORT RECONFIGURABLE DIPOLE ANTENNA

TABLE II LOAD ELEMENT VALUES OF THE LOADED 4 PORT WIDEBAND DIPOLE ANTENNA Fig. 4. Eigenvalue spectrum of the 5-port 1.2 m dipole for both the unloaded case and for the inductive loading case.

the dipole length, while inductive loading extends the effective dipole length. It is interesting to note that if we compute the reactances at each port as a function of frequency between 25 and 400 MHz, and examine their frequency behavior, one can show that they have negative slopes (i.e., non-Foster). The reactance function for port 2 is shown as the dashed line in Fig. 2. If such loads were realized exactly, the dipole antenna would be in continuous resonance at each single frequency between 25 and 400 MHz. For this example, the loads may be satisfactorily approximated by a series of LC circuit where both elements (L and C) have negative values (i.e., non-Foster elements), as shown in Table II. This concept was discussed in a previous paper by the present authors [9]. It is important to emphasize that the loads for the proposed reconfigurable antennas are passive Foster loads. C. Eigenvalue Analysis Between 25 and 400 MHz, Modes 1 and 3 of the unloaded dipole (the first two well-known even dipole modes) are dominant. The eigenvalue magnitudes (in decibels) of these two modes are plotted for the capacitive and inductive loading cases in Figs. 3 and 4, respectively. For clarity, only loaded state (state 0) is shown in Fig. 4. From the eigenvalue spectrum of State 2, it may be observed that Mode 1 resonates at 115 MHz because its corresponding

eigenvalue magnitude approaches zero (i.e., a resonant eigenmode at 115 MHz). Similarly, Mode 3 of State 2 resonates at 355 MHz. Similar conclusions can be drawn for the remaining states. For the unloaded dipole, the modal resonance frequency of Mode 1 is the same as the input impedance series resonance frequency because the other modes are insignificant at this freis much larger than the other ) [8]. quency (i.e., D. Current Analysis To better understand the effect of the loads on the actual antenna current, the normalized magnitude of the antenna current distribution of the reconfigurable antenna was computed at 115, 180 and 300 MHz, respectively, as shown in Fig. 5. The desired current distribution is only specified at the 5 chosen port locations. In all of the curves, the actual current distribution closely follows the desired current distribution, except at 300 MHz (State 6), which demonstrates the greatly increased signifis decreasing. icance of Mode 3 at 300 MHz as the ratio At 300 MHz, the total current contains a noticeable contribution from higher order Mode 3 alongside the desired Mode 1. E. Input Impedance Analysis The next key step in the design is to calculate an envelope at the input feeding port of the antenna as shown in of the

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Fig. 7. The reflection coefficient magnitude S of the reconfigurable dipole antenna, Simulated (dashed) vs. Measured (Solid). Fig. 5. Comparison of the magnitude of the normalized desired current at five ports (dots) with the normalized antenna current distribution for the loading case reported by Momentum simulation at 115 MHz, 180 MHz and 300 MHz.

Fig. 6. The calculated reflection coefficients S of the dipole antenna for the passive reactive loading cases (Solid) all referenced to 50 . The dashed line is an envelope of the S .

Fig. 6 (dashed line). Each frequency point of the envelope is calculated by selecting the appropriate lossless reactive loads so the desired current is resonant at that frequency point. One example of a reactive load obtained to obtain the envelope is depicted by the dashed line in Fig. 2. The next step is to examine the envelope to determine the frequency range where the magnitude is below a desired level. For example, in Fig. 6, is below dB, then the frequency range if the desired would be from 92–246 MHz. The chosen states fall within this frequency range, except for States 0, 1, 5, and 6. The performance of these four states will be improved using a passive matching network. Before discussing matching networks, this antenna was constructed and tested. The calculated reflection of the reconfigurable antenna’s input impedance coefficient seven states are shown in Fig. 6 (solid lines) while the measured results are shown in Fig. 7. The measured results were obtained by loading the antenna at its four ports a set of loads tabulated in Table I. There is generally excellent agreement between the calculated and measured reflection coefficients for low frequency states, while more significant discrepancy between calculation

and measurement is observed at high frequency states. This is mainly attributed to the fact that the 5-port Z network parameters used to derive the actual load inductor and capacitor values were calculated rather than measured. While the loaded dipole layout was carefully modeled, the parasitic effects are becoming more pronounced at the higher frequencies, especially with the lumped loads. The proposed technique ensures a nearly resistive input impedance for frequencies where high order eigenmodes are . However, in the case of very weakly excited is comparable to . Therefore, State 6, the value of Mode 3 will contribute more significantly to the total current and consequently, the total current , as determined by (2), will at high frequencies. This quality not satisfy implies a non-zero reactance at the feed port at high frequencies MHz) and consequently a lower reflection coeffi( cient. More control of Mode 3 can be achieved if additional ports are introduced; however, there has to be trade-off between number of ports/complexity and larger bandwidth. On the other hand, at low frequencies of operation, namely States 0 and 1, the antenna electrical size is small with small radiation resistance; therefore, larger reflection coefficients are expected. Another important advantage of the envelope is in the design of the matching network to enhance the performance of the seven states all at once. A single passive matching network can be designed by considering the envelope as the load. Fig. 11 of the cascaded combinashows the reflection coefficient tion of the measured antenna and the two-port simulated matching network S-parameters. The matching network is a lossless seventh-order passive ladder matching circuit designed from 65–300 MHz using the real frequency technique [10]. F. Radiation Pattern Analysis The radiation pattern at the desired tunable frequency points (states) is also important. Fig. 9 illustrates the measured realized gain of the loaded antenna at 74 MHz (State 0) and the unloaded antenna at 114 MHz (State 2). Fig. 10 illustrates the measured realized gain of the loaded antenna at 290 MHz (State 6) and the unloaded antenna at 114 MHz (State 2). Finally, Fig. 8 illustrates the measured realized gain of the loaded antenna at 290 MHz (State 6) and the unloaded antenna at the same frequency

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Fig. 11. The reflection coefficient magnitude S of the reconfigurable dipole antenna with an ideal passive lossless seventh-order matching network. Fig. 8. The measured realized gain in dB of the loaded dipole antenna (State 6) compared to the unloaded case at 290 MHz.

Fig. 12. PIFA Layout with one microstrip feed port and six other load ports.

Fig. 9. The measured realized gain in dB of the loaded dipole antenna (State 0) at 74 MHz compared to the unloaded dipole antenna (State 2) at 114 MHz.

300 MHz (not shown here) still resembles the desired pattern shape, even though the total current has some contribution from Mode 3. This is due to the well-known fact that a first-order error in current manifests itself as a second-order error in the far-field. IV. APPLICATION: FREQUENCY RECONFIGURABLE PIFA

Fig. 10. The measured realized gain in dB of the loaded dipole antenna (State 6) at 290 MHz compared to the unloaded dipole antenna (State 2) at 114 MHz.

point. As far as the pattern is concerned, loading the antenna has extended the desired pattern shape seen at 70 MHz up to 300 MHz by suppressing higher-order modes. The pattern at

To show the versatility of the method, this section will briefly discuss the design of a more complex frequency reconfigurable planar inverted-F antenna (PIFA). The geometry for the PIFA is shown in Fig. 12. All metal is 0.047 mm thick copper. The substrate separating the microstrip ground plane from the top metal layer is 1.5 mm thick FR4. Specifically, it has a relative permittivity of 4.4 and a loss tangent of 0.016. The whole layout is 50 mm by 50 mm. Seven loading sites, including the feed point, were chosen. The number of loads and their location were chosen at points of high magnitude current at the PIFA’s “natural” operating frequency of 2.4 GHz. In other words, the desired current is the mode commonly used to operate a PIFA. Six different states were chosen and are listing below in Table III. The seven port Z matrix of the unloaded antenna was extracted from a simulation using Agilent Momentum [7] from 1 GHz to 5 GHz. The NCM spectrum was computed over the whole frequency range and is shown in Fig. 13.

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TABLE III PIFA FREQUENCY STATES

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TABLE IV LOADED PIFA OPERATING FREQUENCIES

TABLE V UNLOADED PIFA PEAK REALIZED TOTAL GAIN VALUES

TABLE VI LOADED PIFA PEAK REALIZED TOTAL GAIN VALUES

Fig. 13. Eigenvalue spectrum of the unloaded PIFA.

Fig. 14. The calculated reflection coefficients S of the PIFA for the passive reactive loading cases (Solid), all referenced to 50 . The dashed line is an envelope of the S .

As mentioned above, we chose the desired current to be the dominant mode at 2.425 GHz of the unloaded PIFA, which is Mode 2. This current was resonated using reactive loading to envelope following the same procedure degenerate the envelope was calculated scribed above for the dipole. The at 401 frequencies, as shown in Fig. 14 as a dashed line. In other words, at each frequency, the eigenvalue of Mode 2 was forced to be zero by properly choosing the values of the reactive loads (3). It should be noted that for the PIFA example, there is no input matching network used, so the observed input reflection levels would be improved if such a network were attached.

Observing the envelope, we can choose states between 1.96 of these states would and 3.45 GHz and be assured that the dB. States 2–6, which fall within this frequency be below , as shown in Fig. 14. Unforrange, will have a satisfactory tunately, State 1 falls below this range. A set of reactive loads is also was designed for this state, and its corresponding of State shown in Fig. 14. Although it is not guaranteed, the 1 happens to be acceptable at 1.65 GHz for this antenna and its given location and number of ports. If that were not the case, a matching network would be necessary. Furthermore, State 5, which was designed for 3 GHz, also happens to provide an additional “extra” state at 1.82 GHz. curves associated with a particIt is noted that a few of the at a frequency slightly ular state in Fig. 14 have a minimum different from the design. An additional iteration to fine tune the reactive loads is necessary to exactly achieve the desired frequency states. This in no way is a limitation of the method. The fine tuning was simply not performed here. The operating frequencies achieved with only one iteration are given below in Table IV. The unloaded and passively loaded peak realized gains (including impedance mismatch lossless) at the operating frequencies listed in Table IV are given in Tables V and VI, respectively. By comparing the peak gains in both tables, it is observed that the loaded PIFA features improved peak gain at each operating frequency compared to the unloaded PIFA. While this design also enjoys fairly wide tunability in input reflection coefficient level, the pattern does not maintain its shape after approximately 3 GHz, as can be seen by comparing the patterns in Figs. 15 and 16. If more states were desired for frequencies below 1.6 GHz or above 3.25 GHz, additional

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Fig. 15. The calculated total realized gain radiation patterns at  unloaded PIFA at 1.65, 2.0, 2.4, 2.65, 3, and 3.25 GHz.

=0

of the

successfully controlled at the tunable frequency points over a wide frequency range. The method was also applied to a PIFA originally designed to resonate at 2.4 GHz. Similar to the dipole, it was loaded with another set of reactive elements at many states between 1 and 3.5 GHz to achieve tunable performance over the band. The pattern was stable between 1 and 3 GHz. Further work on this PIFA, as described in the previous section, will enhance the overall tunable bandwidth. A key concept introduced here to facilitate the design of the envelope. reconfigurable antennas is the calculation of the envelope was computed and used for two different The purposes. It was used to determine the minimum frequency levels. Also, in the range of states which yield acceptable case of the loaded dipole antenna, the envelope was used as the load to design a single passive matching network which would improve the input reflection coefficient levels of all states. Finally, it should be observed that although the example presented in this work was used to design frequency tunable reconfigurable antennas, it can also be used to design pattern tunable antennas or a mix of pattern and frequency tunable antennas. REFERENCES

Fig. 16. The calculated total realized gain radiation patterns at  passively-loaded PIFA at 1.65, 2.0, 2.4, 2.65, 3, and 3.25 GHz.

=0

of the

iterations would be required. It is very likely that more ports would be required to keep tight control on the desired current distribution over a larger frequency band to achieve both satislevels and pattern shapes. factory V. CONCLUSION In this work, the theory of network characteristic modes was proposed to systematically design widely tunable reconfigurable antennas of arbitrary shape. As an example, the method was applied successfully to a simple 1.2 m dipole antenna from 70 MHz to 300 MHz. The dipole antenna was loaded with a set of different reactive elements at each state. Both pattern and input impedance of the loaded dipole antenna were

[1] R. Harrington and J. Mautz, “Modal analysis of loaded n-port scatterers,” IEEE Trans. Antennas Propag., vol. AP-21, no. 2, pp. 188–199, Mar. 1973. [2] N. Behdad and K. Sarabandi, “A varactor-tuned dual-band slot antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 401–408, 2006. [3] N. Behdad and K. Sarabandi, “Dual-band reconfigurable antenna with a very wide tunability range,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 409–416, 2006. [4] S. Loizeau and A. Sibille, “Reconfigurable PIFA antenna with ultrawideband tuning,” in Proc. EuCAP, 2007, pp. 1–6. [5] C. S. DeLuccia, D. H. Wemer, P. L. Werner, M. F. Pantoja, and A. R. Bretones, “A novel frequency agile beam scanning reconfigurable antenna,” in Proc. Antennas and Propagation Society Int. Symp., 2004, pp. 1839–1842. [6] E. Newman, “The Electromagnetic Surface Patch Code: Version 5,” The Ohio State University [Online]. Available: http://esl.eng.ohiostate.edu [7] Momentum. Santa Clara, CA: Agilent Technologies [Online]. Available: http://www.agilent.com [8] K. Obeidat, B. D. Raines, and R. G. Rojas, “Antenna design and analysis using characteristic modes,” in Proc. Antennas and Propagation Society Int. Symp., 2007, pp. 5993–5996. [9] K. Obeidat, B. Raines, and R. Rojas, “Application of characteristic modes and non-foster multiport loading to the design of broadband antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 203–207, Jan. 2010. [10] H. Carlin and P. Civalleri, Wideband Circuit Design. Boca Raton, FL: CRC, 1997.

Khaled A. Obeidat (M’09) received the B.S. degree from Jordan University of Science and Technology, Irbid, in 1999, the M.S. degree from the University of South Florida, Tampa, in 2003, and the Ph.D. degree from The Ohio State University, Columbus, in 2010, all in electrical engineering. He was with OWSS, from 1999 to 2001, EMAG Technology, from 2003 to 2004, and, from 2009 to 2010, in the Silicon Valley working as an Antenna/RF System Engineer. Currently he is a Research Faculty in the Department of Electrical and Computer Engineering, Brigham Young University, Provo, UT. His main areas of research interest are RF circuits, tunable filters using MEMS, UWB antennas, electrically small antennas and phased array antenna.

OBEIDAT et al.: DESIGN OF FREQUENCY RECONFIGURABLE ANTENNAS USING THE THEORY OF NCM

Bryan D. Raines (M’10) received the B.S.E.E. degree from California Polytechnic State University, San Luis Obispo, in 2005. He is currently working toward the Ph.D. degree at The Ohio State University, Columbus. He was with Apple, Inc., from 2003 to 2006. His main areas of interest are RF integrated circuits, UWB antennas, electrically small antennas, active antennas and characteristic mode theory.

Roberto G. Rojas (S’80–M’85–SM’90–F’01) received the B.S.E.E. degree from New Mexico State University, Las Cruces, in 1979, and the M.S. and Ph.D. degrees in electrical engineering from The Ohio State University (OSU), Columbus, in 1981 and 1985, respectively. He is currently a Professor in the Department of Electrical and Computer Engineering, The Ohio State University. His current research interests include the analysis and design of conformal arrays, active integrated arrays, nonlinear microwave circuits, as well as the analysis of electromagnetic radiation and scattering phenomena in complex environments. Dr. Rojas is an elected Member of the International Scientific Radio Union (URSI), Commission B. He won the 1988 R.W.P. King Prize Paper Award, the 1990 Browder J. Thompson Memorial Prize Award, both given by IEEE, the 1989 and 1993 Lumley Research Awards, given by the College of Engineering at The Ohio State University. He has served as Chairman, Vice-Chairman and Secretary/Treasurer of the Columbus, OH, chapter of the IEEE Antennas and Propagation and Microwave Theory and Techniques Societies.

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Brandan T. Strojny (S’08) received the A.A.S.E.E.T. degree from Fox Valley Technical College, Appleton, WI, in 2004, the B.S.E.E.T. degree from Milwaukee School of Engineering, Milwaukee, WI, in 2006, and the M.S.E.E. degree from The Ohio State University, Columbus, in 2010. He is currently working toward the Ph.D. degree at The Ohio State University under the supervision of Dr. Rojas. His research interests include conformal antennas for unmanned aerial vehicles, characteristic mode analysis of complex antenna structures, electrically small antennas, integration of antennas into conformal surfaces, reconfigurable antennas, ferroelectric materials and material characterization.

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Lower Bounds on the Q of Electrically Small Dipole Antennas Arthur D. Yaghjian, Life Fellow, IEEE, and Howard R. Stuart, Member, IEEE

Abstract—General expressions are obtained for the lower bounds on the quality factor ( ) of electrically small electricand magnetic-dipole antennas confined to an arbitrarily shaped volume and excited by general sources or by global electric-current sources alone. The lower-bound expressions depend only on the direction of the dipole moment with respect to , the electrical size of , and the static electric and magnetic polarizabilities per unit volume of hypothetical perfectly electrically conducting and perfectly magnetically conducting volumes . The lower bounds are obtained directly from the electromagnetic field expressions for with the help of current equivalence principles and the uncoupling of Maxwell’s equations for electrically small volumes into quasi-electrostatic and quasi-magnetostatic fields. Index Terms—Dipole antennas, equivalence principle, lower bounds, quality factor.

I. INTRODUCTION

B

EGINNING with the work of Wheeler [1]–[3] and Chu [4], numerous papers [5]–[16], [37] (to cite a partial list) have been devoted to determining lower bounds on the quality factor of antennas.1 Although nearly all of the past expressions for lower bounds have applied to spherical volumes (the sphere lower bound being less than that of any other volume shape circumscribed by the sphere), the lower-bound expressions on quality factor divided by directivity of Gustafsson et al. [14], [15] based on their “sum rule” can be applied to dipole antennas confined to arbitrarily shaped volumes. However, these lower-bound expressions obtained from the sum rule are proportional to a “generalized absorption efficiency,” equal to the ratio of the integrals over all frequencies of the normalized absorption and extinction cross sections, that must be computed or estimated for each antenna. Also, in the work of [14], [15], no explicit distinction is made between the antenna-proper being exManuscript received July 02, 2009; revised February 06, 2010; accepted March 29, 2010. Date of publication July 19, 2010; date of current version October 06, 2010. This work was supported in part by the U.S. Air Force Office of Scientific Research (AFOSR) through Dr. Arje Nachman. A. D. Yaghjian is a Research Consultant at Concord, MA 01742 USA (e-mail: [email protected]). H. R. Stuart is with LGS, Bell Labs Innovations, Florham Park, NJ 07932 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055790 1Section V-A of [13] determines the effect of highly dispersive, lossy material on the lower bounds of antennas. The references [9] and [10] determine “exact” and accurate approximate expressions for the bandwidth and of general linear antennas, and only peripherally consider lower bounds on . Also, we restrict our attention to the quality factor of linear, passive antennas at isolated resonances and thus ignore increases in bandwidth that can be achieved, in principle, with multiresonances [17]–[19] or with nonlinear and/or active material.

Q

Q Q

Fig. 1. Electric-dipole antenna divided into the antenna-proper of the incident fields.

V and sources

cited by general sources and by global2 electric-current sources alone [12], [16]. Therefore, the primary purpose of the present paper is to obtain through direct derivation, using surface current equivalence principles, simple general expressions for the lower bounds on the of electric- and magnetic-dipole antennas, in arbitrarily shaped electrically small volumes, excited by general sources and by global electric-current sources alone. A secondary purpose is to make explicit the assumptions required for the derivation of these general expressions in order to reveal the conditions under which previous expressions apply as lower bounds on the quality factor of electrically small electric- and magnetic-dipole antennas. Sections II and III derive these lower bounds for electric- and magnetic-dipole antennas, respectively, and the paper ends with a few summarizing conclusions. II. ELECTRIC-DIPOLE ANTENNAS Every antenna can be viewed as a scatterer illuminated by sources of an incident field. (If the antenna sources are connected to the antenna-proper by a feed wave-guide or cable, the sources can, in theory, be separated from the antenna-proper (scatterer) by a closed surface that encloses the power supply and cuts through the reference plane of the feed wave-guide or cable.) Therefore, consider an electric-dipole antenna consisting of an electrically small, arbitrarily shaped volume (the scatterer or antenna-proper) illuminated by time-harmonic sources outside of , as shown in Fig. 1. (Throughout, the symbol for volume will be used to designate the shape as well as the value of the volume of a three-dimensional (3D) region.) The volume of the antenna-proper (scatterer) may contain both electric and magnetic material. How2By “global” electric-current sources, we mean electric-current sources other than small Amperian current loops (or possibly slots in electrically conducting surfaces) that produce effective magnetic-current/polarization sources.

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YAGHJIAN AND STUART: LOWER BOUNDS ON THE Q OF ELECTRICALLY SMALL DIPOLE ANTENNAS

Fig. 2. Electric-dipole antenna-proper in the volume V excited by equivalent surface currents producing the incident fields.

ever, for purely electric or purely magnetic dipole antennas, there is no loss of generality in assuming that the relative permeability or the relative permittivity of the material in is unity, respectively. Next, replace the sources by their equivalent electric and magand , respectively, netic surface currents ( and are the incident source electric and where magnetic fields, and is the unit normal out of ) on a surjust outside the surface of the volume of the anface tenna-proper [20, sec. 8–14], [21, sec. 2.3.8]; see Fig. 2.3,4 In the of the equivabsence of the antenna-proper, the fields inside alent currents are identical to the incident fields of the sources. in the absence of the antenna-proper, the fields of Outside the equivalent currents are zero (a result sometimes referred to as the “extinction theorem”). Thus, in the presence of the anare tenna-proper, all the sources of the fields external to simply those on the antenna-proper; that is, the transmitted electric-dipole fields of the antenna are produced by the sources on the antenna-proper and not by the electric and magnetic equiv, the energy in the alent surface currents. Moreover, as becomes negligible and all the fields fields between and outside are effectively those of the antenna-proper. It is assumed that the antenna-proper is electrically small, , where and is the specifically radius of the sphere that circumscribes the volume of the antenna-proper, being the free-space speed of light and the free-space wavelength. (It is sufficient for the reasonable accu.) racy of the lower bounds obtained herein to assume In practice an electric-dipole antenna refers to any antenna whose far fields are predominantly those of an elementary electric dipole. If the far fields were exactly those of an elementary 3The idea of using equivalent electric and magnetic surface currents to obtain the lower bounds on the quality factor of a spherical antenna was introduced in [10] and [13]. Here we apply the idea in a way that enables the determination of lower bounds on the quality factor of dipole antennas with their antenna-proper confined to arbitrarily shaped V . 4Since magnetic charge does not exist, the equivalent magnetic surface current surrounding the antenna-proper would have to be produced in practice by a thin layer of magnetic polarization on S ( from natural magnetic material or synthesized from small Amperian current loops (or possibly slots in electrically conducting surfaces)) [22], [23]. Here, however, we are merely postulating surface currents as a means to obtain lower bounds on the quality factor of antennas with various compositions and shapes excited by the most general electric and magnetic surface-current sources. (In the following Section II-A, it is shown that these general-source lower bounds require only a slight modification to apply to antennas whose source excitations are limited to global electric surface currents.)

M

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electric dipole, the analyticity of the Maxwellian electromagnetic fields in free space would imply that the near fields of the antenna up to the material of the antenna would also be exactly those of the elementary electric dipole. For highly eccentric volumes , this would mean that the fields close to the antenna would be very large because they would be equal to the fields close to an elementary electric dipole and, thus, the would be very large. However, for a practical electric-dipole antenna with a far field that approximates well but not perfectly the far field of an elementary electric dipole, only the fields beyond a ) outside the sphere fraction of a wavelength (typically that circumscribes the volume of the antenna-proper need be close to the fields of an elementary electric dipole. Moreover, the fields between the circumscribing sphere and the surface of may differ greatly from and, in particular, the electric field there may be much lower in average squared magnitude than that of the electric field of an elementary electric dipole. Therefore, to obtain the minimum possible for a tuned electric-dipole antenna with the material of the antenna-proper confined to the volume , one would like to choose the incident fields such that the energy stored in the electric field in and between and its circumscribing sphere are minimized. Consequently, to obtain the lower bound on for an electric-dipole antenna-proper confined to the volume , one can first fill the volume with a perfect electric conductor (PEC) so as to force the electric field to zero in (see Fig. 2). Then one is left with choosing the sources of the incident field (the equivalent electric and magnetic surface currents) such that the average squared magnitude of the electric field between the volume and its circumscribing sphere are minimized. Since the fields of the equivalent electric and magnetic surface currents between and its circumscribing sphere are zero, we are left with the problem of minimizing the squared magnitude of the scattered electric field produced by the electric surface currents induced by the incident fields on the surface of the PEC filling . The minimization is done by varying the incident fields (the fields of the equivalent electric and magnetic surface currents). For an electrically small volume , the electric-dipole moment on the PEC is induced by an incident electric field over the volume . Divide this incident electric field within into the sum of a uniform electric field , which is not a function of in , and a nonuniform electric field whose spatial average over is zero. The uniform electric field applied to the PEC in will induce predominantly electric-dipole ) outside the fields a fraction of a wavelength (typically sphere that circumscribes , and reactive fields between and a fraction of a wavelength outside the circumscribing sphere. The nonuniform electric field applied to the PEC will induce predominantly quadrupole and higher-order multipole fields a fraction of a wavelength outside the circumscribing sphere, and reactive fields between and a fraction of a wavelength outside the circumscribing sphere, but with a ratio of the energy stored in the reactive fields to the power radiated by the electric dipole that increases faster with decreasing electrical size of than the same ratio for the fields induced by the uniform electric field [4], [24, sec. 4.2]. In other words, for , the nonuniform incident electric field will produce scattered fields that

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add significantly to the reactive energy while contributing negligible power to the radiated electric-dipole fields. Therefore, for , the minimum possible for an antenna-proper confined to will be the of a PEC scatterer filling subject (which can be chosen to a uniform incident electric field real-valued without loss in generality) produced by the applied equivalent electric and magnetic surface currents on the surface just outside the surface of . The quality factor for the lower-bound, electric-dipole antenna depicted in Fig. 2, after it is tuned to resonance, is given as [9] in terms of the total fields

we can re-express the integral in (2) as

(5)

is a quasi-electrostatic field, it can be written as the Since , and gradient of a scalar potential, that is, with the help of Green’s first identity and (4), we have

(6)

(1) is the spherical volume of radius surrounding and where is the permittivity of free space. The power radiated, , in if the antenna is lossy, where is the (1) is replaced by radiation efficiency of the antenna. (All the expressions for the lower bounds on contained in this paper are for lossless antennas. These lower-bound expressions are merely multiplied by if the antenna is lossy and nondispersive (conductivities nearly independent of frequency over the bandwidth of the antenna); see concluding Section IV.) The integral in (1) of the magnitude squared of the far electric-field pattern subtracts the radiation-field energy in order to obtain stored energy. The fields inside the PEC are zero. Moreover, for electrically small antennas, the predominant contribution from the difference of the two integrals in (1) is from the total quasi-electro. In other words, as , the quasi-elecstatic field trostatic electric-dipole field dominates the contribution to the , (1) can be simplified stored energy. Consequently, for to

where is the induced surface charge density on the surface of and use has been made of the total pothe PEC , tential being constant on S, namely with . (The surface integrals at an infinite radius in Green’s first identity vanish because decays as and thus decays as as .) Insertion from (6) into (5) and the resulting equation into (2) yields (7) Expressing the dipole moment in terms of the electrostatic polarizability dyadic of the PEC volume in a uniform elec5 tric field (8a) or taking the inverse (8b)

(2) further simplifies (7) to denotes the volume outside . In the quasiwhere electrostatic field is that of the scattered field, since the incident fields of the electric and magnetic surface currents are zero outside . The power radiated in (1) has been expressed in terms of the electric-dipole moment of the electric-dipole antenna; specifically [20, p. 437] (3) The volume integral over in the numerator of (2) of the magnitude squared of the quasi-electrostatic field can be converted to an integral over the volume of the antenna-proper in , which equals , is as follows. Noting that of the electric surface equal to the quasi-electrostatic field current on the PEC that produces the electric-dipole moment, and (4)

(9a) with

equal to the unit dyadic and equal to the unit vector . For scalar , (9a) reduces to (9b)

Reciprocity implies that is a real symmetric dyadic [26] and thus the coordinate system of can be oriented to make a diagonal dyadic with three principal directions. With in one of the principal directions of (9c) 5The static polarizabilities for many differently shaped volumes have been determined analytically and numerically, for example, for ellipsoids [20, chs. 3 and 4] and for regular polyhedra [25].

YAGHJIAN AND STUART: LOWER BOUNDS ON THE Q OF ELECTRICALLY SMALL DIPOLE ANTENNAS

where is the value of the diagonal element in that principal has dimensions of volume, so that the direction . Note that in (9) depend only on the electrical size lower bounds on (specifically, ) and the electrostatic polarizability per unit of a PEC volume . Also, note that (9c) implies volume since the quality factor has to be greater than zero, that . that is, the shape factor The result in (9c) is similar though not identical to the inreverse of the “radiation power factor,” namely ported by Wheeler [1], [11] for cylindrical capacitor and inductor antennas, where is a “capacitor” or “inductor” shape factor in the axial direction of the cylinder. Of course, the two expressions can be made identical in form if is set equal to . The lower bounds in (9) are attained in principle by magnetic placed over a PEC in that produce surface currents in in the absence of the PEC. a uniform electric field For a spherical volume of radius ; see, for example, [20, eq. (PEC) (32), p. 206] as the relative permittivity (10)

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Choosing a ratio of the lengths of the major to minor axes of the [15], (13c) gives prolate spheroid equal to (14) an upper bound on the half-power VSWR bandwidth for this that is slender prolate spheroid with times the upper bound for the half-power VSWR bandgiven in (14) width of a sphere of radius . The value of is close to that computed numerically for an electric-dipole antenna formed by a realistic electrically small, thin, straight, conducting wire [9]. polarFor a very flat oblate spheroid ized in its plane, Osborn [28] finds that (15a) which gives a polarizability of (15b) and a lower-bound quality factor from (9c) of

so that (15c) (11) which is the Chu lower bound for an electrically small electric-dipole antenna. Although it is not obvious from (9), the lower bound for an electric-dipole antenna confined to a sphere circumscribed by the is less than that of any other volume sphere because of the extra stored energy between and its cir, the term in the more cumscribing sphere. (For [4], [8] accurate Chu lower bound given by adds an amount equal to 25% of the term. This percentage decreases rapidly with decreasing . For other more eccentric volumes, one would expect that the higher-order terms would contribute relatively less because the lower-bound of a sphere is smaller than that of any other volume circumscribed by the sphere.) The polarizability of a PEC ellipsoid for the direction of the dipole moment parallel to one of its principal axes can be found from [27, sec. 5.3] as (12) is the depolarization factor for a given principal-axis where with , , and equal to the direction and lengths of the principal semi-axes. For a very slender prolate polarized along its long axis, spheroid Osborn [28] finds that (13a) which gives a polarizability of (13b) and a lower-bound quality factor from (9c) of (13c)

This value of is times the Chu lower bound for the sphere in (11) with the same maximum dimension, that is, the upper bound on half-power VSWR bandwidth [9] for a very flat oblate spheroid of radius is times the upper bound for the half-power VSWR bandwidth of a sphere of radius . As their thicknesses approach zero, we find from Osborn [28] that the polarizability perpendicular to either the long axis of a or to the plane of a very slender prolate spheroid approaches zero, so flat oblate spheroid . that in either case the A. Electric-Dipole Antennas Fed With Global Electric Surface Currents Only The lower bounds on the quality factor given in (9) for an electric-dipole antenna with material (except for that of the exciting sources) confined to an electrically small volume ( denoting the shape of the 3D region as well as the value of the volume of the 3D region) were derived assuming this antenna-proper could be excited by magnetic as well as electric surface currents outside the antenna-proper. Since magnetic charge does not exist, as mentioned in Footnote 4, applied magnetic surface currents would have to be produced by thin layers of magnetization from natural magnetic material or synthesized from small Amperian current loops (or possibly slots in electrically conducting surfaces). Recently, Thal [12], [16] determined the lower bounds of electrically small spherical antennas allowing only global applied electric surface currents in free space. In this section, the lower-bound expressions in (9) will be modified to obtain the lower bounds on the quality factor for electric-dipole antennas with the antenna-proper confined to an arbitrarily shaped volume and excited by applied electric surface currents alone outside .

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To do this, remove the magnetic surface current and replace the electric surface current on a surwith a new electric surface current on that proface duces the same fields outside . This electric surface current will no longer be equal (or proportional) to and the fields will no longer be zero inside . However, keeping to bring the fields to zero inside would a PEC just inside induce electric surface currents opposite and nearly equal to the original surface currents such that the total electric surface cur. rents and the fields they produce would be proportional to Unfortunately, it can be proven that the ratio of the stored enand the PEC to the radiated power becomes ergy between approaches the surface of the PEC and, thus, the larger as conductor increases the quality factor.6 Also, introducing matewithout exciting rial with high magnetic permeability inside an appreciable magnetic-dipole moment would not lower the quality factor of the electric-dipole antenna, because a high-permeability material differs from a perfect magnetic conductor (PMC) in that it does not lower the interior electric field. In short, we are left with the problem of determining the quality factor of an antenna consisting of electric surface current on (which is arbitrarily close to the surface of ) in free space electric-dipole fields outside that produces the same and nonzero fields in the free space inside . are the same as the original fields, Since the fields outside the lower bound on the quality factor is given by the right-hand side of (2) with the electric-field energy from inside added , the energy to the stored electric-field energy from outside is given in the numerator of (2). Now the electric field inside simply the negative of the original incident field because on producing the original fields outside of the the PEC antenna-proper in a uniform incident field is identical to the electric surface current excited by a uniform incident field on a PEC. Thus, the electric-field energy inside adds the integral (16)

to the value of in (2). This term ends up canterm in (9a), thereby yielding the following celing the lower bound on the quality factor of electric-dipole antennas restricted to global electric surface current excitations outside the material of the antenna-proper confined to electrically small volumes (17a)

and for

in a principal direction with polarizability element

(17c) which compares closely in form with the inverse of the “radiaof Wheeler’s [1], [11] capacitor tion power factor” or inductor cylindrical antennas where is the axial “capacitor” or “inductor” shape factor, respectively. For the sphere with radius , the very flat oblate spheroid with maximum semi-axis length , and the very slender prolate spheroid with (18) The for very flat oblate spheroids and very slender prolate spheroids do not differ appreciably from the least lower bounds in (9) because or for these small of an electrically small volumes. The factor of 3/2 for the electric-dipole antenna restricted to a sphere, excited by global electric surface current only, agrees with the lower bound obtained by Thal [12]. The corresponding upper bounds of on the half-power VSWR bandwidth of electric-dipole antennas excited by global electric surface current only on a sphere, a very flat oblate spheroid, and a prolate spheroid agree with the very slender upper-bound half-power VSWR bandwidth ratios of (1, .42, .05) obtained by Gustafsson et al. [14], [15] for a sphere, thin disk, and prolate spheroid using a derivation based on their “sum rule” with their “generalized absorption chosen equal to 0.5 and the magnetic-dipole efficiency” moment of their antenna-proper equal to zero. III. MAGNETIC-DIPOLE ANTENNAS of The least lower bound on the quality factor an electrically small magnetic-dipole antenna-proper in a volumetric region can be found by repeating the derivation in Section II with a hypothetical perfect magnetic conductor (PMC) rather than a PEC to get the following results analogous to (9) (19a) and for scalar (19b) or for

in the direction of a principal element (19c)

For a scalar

, (17a) reduces to

placed a distance

where or is the magnetostatic dyadic or scalar polarizability of the PMC volume ; that is, the magnetic-dipole moof the PMC volume is given in terms of a ment as uniform incident magnetostatic field

 above a PEC sphere of radius a gives a quality factor obtained from (1) equal to (:5a= + 1)=(ka) for k ka 1.

(20a)

(17b) 6For example, the electric-dipole electric surface current

 

K

YAGHJIAN AND STUART: LOWER BOUNDS ON THE Q OF ELECTRICALLY SMALL DIPOLE ANTENNAS

or taking the inverse (20b)

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a result that can be proven by solving the boundary value problem of an electric surface current of a magnetic dipole at the surface of a sphere of permeability [29], [30], [38].8 IV. CONCLUSION

Because the problems of a PMC in a magnetostatic field and a PEC in an electrostatic field are duals of each other for the same . direction of the dipole moments (21) In other words, the least lower bounds for the quality factors of electric-dipole antennas and magnetic-dipole antennas in a given are the same. Moreover, the lower bounds on the quality factors of magnetic-dipole antennas excited by global electric surface current only, unlike the lower bounds on the quality factors of electric-dipole antennas excited by global electric surface current only, are identical to the least lower bounds in (19) for magnetic dipoles excited by both electric and magnetic surface currents; that is (22) The reason for this identity can be understood by applying the argument in Section II-A for the electric surface currents of electric-dipole antennas to the electric surface currents of magnetic-dipole antennas. When the argument is applied to magnetic dipoles, and the volume is filled with material with a high magnetic permeability, the induced equivalent magnetic surface current on the high-permeability material approximately outside while nulling the total magdoubles the fields of netic-field energy inside . Therefore, the quality factor of this high-permeability magnetic-dipole antenna excited by electric surface currents alone approaches the same lower bound as the antenna excited by both electric and magnetic surface currents Chu lower bound if is a sphere). (the The lower bounds in (19) and (22) are attained in principle by placed over a PMC in electric surface currents that produce a uniform magnetic field in in the absence of the PMC. The result that the lower bound for spherical magnetic-dipole antennas with high-permeability spherical cores could approach the Chu lower bound was apparently first discovered by Wheeler in his innovative work with electrically small antennas [2], [37]. He found that the quality factor of an electrically small, spherical, magnetic-dipole antenna of radius with a lossless core (and ) is given by7 relative permeability (23) 7It can be shown [29], [30], [38] that the formula in (23) for the quality factor assumes that the value of ka is below any resonant value of the magnetic core and that the dispersion of  is negligible over the relevant bandwidth (d(! )=d!  ) [9], a condition that can generally be satisfied if  1 or, allowing for artificial diamagnetic material,  > 0 [31].





Lower bounds on the quality factor of electrically small dipole antennas with the material of the antenna-proper (which excludes the material of the sources) confined to an arbitrarily shaped volume are determined from the general electromagnetic field expression for the quality factor of an antenna. The derivations of these lower bounds in terms of the direction of the dipole moment, the electrical size of the volume , and the per unit volume static PEC and PMC electric and magnetic polarizabilities of are enabled by equivalent surface-current representations for the fields exciting the antenna-proper, and by the uncoupling of Maxwell’s equations into quasi-electrostatic and quasi-magnetostatic fields for sources and media confined to electrically small volumes. As one would expect from the duality in Maxwell’s equations, the lower bounds for either electric-dipole or magnetic-dipole antennas with the material of the antenna-proper confined to a volume are identical. For a spherical antenna, this common lower bound equals the Chu , , a value lower than that of lower bound of any other volume shape circumscribed by the sphere. If only global electric surface current excitations are allowed to excite the antenna-proper, the lower bounds on the of magnetic dipoles do not change (if high-permeability materials in the antenna-proper are allowed), whereas the lower bounds on , the of electric dipoles increase, equaling , for a spherical volume—this latter result obtained by Thal [12]. The reason for this difference is that true PMC material with vanishing interior electric as well as magnetic fields does not exist and high-permeability materials do not lower the interior electric-field energy as they do the interior magnetic-field energy. The general expressions in (9), (17), (19), and (22) for the lower bounds of electrically small electric- and magnetic-dipole antennas with unrestricted sources ((9) and (19)) and sources restricted to global electric surface currents ((17) and (22)) are apparently new. However, the global electric surface-current lower bounds in (17) are closely related to the lower bounds obtained 8The determination of the lower bound on the Q of a magnetic dipole produced by global electric surface currents confined to a volume V in free space ( = 1) is equivalent to determining the Q of the scattered magnetic-dipole fields of a PEC volume V in a uniform incident magnetic field (since the electric surface current on the PEC volume V is that which produces a uniform field inside V and a magnetic-dipole moment outside V ). For the particular case of a spherical V , a unit incident magnetic field produces a magnetic-dipole moment that is one-half the value of the electric-dipole moment of a PEC sphere in a unit electric incident field. Thus, for a unit of stored magnetic scattered energy inside the free-space sphere, the stored scattered energy outside the sphere and the radiated power of the scattered magnetic-dipole fields are one-fourth that of the corresponding electric-dipole problem. In the electric-dipole problem, the ratio of the quality factor of the stored energy inside the sphere to the quality factor of the stored energy outside the sphere was found to be 1/2, giving a quality factor of (1=2 + 1)=(ka) = 1:5=(ka) . Thus, for the magnetic dipole produced by global electric currents in free space, the quality factor is

Q

=

1=2 + 1=4

1

1=4

(ka)

=

3 (ka)

(24)

which agrees with (23) for  = 1 and with Thal’s result for global electric surface currents in free space producing a magnetic dipole [12], [32], [33].

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by Gustafsson et al. [14], [15] using a “sum rule.” The evaluation of our lower-bound expressions that apply to electric-dipole antennas excited by global electric surface currents alone for the sphere, very flat oblate spheroid (disk), and very slender prolate spheroid agree with the lower bounds obtained by Gustafsson et al. using their “sum rule” with their magnetic-dipole moment equal to zero and their “generalized absorption efficiency” equal to 0.5. The lower-bound expressions in (9), (17), (19), and (22) apply to lossless antennas. As mentioned in Section II, for lossy, nondispersive antennas (conductivities nearly independent of frequency over the bandwidth of the antenna), these lower bounds are merely multiplied by the radiation efficiency of the antenna [9], [10], [13]. In [13] it is also shown that if both electric and magnetic dipoles are present in the radiated , then the lower bounds power ratio equal to with . In addition, it on the are reduced by a factor of is shown in [13] that frequency dispersion in low-loss isotropic materials (with negative or positive real values of permittivity and permeability) cannot further reduce the lower bounds on , whereas very lossy dispersive materials appear to allow for the possibility of further reducing the lower bounds by as much as a factor of two (in addition to the reduction from the efficiency factor ). Lastly, we mention some electrically small, single-resonance antennas found in the literature that have quality factors close to the lower bounds. Best [34] has designed, simulated, built, and tested an efficient, self-resonant, impedance matched, electrically small, four-arm, spherical-helix electric-dipole antenna , at about 300 MHz with a approximately equal to the Thal lower bound for electric dipoles excited by global electric surface currents on a sphere (a lower bound predicted by the general global electric surface current lower-bound formulas in (17), (18)). Adams and Bernhard [35] later designed and simulated alternative wound-wire, spherical, electric-dipole antennas that offer comparably low values. Stuart and Pidwerbetsky [36] have designed and simulated an efficient, self-resonant, impedance matched, electrically small, negative-permittivity, spherical, electric-dipole antenna at about 2 GHz with a approximately equal to . This spherical electricdipole antenna does not use magnetic polarization excitation sources and thus attains close to the lowest possible quality factor predicted by the global electric surface current lowerbound formulas in (17), (18). Recently, Kim et al. [29], [30], [38] have designed and simulated an efficient, self-resonant, , four-arm impedance matched, electrically small magnetic-dipole anspherical-helix, magnetic-core tenna with a equal to , a lower bound not too much , the Chu-Wheeler lower bound and the greater than lower bound predicted by the general formulas in (19) for magnetic-dipole antennas. Best [32] and Kim [33] have designed and simulated air-core spherical magnetic-dipole antennas with quality factors that approach the Thal free-space global electric surface-current lower bound for magnetic-dipole antennas ; see Footnote 8. of The derivation of the lower-bound formulas in (9) indicates that the Chu lower bound for spherical electric-dipole antennas can be approached if thin-shell magnetic polarization

sources are used to excite the electric dipole. Although this may appear to be a formidable engineering task, in fact, spherical electric-dipole antennas have been recently designed and simulated using thin, high permeability shells that reduce the to a [22], [23]. value of REFERENCES [1] H. A. Wheeler, “Fundamental limitations of antennas,” Proc. IRE, vol. 35, pp. 1479–1484, Dec. 1947. [2] H. A. Wheeler, “The spherical coil as an inductor, shield or antenna,” Proc. IRE, vol. 46, pp. 1595–1602, Sep. 1958. [3] H. A. Wheeler, “Small antennas,” in Antenna Engineering Handbook, R. C. Johnson and H. Jasik, Eds., 3rd ed. New York: McGraw-Hill, 1993, ch. 6. [4] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [5] R. F. Harrington, “Effect of antenna size on gain, bandwidth, and efficiency,” J. Res. Nat. Bureau Stand., vol. 64D, pp. 1–12, Jan. 1960. [6] R. E. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Trans. Antennas Propag., vol. AP-12, pp. 23–27, Jan. 1964. [7] R. L. Fante, “Quality factor of general ideal antennas,” IEEE Trans. Antennas Propag., vol. AP-17, pp. 151–155, Mar. 1969. [8] J. S. McLean, “A re-examination of the fundamental limits on the radiation of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, pp. 672–676, May 1996. [9] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 1298–1324, Apr. 2005. [10] A. D. Yaghjian, “Improved formulas for the Q of antennas with highly lossy dispersive materials,” IEEE Antennas Wireless Propag. Lett., vol. 5, Aug. 2006, Online. [11] A. R. Lopez, “Fundamental limitations of small antennas: Validation of Wheeler’s formulas,” IEEE Antennas Propag. Mag., vol. 48, pp. 28–35, Aug. 2006. [12] H. L. Thal, “New radiation Q limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 2557–2761, Oct. 2006. [13] A. D. Yaghjian, “Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material,” IEEE Trans. Antennas Propag., vol. 55, pp. 1495–1505, Jun. 2007. [14] M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape,” Proc. Roy. Soc. A, vol. 463, pp. 2589–2607, 2007. [15] M. Gustafsson, C. Sohl, and G. Kristensson, “Illustrations of new physical bounds on linearly polarized antennas,” IEEE Trans. Antennas Propag., vol. 57, pp. 1319–1327, May 2009. [16] R. C. Hansen and R. E. Collin, “A new Chu formula for ,” IEEE Antennas Propag. Mag., vol. 51, pp. 38–41, October 2009. [17] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,” J. Franklin Inst., vol. 249, pp. 57–154, Jan. 1950. [18] A. Hujanen, J. Holmberg, and J. C.-E. Sten, “Bandwidth limitations of impedance matched ideal dipoles,” IEEE Trans. Antennas Propag., vol. 53, pp. 3236–3239, Oct. 2005. [19] H. R. Stuart, S. R. Best, and A. D. Yaghjian, “Limitations in relating quality factor to bandwidth in a double resonance small antenna,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 460–463, 2007. [20] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [21] T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications. Piscataway, NJ: IEEE, 1999. [22] H. R. Stuart and A. D. Yaghjian, “Approaching the lower bounds on Q for electrically small electric-dipole antennas using high permeability shells,” IEEE Trans. Antennas Propag., 2010, accepted for publication. [23] H. R. Stuart and A. D. Yaghjian, “Using high permeability shells to improve the Q of electrically small electric-dipole antennas,” presented at the IEEE-APS/URSI Symp., Toronto, ON, Canada, 2010. [24] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1999.

Q

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[25] A. Sihvola, P. Ylä-Oijala, S. Järvenpää, and J. Avelin, “Polarizabilities of platonic solids,” IEEE Trans. Antennas Propag., vol. 52, pp. 2226–2233, Sep. 2004. [26] C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on broadband scattering by heterogeneous obstacles,” J. Phys. A: Math. Theor., vol. 40, pp. 1165–1182, 2007. [27] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983. [28] J. A. Osborn, “Demagnetizing factors of the general ellipsoid,” Phys. Rev., vol. 67, pp. 351–357, Jun. 1945. [29] O. Breinbjerg and O. S. Kim, “Minimum Q electrically small spherical magnetic dipole antenna—Theory,” presented at the ISAP, Bangkok, Thailand, Oct. 2009. [30] O. S. Kim, O. Breinbjerg, and A. D. Yaghjian, “Electrically small magnetic-dipole antennas with quality factors approaching the Chu lower bound,” IEEE Trans. Antennas Propag., vol. 58, pp. 1898–1906, Jun. 2010. [31] A. D. Yaghjian, “Power-energy and dispersion relations for diamagnetic media,” in Proc. Antennas and Propagation Society Int. Symp., Charleston, SC, Jun. 2009, pp. 1–4. [32] S. R. Best, “A low-Q electrically small magnetic (TE mode) dipoles,” IEEE Antennas Wireless Propag. Lett., vol. 8, 2009. [33] O. S. Kim, “Low-Q electrically small spherical magnetic dipole antennas,” IEEE Trans. Antennas Propag., 2010, accepted for publication. [34] S. R. Best, “The radiation properties of electrically small folded spherical helix antennas,” IEEE Trans. Antennas Propag., vol. 52, pp. 953–960, Apr. 2004. [35] J. J. Adams and J. T. Bernhard, “Tuning method for a new electrically small antenna with low ,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 303–306, 2009. [36] H. R. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antennas Propag., vol. 54, pp. 1644–1653, Jun. 2006. [37] H. A. Wheeler, “Correction: The spherical coil as an inductor, shield or antenna,” Proc. IRE, vol. 48, p. 328, Mar. 1960. [38] O. S. Kim and O. Breinbjerg, “Minimum Q electrically small spherical magnetic dipole antenna—Practice,” presented at the ISAP, Bangkok, Thailand, Oct. 2009.

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Arthur D. Yaghjian (S’68–M’69–SM’84–F’93– LF’09) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Brown University, Providence, RI, in 1964, 1966, and 1969. During the spring semester of 1967, he taught mathematics at Tougaloo College, MS. After receiving the Ph.D. degree he taught mathematics and physics for a year at Hampton University, VA, and in 1971 he joined the research staff of the Electromagnetics Division, National Institute of Standards and Technology (NIST), Boulder, CO. He transferred in 1983 to the Electromagnetics Directorate, Air Force Research Laboratory (AFRL), Hanscom AFB, MA, where he was employed as a Research Scientist until 1996. In 1989, he took an eight-month leave of absence to accept a Visiting Professorship in the Electromagnetics Institute, Technical University of Denmark. He presently works as an independent consultant in electromagnetics. His research in electromagnetics has led to the determination of electromagnetic fields in materials and “metamaterials,” the development of exact, numerical, and high-frequency methods for predicting and measuring the near and far fields of antennas and scatterers, the design of electrically small supergain arrays, and the reformulation of the classical equations of motion of charged particles. Dr. Yaghjian is a Life Fellow of the IEEE, has served as an Associate Editor for the IEEE and URSI, and is a member of Sigma Xi. He has received best paper awards from the IEEE, NIST, and AFRL.

Howard R. Stuart (M’98) received the S.B. and S.M. degrees in electrical engineering from the Massachusetts Institute of Technology, Cambridge, in 1988 and 1990, respectively, and the Ph.D. degree in optics from the University of Rochester, Rochester, NY, in 1998. From 1990 to 1993, he worked as a Research Scientist for the Polaroid Corporation in Cambridge, MA. In 1998, he joined Bell Laboratories, Lucent Technologies, as a Member of Technical Staff in the Advanced Photonics Research Department in Holmdel, NJ. Since 2003, he has worked in the Bell Labs Government Communications Laboratory, Florham Park, NJ, which became part of LGS Innovations in 2007. He has published papers on a variety of research topics, including small resonant antennas, metal nanoparticle enhanced photodetection, multimode optical fiber transmission, optical waveguide interactions and devices, optical MEMS, and optical performance monitoring Dr. Stuart served as the Integrated Optics Topical Editor for the OSA journal Applied Optics from 2002 to 2008.

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Analysis and Design of a Differentially-Fed Frequency Agile Microstrip Patch Antenna Sean Victor Hum, Member, IEEE, and Hui Yuan Xiong

Abstract—The design of a frequency agile microstrip patch antenna is described that is readily interfaced with differential RF transceivers. By integrating three pairs of varactor diodes with the patch antenna, and tuning them in unison, frequency tuning ratios approaching 2 are possible with the design. This is made possible by the intrinsically broadband nature of the differential feeding scheme used. An intuitive equivalent circuit for predicting the port characteristics of the antenna is presented. Moreover, the circuit is shown to be highly accurate in predicting losses produced by the varactor diodes and the consequent radiation efficiency of the antenna. To the best of the authors’ knowledge, this is the first time a detailed analysis of the effect of varactor diode losses has been undertaken in frequency agile antennas using an equivalent circuit. The equivalent circuit model is subsequently validated using full-wave simulations and experimental measurements of an antenna operating in the 2–4 GHz range. Index Terms—Frequency agile antennas, microstrip antennas, multifrequency antennas, reconfigurable antennas, varactors.

I. INTRODUCTION ODAY’S mobile wireless devices contain a multitude of different radios providing numerous services, spread across an ever-widening range of frequency bands. Future devices will undoubtedly need to support additional emerging wireless standards, and potentially cognitive-radio levels of agility [1] creating an enormous engineering challenge at the physical layer of these systems. While Moore’s Law has allowed the electronics to scale so that all these radios can fit on a few chips (or less), antennas do not scale in the same way. Designers must therefore create either a device incorporating multiple antennas to cover the frequency band associated with each service, or a single multi-band antenna that adequately covers them all. The latter approach is generally favoured in applications where space is a major constraint, but fixed multi-band antennas can be challenging to design. Furthermore, multi-band antennas require a significant amount of filtering circuitry so the receiver can handle increased interference from other bands, and crosstalk from transmit circuits using the same antenna. Frequency agile antennas (FAAs) are a unique class of antennas that have an operating frequency (or multiple frequen-

T

Manuscript received October 17, 2009; revised February 20, 2010; accepted March 27, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are with Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055805

cies) that can be dynamically manipulated in accordance with the bands that need to be serviced by the system’s radios. As such, the antenna can be shared for each of the services on the user’s terminal thus reducing the space required for the multiband antenna while potentially reducing out-of-band interference. Such antennas have attracted significant interest in recent years, owing mainly to advances in semiconductor materials, and micro-electro-mechanical system (MEMS) technologies. There have been significant developments in designing planar FAAs, since planar antennas such as patches typically suffer from low instantaneous bandwidths. Initial work on patch antennas [2], [3] showed that loading the radiating edge of a patch antenna with varactor diodes can produce frequency tuning over a 10–20% (1.1–1.2 frequency ratio) range. This ratio can be increased to 1.6 by loading both edges of the patch with diodes, and increasing the number of diodes used [4]. It is worth noting that diodes need not load the radiating edge to achieve these tuning ratios; recently it has been shown that diodes loading the non-radiating edges of a patch can achieve a tuning ratio of 1.6 while having the benefit of minimal perturbation of the radiation pattern [5]. Also, RF (e.g., PIN diode) switches can also be used to control the current path on a patch antenna [6]–[8], and produce frequency tuning ratios as high as 2 due to more pronounced changes in the current paths on the antenna. Work has also gone into developing dual-band frequency-tunable antennas using similar concepts. These tend to be based on the well-known planar inverted L (PIL) antenna and planar inverted F antenna (PIFA), with the notion of reconfigurability first suggested in [9]. Single-band designs employing varactor diodes followed shortly [10] with tuning ratios of 1.5. Most recently, dual-band PIFA designs based on varactor diodes [11]–[14] as well as switches [15]–[18] have been approached, with some designs supporting independent control over two or more bands. The slot antenna, too, provides an excellent platform for realizing tunable multi-band antennas. A dual-band slot antenna based on loading with varactor diodes has been experimentally demonstrated to provide two tunable bands, where one band had enough frequency agility to give a tuning ratio of about 1.5. Other slot-based designs have explored the use of slot-ring antennas for dual-polarization applications [19] and also demonstrate excellent tuning ratios. Varactor diodes and switches clearly make up the bulk of the approaches to implementing frequency agility. A combination of these approaches can yield even greater degrees of frequency agility [20]. It is also worth mentioning that there are many emerging technologies that promise to change the landscape of frequency tunable antennas. MEMS technology is particularly promising. MEMS switches have already been explored for use in patch-based [21] and slot-based [22] designs, and pos-

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sess many advantages because of their low loss and high linearity. Interesting MEMS antennas based on a variable-air-gap patch [23] and unfurling patch antenna designs [24] have also been presented, with particularly large tuning ratios offered by the latter approach. Finally, exotic technologies such as magnetoelectric substrates [25] and liquid-crystal substrates [26] promise to make distributed tuning a reality for FAAs. While there have been significant advances in the design of FAAs, there is room for improvement in two major areas related to planar designs: achieving larger tuning ratios, which will be necessary as services are deployed over a wider frequency band, and amenability for integration with radio frequency integrated circuits, which often employ differential inputs/outputs. This paper presents a differentially-fed frequency agile patch antenna that uses the combination of these two features to achieve high levels of frequency agility using only varactor diodes as the tuning mechanism. The paper is organized as follows. Section II introduces the fundamental FAA design. Section III describes how the design can be modelled using an equivalent circuit. Section IV presents the tuning characteristics of the antenna, as evaluated using the equivalent circuit, full-wave simulations, and experimental measurements. Section V describes the effect of varactor diodes on the efficiency of the antenna, and presents a comparison of efficiency predictions using various techniques. Section VI presents the radiating characteristics of the design. Finally, conclusions are put forth in Section VII. II. PROPOSED DESIGN A well-known disadvantage of microstrip patch antennas is their poor bandwidth. Since microstrip antennas are highly resonant structures, their operating bandwidth is usually no more than a few percent. However, as resonators, they are highly amenable to various tuning techniques. The challenges in engineering a patch that can be tuned over a wide frequency range, though, are many-fold. First, most of the loading techniques suggested in the literature are fundamentally limited by the agility of the tuning element itself. For example, hyperabrupt varactor diodes typically do not provide capacitance tuning ratios exceeding 10, which can have corresponding implications on the range over which a patch’s resonant frequency can be tuned. Hence, a way is needed to realize resonators with greater levels of tunability. Second, an equally challenging problem is creating a coupling mechanism that efficiently couples to the patch over a very wide frequency range, because many traditional feeding techniques for patches are inherently narrowband. Finally, ease of integration with RFICs requires in many cases that antenna be differentially-fed. Otherwise, a hybrid must be used to convert a single-ended antenna feed into one compatibile with a differential RFIC input/output stage. A potential solution to feeding patch antennas differentially was recently presented in [27]. Two differentially-fed microstrip lines were proximity-coupled to a patch using a multilayer structure, as shown in Fig. 1(a). The authors of the paper observed that replacing the upper patch layer with patches of different , allowed them to realize a large range of dimensions resonant frequencies from the antenna without modifying the feed dimensions or geometry. For the geometry presented in

Fig. 1. Topology of a differentially-fed patch antenna (a) fixed-frequency design, (b) tunable-frequency design.

[27], patches resonating between 1.46 to 2.77 GHz could be efficiently coupled to. This raises the question of how well such a coupling structure would work with a frequency agile patch. The effective tuning ratio with fixed patches has already been shown to be 1.89, which is substantial. Replacing the fixed patches with a single tunable patch can potentially offer a comparably large tuning ratio. The design of such a patch structure is shown in Fig. 1(b). The essence of the FAA design is how two groups of varactor diodes are introduced at different locations of the patch and how together they produce a very wide usable frequency range. The shunt group consists of two pairs of varactor diodes at the patch corners that are connected to the ground plane through vias. These diodes provide a large amount of controllable capacitive loading equally on both sides of the patch antenna. The symmetry of the antenna is preserved while the large capacitive loading introduces a significant level of frequency agility (tuning ratios close to 2 can be easily achieved). They load the radiating edge of the patch similar to traditional FAA designs [2], [4]. However, a major problem is that the antenna bandwidth becomes very small and the antenna patch becomes difficult to feed, because the antenna becomes electrically smaller as the

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Fig. 2. Top and side views of frequency agile patch antenna.

varactor diode capacitance is increased. Also, at high capacitance levels, increased current flow through the diodes and loss in the varactor diode becomes a very significant problem in terms of antenna efficiency. To address these problems, a gap is cut transversely in the middle of the antenna as shown in Fig. 1(b). This gap significantly increases the working frequency of the patch, since the gap serves to load the patch in a serial fashion which results in miniaturization [28]. The resulting larger electrical size of the antenna improves its bandwidth and improves efficiency due to the diminished current flow through the diodes due to the lower tuning capacitance used at higher working frequencies. Diodes are added across the gap that serially connect the two patch-halves, similar to the patches presented in [29]. This extends the frequency agility of the patch while maintaining the higher operating frequency of the antenna. For simplicity of design and analysis, it will be assumed that the series and shunt group capacitors are identical. III. EQUIVALENT CIRCUIT MODELING OF THE PROPOSED DESIGN We now derive and explain an equivalent circuit model that can be used to aid in analyzing the behavior this type of FAA. Such a circuit model can be used to produce insights into the operation of the FAA while simultaneously providing designers with a tool that could be used to design FAAs quickly, using simple circuit analysis techniques. The interest in the model is primarily to use it to explain experimental observations. In Section V, it will be shown that the circuit model is particularly valuable in predicting the radiation efficiency of this type of antenna. The antenna structure shown in Fig. 2 has three distinct regions to the patch structure, which are labelled with numerals along the bottom side of the figure. Regions 1 and 3 are standard microstrip-line sections. Section 2 is a coupled-line region whereby power is exchanged between layers, which is responsible for coupling into the patch. We also note that since the structure is differentially driven, i.e., an inverted version of the excitation at port S1 appears at port S2, the circuit possesses odd symmetry in the longitudinal direction which leads to a virtual ground plane located along the plane of symmetry. This is indicated as a line of symmetry in Fig. 2. Hence, the analysis of this circuit can be simplified to considering the behavior of a simple one-port circuit, driven from one of ports S1 or S2. That

Fig. 3. Top/side view of half-patch and equivalent circuit. Circled numerals indicated port numbers for the coupled-line section.

is, the differential input reflection coefficient of the antenna can be found simply by measuring the single-ended reflection coefficient at ports S1 or S2 of the circuit in Fig. 2 with the virtual ground plane in place. The half-circuit of Fig. 2 can be analyzed using established techniques for proximity-coupled patches, e.g., [30]. The feed line section, region 1, does not directly factor into the analysis other than adding delay to the response, assuming the feed line is impedance-matched to the patch and lossless. Hence, we focus our attention on regions 2 and 3. The structure under analysis, and the associated equivalent circuit, are shown in Fig. 3. The feed, originally at port S1, has been transformed to the input to region 2, at port S1’. Transmission line theory predicts that the three-conductor line in region 2 can be modelled as a set of two coupled transmission lines, as shown in Fig. 3. These lines support two modes of propagation, commonly named the -mode and the C-mode, with distinct propagation constants and characteristic impedances [31]. For the geometry shown, the two propagation modes are found to be the well-known stripline mode (C-mode) and the microstrip line mode ( -mode). Meanwhile, region 3 is a microstrip line which incidentally has the same characteristic impedance and propagation constant as the -mode of the coupled-line region. The rest of the equivalent circuit in Fig. 3 consists of a number of lumped-element components. A number of transmission line discontinuities exist in the equivalent circuit. The open-ended line representing the edge of the patch can be modeled using a fringing capacitance designated . Similarly, the gap in the centre of the patch has an associated capacitance which has been symmetrically split between two series capacitors of value . The varactor diodes are modeled using a simple series RLC cir, and varicuit consisting of a package inductance , loss able capacitance . Note the reactances associated with the varactor diode on the right hand side of the circuit have been adjusted by a factor of two as the varactor has effectively been represents split by the virtual ground in Fig. 2. The inductor additional electrical length added to the circuit because of currents meandering through the conductors to reach the varactor diodes in the series group. The inductor , which models the corresponding current crowding for the shunt varactor group as

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TABLE I RELEVANT FAA DESIGN PARAMETERS

Fig. 4. Equivalent circuit of coupled-line section.

well as the via inductance. The radiation resistance appears across the radiating edge of the half-patch. Finally, note that the middle conductor of the coupled-line section has been left open-circuited, although even- and odd-mode fringing capacitances could be added there for improved accuracy. However, evaluation of the circuit model showed that including these capacitances did not significantly alter the circuit’s behavior, and hence they have been omitted. The values of the inductances and capacitances in Fig. 3 can be found from full-wave simulations of corresponding microstrip discontinuities, such as open-ended lines, and via-loaded lines. Similarly, the radiation resistance of the patch can be found as a function of frequency by evaluating the characteristics of the patch in the far field. While these individual simulations do constitute some up-front effort, the effort does produce a model that accurately captures the behavior of the antenna for analysis purposes. Many of the quantities could also be estimated using empirical formulas. For ease of analysis, the substrate is considered to be homoge), allowing established techniques to be used nous (i.e., for determining the propagation constants and characteristic impedances associated with each mode. Knowing the parameters of the coupled-line section, a standard model for general coupled-lines can be employed to model the coupled-line section as a four-port network [32]. This allows the equivalent circuit to be reduced by substituting the equivalent circuit of Fig. 4 for the coupled-line section. The ports 1–4 shown in the diagram correspond to the ports indicated by circled numerals in Fig. 3. IV. TUNING RANGE ASSESSMENT With the parameters of the equivalent circuit completely described, we now present a candidate FAA design and compare the circuit model to full-wave simulations as well as experimental measurements. The design proposed here differs considerably from that presented in [33]. It was noted in [33] that while simulated and experimental fixed-capacitor versions of the prototype exhibited large frequency tuning ratios (more than 2.4), the introduction of real varactor diodes significantly reduced the tuning range. This was initially attributed to the simplified biasing scheme that was used, whereby the series group of diodes were biased at double the voltage compared to the shunt group, which is not ideal. This has been corrected in the new experimental antenna through the use of an improved biasing scheme, discussed in more detail in Section IV.B. However, simulations also revealed that the introduction of losses into the tuning diode seriously degraded the impedance match, despite

the fact that the varactor diodes used only have a series resistance of . Hence, the design was modified to produce a better impedance match with real varactor diodes. The design also uses two substrates with identical dielectric constants (unlike [33]), in order to simplify the evaluation of the equivalent circuit model (particularly, the calculation of coupled-line parameters and open-circuit line discontinuities). The design was optimized in full-wave simulations using the Simulation platform for EMC, Antenna Design and Dosimetry (SEMCAD) X [34] FDTD simulation package to work with Rogers RO3203 substrates and to produce tunability in the 2–4 GHz range. The relevant parameters of the patch are shown in Table I, and where appropriate, electrical sizes in free-space wavelengths at the tuning range extremities have been shown. The frequency tunability of this design was characterized by examining the differential reflection coefficient produced from three methods: 1) full-wave simulations in SEMCAD; 2) calculations using the equivalent circuit model proposed in Section III; and 3) experimental measurements. For simulations and circuit predictions, all antenna components (dielectric and conductors) are assumed to be lossless for illustrative purposes; the effect of diode losses will be discussed shortly. A. Evaluation of Equivalent Circuit To validate the equivalent circuit model, the associated lumped element component values extracted and de-embedded from full-wave discontinuity simulations are shown in Table II. The constants associated with the coupled-line section are also summarized in this table. The transformer turns ratios and in the last two rows were found by even/odd mode analysis. nH Finally, a tuning capacitor package inductance of and a diode resistance of were used for the varactor diode, which are the same values as those of an experimental diode used later on. , being a significant function The radiation resistance of frequency over the operating frequency range of the FAA, can be evaluated using the closed-form empirical formulas [35] or calculated from full-wave simulations of the patch. It was found that the aperture blockage introduced by the feed produced some deviations from standard empirical formulas, therefore, the latter approach was adopted to accurately determine the value of the radiation resistance. This was done by monideveloped across the radiating edges of toring the voltage the patch and the radiated power from the patch, and using the

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TABLE II EQUIVALENT CIRCUIT PARAMETERS OF DISCONTINUITIES AND COUPLED-LINES

relation , where it is noted the the radiated power is divided equally between the two radiating edges of the patch. The differential reflection coefficient of the antenna is readily determined by determining the two-port, single-ended scatof the patch, and then determining the tering parameters reflection coefficient seen from a differential source as [27] (1) A plot of the differential reflection coefficient, as predicted by SEMCAD and the equivalent circuit model, is plotted in Fig. 5. First, we note an excellent correlation between the reflection coefficient produced by the two techniques. This validates the ability of the equivalent circuit model derived in Section III to accurately predict quantities measured at the antenna ports, though we will extend its utility to radiation characteristics shortly. Most significantly, the results demonstrate a large frequency tuning range is predicted by both methods. Considering the full-wave simulations, and initially considering tuning capacitances in the 0.1–2.0 pF range, a tuning range between 1.859 GHz and 3.672 GHz, corresponding to a frequency ratio of 2, can be easily achieved while maintaining a reflection dB. This tuning ratio value coefficient magnitude below is consistent with earlier measurements of an experimental antenna using fixed, high-Q capacitors [33], and is achieved with capacitances only in the 0.1–1.0 pF range to meet the 10 dB return loss threshold. The tuning range also compares favourably to designs in the literature, which generally have tuning ratios of 1.6 or less, though recently have approached 2 [19]. The resonant frequency prediction between the circuit model and full-wave simulations is excellent. The circuit seems to predict well in the middle of the capacitance range, but the worst-case error in the resonant frequency is only 1.7% which pF. The correlation could be improved if occurs for circuit models predicting higher-order effects from the discontinuities were used, since lumped elements associated with microstrip discontinuities are in general frequency-dependent, and the circuit model is being exercised over a large frequency range. Some discontinuities were also ignored, such as the fringing capacitances produced by the feedline. Nevertheless, for a simple circuit model, it does well in tracking both the resonant frequency, as well as the bandwidth and return loss achievable with the designs. Hence, the circuit can be readily used by designers to predict tuning ranges of FAAs, even

Fig. 5. Comparison of reflection coefficient predicted by full-wave simulations and the equivalent circuit model.

without accurate knowledge of circuit parameters, if maximizing the tuning range of the circuit is of primary interest. B. Experimental Measurements of Tuning Range For experimental measurements, a prototype of the patch was fabricated on Rogers RO3203 substrates and the two substrates aligned and secured using nylon screws. Aeroflex MGV-100-21-E28X varactor diodes were used for the tuning diodes, which provide a minimum tunable capacitance range of 0.65–2.2 pF. To bias the diodes, a new biasing scheme was devised so that the same voltage could be developed across all three pairs of diodes, since the design assumes that the tuning capacitances are varied in unison. Previously, for simplified biasing, only the shunt group of diodes were biased in this manner, and lack of DC isolation of the series group meant the diodes were biased at twice the voltage of the shunt group [33]. In this design, a pair of RF shorts (RFSs) has been introduced where one pair of diodes connects to the ground plane, in order to provide DC isolation of all three pairs of diodes. The scheme is shown in more detail in Fig. 6. A positive DC bias voltage is applied to the terminal labelled V while an equal and opposite voltage is applied to the terminal labelled V, with respect to the ground plane. 10 k resistors were used for the bias resistors. The RFS consisted of a 5.6 pF capacitor self-resonant at 3.5 GHz, though in the experiment four were placed in a star arrangement to provide a better connection with the ground plane. It was confirmed, through the circuit model, that the RFS had a negligible effect on the tuning range and impedance match of the patch, an observation that was also corroborated with full-wave simulations. A photograph of the complete design is shown in Fig. 7. The two-port scattering parameters of the patch were measured using a network analyzer as function of frequency and bias voltage, and the differential reflection coefficient computed using (1). The results of this measurement are shown in Fig. 8. Each curve corresponds to a different reverse bias voltage across the varactor, in increments of 1 V. The associated diode capacitance, as determined from the diode’s data sheet parameters, has

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Fig. 6. DC biasing scheme.

Fig. 8. Measured reflection coefficient as a function of tuning diode bias voltage and frequency. Each curve is plotted in increments of 1 V, and the corresponding diode capacitance value appears in parentheses.

Fig. 7. Photograph of FAA prototype.

also been shown in parentheses, for ease of comparison to Fig. 5. The experimental results align very closely to predictions from the other two techniques. We see that the return loss drops below 10 dB at around 1.8 GHz which is precisely where it crossed this threshold in the simulations. The upper frequency limit of the tuning is 3.15 GHz, corresponding to an overall tuning factor of 1.75. The trends in the change in return loss and bandwidth also track well with the full-wave and circuit predictions. In fact, the upper frequency limit is primarily imposed by the tuning limitations of the capacitor, rather than the patch and feed, since capacitances less than about 0.2 pF were not experimentally found to be achievable with this particular diode model. Also, at large diode capacitances where diodes were biased near zero volts, the FAA was not well impedance-matched according to the original design. Nevertheless, the frequency tunability of the design is impressive, and correlation with predictions extremely good. Fig. 9 compares the resonant frequencies as a function of diode capacitance, as derived from the equivalent circuit model, SEMCAD simulations, and measured results. The plot shows excellent correlation between the three techniques used to characterize the FAA. The minor discrepancy between the measured results and simulations is likely attributable to a slight deviation in the diode’s capacitance-versus-voltage characteristic from that provided on the diode’s data sheet. The plot further validates the accuracy of the circuit model. V. EFFECT OF VARACTOR DIODE LOSSES ON EFFICIENCY The effect of tuning component losses on the performance of the circuit can be significant when semiconductor devices are used to implement the tunable capacitor. To the best of our knowledge, the effect of this component on the efficiency has not been analytically studied in FAA designs. For this study,

Fig. 9. Resonant frequency of the FAA as a function of tuning capacitance.

other losses such as conductor and dielectric losses, are omitted to highlight the effect of the diode loss, which is the most significant source of loss in frequency agile antenna designs. This is particularly the case when the diode is biased for relatively are two-fold. large capacitances. The effects of a nonzero First, and most obvious, is that the efficiency of the antenna decreases. Second, lossy diodes seriously degrade the impedance match of this antenna design, if they are not taken into account in the selection of the patch parameters. Designers should be aware of this when integrating real diodes with frequency agile antennas. In terms of efficiency, more current flows through the diode losses can be significant as the varactor diode caand hence pacitance increases (frequency decreases). In fact, the radiation efficiency of the antenna can be easily predicted using the equivalent circuit model of Fig. 3. By exciting the circuit with a 50 voltage source of known amplitude, the power dissipated by the radiation resistance can be easily evaluated and divided

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TABLE III COMPARISON OF FULL-WAVE SIMULATION AND MODEL PREDICTIONS OF RADIATION EFFICIENCY

by the input power to yield the radiation efficiency of the antenna. The result of this calculation, using the equivalent circuit model, is compared with a efficiency predicted by SEMCAD using far-field simulations of the antenna, in Table III. From the table, we notice an excellent correlation between the predictions of the equivalent circuit model and the radiation efficiency predicted by full-wave simulations. Generally, the % circuit model underestimates the efficiency, producing a error in the worst case, which is a very acceptable amount of error. More importantly, both techniques underscore the relationship between tuning capacitance and radiation efficiency described above: that increases in capacitance promote increased losses through the diode, producing significant reductions in efficiency. Hence, in a practical design, the desired efficiency range of the antenna would have to be weighed carefully against the desired tuning range of the antenna. The equivalent circuit model can be used in evaluating this balance, to a large degree of accuracy. To further validate the efficiency prediction method, the efficiency of the constructed FAA was evaluated using two experimental techniques. First, the gain and directivity of the FAA was measured in a far-field anechoic chamber. Experimental pattern measurements are presented in more detail in Section VI. Using the measured gain and directivity the radiation efficiency can be determined. However, since the far-field gain was limited in dB, additional efficiency measurements accuracy to about were taken using the Wheeler Cap method [36] within various closed conducting spheres at multiple bias voltages. The results of these measurements and calculations are shown in Fig. 10. The upper plot shows the efficiencies determined from circuit model predictions, SEMCAD simulations, and the two experimental techniques. For the SEMCAD predictions in this plot, two models were evaluated. The first was the model described earlier which was lossless except for the diode losses, and is labelled as lossless in the legend. The second incorporates di, which electric and conductor losses assuming corresponds to the loss tangent for the Rogers RO3203 material, S/m for the conand 35 m rolled copper with ductor losses. It can be seen that even once dielectric and conductor losses are incorporated, the efficiency of the antenna drops only slightly, mainly at higher frequencies due to the greater impact of dielectric loss as frequency increases. The circuit model still does a good job of predicting the antenna efficiency, and furthermore could be adapted to incorporate lossy transmission lines to improve its accuracy. Comparisons to experimental

Fig. 10. Efficiency and gain as a function of operating frequency/tuning voltage.

results also show very good correlation. The measured efficiencies from the far-field techniques tend to be more dispersed, owing to small errors in the gain measurements, but otherwise follow the expected trends. The Wheeler Cap efficiencies are much closer to the lossy SEMCAD results. Overall all four techniques for determining efficiency are in excellent agreement. The lower plot of Fig. 10 compares the simulated gain from lossy SEMCAD simulations and the measured far-field gain, as a function of operating frequency. Here, too, excellent correlation between the two techniques is observed. The patterns are discussed in further detail in the following section. VI. RADIATION CHARACTERISTICS This antenna design is known for having good patch-like radiation characteristics over the operational bandwidth of the antenna [33]. The co-polarization and cross-polarization radiation and H-plane , patterns of the antenna, in the E-plane are shown for three different frequencies in Fig. 11. The patterns were computed from far-field simulations in SEMCAD, and from experimental measurements of the antenna in a far-field chamber. For the latter, ports 1 and 2 were driven separately, and the far field pattern derived synthetically using superposition, to simulate the antenna being driven differentially. The experimental patterns were also measured using a broadband 180 splitter used to split a single-ended driving signal into differential feeding signals for the antenna. No significant differences in the pattern were produced using the second technique. Also, only measured cross-polarization levels are included as simudBi. lated levels were below Overall, the simulated and measured gains agree very well with each other. The pattern shapes and backlobe levels are comparable for all frequencies tested, and the observed gain of the antenna drops with frequency in the same way as predicted by simulations. In the worst case, there is about 0.8 dB of error between the measured and simulated gains, which is considered acceptable given the accuracy of the gain of the chamber used

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tuned to different capacitances. The equivalent circuit not only predicts the frequency agility and bandwidth characteristics of the antenna, but also achieves high accuracy in its ability to predict the radiation efficiency of this particular type of FAA, when compared to both full-wave and experimental measurements. To the best of our knowledge, an efficiency analysis based on a circuit description of a frequency agile antenna has not been performed. This can be used to aid in the design of FAAs so that a suitable balance can be achieved between tuning range, bandwidth, and efficiency. This model also generates insight into the effect of tuning component loss on the antenna’s performance, and that if dimensions are not chosen carefully, FAAs can suffer from poor efficiencies due to the impedance-transforming nature of the feed. The proposed FAA achieves a good compromise between tuning range and efficiency, using a relatively simply bias scheme. In the future, it may be possible to address the efficiency of this antenna through developments in tuning components, particularly those based on MEMS and other high-Q tuning technologies. Additionally, these technologies may be able to improve the linearity of FAAs, while simultaneously providing greater tuning ranges than conventional semiconductor devices. This FAA design is amenable for integration with MEMS, and moreover, will benefit significantly from devices with larger tuning ranges as experiments have revealed an untapped frequency range due to limitations in the varactor diodes. This could enable antennas with even greater frequency agility to be developed in the future. ACKNOWLEDGMENT

Fig. 11. Simulated co-pol gain (solid), measured co-pol (dashed) gain of FAA, and measured cross-pol (dash-dotted), all in dBi (a) E-plane, 1.937 GHz, (b) H-plane, 1.937 GHz, (c) E-plane, 2.492 GHz, (d) H-plane, 2.492 GHz, (e) E-plane, 3.212 GHz, (f) H-plane, 3.212 GHz.

was only about 1 dB. Most importantly, the patterns are relatively invariant of frequency over the realized tuning range of the FAA, making the antenna an attractive candidate for applications requiring such characteristics. Cross-polarization levels are also very low, as expected from simulations. VII. CONCLUSION This paper shows the design of a frequency agile antenna that exhibits a large tuning ratio while simultaneously being able to be differentially fed, which can be an advantage for interfacing with differential transceivers. The design is based on integrating six varactor diodes with the structure of a microstrip patch and combining the antenna with an intrinsically broadband proximity-coupled microstrip feed. Experimentally, the resulting antenna exhibits a tuning factor of 1.75 while maintaining 10 dB of return loss and a frequency-invariant radiation pattern. Simulations of the antenna show that with lower loss tuning components, tuning ratios in excess of 2 could be potentially achieved. This paper has described an equivalent circuit that accurately captures the behavior of the FAA as the varactor diodes are

The authors are grateful to J. Martinko for his assistance in the fabrication of the antenna, to Y. Zhou for the radiation pattern measurements of the antenna, and to M. Antoniades for the Wheeler Cap efficiency measurements. REFERENCES [1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [2] P. Bhartia and I. J. Bahl, “Frequency agile microstrip antennas,” Microw. J., pp. 67–70, Oct. 1982. [3] R. Waterhouse and N. Shuley, “Full characterisation of varactor-loaded, probe-fed, rectangular, microstrip patch antennas,” IEE Proc. Microw. Antennas Propag., vol. 141, no. 5, pp. 367–373, Oct. 1994. [4] E. Bhuiyan, Y.-H. Park, S. El-Ghazaly, V. Nair, and H. Goronkin, “Active tuning and miniaturization of microstrip antennas,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., 2002, vol. 4, pp. 10–13. [5] E. Nishiyama and T. Itoh, “Dual polarized widely tunable stacked microstrip antenna using varactor diodes,” in 2009 IEEE Int. Workshop Antenna Tech. (iWAT 2009)., Mar. 2009, pp. 1–4. [6] L. Le Garrec, R. Sauleau, and M. Himdif, “A 2:1 band frequency-agile active microstrip patch antenna,” in Proc. Eur. Conf. Antennas Propag., (EuCAP 2007)., Nov. 2007, pp. 1–6. [7] A.-F. Sheta and S. Mahmoud, “A widely tunable compact patch antenna,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 40–42, 2008. [8] J. L. A. Quijano and G. Vecchi, “Optimization of an innovative type of compact frequency-reconfigurable antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 9–18, Jan. 2009. [9] K. Virga and Y. Rahmat-Samii, “Low-profile enhanced-bandwidth PIFA antennas for wireless communications packaging,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 10, pp. 1879–1888, Oct. 1997. [10] P. Panayi, M. Al-Nuaimi, and I. Ivrissimtzis, “Tuning techniques for planar inverted-F antenna,” Electron. Lett., vol. 37, no. 16, pp. 1003–1004, Aug. 2001.

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[11] S.-K. Oh, H.-S. Yoon, and S.-O. Park, “A PIFA-type varactor-tunable slim antenna with a PIL patch feed for multiband applications,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 103–105, 2007. [12] A. Sheta and M. Alkanhal, “Compact dual-band tunable microstrip antenna for GSM/DCS-1800 applications,” IET Proc. Microw. Antennas Propag., vol. 2, no. 3, pp. 274–280, Apr. 2008. [13] V.-A. Nguyen, R.-A. Bhatti, and S.-O. Park, “A simple PIFA-based tunable internal antenna for personal communication handsets,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 130–133, 2008. [14] Y. Zheng, A. Giere, and R. Jakoby, “A compact antenna with two independently tunable frequency bands,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2008, pp. 1–4. [15] N. Karmakar, “Shorting strap tunable stacked patch PIFA,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2877–2884, Nov. 2004. [16] K. R. Boyle and P. G. Steeneken, “A five-band reconfigurable PIFA for mobile phones,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3300–3309, Nov. 2007. [17] A. C. K. Mak, C. R. Rowell, R. D. Murch, and C.-L. Mak, “Reconfigurable multiband antenna designs for wireless communication devices,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 1919–1928, Jul. 2007. [18] M. Komulainen, M. Berg, H. Jantunen, E. Salonen, and C. Free, “A frequency tuning method for a planar inverted-F antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 944–950, Apr. 2008. [19] C. R. White and G. M. Rebeiz, “Single- and dual-polarized tunable slot-ring antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 19–26, Jan. 2009. [20] C. Jung, Y. Kim, Y. Kim, and F. De Flaviis, “Macro-micro frequency tuning antenna for reconfigurable wireless communication systems,” Electron. Lett., vol. 43, no. 4, pp. 201–202, 15, 2007. [21] E. Erdil, K. Topalli, M. Unlu, O. Civi, and T. Akin, “Frequency tunable microstrip patch antenna using RM MEMS technology,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1193–1196, Apr. 2007. [22] K. Van Caekenberghe and K. Sarabandi, “A 2-bit Ka-band RF MEMS frequency tunable slot antenna,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 179–182, 2008. [23] R. V. Goteti, R. E. Jackson, and R. Ramadoss, “MEMS-based frequency switchable microstrip patch antenna fabricated using printed circuit processing techniques,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 228–230, Dec. 2006. [24] P. Blondy, D. Bouyge, A. Crunteanu, and A. Pothier, “A wide tuning range MEMS switched patch antenna,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, pp. 152–154. [25] G.-M. Yang, X. Xing, A. Daigle, M. Liu, O. Obi, J. Wang, K. Naishadham, and N. Sun, “Electronically tunable miniaturized antennas on magnetoelectric substrates with enhanced performance,” IEEE Trans. Magn., vol. 44, no. 11, pp. 3091–3094, Nov. 2008. [26] L. Liu and R. Langley, “Liquid crystal tunable microstrip patch antenna,” Electron. Lett., vol. 44, no. 20, pp. 1179–1180, 25, 2008. [27] E. Lee, K. M. Chan, P. Gardner, and T. E. Dodgson, “Active integrated antenna design using a contact-less, proximity coupled, differentially fed technique,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 267–276, Feb. 2007. [28] D. H. Lee, A. Chauraya, Y. Vardaxoglou, and W. S. Park, “A compact and low-profile tunable loop antenna integrated with inductors,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 621–624, 2008. [29] S. V. Hum, M. Okoniewski, and R. J. Davies, “Modeling and design of electronically tunable reflectarrays,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2200–2210, Aug. 2007.

[30] Q. Zhang, Y. Fukuoka, and T. Itoh, “Analysis of a suspended patch antenna excited by an electromagnetically coupled inverted microstrip feed,” IEEE Trans. Antennas Propag., vol. 33, no. 8, pp. 895–899, Aug. 1985. [31] V. Tripathi, “Asymmetric coupled transmission lines in an inhomogeneous medium,” IEEE Trans. Microw. Theory Tech., vol. 23, no. 9, pp. 734–739, Sep. 1975. [32] C.-M. Tsai and K. Gupta, “A generalized model for coupled lines and its applications to two-layer planar circuits,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 12, pp. 2190–2199, Dec. 1992. [33] H. Y. Xiong and S. Hum, “A differentially-fed microstrip patch antenna with enhanced frequency agility,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2008, pp. 1–4. [34] SPEAG. SEMCAD X 2009 [Online]. Available: http://www. speag.com [35] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, Apr. 2005. [36] W. E. McKinzie, III, “A modified Wheeler cap method for measuring antenna efficiency,” in Antennas Propag. Soc. Int. Symp. Dig., Jul. 1997, vol. 1, pp. 542–545.

Sean Victor Hum (S’95–M’03) was born in Calgary, Alberta, Canada. He received the B.Sc., M.Sc., and Ph.D. degrees from the University of Calgary, in 1999, 2001, and 2006 respectively. In 2006, he joined the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada, where he currently serves as an Assistant Professor. His present research interests lie in the area of reconfigurable RF antennas and systems, antenna arrays, and ultrawideband communications. Prof. Hum received the Governor General’s Gold Medal for his master’s degree work on radio-on-fiber systems in 2001. In 2004 he received the IEEE Antennas and Propagation Society Student Paper award for his work on electronically tunable reflectarrays. In 2006, he received an ASTech Leaders of Tomorrow award for his work in this area. He is also the recipient of three teaching awards. He served on the Steering Committee and Technical Program Committee for the 2010 IEEE AP-S International Symposium on Antennas and Propagation. In August 2010, he was appointed as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Hui Yuan Xiong was born in China in 1986. He received the B.A.Sc. degree (with distinction) in engineering science from the University of Toronto, Toronto, ON, Canada, in 2009, where he is currently working towards the M.A.Sc. degree. From 2007 to 2009, he worked in the Electromagnetics Group, University of Toronto, with an engineering science research scholarship and an NSERC Undergraduate Research Scholarship. He assisted in the design of reconfigurable antennas and ultrawideband antenna arrays. Mr. Xiong is a recipient of the 2008 IEEE Antennas and Propagation Society Research Award.

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Coplanar Capacitively Coupled Probe Fed Microstrip Antennas for Wideband Applications Veeresh G. Kasabegoudar and K. J. Vinoy, Senior Member, IEEE

Abstract—The design and analysis of a coplanar capacitive fed microstrip antenna suspended above the ground plane is presented. It is demonstrated that the proposed approach can be used for designing antennas with impedance bandwidth of about 50% and a good gain to operate in various microwave bands. The model of the antenna incorporates the capacitive feed strip which is fed by a coaxial probe using equivalent circuit approach, and matches simulation and experimental results. The capacitive feed strip used here is basically a rectangular microstrip capacitor formed from a truncated microstrip transmission line and all its open ends are represented by terminal or edge capacitances. The error analysis was carried out for validity of the model for different design parameters. The antenna configuration can be used where unidirectional radiation patterns are required over a wide bandwidth. Index Terms—Coplanar capacitive feed, microstrip antennas, wideband and input impedance.

I. INTRODUCTION ICROSTRIP antennas are suitable for modern broadband applications because of their desirable characteristics [1]–[3]. Although microstrip antennas in their basic form exhibit limited bandwidth, it has been shown by several researchers that the bandwidth can be significantly improved by altering the basic geometry and/or feed or by using impedance matching techniques [2]. However most of these geometries employ stacked multiple metal/dielectric layers [4], or use modified probe shape (L-, T-, or meander-shaped probes) [5]–[7], which elude the primary advantages of microstrip antennas such as ease of fabrication and assembling [8]. On the other hand, the antenna reported in [3] is a single layer coplanar capacitive fed wideband microstrip antenna. This antenna is simple to fabricate and assemble, and the 28% bandwidth reported there could be enhanced by optimizing the dimensions of feed strip and its placement with respect to the radiator patch [8]. Radiation patterns of this antenna with bandwidth of almost 50% can be improved by modifying the edge of the

M

Manuscript received March 19, 2009; revised November 12, 2009; accepted April 08, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. V. G. Kasabegoudar is with the Electronics and Telecommunication Engineering Department, College of Engineering, Ambajogai-431517, India (e-mail: [email protected]). K. J. Vinoy is with the Microwave Laboratory, Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560 012, India (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055781

radiator patch close to the feed strip [9]. The maximum bandwidth can be achieved for this antenna with an overall substrate height of about 0.16 [8]. In the above geometries, the antenna has a unidirectional radiation pattern because of the presence of the ground plane. The capacitive feed strip placed along one of the radiating edges of the patch compensates for the probe inductance. This approach for compensating probe inductance is widely known [10], but the impedance bandwidth of the antenna configuration in the present work is significantly higher than previously reported [3], [10]. Input impedance of microstrip antennas plays a major role in determining the matching between antenna terminals and transmission lines. Extensive studies have been reported in the literature on input impedance calculations of microstrip antennas [11]–[22]. A few of these studies include transmission line model [11], cavity model [12], moment method solutions [13]–[17], equivalent circuit (extracted from cavity model i.e., treating volume below the patch as a cavity) approaches [18]–[21], some of which involve solving complicated integrals. An equivalent circuit based approach provides a clear physical insight into the operation of the antenna. The equivalent circuit approaches reported in [18]–[21] are simple and reasonably accurate but these consider that the probe feed is directly connected to the radiator patch geometries. The impedance analysis of coplanar capacitive fed wideband microstrip antenna has been reported by [22] based on full wave analysis. However the suggested method involves numerically intensive calculations for the basic antenna geometry as mentioned above. In this paper we present the design, optimization, and analysis of input impedance of coplanar capacitive fed wideband microstrip antennas. Input impedance analysis presented here is based on equivalent circuit approach which also incorporates the probe feed connected outside the radiator patch into the complete model. The model developed here is simple to analyze and suitable for CAD implementation. The basic geometry of the antenna and its optimization are presented in Section II. Input impedance analysis of the antenna is presented in Section III. This is followed by experimental validation and conclusions in Sections IV and V, respectively. II. ANTENNA GEOMETRY AND ITS OPTIMIZATION The basic geometry of the antenna is shown in Fig. 1 [8] and optimized dimensions are listed in Table I. The configuration is basically a suspended microstrip antenna in which radiating patch and the feed strip are placed above the substrate of thickness “ ” mm. A long pin SMA connector is used to connect the feed strip which capacitively couples the energy to a radiating patch. The detailed parametric studies have been reported

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Fig. 1. Basic geometry of coplanar capacitive fed UWB microstrip antenna. (a) Top view. (b) Cross sectional view.

TABLE I DIMENSIONS FOR THE ANTENNA DESIGNED FOR 5.9 GHz

Fig. 2. Effect of feed strip dimensions on impedance bandwidth for an antenna designed to operate at 5.9 GHz as the center frequency. (a) Feed strip length versus bandwidth. (b) Feed strip width versus bandwidth.

earlier [8], [9] for the optimization of this geometry. In this section, we briefly discuss the effect of key design parameters on , separation the antenna performance. These include air gap , and the length between feed strip and the radiator patch and width (feed strip dimensions). The substrate used for antenna fabrication is a RO3003 with dielectric , and thickness . All paramloss eters are optimized using IE3D which is a method of moments (MoM) based electromagnetic (EM) software. The basic patch design starts from the selection of center frequency of the band of operation. For demonstration purpose 5.9 GHz is chosen in the present study. Radiator patch dimensions can be calculated from standard design expressions after making dielectric [1], necessary corrections for the suspended [23]. These corrections incorporate the total height above the ground and effective dielectric constant of the suspended microstrip [24]. It has been shown that the impedance bandwidth of the antenna may be maximized by using the design expression (1) Where is the height of the substrate above the ground, and and are the thickness and dielectric constant of the substrate respectively. However, it should be noted that this equation enables us to predict the initial value but the final optimum value of this [8]. would be within

As an example, the air gap was varied from 5.0 mm to 7.5 mm to study its effect on the impedance bandwidth. It can be seen from Table II that the optimum bandwidth solution (set of key design parameters) is not unique and for each air gap value, any of the key design parameters can be used to maximize the antenna bandwidth. For example when air gap is equal to 5.5 mm, there are two possible sets of parameters to get the best possible BW of 3.07 GHz for this height. From these studies, the optimum air gap is found to be 6.0 mm for an antenna operating with a center frequency of 5.9 GHz. However it should be noted that the air gap of 6.0 mm is optimum only for the present operating band and dielectric sheet properties used. Hence the should be recalculated optimum height above ground plane with the help of (1) as discussed above. This is illustrated by considering some arbitrary frequencies (2.0, 4.5, 8.0 and 10.0 GHz) on either side of the present 5.9 GHz. Boresight gain and efficiency for the corresponding center frequency of the operating band for each case are listed in Table II. From Table III it can be seen that nearly 50% bandwidth can be obtained for almost all frequencies considered from the proposed design except for 2.0 GHz case for which the BW is 46%. For small variations in dielectric constant of the suspended substrate the effective dielectric properties would remain largely unaffected and hence the bandwidth. However there could be a shift in center frequency due to the change in feed strip reactance. A similar effect is also observed with the variation in the . thickness of the dielectric layer

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TABLE II OPTIMIZATION OF AIR GAP USING DIFFERENT KEY DESIGN PARAMETERS AND OTHER PARAMETERS ARE AS LISTED IN THE TABLE I

TABLE III SCALING OF ANTENNA FOR DIFFERENT CENTER FREQUENCIES WITH THEIR OPTIMUM DIMENSIONS

Finally, the effect of feed strip dimensions on impedance bandwidth was investigated. These two dimensions ( and ) control the reactive part of the antenna’s input impedance. It can be noticed from Fig. 2 that impedance bandwidth decreases with the increase in or . In other words, the bandwidth lost in increasing one parameter can be regained by decreasing the other. For example changing “ ” from 1.2 mm to 1.8 mm results in the reduction of bandwidth from 2.99 GHz to 2.80 GHz, which can be restored by reducing the dimension “ ” from 4.1 mm to 3.1 mm (Fig. 2(b)). However it should be noted that use may pose difficulty in solof lower dimensions dering the SMA probe pin while higher values cause feed strip radiations to disturb the radiation field [8]. For further details on parametric studies, our earlier works [8], [9] may be useful. III. EQUIVALENT CIRCUIT MODELING To find the input impedance of the antenna, analysis is made in following paragraphs. A. Radiator Patch The equivalent model of a rectangular microstrip antenna is basically a parallel tuned circuit for the antenna operating in the fundamental mode [1]. For the fundamental mode of operation its equivalent circuit consists of well known parallel combination of RLC elements. However it should be noted that for

DC and extremely high frequency operations, a capacitor and an inductor, respectively, can be placed in series with the equivalent circuit of fundamental mode. The radiating patch may be treated as an edge fed microstrip antenna for calculating the patch equivalent capacitance (2) It may be noted that height of the dielectric substrate here is the total height of the geometry including air gap and the effective dielectric constant for the suspended or airdielectric configuration calculated from the equations given in [24]. In the present case, the antenna was designed to operate in the mode and other values ( and ) of the patch equivalent can be calculated from equations given in [23]. B. Feed Section As discussed in earlier sections, in the present geometry (Fig. 1), the coaxial probe is not directly connected to the radiator patch instead it is connected to a small rectangular strip placed near the radiator patch which excites it by capacitive coupling. Feed strip can be treated as a rectangular microstrip capacitor as its dimensions are much smaller in comparison with the wavelength of operation. It can also be considered as an abruptly terminated microstrip line and all of its sides (open ends) that can be represented by terminal capacitances. The

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Fig. 4. Complete circuit model of antenna shown in Fig. 1. Fig. 3. Equivalent circuit of probe with feed strip.

parallel plate capacitance of the feed strip (3)

is given in

(3) It may be noted that the area of the strip is much smaller than the patch. The perimeter to area ratio is much larger in this case. Hence the fringing capacitances from the edges may be significant in this case. These capacitances, often known as terminal capacitances of the line, can be calculated from [20]:

width, separated by a short distance . The separation between the radiator patch and feed strip can be modeled by the -net, , and shown in Fig. 4) given in [25]. Two work ( and ) represent the terminal capaciparallel capacitors ( tances of the two microstrip sections and the series capacitance represents the gap. These capacitances can be calculated by a rigorous spectral domain method [25] after making necessary corrections for two layer air dielectric medium (7) (8)

(4) (5) In (4) and (5) and, and are microstrip line impedances in air and substrate medium. A factor of 2 is used here to include opposite sides (Fig. 3). The dissipation and radiation losses from the feed strip are modeled with a resistance . At lower frequencies radiation losses from the feed strip can be neglected. However with the increase in the operating frequency, radiation losses should be taken into account. The strip resistance can be calculated similar to the patch equivalent resistance. The probe feed can be represented by an inductive reactance in series with the feed strip equivalent circuit element [20]. The probe reactance can be calculated from [20], [21] (6) Where is the probe diameter, is the operating frequency; is the velocity of wave through a free space, is the total is the equivheight of mixed air substrate combination, and alent dielectric constant of the composite air-dielectric medium and can be calculated by

(9) In (7)–(9) [25],

(10)

(11)

(12) (13) (14)

Based on the above, the equivalent model of the feed is shown in Fig. 3. This circuit models the effects in the region below the feed strip. C. Separation Between Radiator Patch and Feed Strip The separation between radiator patch and the feed strip is essentially an asymmetrical gap i.e., two conductors of unequal

Some additional correction factors have been added to few terms of these equations based on data obtained from a number of cases studied. However it should be noted that, these correction factors are required only outside the parametric range ( , and ) defined in [25] and for suspended configurations. For example in the present case, (0.22) and (0.066) are outside the

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range mentioned above and hence need corrections. With these corrections, the model works for different values of feed strip dimensions considered in [8] to obtain best possible bandwidth . In and , and are the for this antenna correction factors for suspended substrates, whose values are defined as

And for

(15)

Fig. 5. S characteristics obtained from different techniques for the antenna shown in Fig. 1.

IV. VALIDATION OF THE MODEL AND DISCUSSIONS

(16) In all above expressions, the substrate height is replaced by the total height to include air-dielectric medium . In (8), the static edge capacitance of the radiator patch capacitance can be computed from (17) and are microstrip line impedances in air In (17), and substrate medium respectively. It should be noted that the dynamic radiating edge capacitance of the patch is equal to its static edge capacitance as the antenna is designed to operate in mode. The strip width fringing capacitance the in (7) can be calculated as explained in Section III-B. D. Complete Equivalent Circuit The complete circuit after combining all individual parts of entire geometry developed in the above paragraphs is shown in Fig. 4. The input impedance of the complete antenna geometry shown in Fig. 4 can be calculated from

(18) In (18),

is the input impedance of patch equivalent and is the input impedance of feed strip.

Above designed prototype with dimensions given in Table I was fabricated and its characteristics ( (dB) (input reflection coefficient), gain and radiation patterns) were measured. The input impedance model given by (18) was implemented in MATLAB. The characteristics obtained from simulation, measurement and our model are compared in Fig. 5. It can be noticed from Fig. 5 that model is in good agreement with the simulated and measured results. For other values of feed strip dimensions, the characteristics from the model do not deviate much with the simulated results obtained from the IE3D. All geometries were simulated with the probe diameter of 1.4 mm of a commercially available SMA connector. The substrate used for all the simulations, analysis and fabrication of the antenna is Roger’s made RO3003 with dielectric constant of 3.0, loss tangent of 0.0013 and thickness of 1.56 mm. Tables IV and V show the performance for different sets of values of feed strip length from 1.2 mm to 1.8 mm and width from 3.1 mm to 4.1 mm and found good matching with the simulated results. The model was tested further for bands of frequencies (L, S, C, and X) as the microstrip antenna is redesigned for almost any frequency by appropriate scaling in Table VI. The frequencies chosen within these bands are 2, 4.5, 5.9, 8, and 10 GHz as demonstrated in [8]. For all these designs, we found good match between the simulated and the calculated characteristics. In Tables IV–VI, a comparison of frequency deviation between the measured and computed values of the two resonant frequencies of antenna is given. The resonant frequencies ( and ) correspond to the peak negative values of the characteristics. From these studies, it can be noted that the total relative percentage error does not exceed 3.5% in all cases considered (Tables IV–VI). Since the error is much smaller than the bandwidth, we conclude that the percentage of error is acceptably low. Radiation patterns were measured in an anechoic chamber. Comparisons of simulated and measured radiations patterns of E and H plane patterns at the start, center, and end frequencies

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ERROR ANALYSIS FOR TWO RESONANT FREQUENCIES IN THE S

TABLE IV BAND ( 10 dB) WITH FEED STRIP WIDTH (s = 3:7 mm) CONSTANT

ERROR ANALYSIS FOR TWO RESONANT FREQUENCIES IN THE S

TABLE V BAND ( 10 dB) WITH FEED STRIP LENGTH (t = 1:2 mm) CONSTANT

0

0

TABLE VI ERROR ANALYSIS FOR TWO RESONANT FREQUENCIES IN THE S BAND ( 10 dB) FOR DIFFERENT BANDS OF FREQUENCIES

0

of the useful frequency band are shown in Fig. 6. These emphasize the unidirectional nature of the radiations. It may be noted that the H-plane patterns are symmetrical throughout the band of operation whereas E-plane patterns are symmetrical at lower frequencies and the degree of asymmetry increases at the higher end of the operating band. Furthermore, these also indicate minor beam squinting and increased cross polarization levels. These may be attributed to the excitation of unwanted higher order modes and/or spurious direct radiations from the feed [26]–[29]. Approaches suggested to address these problems are based on modifying the probe feed of the patch such as by employing a dual feed arrangement with 180 phase shift [26], or a balanced feeding technique [27]. In a separate effort,

the geometry of the presently used capacitively coupled patch has been modified to achieve symmetrical E-plane radiation patterns across the band of operation [9]. However, it can be noted that the measured radiation patterns show more than cross polarization level in the boresight direction and a comfortable back lobe radiation levels. Comparisons of simulated and measured gain versus frequency characteristics of the antenna are plotted in Fig. 7. The gain was measured by comparison (three antennas) method. The gain is above 6 dBi at the center of the operating band. From all above discussions, it can be noted that all results (simulated and calculated from equations presented in this paper), fairly agree with the measured results.

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Fig. 6. Radiation patterns comparisons of antenna shown in Fig. 1. (Left-hand side: E-plane patterns and right-hand side: H-plane patterns). Solid line: Simulated co-polarizations; Dashed line: Measured co-polarizations; Dotted Line: Simulated cross polarizations; Dashed-dotted line: Measured cross-polarizations. Note: H-cross (simulated) cannot be seen in patterns as it is well below 30 dB. (a) E and H-plane patterns at 4.5 GHz. (b) E and H-plane patterns at 6.0 GHz. (c) E and H-plane patterns at 7.5 GHz.

0

V. CONCLUSIONS

Fig. 7. Gain versus frequency plots.

A coplanar capacitively coupled probe fed microstrip antenna suitable for wideband applications has been presented. After presenting the basic configuration, an equivalent circuit based approach to calculate the input impedance is discussed. This unified model predicts the input impedance of the antenna, including the effects of the feed strip and the probe pin, over a wide characteristics range of frequencies. Input impedance and obtained using the developed equations are found to be in good agreement with the IE3D simulated and experimental results. Some of the expressions for the constituent models have been modified to suit the proposed antenna configuration. With this

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approach, conformal wideband antennas have been designed with different center frequencies from 2 GHz to 10 GHz and all of these showed nearly similar radiation performance.

REFERENCES [1] R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip Antenna Design Handbook. Norwood, MA: Artech House, 2001. [2] G. Kumar and K. P. Ray, Broadband Microstrip Antennas. Norwood, MA: Artech House, 2003. [3] G. Mayhew-Ridgers, J. W. Odondaal, and J. Joubert, “Single-layer capacitive feed for wideband probe-fed microstrip antenna elements,” IEEE Trans. Antennas Propag., vol. 51, pp. 1405–1407, 2003. [4] D. M. Kokotoff, J. T. Aberle, and R. B. Waterhouse, “Rigorous analysis of probe fed printed annular ring antennas,” IEEE Trans. Antennas Propag., vol. 47, pp. 384–388, 1999. [5] B. L. Ooi and I. Ang, “Broadband semicircle fed flower-shaped microstrip patch antenna,” Electron. Lett., vol. 41, no. 17, 2005. [6] C. L. Mak, K. F. Lee, and K. M. Luk, “A novel broadband patch antenna with a T-shaped probe,” Proc. Inst. Elect. Eng.. Microw., Antennas Propag., vol. 147, pp. 73–76, 2000. [7] H. W. Lai and K. M. Luk, “Wideband stacked patch antenna fed by meandering probe,” Electron. Lett., vol. 41, no. 6, 2005. [8] V. G. Kasabegoudar, D. S. Upadhyay, and K. J. Vinoy, “Design studies of ultra wideband microstrip antennas with a small capacitive feed,” Int. J. Antennas Propag., vol. 2007, pp. 1–8. [9] V. G. Kasabegoudar and K. J. Vinoy, “A wideband microstrip antenna with symmetric radiation patterns,” Microw. Opt. Technol. Lett., vol. 50, no. 8, pp. 1991–1995, 2008. [10] P. S. Hall, “Probe compensation in thick microstrip patches,” Electron. Lett., vol. 23, pp. 606–607, 1987. [11] A. K. Bhattacharjee, S. R. B. Chaudhuri, A. Mukherjee, D. R. Poddar, and S. K. Chowdhury, “Input impedance of rectangular microstrip antennas,” Proc. Inst. Elect. Eng. Pt. H., vol. 135, no. 5, pp. 351–352, 1988. [12] Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and experiment on microstrip antennas,” IEEE Trans. Antennas Propag., vol. 27, pp. 137–145, 1979. [13] E. H. Newman and T. Pravit, “Analysis of microstrip antennas using moment methods,” IEEE Trans. Antennas Propag., vol. 29, pp. 47–53, 1981. [14] M. D. Deshpande and M. Bailey, “Input impedance of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 30, pp. 645–650, 1982. [15] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstrip antennas,” IEEE Trans. Antennas Propag., vol. 30, pp. 1191–1196, 1982. [16] J. P. Damiano, J. Bennegueouche, and A. Papiernik, “Study of multilayer microstrip antennas with radiating elements of various geometries,” Proc. Inst. Elect. Eng. Pt. H., vol. 137, no. 3, pp. 163–170, 1990. [17] J. P. Damiano and A. Papiernik, “Survey of analytical and numerical models for probe-fed microstrip antennas,” Proc. Inst. Elect. Eng.. Microw. Antennas Propag., vol. 141, no. 1, pp. 15–22, 1994. [18] V. K. Pandey and B. R. Vishvakarma, “Analysis of an E-shaped patch antenna,” Microw. Opt. Technol. Lett., vol. 49, no. 1, pp. 4–7, 2007. [19] J. A. Ansari and R. B. Ram, “Analysis of a compact and broadband microstrip patch antenna,” Microw. Opt. Technol. Lett., vol. 50, no. 8, pp. 2059–2063, 2008. [20] F. Abboud, J. P. Damiano, and A. Papiernik, “Simple model for the input impedance of the coax-feed rectangular microstrip patch antenna for CAD,” Proc. Inst. Elect. Eng., vol. 135, no. 5, pp. 323–326, 1988. [21] A. K. Verma, N. V. Tyagi, and D. Chakraverty, “Input impedance of probe fed multilayer rectangular microstrip patch antenna using the modified Wolff model,” Microw. Opt. Technol. Lett., vol. 31, no. 3, pp. 237–239, 2001.

[22] G. Mayhew-Ridgers, J. W. Odondaal, and J. Joubert, “Efficient full-wave modeling of patch antenna arrays with new single-layer capacitive feed probes,” IEEE Trans. Antennas Propag., vol. 53, pp. 3219–3228, 2005. [23] I. J. Bahl and P. Bhartia, Microstrip Antennas. Boston, Ma: Artech House, 1980. [24] J. M. Schellenberg, “CAD models for suspended and inverted microstrip,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 6, pp. 1247–1252, 1995. [25] M. Kirschning, R. H. Jansen, Jansen, and N. H. L. Koster, “Measurement and computer aided modeling of microstrip discontinuities by an improved resonator method,” IEEE MTT-S Digest, pp. 495–497, 1983. [26] C. H. K. Chin, Q. Xue, and H. Wong, “Broadband patch antenna with a folded pair as a differential feeding scheme,” IEEE Trans. Antennas Propag., vol. 55, pp. 2461–2467, 2007. [27] Z. N. Chen and M. Y. W. Chia, “Experimental study on radiation performance of probe-fed suspended plate antennas,” IEEE Trans. Antennas Propag., vol. 51, pp. 1964–1971, 2003. [28] X. N. Low, Z. N. Chen, and W. K. Toh, “Ultrawideband suspended plate antenna with enhanced impedance and radiation performance,” IEEE Trans. Antennas Propag., vol. 56, pp. 2490–2495, 2008. [29] C. Y. D. Sim, C. C. Chang, and J. S. Row, “Dual feed dual polarized patch antenna with low cross polarization and high isolation,” IEEE Trans. Antennas Propag., vol. 57, pp. 3405–3409, 2009. Veeresh G. Kasabegoudar received the Bachelor’s degree from Karnatak University Dharwad, India, the Masters degree from the Indian Institute of Technology (IIT) Bombay, India, and the Ph.D. degree from the Indian Institute of Science (IISc), Bangalore, in 1996, 2002, and 2009, respectively. From 1996 to 2000, he worked as a Lecturer in the Electronics and Telecommunication Engineering Department, College of Engineering, Ambajogai, India, where, from 2002 to 2006, he worked as an Assistant Professor and, since 2008, he has been a Professor and Dean of Research and Development. He has published over 15 papers in technical journals and conferences. His research interests include microstrip and CPW fed antennas, and microwave filters.

K J. Vinoy (M’02–SM’06) received the Bachelor’s degree from the University of Kerala, India, the Masters degree from Cochin University of Science and Technology, Cochin, India, and the Ph.D. degree from the Pennsylvania State University, University Park, in 1990, 1993, and 2002, respectively. From 1994 to 1998, he worked in various positions at the National Aerospace Laboratories, Bangalore, India, where he was attached to the Computational Electromagnetics Group. He was a Research Assistant at the Center for the Engineering of Electronic and Acoustic Materials and Devices (CEEAMD) at the Pennsylvania State University, from 1999 to 2002. He continued there for Postdoctoral research until 2003. Since 2003, he has been with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, where he is currently an Associate Professor. His research interests cover several aspects of microwave engineering including, microwave antennas, microwave circuits, metamaterials and RF MEMS. He has published over 100 papers in technical journals and conferences. He has also published three books: Radar Absorbing Materials: From Theory to Design and Characterization (Kluwer, 1996), RF MEMS and their Applications (Wiley, 2002), and Smart Material Systems and MEMS: Design and Development Methodologies (Wiley, 2006). He is serving on the editorial board of the Journal of the Indian Institute of Science. Prof. Vinoy is currently the Chairman of the IEEE joint MTT/AP Societies Bangalore Chapter.

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Theory of Electromagnetic Time-Reversal Mirrors Julien de Rosny, Member, IEEE, Geoffroy Lerosey, and Mathias Fink

Abstract—The theory of monochromatic time-reversal mirrors (TRM) or equivalently phase conjugate mirrors is developed for electromagnetic waves. We start from the fundamental time-symmetry of the Maxwell’s equations. From this symmetry, a differential expression similar to the Lorentz reciprocity theorem is deduced. The radiating conditions on TRM are expressed in terms of 6-dimension Green’s functions. To predict the time reversal focusing on antenna arrays, a formalism that involves impedance matrix is developed. We show that antenna coupling can dramatically modify the focal spot. Especially, we observe, that in some circumstances, sub-wavelength focusing on a bi-dimensional array may arise. Index Terms—Antenna arrays, diffraction, microwaves, phase conjugate mirrors, plasmonic, sub-wavelength focusing, time-reversal, time-symmetry.

I. INTRODUCTION

A

time-reversal mirror (TRM) is a device that produces an outgoing wave which is the time symmetric of an incoming wave. A TRM is made of an array of transceivers and dedicated electronics. In the first step, on each transceiver, the time dependence of an incident field generated by a source is recorded and memorized. During the second step, each transceiver plays back the signals but in a reverse order. Then, in case of a time symmetric propagation medium, the wave back-propagates and finally focuses on the initial source location. The first demonstration of such TRM has been performed with ultrasounds in 1989 thanks to the development of fast digital to analog and analog to digital converters that are the basic elements of TRM electronics [1]. Since then, TRM have been developed for many acoustic applications such as sound focusing, ultrasound non destructive testing, ultrasound hyperthermia for medical therapy, seismology, etc. A good review can be found in [2]. At one frequency, TRM behave like phase conjugate mirrors (PCM). Works on microwave PCM started during the sixties [3], [4]. Since then, many works have been performed on this subject. It is only a few years ago that time-reversal of waves has been transposed to microwaves. The main experimental difficulty compared to ultrasonic experiments lies in the much higher carrier frequency. In 2004, a first narrow band prototype was developed [5]. The TRM was only composed of one antenna. Nevertheless, it successfully succeeded in time-reversing a 1

Manuscript received August 28, 2009; revised January 28, 2010; accepted March 20, 2010. Date of publication June 14, 2010; date of current version October 06, 2010. This work was supported in part by Orange Labs (Grant 292743FT). The authors are with the Institut Langevin, ESPCI ParisTech, CNRS UMR 7587, 75231 Paris Cedex 05, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2052567

long burst at 2.4 GHz in a high Q reverberant chamber. Concurrently, other authors showed [6] that phase conjugation arrays in a cluttered environment achieve much better spatial resolution than in free space. Later, a wideband TRM made of 8 channels was developed [7]. There are at least three promising applications for microwave time-reversal mirrors : radio-communications [8], [9], terahertz imaging [10], [11], medical imaging [12]. Recently sub-wavelength focal spots were obtained with time reversal or phase conjugation mirrors. In [13], the time reversed wave focuses in a micro-structured medium. The scattering of the time reversed wave on the micro-structure generates the evanescent components (sub-wavelength spatial variations of the field) at the focus. More recently, this effect was also observed with active phase conjugating lens and wire scatterers close to the focus [14], [15]. From a theoretical point of view, soon after the first ultrasonic time-reversal experiments, acoustic theory of time-reversal was proposed [16], [17]. Later, extensions of the theory have been developed, especially to explain the efficiency of time-reversal through disordered media [18]–[22]. Only a few works are specifically devoted to the theory of time-reversal of electromagnetic waves. In optics, Carminati et al. [23], developed a theory based on Helmholtz’s equation. With microwaves, the works are related to the decomposition of the time reversal operator [24]–[27] introduced by Prada et al. [28]. Here, our goal is to come back, as it has been done in acoustics [16], on fundamental aspects of time-reversal of electromagnetic waves. To begin with, some generalities about timereversal symmetry and electromagnetic fields are considered. Then the link between time-reversal and phase conjugation is established. We elaborate an analog of the Lorentz reciprocity theorem that involves time reversed fields. In order to write concise expressions, we introduce six-dimension vectors which contains the electric and the magnetic fields. The corresponding Green’s functions are defined. With these tools, we derive an integral equation of time-reversal in terms of Green’s functions. Both sides of the equation are studied with care. As an example, the case of the homogeneous medium is treated. We also study the cases of far-field time-reversal and time-reversal by only electrical dipoles. Practically speaking, the Green’s function is never directly measured. We record or apply voltages or currents on antennas. Consequently, it is important to develop a theory in terms of impedance. From the Green’s function relation, we derive a new time-reversal relation that only involves impedance and impedance matrices. This relation is valid for any arrays made of resonant antennas. From this relation, we show that the focusing greatly depends on the load impedance of the antennas. Especially, in some specific cases, sub-wavelength focusing, similar to the one observed experimentally in [13] is obtained. We explain this results in terms of plasmonic-like modes.

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II. TIME-REVERSAL OPERATOR A propagation medium is said “reversible” if a field and its time symmetric can both propagate in it. An electromagnetic wave is described by 4 vectorial fields, the electric field , the magnetic field , the electric displacement and the magnetic induction . Due to the intrinsic definition of these fields, and are even under time-reversal transformation while and are odd [29]. Introducing the time-reversal operator , this property can be concisely written as follows:

occasionally applied by various authors [31]–[36]. In [37], the authors apply this formalism to simplify the expression in bi-anistropic media. Here, the definition of the six-vector of the electromagnetic field is slightly different from the previous one. It is expressed in terms of the magnetic and electric fields as

(3) Hence, the dimension of the six-vector way, the source vector is defined by

equals 6. In the same

(1) With these transformations, it is easy to show that the four Maxwell’s equations are time-symmetric. It means that if the and are solution of the Mawxell’s four fields , , equations, , , and are also solution of the same set of equations. Now, we show the link between time reversal operator and phase conjugation. Let us introduce a time depend field (scalar or vectorial) which is even or odd under time re). As we have seen, versal operator ( the choice of the sign depends on the nature of the field. It for electric field, charge distribution, electric displaceis ment field, etc. and – for magnetic field, current density, etc. of the field is [30]. The Fourier transform . Consequently, deduced from . For odd fields, time reverse the monochromatic field is not equivalent to phase conjugate it. That is why we prefer to use the term “time-reversal” rather than phase conjugate, even if the following mathematical developments are performed in the frequency domain.

(4) In terms of antenna theory, represents a distribution of infina distribution of infinitely small curitely small dipoles and rent loops. These two definitions have been chosen to concisely write the action of the time reversal operator and the reciprocity theorem. The time symmetric of the 6-vector field is the conjugate and (this is not the case of the field, i.e., with the six-vector field defined in [37]). As for the reciprocity theorem ([38, p. 145]) which is written

(5) it becomes

III. INTEGRAL EQUATIONOF TIME-REVERSAL

(6)

A. Integral Equation Between a Field and a Conjugate One From now on, for conciseness, the frequency dependence is omitted. When and are hermitian, the following expression can be deduced from the Maxwell’s equations (see (48) in Appendix 2):

C. Green’s Functions The Green’s function is an operator that linearly relates the field vector distribution to the source vector distribution, namely

(7) (2) where the magnetic density and electric density (respect. and ) generate the fields , (respect. , ). The upper scripts , and represent the conjugate, transpose and transpose-conjugate operators, respectively. To deduce from (2), a simple integral relation of the TRM, we need to introduce some mathematical formalisms.

For the sake of simplicity, when no confusion can occur, the previous volume integration is kept implicit and written

(8) The dimension of Green’s function

is 6 by 6.

B. Six-Vector Formalism

D. Time-Reversal Integral Equation

To handle at the same time electric and magnetic fields, we use six-vector formalism. Six-vector notation has been

From (2), an integral relation ((54) in Appendix 3 is derived using the Green’s function formalism. Moreover, in Appendix 4,

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it is shown that from this relation and the reciprocity theorem, the following integral equation can be worked out

In order to obtain this relation, it has been assumed that the tenand are both hermitian and symmetric. Obviously, sors these two properties imply that and are real. This is consistent with the time-symmetry assumption. In the next two sections, we discuss the physical meaning of the left and right hand sides of (9).

The first term is equal to , namely the time-reversal of . Thus as expected, time-reverse the field from a close surface leads to the time-reversed field inside the volume . However this field is not created alone, it adds , term which is a “diverging” field (causal Green’s function). This extra term “breaks” the time-reversal process. As shown in ref [39], in order to get a “full” time-reversal, i.e., only gen, one has to time-reverse the field on the boundaries erate but also to time reverse the initial source, i.e., to emit the source . In such a case, using Green’s function formalism, the field is added to the time-reversed field ( ) generated by the TRM. Due to the linearity of Maxwell’s equations, only remains, which is the expected result. However, in most practical cases, during time reversal step, the initial source is turned off. Then, the interference between the converging and the diverging waves gives rise to the focal spot.

E. Left Hand-Side of (9)

G. Focal Spot in Homogeneous Medium

The left-hand side field, can be interpreted as follows: during the first step, generates the field ( ). The electric and magnetic components are recorded on a close is phase conjusurface . This field is time reversed, i.e., gated. Now, from this phase-conjugate field, a new distribution of dipolar/loop current sources on the close surface is deduced

In acoustics, it has been shown that the interference between the two waves inside an homogeneous medium, produces a halfwavelength focal spot. Here we show that the same effect occurs for electromagnetic waves. For the sake of simplicity, we introduced the electric-electric, electric-magnetic, magnetic-electric and magnetic-magnetic Green’s functions. This set of four Green’s functions forms the Green’s function

(10)

(13)

(9)

This expression of can be simplified when the sources are far from the close surface. In such a case the electromagnetic wave is locally plane near the surface. Hence the electric and where is magnetic fields are orthogonal, the wave vector and is the characteristic impedance of vacuum ( ). Assuming a spherical close surface, a surrounding TRM, the local wave vector is normal to the surface ( ), i.e., and the time-reversed source distribution becomes

In a homogeneous medium, the propagation is invariant under translation, consequently the Green’s functions only depend on . The expressions of the electric-electric and magnetic-electric Green’s function in an homogeneous medium ([40, Ch. 14, p. 578]) are given by

(11)

(14)

Hence, in the far field approximation, the electric and magnetic and current distributions given by on the surface of the TRM generate the time reversed field within the volume. In other words, in the far field approximation, time reverse an electromagnetic field simply consists in recording the currents induced in small wire and loop antennas, and time-reversing them.

(15) where , and is the scalar Green’s ). Due to symmetries befunction ( tween electric and magnetic fields, it comes

(16)

F. Right Hand-Side of (9) inside the The sources at the surface generate the field volume . This field is directly related to the imaginary part of the Green’s function. It can be rewritten using the implicit integration notation (12)

(17) Contrary to the real parts of the Green’s function, the imaginary because parts do not show singularities at

(18)

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and

(19) Hence, the imaginary part is continuous and the typical fluctuation length scale is given by , i.e., the wavelength. The typical size of the focal spot is the wavelength. H. Far Field Time-Reversal by Dipoles Here we assume that only the dipoles radiate in the far field of the close surface (i.e, the small current loops are turned off). In such a case, the source distribution can be written as the sum of two source terms with opposite magnetization currents

(20)

The first source term, , generates /2 inside the volume . , it generates local plane waves that are opposite in diAs for . Thus it consists of a wave that rection to ones produced by propagates outside the volume. Unless a very special geometry for the close surface, such as two infinite parallel planes, this second radiated field has no particular focusing property. The simplified sketch of time-reversal with dipolar sources is shown in Fig. 1. When, the source is electric and we are only interested in far field time reversal, (9) can be simplified into

Fig. 1. Schematic illustration of the far-field time-reversal by dipolar sources. (a) A source q generates a wave. The electric components are recorded on a close surface. (b) This electric field is time-reversed and remitted by dipolar source and the back-propagated wave is generated. (c) The converging wave is followed by a diverging one.

(21) This expression is valid only inside volume . I. Numerical Validation The numerical computation is performed with a TRM made of 124 couples of cross-polarized dipoles uniformly distributed over a 6 wavelength radius sphere. The initial source, a dipole, is at center of the sphere and its polarization is parallel to unit vector. At first, amplitude, phase the electric field radiated by this dipole are recorded on the 124 couples of cross-polarized dipoles. Then the 124 fields are phase conjugated, and re-emitted back by the cross-polarized dipoles. On Fig. 2 the -component of the electric field is plotted. To compare with, the imaginary part of the Green’s function is computed and plotted. The expression of the imaginary part of the component of deduced from (14) is

Fig. 2. Amplitude of the z-component of the time reversed electric field with respect to the distance to the initial emitting dipole along the x-axis. The plus signs (respect. dots) corresponds to the real (respect. imaginary) part of the field generated by a time reversal [left-hand-side of (21)] made of 124 emitters uniformly distributed on a 6-wavelength radius sphere. The continuous line corresponds to the imaginary part of the Green function [right hand side of (21)].

function is never directly measured. We measure or apply voltages or currents on antennas. Consequently, it is important to develop a theory in terms of impedance. IV. TIME-REVERSAL AND MUTUAL IMPEDANCE A. Single Time-Reversal Antenna

(22) On Fig. 2, we observe that the imaginary part of Green’s function corresponds to focal spot generated by the TRM. This result is in agreement with (21). Practically speaking, the Green’s

Before estimating the voltage induced by a surrounding TRM, we first analyze a very simple case : a wave that is generated by an antenna is time reversed by another antenna . During the first step, the forward field is recorded by an antenna

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(see Fig. 3(a)). From the electrical representation shown on Fig. 3(b), the voltage produced on antenna is given by

(23) , are the self impedance of antenna and , where and are the murespectively. The two impedances tual impedances between antenna and . During the backward step, is flipped in time, i.e., phase conjugated and feeds antenna . In a standard configuration, the same antenna, antenna is used for recording the TR field. However, in some cases, we may have to replace antenna by another one, let say (see Fig. 3(c)). antenna with the same load impedance Especially, this substitution is useful to estimate the focal spot. , on antenna is given by (23) after substitution The field, by , , , , respectively. The expresof , , , , which is very general, is not written here for the sion of sake of simplicity. It can be dramatically simplified using the following approximations. First we assume that the TR antenna ( ) is far from the initial one ( and ). In this paper, we are not concerned by the coupling between the initial antenna and the TRM. Second, is the same antenna as but at a different position. These approximations lead to

(24) is the product . This last The key parameter of is estimated using the induced electromechanical force method ([38, p. 471]). This theory is well adapted for wire antennas. Based on this approach, the impedance is given by

Fig. 3. (a),(b) Forward propagations. (c),(d) Backward propagations. (a),(c) schematic representations. (b),(d) Electrical representations.

B. Surrounding Time-Reversal Mirror Now, we assume a time-reversal mirror that is made of many small identical dipoles distributed on a close surface. Due to the linearity of the system, the time reversed field is proportional to

(28) antenna of the TRM. This index where corresponds to the describes, at the same time, the 2 polarizations and the positions of the dipoles. Indeed, to measure the 2 far-field components, there are 2 cross-polarized dipoles at each position on the close surface. Introducing as the density of dipole couples, is well approximated by

(25) where (respect., ) is the current that feeds antenna (reis the elecspect. ) and is the transpose operator. Vector trical field produced by antenna . Using the Green’s function formalism, the previous expression can be rewritten as

(26) where is the electric-electric Green’s function. We assume that antenna (TR antenna) is a small dipole, i.e., , where , , and are the current, the position, the effective length and the direction of the dipole, respectively. In such a case, the product of the conjugated impedances is given by

(29) Thanks to the far-field relation ((21)), the previous equation becomes

(30) and are real because we only The ratios deal with resonant antennas. Consequently, using the induced electromechanical force method ((26)), the previous equation is expressed in terms of the mutual impedance between antennas and ,

(31)

(27)

To validate (31), we simulated the TR of electromagnetic waves with NEC2 [41]. An array of 19 half-wavelength parallel

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each antenna of the TRM, the potentials of the TRM is deduced from the relation

at the antenna ports

(33) To obtain this result, it is assumed that the coupling between antennas of the TRM is small. After time-reversal, the voltage on the load impedance of the antennas of the array is

(34) Fig. 4. (a) Field recorded on a linear array made of 19 half-wavelength dipoles after time-reversal of the field emitted by the dipole at the middle of the array. The continuous line corresponds to the field after time-reversal (28) and the (31). The fields have dash dotted line to the real part of the impedance Z been divided by the l  constant. (b) Same configuration but with parasitic radiators between dipoles.

is given by (31). The superscript The matrix product behind a matrix means that the matrix is first inversed and then conjugate-transposed. Then it comes

(35) antennas is set at the center of the surrounding TRM. The radius of the TRM is . The field is recorded over 124 positions. At each position, 2 orthogonal half-wavelength antennas record the 2 electrical polarizations. The radius of the half-wavelength antennas equals 1 mm and the simulation is performed at 300 MHz. On Fig. 4(a), (28) and (31) are compared for such a configuration. Note that (31) is still valid in heterogeneous media; no assumption has been made about the Green’s functions. To test the relation in such configuration, we added 18 half-wavelength wires (parasitic radiators) to the 18 element array. The parasitic radiators are parallel and set between the antennas. The results are still in good agreement (Fig. 4(b)).

. Expression has been replaced by because the matrix is symetric. is much larger than We immediately observe that when , is given by (31). Indeed, when the load impedances are very large, the induced currents in the array’s antenna are small enough to suppress the coupling effects. Most of the time, the array is composed of identical antennas that are equally spaced. In such a case, the self-impedance matrix is Toeplitz, i.e., the coupling between two antennas and only depends on the difference between indexes and . Moreover, when the array is large enough, a good approximation of the TR field is where

C. Antenna Coupling Practically speaking, we often want to focus a TR wave on one antenna of an array. For instance, this issue is very important for time-reversal MIMO communications [42], [43]. In such a case, the focusing pattern depends on the coupling between antennas. To quantify this effect, we introduce the mutual coupling of the array where one wants to focus impedance matrix on. During the first step of a TR experiment, the initial excitation on the antenna array is a vector . Its dimension is equal to , i.e., the number of antenna of the array. Assuming that the load impedance on each array are identical ( ), the current vector on the antenna array is given by

(36) represents the discrete convolution product1, and (Lower case is the generator vector of the Toeplitz matrix represented by the upper case ). Symbol is a vector with only one non-zero element (equal to 1). Then a simple expression can be obtained in the Fourier domain where

(37) where

represents the discrete Fourier transform2

(32) D. Resistive Load Impedances by unity matrix and is the mutual where is the impedance matrix of the array. As seen in the previous section, the propagation between the array and the TRM can be also described in terms of impedances. Introducing the by impedance matrix that relies each antenna of the array to

and 50 of the resisThe case of two different values 1 tive load impedance on a bidimentional array (see Fig. 5) made 1(f

F

2

g)(k) = k

pf (l)g(k 0 l) x= 2) f (l)exp(ilkx).

[f ]( ) = (

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Fig. 5. Schematic view of a 5 by 5 array of =2 dipoles inside a “perfect” time reversal mirror. Each dipole is loaded with an impedance Z . The green dipole at the center of the 2D array is the one that emits the initial wave before time reversal. Note that in the numerical calculus the array is composed of 31 by 31 dipoles.

Fig. 7. Slices (k = 0) of 2D Fourier transforms of the real (top plot) and the imaginary part (bottom plot) of the impedance matrix.

We clearly observe that due to the diffraction limit, the spectrum . Hence, the resistive load impedance is almost null for modifies the spectrum distribution but not its support. E. Reactive Load

Fig. 6. Time-reversal on a two dimensional array. The continuous line with circle is obtained from (35) with  arbitrary set to 1. This relation is valid for a time reversal mirror that forms a closed cavity (see Section IV-B). The dash line is given by the imaginary part of the Green’s function [(22)]. The focusing is measured on a line of dipole the goes thought the center of the array. (a) (respect. b) Obtained with 1 M (respect. 50 ) load impedance.

of 31 by 31 parallel dipoles is studied. The impedance matrix is computed using Orfanidis’s matlab script “imped.m” [40, and the dipole radius p. 907]. The dipoles are separated by (e.g., 1 mm @ 3 GHz). On Fig. 6, the focusing deduced is from (35) is compared to the imaginary part of the Green’s function given by (22). We observe that when the load impedance is very large, the imaginary part of the Green’s function fits very well the TR focal spot. Indeed, in such a case, as there is no coupling between antennas of the array, the dipoles of the array measure directly the vertical components of the electric field. For a load impedance of 50 , the time reversed field differs slightly from the Green’s function. This effect is due to the coupling terms, i.e., that slightly modifies the spectrum of the TR field. Nevertheless, in both cases we observe . This result can be exthat the focal spot width is limited to , i.e., plained from (37). The Fourier transform of the the imaginary part of the Green’s function is plotted on Fig. 7.

A close look on Fig. 7 shows that the components of the spectrum for larger than is weak but non-null. This is due to the finite size of the array. Indeed, for an infinite array, the spectrum would be strictly confined between 0 and . The finite size yields radiation of a weak part of the subwavelength oscillating field on the array. Usually, this part is very low compared to the . One way to significantly enhance this components contribution is to tune the load impedance in such a way that the denominator in (37) becomes very small for . As the imaginary part of the Fourier transform is positive for , a capacitive load may induce a sub-wavelength field oscillation. One difficulty, not discussed here, is that the Fourier transform of a field of a bi-dimensional array is not isotropic. Consequently, for a given load impedance and a given frequency, due to the square symmetry, only four wavevectors , i.e., directions, correspond to a network resonance. Four plane waves is not sufficient to generate a localized spot. One way to overcome this difficulty consists in looking for large wavevectors and adding also a resistive part to the loading impedance in order to broaden the resonance. A good trade off has been found with a complex equals to . For such a value, the denomimpedance inator is the smallest for . This value is completely in agreement with the focal spot plotted in Fig. 8. The focal spot is roughly 2 times smaller than the one obtained with a very large load impedance. To show that the subwavelength focusing is not only due to the antenna coupling but also to time reversal, we compute the currents induced at the 2D array when the electric field recorded on the close surface given by (33) is emitted without time-reversing it (see Fig. 8). In this case, we do not observe sub-wavelength focusing.

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along the array. To generate a focal spot with monochromatic waves, one needs more spatial diversity. Interferences between waves that comes from all the directions of the space give rise to a focal spot described by a sinc function. The argument of the sinc function is time the distance from the focus. The typical width of a sinc focal spot is equal to half a wavelength. In 2D, the sinc function is replaced by an Airy function which is 1.22 times wider than the sinc. Here, the different angles of arrival of the time-reversed wave generate an airy like focal spot. But is ( ) corresponding to the plasmonic mode. replaced by Sentenac and Chaumet were the first to take benefit of this principle to obtain subwavelength focusing with a bi-dimensional grating in optics [51] V. CONCLUSION

Fig. 8. Induced voltage amplitude on an array of 31 by 31 dipoles after timereversal of the field from a close surface. The load impedance of dipoles equals 3 81i . (a) Line with circles is the induced field on the array generated by the TRM [(35) with  arbitrary set to 1]. The line with diamonds is given by the imaginary part of the Green’s function [(22)] and the line with plus signs is the field emitted by a the close surface (10 wavelength away) without time reversing the signals. The focusing is plotted on a horizontal line of 31 dipoles that goes thought the center of the 2D array (y = 0). The dimension is due to  = 1 (see text). (b) Gray-level image of the voltage induced by time reversal on the 2D array.

0

F. Discussion Many works are related to the subwavelength focusing with plasmonic materials. Examples of diffraction-beating devices based on plasmonics include superlenses [44]–[47], coupledsphere waveguides [48], sharp focusing tips [49], and resonantly excited arrays of metal wires [50]. In 2001, Mayer et al. showed that the energy transport along arrays of closely spaced metal rods is an analogue to plasmonic devices [48]. In the same way, our array of loaded dipoles can be considered as a plasmonic device. Very strong coupling between the antenna of the array occurs for sufficiently small load impedances. Even more, we have seen that for specific values of load impedance, the coupling is sufficiently high and local to generate “plasmonic” modes, i.e., modes with spatial field fluctuations that are smaller than . If the array is infinite, the mode is completely confined on the array without radiating. But due to the finite size of the array, some energy is radiated from the extremities of the array (see [48]). These two properties are the sine qua non conditions for far field sub-wavelength time-reversal. However, they are not sufficient conditions to obtain a subwavelength focusing. Indeed, work with a one dimensional array only gives rise to a standing wave. This last results from the interference between a right to left ( ) and a left to right ( ) propagating waves

In this paper, we have developed a complete approach of the time-reversal of electromagnetic waves with aspects ranging from very fundamental ones to very practical ones. We started from the time-reversal symmetry of the Maxwell’s equations. Then using an integral approach, a general expression for timereversal has been obtained that is always valid. Thanks to a far field approximation, the concept of surrounding TRM is introduced. To be closer to microwave experiments, we completely re-expressed the time-reversal in terms of impedance. Impedance matrix formalism is very useful because it naturally takes into account the coupling between radiating antenna. From this theory we have shown that a sub-wavelength focal spot can be obtained on a bidimentionnal array of loaded dipoles by the way of plasmonic-like modes. There are many theoretical aspects that have not yet been covered with electromagnetic waves. Indeed, here we assumed a perfect TRM that forms a close surface that is at several wavelength away from the focus. But what is the focal spot for a TRM which is closer than a wavelength from a source? In acoustics, it has been shown that sub-wavelength focal spot can be obtained [52] in a homogeneous medium but with a non-perfect TRM. Does this still hold for electromagnetic waves? Especially, in acoustics, it is wellknown that for a finite aperture TRM, the focal spot width is larger and roughly equal to the F-number times the wavelength. How does the polarized nature of the microwaves modify the focus? Finally, here, only resonant-electrical antennas have been considered. do magnetic or traveling waves antennas modify the impedance equation? APPENDIX I RECIPROCITY The integral form of the reciprocity theorem ([38, p. 145]) is given by

(38)

where (resp. ), (resp. ) are the electric and magnetic fields induced by the electric and magnetic current distri(respect. and (respect. . Superscript butions

DE ROSNY et al.: THEORY OF ELECTROMAGNETIC TIME-REVERSAL MIRRORS

is the transpose operator. This expression can be rewritten introducing 6 dimension vectors

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The sum of (46) and (47) leads to the differential form

(39) Using the Green’s function formalism introduced in Section III-C, it finally comes

(48) This expression has been obtained thanks to the relation

(49) (40) Because this relation is valid for any vectors

and

,

(41)

Moreover, it has been assumed that and . In other words (48) is obtained when the permittivity and permeability tensors are hermitian. Both sides of (48) are integrated over a volume delimited by a close surface . Using the Green’s theorem on the left hand side, it comes

APPENDIX II INTEGRAL RELATION IN A LOSSLESS MEDIUM

(50)

Let us considerer 2 sets of electromagnetic fields , and , that are respectively generated by the current densiand , . ties and the magnetization current densities , These fields are solution of the Maxwell’s generalization of Ampère’s law

where is the normal to surface . Using, the mixed product properties, the previous expression can be rewritten

(42) (43) and solution of Faraday’s law

(51)

APPENDIX III GREEN’S FUNCTION RELATION IN A LOSSLESS MEDIUM (44) (45)

In most of propagation media, the electric flux density linearly where is depends on the electric field density, i.e., the permittivity tensor. In the same way where is the permeabillity tensor. Both tensors and can depend on position. times (43) plus times (42) Using these properties, leads to

Replacing the fields by their 6 dimension vector equivalents (see Section III-B), it comes

(52) where

product operator is defined as following

(53) (46) and

times (44) plus

Finally using the Green’s function formalism introduced in Section III-C, it comes

times (45) writes

(47)

(54)

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APPENDIX IV TIME REVERSAL MIRROR EQUATION The right and left hand sides of (54) can be transformed using the reciprocity relation (41). We are going to transform of the left hand side (55) (56) The last equation is obtained thanks to the reciprocity theorem (41) and the fact that (54) is valid when all the sources are contained inside volume . In other words the source distributions and equal 0 outside . Therefore, the integration over all ) can be replaced by an integration over the the space ( ) volume ( (57) Same properties lead to

(58) The last equation is simply obtained by switching the integration variables and . Setting (55) and (58) in (54) gives rise to

(59) This relation is true whatever

inside

, so

can be omitted

(60) This expression is the starting point of mathematical justification of the time-reversal of electromagnetic waves. ACKNOWLEDGMENT We would like to thanks A. Sentenac for fruitful discussions. REFERENCES [1] M. Fink, C. Prada, F. Wu, and D. Cassereau, “Self focusing in inhomogeneous media with time reversal acoustic mirrors,” in Proc. IEEE Ultrasonics Symp., Oct. 3–6, 1989, pp. 681–686. [2] M. Fink and C. Prada, “Acoustic time-reversal mirrors,” Inverse Problems, vol. 17, no. 1, pp. R1–R38, 2001.

[3] M. Skolnik, J. I. Sherman, and F. J. Ogg, “Statistically designed density-tapered arrays,” IEEE Trans. Antennas Propag., vol. 12, pp. 408–417, Jul. 1964. [4] B. Sichelstiel, W. Waters, and T. Wild, “Self-focusing array research model,” IEEE Trans. Antennas Propag., vol. 12, pp. 150–154, Mar. 1964. [5] G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett., vol. 92, no. 19, p. 193904, 2004. [6] B. E. Henty and D. D. Stancil, “Multipath-enabled super-resolution for RF and microwave communication using phase-conjugate arrays,” Phys. Rev. Lett., vol. 93, no. 24, p. 243904, 2004. [7] G. Lerosey, J. de Rosny, A. Tourin, A. Derode, and M. Fink, “Time reversal of wideband microwaves,” Appl. Phys. Lett., vol. 88, no. 15, p. 154101, 2006. [8] G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves and telecommunication,” Radio Sci., vol. 40, no. 6, p. RS612, 2005. [9] T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” in Proc. IEEE Global Telecommunications Conf., Dec. 2004, vol. 59, pp. 3123–3127, 1. [10] A. B. Ruffin, J. Van Rudd, J. Decker, L. Sanchez-Palencia, L. Le Hors, J. F. Whitaker, and T. B. Norris, “Time reversal terahertz imaging,” IEEE J. Quantum Electron., vol. 38, pp. 1110–1119, Aug. 2002. [11] T. Buma and T. B. Norris, “Time reversal three-dimensional imaging using single-cycle terahertz pulses,” Appl. Phys. Lett., vol. 84, no. 12, pp. 2196–2198, Mar. 2004. [12] P. Kosmas and C. M. Rappaport, “Time reversal with the FDTD method for microwave breast cancer detection,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 7, pp. 2317–2323, 2005. [13] G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science, vol. 315, p. 1120, 2007. [14] V. Fusco and O. Malyuskin, “Active phase conjugating lens with subwavelength resolution capability,” IEEE Trans. Antennas Propag., to be published. [15] O. Malyuskin and V. Fusco, “Subwavelength source resolution using phase conjugating lens assisted with evanescent-to-propagating spectrum conversion,” IEEE Trans. Antennas Propag., 2010, to be published. [16] D. Cassereau and M. Fink, “Time-reversal of ultrasonic fields. III. Theory of the closedtime-reversal cavity,” IEEE Trans. Ultrason. Ferroelect. Freq. Control, vol. 39, no. 5, pp. 579–592, 1992. [17] D. Dowling and D. Jackson, “Narrow-band performance of phase-conjugate arrays in dynamic random media,” J. Acoust. Soc. Am., vol. 91, no. 6, pp. 3257–3277, 1992. [18] K. Solna, “Focusing of time-reversed reflections,” Waves Random Media, vol. 12, no. 3, pp. 365–385, 2002. [19] B. A. van Tiggelen, “Green function retrieval and time reversal in a disordered world,” Phys. Rev. Lett., vol. 91, p. 243904, 2003. [20] C. Oestges, A. Kim, G. Papanicolaou, and A. Paulraj, “Characterization of space-time focusing in time-reversed random fields,” IEEE Trans. Antennas Propag., vol. 53, pp. 283–293, 2005. [21] P. Blomgren and G. Papanicolaou, “Super-resolution in time-reversal acoustics,” J. Acoust. Soc. Am., vol. 111, no. 1, pp. 230–248, 2001. [22] J. de Rosny, A. Tourin, A. Derode, B. van Tiggelen, and M. Fink, “Relation between time reversal focusing and coherent backscattering in multiple scattering media: A diagrammatic approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys., vol. 70, no. 4, pt. 2, p. 046601, Oct. 2004. [23] R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett., vol. 32, no. 21, pp. 3107–3109, 2007. [24] D. H. Chambers and J. G. Berryman, “Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field,” IEEE Trans. Antennas Propag., vol. 52, pp. 1729–1738, 2004. [25] A. J. Devaney, “Time reversal imaging of obscured targets from multistatic data,” IEEE Trans. Antennas Propag., vol. 53, pp. 1600–1610, May 2005. [26] M. E. Yavuz and F. L. Teixeira, “Full time-domain dort for ultrawideband fields in dispersive, random inhomogeneous media,” IEEE Trans. Antennas Propag., vol. 54, pp. 2305–2315, 2006.

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[27] G. Micolau, M. Saillard, and P. Borderies, “Dort method as applied to ultrawideband signals for detection of buried objects,” IEEE Trans. Geosci. Remote Sensing, vol. 41, pp. 1813–1820, Aug. 2003. [28] C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decomposition of the time reversal operator: Detection and selective focusing on two scatterers,” J. Acoust. Society Amer., vol. 99, no. 4, pp. 2067–2076, 1996. [29] J. Jackson, Classical Electrodynamics. New York: Wiley, 1975, ch. 6, pp. 245–250. [30] C. Altman, A. Schatzberg, and K. Suchy, “Symmetry transformations and time reversal of currents and fields in bounded (bi)anisotropic media,” IEEE Trans. Antennas Propag., vol. 32, pp. 1204–1210, Nov. 1984. [31] A. Sommerfeld, Elektrodynamik. Leipzig: Geest & Portig, 1961, ch. 26B. [32] L. B. Felsen and N. Marcuvitz, “Radiation and scattering of waves,” in Series on Electromagnetic Waves, I. Press, Ed. New York: IEEE Press, 1994, ch. 1, p. 31. [33] K. Suchy, “The velocity of a wave packet in an anisotropic absorbing medium,” J. Plasma Phys., vol. 8, p. 33, Aug. 1972. [34] J. A. Bennett, “Complex rays for radio waves in an absorbing ionosphere,” IEEE Proc., vol. 62, pp. 1577–1585, Nov. 1974. [35] C. Krowne, “Electromagnetic theorems for complex anisotropic media,” IEEE Trans. Antennas Propag., vol. 32, pp. 1224–1230, Nov. 1984. [36] I. V. Lindell, “Variational method for the analysis of lossless bi-isotropic (nonreciprocal chiral) waveguides,” IEEE Trans. Microw. Theory Tech., vol. 40, pp. 402–405, Feb. 1992. [37] I. V. Lindell, A. Sihvola, and K. Suchy, “Six-vector formalism in electromagnetics of bianisotropic media,” J. Electromagn. Waves Appl., vol. 9, pp. 887–903, 1995. [38] C. A. Balanis, Antenna Theory Analysis and Design. New York: Wiley-Interscience, 2005, ch. 3. [39] J. de Rosny and M. Fink, “Overcoming the diffraction limit in wave physics using a time-reversal mirror and a novel acoustic sink,” Phys. Rev. Lett., vol. 89, no. 12, p. 124301, 2002. [40] S. J. Orfanidis, “Electromagnetic Waves and Antennas,” 2009 [Online]. Available: http://www.ece.rutgers.edu/~orfanidi/ewa [41] T. Marshall, “Numerical Electromagnetics Code,” [Online]. Available: http://www.nec2.org/ [42] A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, and M. Fink, “Taking advantage of multiple scattering to communicate with timereversal antennas,” Phys. Rev. Lett., vol. 90, no. 1, p. 014301, Jan. 2003. [43] H. T. Nguyen, J. B. Andersen, and G. F. Pedersen, “The potential use of time reversal techniques in multiple element antenna systems,” IEEE Commun. Lett., vol. 9, no. 1, pp. 40–42, Jan. 2005. [44] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, no. 18, pp. 3966–3969, Oct. 2000. [45] N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science, vol. 308, no. 5721, pp. 534–537, 2005. [46] D. Melville and R. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express, vol. 13, no. 6, pp. 2127–2134, 2005. [47] T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science, vol. 313, no. 5793, p. 1595, 2006. [48] S. A. Maier, M. L. Brongersma, and H. A. Atwater, “Electromagnetic energy transport along arrays of closely spaced metal rods as an analogue to plasmonic devices,” Appl. Phys. Lett., vol. 78, no. 1, pp. 16–18, 2001. [49] M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett., vol. 93, no. 13, p. 137404, Sep. 2004.

[50] K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature, vol. 432, no. 7015, pp. 376–379, Nov. 2004. [51] A. Sentenac and P. C. Chaumet, “Subdiffraction light focusing on a grating substrate,” Phys. Rev. Lett., vol. 101, no. 1, p. 013901, 2008. [52] J. de Rosny and M. Fink, “Focusing properties of near-field time reversal,” Phys. Rev. A: Atomic, Molecular, Opt. Phys., vol. 76, no. 6, p. 065801, 2007.

Julien de Rosny (M’01) was born in 1972 in Conflans Sainte Honrine, France. He received the B.S., M.S., and Ph.D. degrees in fundamental physics from Université Pierre et Marie Curie (Paris 6), France, in 1993, 1995, and 2000, respectively. In 2001, he worked as a Postdoctoral Researcher in the Marine Physical Laboratory, SCRIPPS, University of California, San Diego. He is currently a Researcher at CNRS (National Center for Scientific Research). He works at the Institut Langevin, Paris, France. His current research interests include signal processing, microwaves, wave propagation in complex media and time-reversal of waves.

Geoffroy Lerosey was born in February 1979 in Avignon, France. He received the Engineer degree in physics from Ecole Supérieure de Physique et de Chimie Industrielles de la ville de Paris (ESPCI), Paris, France, in 2003 and the M.S. and Ph.D. degrees in physics from the University Paris, in 2003 and 2006, respectively. In 2007, he worked as a Postdoctoral Researcher in Dr. Xiang Zhang’s laboratory at the University of Berkeley, Berkeley, CA. He is currently a Researcher at CNRS (National Center for Scientific Research). He works at Institut Langevin, Paris, France. His research interests include time reversal of electromagnetic waves, time reversal telecommunication, propagation of electromagnetic waves in complex media and plasmonic.

Mathias Fink received the M.S. degree in mathematics, the Ph.D. degree in solid state physics, and the Doctorat es-Sciences degree from Paris University, Paris, France, in 1967, 1970, and 1978, respectively. His Doctorat es-Sciences research was in the area of ultrasonic focusing with transducer arrays for real-time medical imaging. He is a Professor of physics at the Ecole Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI), Paris, France, and at Paris 7 University (Denis Diderot), France. In 1990, he founded the Laboratory Ondes et Acoustique at ESPCI. In 2002, he was elected to the French Academy of Engineering and in 2003 to the French Academy of Science. His current research interests include medical ultrasonic imaging, ultrasonic therapy, nondestructive testing, underwater acoustics, telecommunications, seismology, active control of sound and vibration, analogies between optics, quantum mechanics and acoustics, wave coherence in multiply scattering media, and time-reversal in physics. He has developed different techniques in acoustic imaging (transient elastography), wave focusing in inhomogeneous media (time-reversal mirrors), speckle reduction, and in ultrasonic laser generation. He holds 40 patents, and he has published more than 300 articles.

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Microstrip-Fed Wideband Circularly Polarized Printed Antenna Xiu Long Bao, Member, IEEE, Max J. Ammann, Senior Member, IEEE, and Patrick McEvoy, Member, IEEE

Abstract—A wideband circularly-polarized printed antenna is proposed, which employs an asymmetrical dipole and a slit in the ground plane which are fed by an L-shaped microstrip feedline using a via. The proposed antenna geometry is arranged so that the orthogonal surface currents, which are generated in the dipole, feedline and ground plane, have the appropriate phase to provide circular polarization. A parametric study of the key parameters is made and the mechanism for circular polarization is described. The measured results show that the impedance bandwidth is approximately 1.34 GHz (2.45 GHz to 3.79 GHz) and the 3 dB axial ratio bandwidth is approximately 770 MHz (2.88 GHz to 3.65 GHz) which represent fractional bandwidths of approximately 41% and 23%, respectively, with respect to a centre frequency of 3.3 GHz. Index Terms—Asymmetrical dipole, circular polarization, printed dipole antenna, slot antenna. Fig. 1. Geometry and coordinate system for the proposed printed antenna. (a) The asymmetrical dipole and microstrip feedline network with a slit in the ground plane. (b) Edge profile.

I. INTRODUCTION IRCULARLY polarized (CP) radio propagation links in satellite communications, satellite positioning and radio frequency identification (RFID) [1]–[3] systems are preferred to linear polarization schemes which are subject to losses when arbitrary polarization misalignment occurs between the transmitter and receiver. With CP antennas at both radios, the enhanced gain and cross-polar discrimination improve the resilience of the system to multipath propagating effects. To create circular polarization, the antenna must radiate from modes of equal magnitude that are orthogonal in space and in phase quadrature. Several techniques have been used in various types of circularly polarized antennas that have been reported in recent decades. Printed circular or square patch geometries with perturbing narrow slots or truncated stubs [4], [5] achieve CP by introducing degenerate modes with 90 phase difference. To create broader operating bandwidths, designs have included paired rectangular wire loops above an infinite ground plane [6], a coupled loop with parasitic loop [7] and a two-layer substrate [8], where axial ratio (AR) bandwidths of 18%, 16% and 9.6% respectively, were achieved. Furthermore, balanced

C

Manuscript received December 08, 2008; revised March 18, 2009; accepted April 13, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by Science Foundation Ireland under Grants 08/CE/I1523 and 09/SIRG/I1644. The Telecommunications Research Centre, CTVR, provided the research infrastructure. X. L. Bao is with the Antenna and High Frequency Research Centre, Dublin Institute of Technology, Dublin 8, Ireland M. J. Ammann and P. McEvoy are with the Antenna and High Frequency Research Centre, Dublin Institute of Technology, Dublin 8, Ireland and also with the Telecommunications Research Centre, Dublin 8, Ireland (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055776

dual circular and rhombic loops with small parasitic elements [9] use gaps in the structure to achieve circular polarization with AR bandwidths up to 40%, but at a cost of the antenna profile being greater than a quarter-wavelength. Alternatively, lower profile slot antennas [10]–[13] have achieved CP bandwidth characteristics that have AR bandwidths that range from 4% to 25%. The literature also reports coupled feed designs for CP performance. An annular-ring patch element with a parasitic fan-shaped patch [14] is excited by a single port; and a two-port U-shaped microstrip feedline couples into square and circular annular ring patches to provide reconfigurable polarization sense, depending on which port is driven [15]. In both cases, the operational bandwidths are less than 2%. These performances contrast with various printed dipole antennas with broad bandwidths, low-profiles and light weights, but linear polarization. In [16]–[19], the printed dipole antennas have integrated microstrip baluns and in [20] the dual-band printed dipole antennas comprise a pair of arms with two parallel strips. In [21], by using four sequentially rotated configurations of the crossed dipole, circularly polarized characteristics are realized. In this paper, a simple single layer printed dipole with a pair of asymmetrical arms is combined with a slot antenna to provide circular polarization. A parametric study is made to optimize the performance of the small structure to realize a 23% AR bandwidth. II. GEOMETRY OF THE PROPOSED PRINTED CP ANTENNA The geometry and coordinate system of the proposed printed circularly polarized antenna are shown in Fig. 1.

0018-926X/$26.00 © 2010 IEEE

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Fig. 2. The simulated and measured S

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for the proposed dipole antenna. Fig. 4. Simulated and measured radiation patterns in the xz Elevation Plane (' = 0 ) at (a) 2.9 GHz, (b) 3.1 GHz, (c) 3.3 GHz and (d) 3.5 GHz. Simulated ; Simulated LHCP | | ; MeaRHCP | | { ; Measured RHCP sured LHCP .

11111111111

000000

1

1

Fig. 3. The simulated and measured axial ratio for the proposed dipole antenna.

The L-shaped microstrip feedline is printed on one side of the substrate and an asymmetrical dipole with ground plane is printed on the rear side. The narrow dipole gap is continued between a pair of coplanar strip lines and it extends as a slit into the small ground plane. The microstrip feedline extends above the groundplane to an orthogonal stub which is connected to the short arm of the dipole by means of a via. The microstrip feedline is wider than the dipole gap and it is centered opposite the groundplane slit, which is also centered on the substrate. The feed configuration differs from a conventional microstrip-via balun which simply couples to one dipole feedline [17]. In our case, the overlap between the microstrip edges and the slit edges introduces coupling to alter the surface current densities and phasing along the feed network. The feed network functions in two ways. Firstly, it enables an impedance transformation, between 50 at the microstrip feed port and the higher dipole impedance, across a wide frequency range. Secondly, it supports orthogonal radiating currents of similar magnitude but

Fig. 5. Simulated and measured radiation patterns in the yz Elevation Plane (' = 90 ) at (a) 2.9 GHz, (b) 3.1 GHz, (c) 3.3 GHz and (d) 3.5 GHz. Simu; Simulated LHCP | | ; lated RHCP | | { ; Measured RHCP Measured LHCP .

11111111111

000000

1

1

in phase quadrature to those on the asymmetrical dipole, thereby stimulating radiation in the and directions. The longer arm of the printed dipole, the ground plane slit and the microstrip feedline are approximately a quarter wavelength. The proposed antenna is printed on a Taconic RF35 substrate, with a relative permittivity of 3.5, a thickness of 1.57 mm and a loss tangent of 0.0018. The substrate size is 30 mm 36 mm. The geometric parameters and modeled features are detailed in Fig. 1. The diameter of the via is 1.4 mm and it is centered with respect to both the microstrip and the dipole feedline.

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TABLE I

(a)

(b)

!

Fig. 6. Simulated surface current distributions for antenna orthogonal phases. (a) 0 , (b) 90 , (c) 180 and (d) 270 . ( solid line of arrows on the microstrip line; { dash line of arrows on the back of the substrate).

!

III. NUMERICAL AND MEASURED RESULTS Numerical work was carried out using the time domain solver in CST Microwave Studio and included a SMA feed connector model. The dimensions for the dipole, ground plane and feedline parameters are given in Table I. The simulated and meavalues for the proposed antenna are given in Fig. 2 sured bandand show good agreement. The measured width was found to be approximately 1.34 GHz (i.e., 2.45 GHz to 3.79 GHz). In this case, the antenna was designed to realize direction. right-hand circular-polarization (RHCP) in the The RHCP and left-hand circular-polarization (LHCP) radiation patterns and AR were measured in an anechoic chamber using a standard gain horn antenna as a reference and computed using and using formulae data from their far field components from [22].The measured AR is shown in Fig. 3 along with numerical predictions. It can be seen that the measured 3 dB AR bandwidth is approximately 770 MHz (i.e., 2.88 GHz to 3.65 GHz) and is in agreement with numerical values. The RHCP

W

Fig. 7. (a) Comparison of the S with different widths of printed arms = 8 0 mm, = 12 0 mm, = 2 0 mm, = 4 0 mm. and with (b) Comparison of the AR with different widths of printed arms and with = 8 0 mm, = 12 0 mm, = 2 0 mm, = 4 0 mm.

L

L :

: L

L :

: W

: W : S W

S :

:

and LHCP radiation patterns were measured in the , 90 planes (antenna co-ordinates, and respectively) at frequencies of 2.9 GHz, 3.1 GHz, 3.3 GHz and 3.5 GHz. These were normalized and compared to simulated patterns and the agreement is good in the vicinity of maximum radiation for both RHCP and LHCP as indicated in Figs. 4 and 5. The measured peak gain was found to be approximately 2.0 dBic and the simulated efficiency was 93% at the center frequency. To illustrate the circular polarization mechanism, which requires modes of equal magnitude that are phase orthogonal, the simulated surface current distributions viewed from the microstrip side are illustrated in Fig. 6. The direction of the surface currents on the dipole arms and the microstrip feed network is shown at 3.3 GHz as the phase changes from 0 through 270 . The 0 phase reference shows that the dominant radiating curdirected, while on the -axis, the microstrip feedrents are line current is phase opposed to each of the groundplane slit

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(a) (a)

(b) Fig. 8. (a) Comparison of the S for different arm length L with W = 1:0 mm, L = 12:0 mm, W = 2:0 mm, S = 4:0 mm. (b) Comparison of the AR for different arm length L with W = 1:0 mm, L = 12:0 mm, W = 2:0 mm, S = 4:0 mm.

edge currents. For the 90 phase, the dominant surface current direction. Currents on the left and right edges of flow is in the the groundplane slit are in phase opposite directions but the left edge direction is phase aligned with the microstrip feedline. The dipole currents are phase opposed to their respective adjacent directed groundplane edges. For the 180 phase, a dominant current flow is observed, which is an inverted current phase arrangement to the 0 phase reference. Finally, for the 270 phase, the currents are directed (phase inverted with respect to the 90 phase), hence the polarization sense is right-hand in the ( ) direction. Furthermore, LHCP may be achieved by interchanging the dipole arms and the microstrip L-shaped feedline via connection, or by simply flipping the antenna 180 about the vertical centerline parallel to the -axis. IV. PARAMETRIC STUDY OF THE PROPOSED PRINTED ANTENNA The performance of the proposed printed antenna structure shows greater sensitivity to variation in some parameters, such

(b) Fig. 9. (a) Comparison of the S for different values of S with W = 1:0 mm, L = 8:0 mm, L = 12:0 mm, W = 2:0 mm. (b) Comparison of the AR for different values of S with W = 1:0 mm, L = 8:0 mm, L = 12:0 mm, W = 2:0 mm.

as the width and length of the printed dipole arm, the and length of the slit between the two feedlines and width the location of the microstrip via. A parametric study of these key parameters presented below provides a useful evaluation of their effects on antenna performance. The other parameters for , the proposed antenna are listed as follows: , , , , , (Fig. 1). A. The Length and Width of the Printed Dipole Arms and AR against freFig. 7(a) and (b) illustrate plots of quency. The variation of the width of the two printed arms has little effect on the impedance bandwidth and the frequency is reduced, the AR of operation. However, as the arm width is seen to increase as shown in Fig. 7(b). Fig. 8(a) and (b) show and AR against frequency for different values of the the

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(a)

(a)

(b)

(b) Fig. 10. (a) Comparison of the S for various values of slit length L , with W : ,L : ,W : ,S : . (b) : , Comparison of the AR for various values of slit length L , with W : ,W : ,S : . L

= 1 0 mm = 8 0 mm

= 8 0 mm = 2 0 mm = 2 0 mm = 4 0 mm

= 4 0 mm = 1 0mm

short dipole arm length . In this case, the longer arm of the . As the length of the printed dipole is fixed at is shortened from 12 mm to 8 mm, the arm bandwidth is significantly increased. It is noted from Fig. 8(b) that the CP bandwidth is shifted downwards as the arm length is increased. B. The Location of the Feedpoint Via Fig. 9(a) and (b) illustrate the plots of and AR for different feedline via locations. The via connection point to the dipole while keeping feedline is varied by changing the parameter constant. The AR values are found to be sensitive to the . It is found that the widest AR and variation of the value of . impedance bandwidth is obtained for

Fig. 11. (a) Comparison of the S for various value of slit width : : : : W ,L ,L ,S Comparison of the AR for various value of slit width W , with W : : : ,L ,S . L

= 1 0 mm = 8 0 mm

= 8 0 mm = 12 0 mm

= 12 0 mm = 4 0 mm

W , with

= 4 0 mm. (b) = 1:0 mm,

C. The Length and Width of the Slit Between Dipole Feedlines The length and width of the slit between the dipole feedlines was found to have a significant effect on the performance. Fig. 10(a) illustrates the sensitivity of the impedance to (all other parameters are fixed). variation in the slit length It is seen that for greatest impedance bandwidth, a value of is optimum. A wide AR bandwidth is realized for this value of as shown in Fig. 10(b). . Fig. 11 illustrates the sensitivity to the width of the slot , This parameter was varied from 1 mm to 3 mm (while and remain constant) and it was found that the widest AR and impedance bandwidth occurred for . This parametric study has helped to identify dimensional trends of the most important parameters and facilitated the design of a wideband circularly polarized antenna.

BAO et al.: MICROSTRIP-FED WIDEBAND CIRCULARLY POLARIZED PRINTED ANTENNA

V. CONCLUSIONS A novel approach to generate circular polarization using a microstrip-via feedline is presented. The proposed printed antenna element comprises a dipole and ground plane slit which are fed by an L-shaped microstrip-via feed. By the appropriate adjustment of the key dimensional parameters, circular polarization with an axial ratio of 3 dB or less is realized over a wide bandwidth. The circular polarization mechanism was described using surface currents. Measurements show the 3 dB axial ratio bandwidth to be approximately 770 MHz, representing a fractional bandwidth of approximately 23% with respect to a centre frequency of 3.3 GHz. An impedance bandwidth of 1.34 GHz corresponding to a fractional bandwidth of approximately 42% with respect to 3.3 GHz, is achieved. REFERENCES [1] F. Ferrero, C. Luxey, G. Jacquemod, and R. Staraj, “Dual-band circularly polarized microstrip antenna for satellite applications,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 13–15, 2005. [2] X. L. Bao, G. Ruvio, M. J. Ammann, and M. John, “A novel GPS patch antenna on a fractal hi-impedance surface substrate,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 323–326, 2006. [3] H. W. Kwa, X. M. Qing, and Z. N. Chen, “Broadband single-fed single-patch circularly polarized antenna for UHF RFID applications,” in Proc. IEEE Antennas Propag. Int. Symp., Jul. 2008, pp. 1–4. [4] A. K. Bhattacharyya and L. Shafai, “A wider band microstrip antenna for circular polarization,” IEEE Trans. Antennas Propag., vol. 36, pp. 157–163, 1988. [5] X. L. Bao and M. J. Ammann, “Comparison of several novel annularring microstrip patch antennas for circular polarization,” J. Electromagn. Wave Applicat., vol. 20, no. 11, pp. 1427–1438, 2006. [6] M. Sumi, K. Hirasawa, and S. Shi, “Two rectangular loops fed in series for broadband circular polarization and impedance matching,” IEEE Trans. Antennas Propag., vol. 52, pp. 551–554, 2004. [7] R. L. Li, J. Laskar, and M. M. Tentzeris, “Wideband probe-fed circularly polarised circular loop antenna,” Electron. Lett., vol. 41, no. 18, pp. 997–999, 2005. [8] S. Gao, Y. Qin, and A. Sambell, “Low-cost broadband circularly polarized printed antennas and array,” IEEE Antennas Propag. Mag., vol. 49, no. 4, pp. 57–64, 2007. [9] R. L. Li, G. DeJean, J. Laskar, and M. M. Tentzeris, “Investigation of circularly polarized loop antennas with a parasitic element for bandwidth enhancement,” IEEE Trans. Antennas Propag., vol. 53, pp. 3930–3939, 2005. [10] S. Shi, K. Hirasawa, and Z. N. Chen, “Circularly polarized rectangularly bent slot antennas backed by a rectangular cavity,” IEEE Trans. Antennas Propag., vol. 49, pp. 1517–1524, 2001. [11] J. S. Row, “The design of a squarer-ring slot antenna for circular polarization,” IEEE Trans. Antennas Propag., vol. 53, pp. 1967–1972, 2005. [12] C. C. Chou, K. H. Lin, and H. L. Su, “Broadband circularly polarised cross-patch-loaded square slot antenna,” Electron. Lett., vol. 43, no. 9, pp. 485–486, 2007. [13] J. Y. Sze and C. C. Chang, “Circularly polarized square slot antenna with a pair of inverted-L grounded strips,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 149–151, 2008. [14] Y. F. Lin, H. M. Chen, and S. C. Lin, “A new coupling mechanism for circularly polarized annular-ring patch antenna,” IEEE Trans. Antennas Propag., vol. 56, pp. 11–16, 2008. [15] K. F. Tong and J. J. Huang, “New proximity coupled feeding method for reconfigurable circularly polarized microstrip ring antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 1860–1866, 2008. [16] L. C. Kuo et al., “A study of planar printed dipole antennas for wireless communication application,” J. Electromagn. Waves Applicat., vol. 21, no. 5, pp. 637–652, 2007. [17] H. R. Chuang and L. C. Kuo, “3-D FDTD design analysis of a 2.4-GHz polarization-diversity printed dipole antenna with integrated balun and polarization-switching circuit for WLAN and wireless communication applications,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 374–381, 2003. [18] Z. G. Fan, S. Qiao, J. T. Huangfu, and L. X. Ran, “A miniaturized printed dipole antenna with V-shaped ground for 2.45 GHz RFID,” Progr. Electromagn. Res., vol. PIER 71, pp. 149–158, 2007.

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[19] B. Edward and D. Rees, “A broadband printed dipole with integrated balun,” Microwave Journal, pp. 339–344, May 1987. [20] F. Tefiku and C. A. Grimes, “Design of broad-band and dual-band antennas comprised of series-fed printed-strip dipole pairs,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 895–900, 2000. [21] K. M. Mak and K. M. Luk, “A shorted cross bowtie patch antenna with a cross dipole for circular polarization,” in Proc. IEEE Antennas Propag. Int. Symp., Jun. 2007, pp. 2702–2705. [22] B. Y. Toh, R. Cahill, and V. F. Fusco, “Understanding and measuring circular polarization,” IEEE Trans. Education, vol. 46, pp. 313–318, 2003. Xiulong Bao (M’09) received the B.Sc. degree in physics from the Huaibei Coal Industry Teachers’ College, Anhui Province, China, in July 1991, and the M.Sc. degree in physics and the Ph.D. degree in electromagnetic field and microwave technology from Southeast University, Jiangsu Province, China, in April 1996 and April 2003, respectively. After graduating, he was a Postdoctoral Researcher at Shanghai Jiaotong University, Shanghai, China, before going to Ireland in 2005. He is currently a Senior Research Associate with the School of Electronic and Communications Engineering, Dublin Institute of Technology, Ireland. His broad research interests include analysis and design of various small and circularly polarized antennas, such as GPS antennas, multiple-band antennas, RFID antennas, a DTV antenna, handset antennas, ultrawideband (UWB) antennas and the design and application of metamaterial/EBG structures. He is also active in the study of electromagnetic scattering, electromagnetic numerical computation (FDTD, PSTD, FDFD and MOM methods) and the study of electromagnetic wave propagation and antenna theory. He has published over 30 peer-reviewed journal papers and 28 conferences articles. Dr. Bao received funding from Science Foundation Ireland to research miniaturization techniques for broadband, circularly-polarized antennas. He was a Technical Program Committee member for the 65th IEEE Vehicular Technology Conference, Dublin, 2007.

Max Ammann (M’96–SM’08) received the Council of Engineering Institution Part II degree in 1980 and the Ph.D. degree in microwave antenna design from Trinity College, University of Dublin, Dublin, Ireland, in 1997. He is a Senior Lecturer in the School of Electronic and Communications Engineering, Dublin Institute of Technology, where he is the Director of the Antenna and High Frequency Research Centre. He also leads the antenna research within Ireland’s Centre for Telecommunications Value-chain Research (CTVR), Dublin. He spent eight years on radio systems engineering and antenna design for TCL/Philips Radio Communications Systems, Dublin, where he was responsible for commissioning the Nationwide Communications Network for Ireland’s national police force. In 1986 he joined the DIT as a Lecturer and was promoted to Senior Lecturer in 2003. His research interests include electromagnetic theory, antenna miniaturization for terminal and ultra wideband applications, microstrip antennas, metamaterials, antennas for medical devices and the integration with photovoltaic systems. He has in excess of 160 peer-reviewed papers published in journals and international conferences. He has served as an expert to industry on various antenna technologies in the communications, medical, aviation and electronic security sectors in Ireland and abroad. The roles have included design assessment, design solutions, technological strategy reporting and assessment of compliance with international standards on human exposure to electromagnetic energy. The industrial contacts also stem from several successful transfers of fundamental design research into applied solutions. Dr. Ammann received the 2006 Best Poster Award at the Loughborough Antennas and Propagation Conference, commercialization awards for work with DecaWave Ltd. and Taoglas Ltd and a 2008 CST Publication Award for work on a “Wideband Reconfigurable Rolled Planar Monopole Antenna.” He sits on the management committee of the EU COST Action IC0603, “Antenna Systems and Sensors for Information Society Technologies” (ASSIST) and he is active in the Antenna Sensors and Systems Work Centre. As a member of the IEEE International Committee for Electromagnetic Safety, he participated in the revision of the IEEE Std. C95.1, 2005 standard for Safety Levels with Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 3 kHz to 300 GHz. He is

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also a member of the URSI Committee for Communications and Radio Science within the Royal Irish Academy, with expertise in Commission K: Electromagnetics in Biology and Medicine. He chaired the IEEE APS Special Session on Antennas for UWB Wireless Communication Systems, Columbus, Ohio, 2003 and was chair for the Antennas and Propagation Track for the 65th IEEE VTC, Dublin 2007. He was the local chair for the October 2008 EU COST IC0603 workshop and meeting in Dublin.

Patrick McEvoy (M’02) received the M.Eng. degree in electronic communications engineering from the University of Hull, Hull, U.K. (partially undertaken at L’Institut Supérieur d’Electronique de Paris, France) in 1998 and the Ph.D. degree in microwave antenna engineering from Loughborough University, Leicestershire, U.K., in 2007. Currently, he is a senior researcher at the Antenna and High Frequency Research Centre, Dublin Institute of Technology (DIT), Dublin, Ireland. Prior to joining the DIT he was a Research Manager at the

Centre for Mobile Communications Research, Loughborough University, where he worked on switched antennas for handheld terminals, applications of metamaterials, low specific absorption rate antenna design and antenna measurement systems. He has twelve years of applied academic research and industrial experience that includes design, high-volume manufacturing and measurement systems for miniaturized microwave antennas. His research focus is currently on body area communications, hyperthermia applicators, ultra wideband antennas for frequency and time-domain applications and the integration of antennas with solar voltaic devices. He has published over 40 scientific papers and has helped to organize four international conferences on antennas and propagation.

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Wideband and High-Gain Composite Cavity-Backed Crossed Triangular Bowtie Dipoles for Circularly Polarized Radiation Shi-Wei Qu, Chi Hou Chan, Fellow, IEEE, and Quan Xue, Senior Member, IEEE

Abstract—A composite cavity-backed antenna excited by crossed triangular bowtie dipoles is proposed and investigated for circular polarization (CP) applications. It is fed by a novel balun, i.e., a transition from a microstrip line to double slot lines, providing symmetrical electric field distributions at the feeding port. Measurements of an optimized antenna prototype show that it can achieve an impedance bandwidth of over 57.6% for SWR 2, a 3-dB axial-ratio bandwidth of 39%, a broadside gain of 8 10 7 dBi, and symmetrical radiation patterns over the whole operating band. The operating principles of the proposed antenna are analyzed carefully and found quite different from crossed thin-wire dipoles with very weak coupling. Problems in the feeding balun, greatly deteriorating the CP performance at the resonance, are clearly addressed and solved. Detailed parametric studies are given in the final part of this paper. Index Terms—Bowtie antennas, cavity-backed antennas, circular polarization, unidirectional patterns.

I. INTRODUCTION

C

IRCULARLY polarized (CP) antennas have been receiving much attention for the applications on wireless communications because they are not only able to reduce the multi-path effects but also to allow more flexible orientation of the transmitting and receiving antennas. Meanwhile, the unidirectional patterns are expected to provide high security and efficiency of the propagating channels in many cases. A conventional method to construct an antenna with CP radiation is to produce two degenerate modes on the radiating element with one feeding point, such as single probe-fed, proximity-coupled and aperture-coupled CP patch antennas [1]–[3]. However, their axial-ratio (AR) bandwidths are always very small, generally a few percent which does not meet the bandwidth requirements in modern wideband communication systems. An improved method is to generate two orthogonal

Manuscript received July 12, 2009; revised January 13, 2010; accepted April 03, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported by the Research Grants Council of Hong Kong SAR, China, under [Project No. CityU122407]. S.-W. Qu was with the State Key Laboratory of Millimeter Waves, City University of Hong Kong, Kowloon, Hong Kong, China. He is now with Tohoku University, Sendai, Japan (e-mail: [email protected]). C. H. Chan and Q. Xue are with the State Key Laboratory of Millimeter Waves, City University of Hong Kong, Kowloon, Hong Kong, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055792

modes separately with dual- or multi-fed mechanism, such as patch antennas fed by L-probe and aperture coupling techbandwidth can niques [4], [5], and although the be enhanced to over 40%, the complicated feeding networks are required and the broadside gains are generally below 8 dBi. Additionally, the sequential arrangement of linearly or circularly polarized antenna elements can also produce CP radiation in a large frequency band, but at the cost of large total dimensions [6], [7]. A straightforward way for CP radiation is to employ two orthogonally crossed dipoles fed by two sources with equal magnitude and 90 phase difference. However, the feeding network also complicates the antenna design and fabrication [8]–[11]. In 1961, it was reported that single-fed crossed dipoles connected in parallel could also generate CP radiation if the lengths of the dipoles were chosen such that the real parts of their input admittances are equal and the phase angles of their input admittances differ by 90 [12]. According to these conditions, several crossed-dipole antennas were developed. A short backfire antenna excited by two crossed slots can present a 4.2% AR bandwidth with a 14-dBi gain [13]. Also reported is a crossed-slot antenna, backed by a metal plate with a quarter wavelength height, which features a 5% AR bandwidth and a 10-dBi gain [14]. Another crossed-dipole antenna with a reactively loaded parasitic loop is developed for land mobile satellite communications [15]. However, all of these antennas show a small AR bandwidth, at most 15%, and are also unsuitable for modern wideband communication systems. In our previous studies, a series of unidirectional cavity-backed bowtie antennas for wideband applications were developed [16]–[20], and the operating principles of this kind of antenna are essentially investigated based on previous outstanding contributions [21], and both their advantages and disadvantages are extensively discussed in these works. However, all of our previous works were focused on linearly polarized (LP) applications. In this paper, cavity-backed structures are also employed to construct a wideband CP antenna. The same composite cavity to [17] is utilized to optimize the electric field distributions for stable radiation patterns. A pair of crossed bowtie dipoles (CBD), formed by crossing two triangular bowtie dipoles together, is employed as the exciter which determines the operating frequency and final bandwidth to some extent. The sharp corners of larger triangular bowtie dipole are rounded based on our previous research [18] and the smaller one is loaded by two overlapped gaps, offering a flexible input capacitance within most part of the interesting

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Fig. 3. Geometry of the transition from the microstrip line to double slot line (in mm). Fig. 1. Geometry of the proposed composite cavity-backed CBD.

Fig. 2. Geometry of the CBD.

frequency band. Measured results of the fabricated antenna prototype verify simulations with reasonably good agreements. II. GEOMETRY OF COMPOSITE CAVITY-BACKED CBD Fig. 1 shows the geometry of the proposed antenna, consisting of a composite cavity, a CBD as an exciter, a transition from a microstrip line to double slot lines (DSL), and an SMA connector placed under the cavity to feed the antenna. The CBD is arranged similar to the exciter in [17] for the lowest profile. Fig. 2 shows the geometry of the CBD and its parameters in detail, and the deep color denotes its shape on the top side of the substrate and the dashed line shows that on the bottom side. The exciter consists of a larger triangular bowtie dipole with rounded corners and a smaller one with two overlapping parts between the top and the bottom layers, corresponding to capacitive loadings. The desired input reactance of smaller triangular bowtie and , representing dipole can be easily achieved by tuning position and value of the capacitive loading, respectively. The smaller dipole is not rounded due to the reasons mentioned in the next section. The CBD is built on a substrate with thickness of

0.5 mm and relative permittivity for easy fabrication and support. For clarity, the central part of the CBD is scaled and given as the inset in the right side of Fig. 2(a). For parallel connection of the two dipoles, a narrow strip with width is utilized to connect one arm of the larger bowtie dipole and that of the smaller one, and a small gap of is located at its vertexes for connection with the DSL of the transition. The geometry of the microstrip-to-DSL transition, with two dielectric and three metallic layers, is given in Fig. 3, and it is mounted vertically on the ground plane and fabricated on a substrate with and 0.381-mm thickness. The DSL section of the transition features a symmetrical structure along both XZ and YZ planes, so the transition can provide symmetrical electric field distributions on the DSL port, which can induce more balanced currents on the CBD than that used in [22]. Several pairs of slots cut on the ground of the transition are employed to lower the resonant frequency of the transition out of the 3-dB AR band, because the resonance of the transition would greatly deteriorate the CP performances at that frequency point, as given in Section III. In the design procedure, the proposed antenna without the feeding transition is optimized firstly using the parameterized model within the HFSS [23], and then the transition is added for impedance matching considerations. The first step is the most critical and time-consuming one, while the impedance matching . The required comis much easier by just tuning , , and puter source may be interesting for antenna designer. As far as our computer is concerned, there are 3-GB memory, Intel Core 2 CPU @ 2.66 GHz, and 32-bit Windows XP Operating System, and each simulation will cost around 1.5-GB memory and 1-hour time (It will require larger memory and longer time for the whole antenna simulation after adding the transition), which is not an unacceptable configuration for modern personal computers. The optimized geometrical parameters are given as follows ( , where is the free-space wave(in mm): frequency band), length at the center of the ( , aperture diameter), ( , height of cylindrical part of the cavity), ( , height

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Fig. 6. Simulated and measured SWRs of the proposed antenna. Fig. 4. Photograph of the fabricated antenna prototype.

Fig. 5. Simulated and measured ARs and broadside gains.

of conical part of the cavity), (flare angle of two (radius of round corners), bowtie dipoles), ( , length of larger dipole), ( , length of (position of loading gaps), small dipole), (size of loading gaps), , , (width of transition), (length of the DSL), (length of (height of the matching the matching stripline), and stripline), and the other detailed dimensions of the transition are given in Fig. 3 in mm. The total electrical dimensions of the proin diameter and in height. posed antenna are III. RESULTS AND ANALYSIS A. Measured and Simulated Results To verify our design, an antenna prototype with optimized dimensions was built, as shown in Fig. 4, and the measured and simulated ARs, broadside gains, and SWRs are shown in Figs. 5 and 6, respectively. Measurements agree reasonably well with simulations, and the whole measured frequency band for is shifted slightly upwards over the simulated one. It can be seen that the proposed antenna presents a measured for , corresponding to a band of fractional bandwidth of 39%, and in this frequency band the . It can be seen from Fig. 6 that broadside gain is ranges from 1.88 to over the frequency band for 3.4 GHz, covering the whole frequency band and

allowing the antenna operate in a circularly polarized manner with low reflection coefficient. The simulated and measured patterns in the XZ, YZ, and diagonal planes, at 2.2, 2.6, and 3 GHz respectively, are shown in Fig. 7, and measurements agree well with simulations along main beams. It can be seen that the antenna’s main beam is fixed in the broadside direction and no side lobes appear in the whole frequency band. Although the cross-polarizations, i.e., left-hand circular polarized (LHCP) components, are sensitive to the fabrication errors and measurements, the measured results also agree reasonably well with the simulated ones. The antenna features good radiation patterns over the whole frequency band, such as large front-to-back ratio (FBR), symmetrical main beam, and small cross-polarization etc. The measured FBR is always over 25 dB, similar to the simulated 20 dB. It can also be observed from Fig. 7 that the antenna shows a very symmetrical right-hand circular polarized (RHCP) component and equal beam width in three planes. B. Analysis of the Cavity-Backed Crossed TBA To accurately design the proposed antenna for practical applications, its operating principles should be studied carefully. First of all, the conditions to produce good CP waves is very stringent for the proposed crossed dipoles topology in [12] and [13], where the input impedance relationships should be satisfied under the condition of no coupling between the two dipoles. For crossed wire dipoles, the coupling can be ignored due to their orthogonal electric field distributions. However, there is strong and complicated coupling between the arms of two bowtie dipoles, so the required relationships of input impedance are also altered compared with those proposed in [12]. It can be assumed that the input impedance can be , where is the input resistance formulated as is the input reactance, and the - diagram of our and proposed antenna, as shown in Fig. 8, can clearly show this change of requirements to produce CP radiation [12], [13]. For example, at 2.2 GHz, the input resistances are 181.7 and 47.7 , , respectively, and the phase angle between them is Ö at 2.5 GHz, and it and the phase angle is as small as can be even smaller as frequency increases, instead of 90 in [12] and [13]. For the cavity-backed antenna, its radiation is determined more directly by the electric field distribution in the aperture than by the current on the exciter. So a straightforward way to

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Fig. 7. Simulated and measured radiation patterns in XZ, diagonal, and YZ planes at 2.2, 2.6, and 3 GHz, respectively.

Fig. 8. R-X diagram of two bowtie dipoles of the exciter.

understand the CP behavior of our proposed antenna is to examine its electric field distribution in the aperture, as given in , 30 , 60 , 90 , 120 , and Fig. 9 with the phase angle 150 , respectively. It can be seen that a right-hand rotated electric field is concentrated within the aperture and strong electric field between the edges of two dipoles proves strong coupling assumed above. It can be concluded that quite a different relationship of input impedances from the crossed thin-wire dipoles should be satisfied when there is strong coupling between the dipoles. Moreover, there is strong electric field near the sharp corners of the loaded dipole, which is why these corners cannot be rounded. C. Issues About Transition In this section, the functions of slots on the ground of transition are analyzed carefully. Fig. 10(a) gives the AR of the proposed antenna with different transitions. It can be seen that there is a resonance with poor AR shifted downwards from 2.43 to increases from 10 to 20 2.12 GHz as the transition’s width mm (without slots), and this resonance will cause great reduction of the AR bandwidth. In order to move this resonance far is further increased away from the 3-dB AR frequency band,

Fig. 9. Electric field distribution in the antenna aperture.

and three pairs of slots are added at the same time because they can increase the length of current path, and finally the simulated resonance is shifted to 1.86 GHz, without any influence on AR bandwidth. Simulations prove that this phenomenon is caused by the resonance of transition. Fig. 10(b) gives the current distribution on one layer of the grounds of transition at 3 and 2.43 GHz, respectively. Ideally, the current should be concentrated very close to the signal line, e.g., the current at 3 GHz in Fig. 10(b). However, the large current density is also observed on the lower part of the ground, similar to a monopole shorted to the cavity’s ground, thus the radiated linearly polarized waves

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Fig. 11. AR of antenna with different gap position L .

Fig. 12. Input impedance of antenna with different gap position L .

Fig. 10. Resonance of transition and its explanations. (a) AR versus different transitions. (b) Current distributions on the ground at 2.43 GHz and 3 GHz.

greatly deteriorate the CP performance at the resonance. Additionally, the air gap between the two layers of substrate dramatically influences the performances, which is one of main reasons causing differences between simulations and measurements. IV. PARAMETRIC STUDIES AND DISCUSSIONS To understand how the dimensions influence the antenna performances, parametric studies are performed by HFSS™. In this procedure, the feeding transition is also replaced by a 70 lumped port to remove its influences on AR and to clearly show the change of the antenna’s input impedance. When one parameter is studied, the others are kept identical to their optimized values. In the parametric studies, the AR as well as input impedance is emphasized, and the broadside gain is almost immune to the parameters of the exciter. The first issue focused is the gap width and its position related to the capacitive loading on the exciter. As shown in Fig. 11, when the loaded gaps are placed with a small disis small, the AR bandwidth will be reduced and tance, i.e., the center frequency will be shifted upward at the same time. beHowever, the circular polarization will be deteriorated as comes too large because the input resistance and input reactance of the capacitive loaded bowtie dipole will be changed simultaleads to neously. For the input impedance in Fig. 12, small large input resistance as well as large input reactance, and large

Fig. 13. AR of the antenna with different gap size g .

input resistance is of benefit for matching, but large input reactance makes the good impedance matching difficult. So, an should be chosen for a better AR and appropriate value of easy matching. The gap width is another critical parameter, and it can be seen from Fig. 13 that the AR is very sensitive to . A smaller causes better in-band AR but narrower 3-dB AR bandwidth, and obviously the largest 3-dB AR bandwidth can be achieved . From Fig. 14, the gap width influences input as reactance more than input resistance because determines the value of capacitive loading of the smaller bowtie dipole. of capacitively loaded bowtie dipole is also The total size a critical parameter that it does not change the second minimum point of the AR but obviously shift the first one at the cost of worse in-band AR as shown in Fig. 15. Thus, an appropriate can bring largest bandwidth, and also value of

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Fig. 14. Input impedance of the antenna with different gap size g . Fig. 17. AR of the antenna with different size L of rounded bowtie dipole.

Fig. 15. AR of the antenna with different L . Fig. 18. Input impedance of the antenna with different L .

Fig. 16. Input impedance of the antenna with different L .

larger causes flatter and larger input resistance and smaller input reactance as shown in Fig. 16, which are both of benefit for good impedance matching. The larger rounded bowtie dipole is determined by three parameters: total length , the flare angle , and radius of round corners. Actually they are relative to each other. According to our previous investigations [18], the largest is of benefit to achieve the best performance, and in this case the two arcs are connected to one part of an inscribed circle of the triangular bowtie dipole. Thus, a formula can be given to describe their relationships

(1) But here they are still studied individually in order to clearly show their influences on antenna performance.

Fig. 19. AR of the antenna with different .

mainly influences From Figs. 17 and 18, it can be seen that causes slightly input resistance of the antenna, and a smaller larger AR bandwidth but worse in-band AR, while a larger one produces smaller AR bandwidth but better in-band AR. Thus, should be selected according to the expected AR a suitable in-band. The flare angle has definitely influences on both AR and input impedance, as shown in Figs. 19 and 20. It can be observed that the second minimum AR point almost disappears as is changed far from 73 . The final parameter studied is , as given in Figs. 21 and 22. When is small, the second AR minimum point is superposed with the first one, and it is is increased from 8 to 14 mm, so the shifted upwards as . Moreover, a large AR bandwidth is also increased versus can also bring both the flattest input resistance and input reactance, which makes impedance matching more easily.

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time to find the optimized solution, especially the resonance phenomenon of the feeding transition.

REFERENCES

Fig. 20. Input impedance of the antenna with different .

Fig. 21. AR of the antenna with different R .

Fig. 22. Input impedance of the antenna with different R .

V. CONCLUSION A composite cavity-backed CBD is investigated in the paper. Good results and good agreement between simulations and measurements prove the success of the proposed antenna as a candidate for wideband CP applications. Operating principles of the proposed antenna, as given in the paper, is quite different from the crossed thin-wire dipoles because of strong coupling between the two bowtie dipoles. After successful relocating the resonance of the transition downwards, the antenna presents the largest 3-dB AR bandwidth of 39.2%, impedance bandwidth of 57.6% for , and symmetrical radiation patterns. Referencing the given design procedure, final results, and parametric studies, it will not be time consuming to design this antenna at the interesting frequencies, although it did take us much

[1] K.-L. Lau, K.-M. Luk, and K.-F. Lee, “Design of a circularly-polarized vertical patch antenna,” IEEE Trans. Antennas Propag., vol. 54, pp. 1332–1335, Apr. 2006. [2] K.-F. Tong and J. Huang, “New proximity coupled feeding method for reconfigurable circularly polarized microstrip ring antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 1860–1866, Jul. 2008. [3] J. R. James and P. S. Hall, Handbook of Microstrip Antennas. London, U.K.: Peter Peregrinus, 1989, ch. 4. [4] K. L. Lau and K. M. Luk, “Novel wide-band circularly polarized patch antenna based on L-probe and aperture-coupling techniques,” IEEE Trans. Antennas Propag., vol. 53, pp. 577–580, Jan. 2005. [5] K. L. Lau and K. M. Luk, “A wide-band circularly polarized L-probe coupled patch antenna for dual-band operation,” IEEE Trans. Antennas Propag., vol. 53, pp. 2636–2644, Aug. 2005. [6] J.-M. Laheurte, “Dual-frequency circularly polarized antennas based on stacked monofilar square spirals,” IEEE Trans. Antennas Propag., vol. 51, pp. 488–492, Mar. 2003. [7] W. K. Lo, C. H. Chan, and K. M. Luk, “Circularly polarized microstrip antenna array using proximity coupled feed,” Eletron. Lett., vol. 34, no. 23, pp. 2190–2191, Nov. 1998. [8] J.-W. Baik, K.-J. Lee, W.-S. Yoon, T.-H. Lee, and Y.-S. Kim, “Circularly polarised printed crossed dipole antennas with broadband axial ratio,” Electron. Lett., vol. 44, no. 13, pp. 785–786, Jun. 2008. [9] R. K. Zimmerman, jr, “Crossed dipoles fed with a turnstile network,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2151–2156, Dec. 1998. [10] J. L. Wong and H. E. King, “A cavity-backed dipole antenna with widebandwidth characteristics,” IEEE Trans. Antennas Propag., vol. 21, pp. 725–727, Sep. 1973. [11] S. Ohmori, S. Miura, K. Kameyama, and H. Yoshimura, “An improvement in electrical characteristics of a short backfire antenna,” IEEE Trans. Antenna Propag., vol. AP-31, no. 4, pp. 644–646, Jul. 1983. [12] M. F. Bolster, “A new type of circular polarizer using crossed dipoles,” IRE Trans. Microw. Theory Tech., vol. 9, no. 5, pp. 385–388, Sep. 1961. [13] R. Li, D. C. Thompson, J. Papapolymerou, J. Laskar, and M. M. Tentzeris, “A circularly polarized short backfire antenna excited by an unbalance-fed cross aperture,” IEEE Trans. Antennas Propag., vol. 54, pp. 852–859, Mar. 2003. [14] K.-L. Lau, H. Wong, and K.-M. Luk, “A full-wavelength circularly polarized slot antenna,” IEEE Trans. Antennas Propag., vol. 54, pp. 741–743, Feb. 2006. [15] Y. Kazama, “One-point feed printed crossed dipole with a reflector antenna combined with a reactively loaded parasitic loop for land mobile satellite communication antennas,” IEE Proc. -H, vol. 140, no. 5, pp. 417–420, Oct. 1993. [16] S.-W. Qu, J.-L Li, Q. Xue, C. H. Chan, and S. Li, “Wideband and unidirectional cavity-backed folded triangular bowtie antenna,” IEEE Trans. Antennas Propag., vol. 57, pp. 1259–1263, Apr. 2009. [17] S.-W. Qu, C. H. Chan, and Q. Xue, “Ultrawideband composite cavitybacked folded sectorial bowtie antenna with stable pattern and high gain,” IEEE Trans. Antennas Propag., vol. 57, pp. 2478–2483, Aug. 2009. [18] S.-W. Qu, C. H. Chan, and Q. Xue, “Ultrawideband composite cavitybacked rounded triangular bowtie antenna with stable patterns,” J. of Electromagn. Waves Applicat., vol. 23, pp. 685–695, 2009. [19] S.-W. Qu, J.-L. Li, Q. Xue, and C. H. Chan, “Wideband cavity-backed bowtie antenna with pattern improvement,” IEEE Trans. Antennas Propag., vol. 56, pp. 3850–3854, Dec. 2008. [20] Q. Xue, S.-W. Qu, and C. H. Chan, “Wideband cavity-backed bowtie antennas,” in 2009 IEEE International Workshop on Antenna Technology on Small Antennas and Novel Metamaterials (iWAT2009), Santa Monica, California, Mar. 2009, pp. 1–4. [21] A. Kumar and H. D. Hristov, Microwave Cavity Antennas. Norwood, MA: Artech House, 1989. [22] R. Li, G. DeJean, J. Laskar, and M. M. Tentzeris, “Investigation of circularly polarized loop antennas with a parasitic element for bandwidth enhancement,” IEEE Trans. Antennas Propag., vol. 53, pp. 3930–3939, Dec. 2005. [23] “HFSS: High Frequency Structure Simulator Based on the Finite Element Method,” Ansoft Corp.

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Shi-Wei Qu was born in He’nan province, China, in October, 1980. He received the B.Eng. and M.Sci. degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2001 and 2006, respectively, and the Ph.D. degree in the City University of Hong Kong (CityU), in October 2009. He is currently a COE (Global Center of Excellence) Fellow and a Postdoctoral Fellow in Tohoku University, Sendai, Japan. From 2001 to 2002, he worked for the 10th Institute of Chinese Information Industry. From 2006 to 2007, he was a Research Assistant in the Department of Electronic Engineering, City University of Hong Kong. His research interests include UWB antennas and arrays, metamaterial applications in antennas, and millimeter-wave antennas and arrays, etc.

Chi Hou Chan (S’86–M’86–SM’00–F’02) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1987. He joined the Department of Electronic Engineering, City University of Hong Kong, China, in 1996 and was promoted to Chair Professor of Electronic Engineering in 1998. From 1998 to 2009, he was first Associate Dean then Dean of College of Science and Engineering. He has been Acting Provost of the university since July 2009. His research interests cover computational electromagnetics, antennas, microwave and millimeter-wave components and systems, and RFICs. Prof. Chan received the US National Science Foundation Presidential Young Investigator Award in 1991 and the Joint Research Fund for Hong Kong and Macau Young Scholars, National Science Fund for Distinguished Young Scholars, China, in 2004. For teaching, he received the outstanding teacher awards in EE Department at CityU in 1998, 1999, 2000 and 2008. Students he supervised also received numerous awards including the Third (2003) and First (2004) Prizes in the IEEE International Microwave Symposium Student Paper Contests, the IEEE Microwave Theory and Techniques Graduate Fellowship for 2004–2005, Undergraduate/Pre-Graduate Scholarships for 2006–2007 and 2007–2008, and the 2007 International Fulbright Science and Technology Fellowship offered by the US Department of State.

Quan Xue (M’02–SM’04) received the B.S., M.S., and Ph.D. degrees in electronic engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 1988, 1990, and 1993, respectively. In 1993, he joined the UESTC as a Lecturer. He became an Associate Professor in 1995 and a Professor in 1997. From October 1997 to October 1998, he was a Research Associate and then a Research Fellow with the Chinese University of Hong Kong. In 1999, he joined the City University of Hong Kong, where he is currently an Associate professor in the Department of Electronic Engineering. He serves as the Deputy Director of State Key Laboratory (Hong Kong) of Millimeter-waves of China. He has authored or coauthored over 160 internationally referred papers and over 60 international conference papers. His current research interests include microwave passive components, active components, antenna, microwave monolithic integrated circuits (MMIC), and radio frequency integrated circuits (RFIC). Dr. Xue was awarded the UESTC “distinguished academic staff” for his contribution to the development of millimeter-wave components and subsystems. He is the Region 10 Coordinator of IEEE MTT-S AdCom.

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Low-Profile Leaky Wave Electric Monopole Loop Antenna Using the = 0 Regime of a Ferrite-Loaded Open Waveguide Toshiro Kodera, Member, IEEE, and Christophe Caloz, Fellow, IEEE

Abstract—A novel low-profile leaky-wave electric monopole loop antenna in the regime of a composite right/left-handed (CRLH) ferrite-loaded open waveguide is presented. Despite its closed configuration, this structure supports a traveling-wave mode due to its non-reciprocity. Moreover, this mode exhibits an regime, achieved at the transition infinite wavelength in the frequency between the backward and forward CRLH bands, which leads to an electric monopole antenna field distribution when the loop is electrically small. The structure is analyzed theoretically and shown to admit, for a given loop width, different groups of discrete modes. The modes within each group converge to the unique mode of the straight waveguide when the radius of the loop becomes very large, as a consequence of the decreased curvature of the structure. A complete antenna, with a wire feed and a stub matching section, is demonstrated by both full-wave and experimental results. The omnidirectional azimuthal gain of the antenna is of 5 dBi with a co-to-cross polarization discrimination in excess of 20 dB. Removal of the ground plane would lead to a unique electric dipole loop antenna.

=0

=0

Index Terms—Composite right/left-handed (CRLH) dispersion, electric monopole antenna, ferrite loaded waveguide, leaky-wave antenna, traveling wave antenna.

I. INTRODUCTION

M

ONOPOLE antennas are ubiquitous antennas providing useful omnidirectional patterns with a ground plane in various wireless communication systems. The most fundamental monopole antennas are the electric monopole, which consists of a quarter-wavelength straight conductor perpendicular to a ground plane, and the magnetic monopole, which consists of a one-wavelength loop conductor above a ground plane [1]. The electric monopole exhibits a gain which is 3 dB higher than the magnetic monopole, but suffers from a non-planar high-profile configuration. Some solutions have been explored to reduce the profile of monopole antennas, such as for instance composite right/left-handed (CRLH) [2] electric and magnetic loops Manuscript received September 26, 2009; revised February 08, 2010; accepted March 24, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by the NSERC Strategic Project under Grant # 1014, in part by Grant-in-Aid for Research Activity Start-up # 22860043 (KAKENHI), and in part by Apollo Microwaves. T. Kodera is with Graduate School of Science and Engineering, Yamaguchi University, Yamaguchi 755-8611, Japan (e-mail: [email protected]). C. Caloz is with the Department of Electrical and Engineering, École Polytechnique de Montréal, Montréal, QC H2T 1J3 Canada (e-mail: christophe. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055799

formed by lumped-element CRLH transmission line loops or mushroom patches operated at the transition frequency between the left-handed and the right-handed CRLH bands where the [3], [4]. This paper propagation constant is zero presents a different approach, using the leaky-wave mode [5] of the uniform ferrite-loaded open waveguide CRLH structure of [6] bent into a closed loop configuration. Although the proposed monopole antenna may be thought a priori to the same operation principle as the monopole antennas reported in [3], [4], since it shares with these antennas both its CRLH dispersive response (backward at low frequenregime cies and forward at high frequencies) and its (uniform—infinite wavelength—field distributions) at the backward-forward transition frequency, its operation principle is fundamentally different: This antenna is a leaky-wave (i.e. traveling-wave) structure, as a result of its non-reciprocity, whereas the monopoles of [3], [4] are resonant structures because the waves traveling in opposite directions in the loop combine to form a standing wave. The consequences of this fundamental difference are the following: i) The wave in the ferrite loop, being traveling by nature, carries energy around the loop, which means that the real part of its Poynting vector is non-zero, whereas the standing-wave regime of the monopoles of [3], [4] is not associated with any power flow around the loop and therefore corresponds to a perfectly null real part of the Poynting vector; ii) Due its leaky-wave nature, the proposed loop radiates progressively as the wave travels around it, until all the energy has leaked out. Therefore, it could theoretically—without ferrite dissipation and with ideal excitation—reach 100% radiation efficiency irrespectively to the smallness of the loop, since the power that has not been radiated in the first trip around the loop, seeing an “infinite structure,” is radiated in the next trips, in a self power recycling fashion similar to that demonstrated in [7]–[9]; iii) Due to its uniformity, the proposed loop structure is much easier to design compared to the lumped-element CRLH loops of [3], [4]; iv) Its operation frequency may be conveniently tuned by varying the magnetic bias field, in contrast to the loop antennas of [3], [4] which require the insertion of varactors or other external tuning elements. It should be noted that other ferrite loops have been reported in the literature [10], [11]. However, these loops were generally used for resonators and non-reciprocal devices unrelated to antennas. The paper is organized as follows. Section II describes the antenna and explains its principle of operation. Section III presents the electromagnetic analysis of the open-ferrite waveguide loop

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Fig. 2. Uniform electric field distribution at the operation frequency ! (where ), illustrating the monopole antenna behavior of the structure.

=0

Fig. 1. Progressive bending of the straight CRLH ferrite-loaded open waveguide leaky-wave antenna reported in [6] into a closed loop for the proposed leaky-wave monopole antenna.

=0

structure, with a detailed description of its modal response. The complete antenna with its feeding and matching structure is described and demonstrated in Section IV. Finally, conclusions are given in Section V. II. DESCRIPTION OF THE STRUCTURE AND PRINCIPLE OF OPERATION OF THE ANTENNA leaky-wave monopole The structure of the proposed antenna is obtained by bending the straight CRLH ferrite-loaded open waveguide leaky-wave antenna structure reported in [6] into a closed loop, as illustrated in Fig. 1. In order to provide half plane, the structure unidirectional radiation in the is is placed on a ground plane. The magnetic bias fields normal to the ground plane. The ferrite-loaded open waveguide structure exhibits essentially the same properties as artificial CRLH transmission line structures [2], including a left-handed (backward phase velocity) band and a right-handed (forward phase velocity) band below and above the transition frequency , where the propagation constant is zero and the guided wavelength is infinite . However, this structure has four unique characteristics: i) it is perfectly uniform and therefore much easier to design, ii) it is inherently balanced, which means that gap-less non-zero group velocity propagation is automatically achieved at the transition where , and iii) due to its non-reciprocity, it frequency supports propagation only in one direction [6], and this is an essential characteristic in the loop antenna proposed in this paper, iv) it can be scanned by bias field tuning, without requiring chip varactors. The loop structure of interest specifically operates at the tran, where the electromagnetic fields are persition frequency fectly uniform along the direction of propagation of the waveguide since the guided wavelength is infinite. In the straight waveguide structure (Fig. 1), this regime leads to a conical or half conical leaky-wave radiation patterns for the isolated and grounded waveguide cases, respectively [6]. In the proposed loop structure, the field exhibits the distribution shown in Fig. 2 where, due to the azimuthal symmetry and assuming a small loop (diameter free space wavelength ), only the component of the electric field exists, which corresponds to an electric monopole configuration. Consequently a monopolar (or dipolar in the ungrounded case) radiation pattern, azimuthally uniform and with a null in the direction perpendicular to the loop, is expected. Note that the classical loop condition

, where represents any field component, is automatically regime, since is satisfied for any loop radius in the perfectly uniform azimuthally. However, this condition is not sufficient and the electromagnetic analysis of Section III will show that restrictions in the radius still exist, due to the curvature of the structure. If the waveguide structure were reciprocal, then two waves traveling in opposite azimuthal directions would combine to produce the classical standing or ] of convenwave regime [ tional loop resonators. Since the waveguide loop is non-reciprocal, only one azimuthally traveling wave is supported, namely propagating wave in Fig. 1, assuming the the . Therefore, despite its closed contime dependence figuration, the loop operates in the traveling-wave regime. This non-reciprocal loop structure is thus quite unique: it has the closed configuration feature of a resonator and the travelingwave feature of a propagating waveguide. As does its straight waveguide counterpart, the loop structure radiates through its opening to free space. Since it also supports a traveling wave, it constitutes in fact a leaky-wave monopole loop antenna. Such an antenna has never been reported before to the knowledge of the authors. Perhaps even more unusual is the fact that the structure is seen as “infinite” by the wave since the wave circles indefinitely around the loop. In this sense, the power, which is typically lost in the load for an open leaky-wave antenna, is automatically recycled within the loop to provide maximal radiation efficiency [7]–[9]. III. ELECTROMAGNETIC ANALYSIS A. Solution to the Wave Equation The leaky-wave monopole loop antenna structure is shown in Fig. 3 along with its geometrical parameters. These parameters are the internal radius , the external radius , the width , and the height . Assuming a low waveguide profile ( , where is the effective wavelength in the ferrite, the fields do not vary and therefore only along the direction modes, with the unique component of the electric field, need to be considered. The corresponding wave equation in cylindrical coordinates reads [12], [13] (1) and admits the solutions (2)

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this PMC approximation, radiation is considered as a perturbation to guidance, which has been shown to be practically approregime, occurring at the frequency , priate [6]. In the the (9) become using (2) and (3) (10a) (10b) Fig. 3. Leaky-wave monopole loop antenna structure operating in the regime with main design parameters.

0

=n=

with (10c)

and

With these boundary relations and the fact that tric and magnetic fields in (2) and (3) reduce to

, the elec-

(11) (3)

and (12)

where is an integer, which corresponds to the number of azimuthal wavelengths of the resonant waveform. In these relations, we have

respectively. In addition, the boundary conditions [(10)] involve the roots of the Bessel functions and as

(4)

(13a)

(5) (6)

(13b) which imposes the following modal restrictions on the allowed parameters of the loop

with the Polder tensor parameters [14] (14a) (7a)

(14b)

(7b) (14c) where (7c) (7d) and with the Lorentz parameters (8b) (8b)

in the regime is Thus, the modal set of loop radii . Since such a restriction does not infinite but discrete exist in the corresponding straight waveguide structure (Fig. 1) is automatically and since the condition satisfied, it must be concluded that this restriction is due to the curvature of the loop structure. Note that, in order to have posi, we must have , since the tive widths, roots of and alternate. B. Modal Analysis

The boundary conditions at the internal and external radii of the loop read (9a) (9b) where the first condition is an exact PEC condition while the second condition is an approximated PMC condition justified by the high permittivity contrast between the ferrite and air. With

Table I gives the parameter values of the loop structure, in reference to Fig. 3, which are used throughout the paper both in the theoretical and experimental results, unless otherwise specified. These values correspond to the parameters of a ferrite material provided by Murata manufacturing Co, which is used in the prototype presented in the following section. Note that the . Since the regime frequency is fixed to corresponds to only one frequency point, no dispersion relation is discussed in following analysis. The resolution of the

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TABLE I PARAMETERS OF THE LOOP STRUCTURE IN REFERENCE TO Fig. 3 FIXED THROUGHOUT THE PAPER UNLESS OTHERWISE SPECIFIED

TABLE III FIRST FIVE LOOP GEOMETRICAL SOLUTIONS OF (14) AND CONVERGENCE FOR THE CASE q p COMPUTED BY (14). THE UNITS FOR a; b; w IS mm

= +1

TABLE II FIRST FIVE LOOP GEOMETRICAL SOLUTIONS OF (14) AND CONVERGENCE FOR THE CASE q p COMPUTED BY (14). THE UNIT FOR a; b; w IS mm

=

Fig. 5. First selected (every two solution to avoid overlaps) normalized fields (or m ). The different distributions as in Fig. 4, but for the case q p segment pairs correspond to different waveguide internal/external radii b =a and widths w b a solutions, the first of which are given in Table III p; q . and identified by their label p; q

= +1

= 0

Fig. 4. First normalized E and H fields distributions computed by (11) and (12), corresponding to segments of the Bessel functions J k and J k , respectively, for the case q p (or m ). The different segment pairs correspond to different waveguide internal/external radii b =a and widths w b a solutions, the first of which are given in Table II and identified by their label p; q p; p .

( )

=

= 0

=1

( )

( )=( )

problem for all frequencies, yielding conventional resonances, would be straightforward but is of no interest for the proposed antenna. Table II gives the first five loop geometrical solutions computed by (14) for the case . As increases, both the internal and external radii of the loop monotonically increase to larger values, whereas the width of the . The reason loop converges to the value of

( )=(

=2

0 1)

is because as the average radius tends to infinity, the radius of curvature of the loop also tends to infinity, so that the structure locally appears to the wave traveling along it as straight. Therefore, a response similar to that of the corresponding straight waveguide structure (Fig. 1) [6] is expected in the case of large radii. This will be demonstrated next. Fig. 4 shows the electric and magnetic field distributions for for different geometrical parameter solutions. the case These distributions, given by (11) and (12), respectively, are and , resimply segments of the Bessel functions, spectively. Due to the asymptotic behaviors of the Bessel funcand tions , we see that the radial field dependence become more and more sinusoidal as the size of the loop is increased. In the limit, the dependence becomes purely sinusoidal and identical to that of the straight waveguide [6], since the radius of curvature becomes infinite. Table III and Fig. 5 show the same results as in Table II and , which corresponds to the Fig. 4 but for the case next mode of the loop. The same comments as for Table II and

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Fig. 7. Transition frequency f versus the waveguide width w for the straight waveguide structure using (17) and for the loop structure for all possible combinations p; q up to the 20th order using (14c). The thick dotted curves are for the straight structure and all the other curves are for the loop structure.

( )

= 3 96 mm = 6 31 mm =

Fig. 6. Normalized fields distributions for fixed internal radius a : (smallest possible radius with the parameters of Table I) and varying external ; m , corresponding to b : radius b . (a) p; q and w : . (b) p; q ; m , corresponding to b : and w . (c) p; q ; m : , corresponding : : and w . to b

( ) = (1 1) ( = 1) = 2 35 mm ( ) = (1 2) ( = 2) = 7 60 mm ( ) = (1 3) ( = 3) 11 56 mm = 16 76 mm = 12 8 mm

Fig. 4 apply, except that the waveguide width is larger and the field variation across the radial direction exhibits an additional modes in terms of their number of node. If we define the corresponds radial nodes, then the mode label for the case to the modal group . It will be shown below that it makes sense to use this mode labeling in comparison to the straight waveguide case. So, Table II and Fig. 4 correspond to , while Table III and Fig. 5 correspond to . The can be easily extrapolated. cases For further insight into the field distributions of the loop structure, Fig. 6 plots the these distributions for a fixed internal radius and three different external radii, corresponding to the modes , and , where the corresponding number of and is clearly apparent. nodes of both Since the loop structure with very large radius may be seen as a limiting case of the straight structure (Fig. 1), it is worth establishing a comparison with it. The dispersion relation for the straight structure reads [6] (15) with (16) From (15), the transition frequency

can be derived as

(17a)

with

(17b) versus the waveguide width for the straight Fig. 7 plots waveguide structure using (17) and for the loop structure for all up to the 20th order using (14c). possible combinations The following observations may be made on this graph: i) as expected, all loop structure curves converge to straight structure curves as and increase; ii) the -index grouping of the modes with identical parameters is meaningful, as it relates all the modes in the group to the corresponding mode of the straight structure; iii) the convergence speed increases as increases, since a larger means a larger radius and therefore a situation closer to the straight structure. IV. ANTENNA DEMONSTRATION A. Design The design of the electric monopole loop antenna consists of two parts: the design of the loop ferrite waveguide and the design of the feeding and matching structure. The complete antenna structure with its feed is shown in Fig. 8. As mentioned in Section II and illustrated in Fig. 3, the loop must be much smaller than the effective wavelength in order to exhibit a monopole antenna behavior. Therefore, the proposed antenna will use the lowest of all modes, namely the mode of the group. For the parameters given in Table I, this mode corresponds to the loop parameters and , according to Table II. The operation frequency of the resulting structure may naturally be tuned by varying the magnetic bias field. As shown in Fig. 8, the structure is fed by a microstrip line at the back of the structure via a conducting wire, crossing the

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Fig. 8. Leaky-wave monopole loop antenna structure fed by a microstrip line at the back of the structure via a conducting wire, crossing the substrate through an isolating slot in the ground plane and connecting the line to the top plate of the structure. (a) Top view. (b) Bottom view.

substrate through an isolating slot in the ground plane and connecting the line to the top plate of the structure. A stub conjugate matching network is appended to the microstrip line for matching at the frequency of 4.38 GHz. After the isolated loop has been designed, the wire was added to the structure in a fullwave simulator (HFSS-FEM) to determine the corresponding parameter. This parameter was then imported to a circuit simulator for the design of the stub matching circuit near the . The loading effect of the ground plane typically frequency is achieved to a slightly decreases the frequency where frequency which may be determined by inspecting the full-wave simulated field distributions.

= = 1 mode = 6 31 mm ( = 2 35 mm)

Fig. 9. Fields distribution computed by HFSS (FEM) for a q p design, corresponding to a : and b : w : at 4.38 GHz. (a) Electric field. (b) Magnetic field.

= 3 96 mm

B. Fields Distributions Inside the Structure Fig. 9 shows the full-wave fields distribution along the loop structure for the mode and the parameters mentioned above, except for the frequency, which was decreased from the initial target of 4.5 GHz to 4.38 GHz, as the effect of the ground plane. Quasi perfect azimuthal field uniformity may be observed at this frequency. Inspection of the magnetic field distribution in Fig. 9(b) also confirms the validity of the is esPMC approximation at the ferrite-air interface, since sentially null at this interface. Moreover, Fig. 10 compares the radial distributions of the electric and magnetic fields computed analytically using (11) and (12) and by full-wave analysis. Good agreement is found between the two cases. The fields distributions of Fig. 9 could be those of a standing wave and do not prove the expected traveling wave operation of the structure. Therefore, Fig. 11 shows the Poynting vector distribution. In this case, it clearly appears that the wave is traveling, in the direction allowed by the non-reciprocal

Fig. 10. Comparison of the fields distributions obtained analytically by Eqs. : and by HFSS at f : (lower (11) and (12) at f frequency due to the ground plane loading effect) for the design of Fig. 8.

= 4 5 GHz

= 4 38 GHz

nature of the structure. Thus, the structure really supports a traveling wave, as anticipated in Section II. The magnitude of the Poynting vector progressively decreases as the wave propagates around the loop, due to the leaky-wave radiation of the structure. In contrast to conventional leaky-wave antennas, where the non-radiated power is wasted in the load, this antenna automatically recycles the non-radiated power after one or more turns

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Fig. 13. Full-wave (HFSS-FEM) 3D Radiation pattern for the design of Fig. 8. Fig. 11. Poynting vector corresponding to the design of Fig. 9, showing the traveling wave nature of the mode excited and the leaky-wave attenuation of the wave from the excitation point.

Fig. 14. Full-wave elevation radiation patterns corresponding to Fig. 13 in the plane and 90 (cross section in Fig. 13).

=0

1H =

Fig. 12. Dispersion and attenuation diagram under a ferrite loss of obtained by Eq. (15), which corresponds to the dispersion relation for the straight waveguide and which considers only ferrite dissipation (excluding leakage loss).

20 Oe

back into the structure due the loop configuration. It is in this sense a power-recycling leaky-wave antenna, similarly to the antennas reported in [7]–[9]. In order to estimate how much power is left after one loop turn, corresponding to the amount of power to be recycled, let us compute the total loss for one turn. This loss is composed of ferrite dissipation, which irreversibly limits the radiation efficiency, conduction and dielectric losses, which are assumed negligible compared to ferrite losses, and leakage “loss,” which represents in fact the usefully radiated power of the antenna. The approximate PMC closed waveguide approach used in this paper (Section III), although very accurate from the viewpoint of dispersion, allows one to compute only the ferrite dissipation , the loss. As shown in Fig. 12 for a ferrite loss of design frequency attenuation factor is of 3.83 Np/m at the ( 4.51 GHz). Using for the single-trip length the average circumference of the loop, which corresponds to 32.28 mm, the fraction of the non-dissipated power after one turn propa. By using a generalgation is of ized expression of the dispersion relation taking into account

Fig. 15. Full-wave azimuthal radiation pattern corresponding to Fig. 13 in the plane (cross section in Fig. 13).

 = 90

the open nature of the ferrite waveguide [15], the total amount of remaining power after one turn, which is to be recycled, is . found to be of C. Radiation Performance Figs. 13, 14 and 15 show the full-wave radiation patterns of the antenna design of Fig. 8. These graphs confirm the expected monopole operation of the antenna, with a pronounced dip in the broadside direction (Fig. 14) and an omnidirectional pattern in the horizontal plane (Fig. 15) with more than 20 dB co-to-cross polarization discrimination. The input (taking into account matching) gain is of around 5 dBi, and is essentially limited by the loss of the ferrite material considered (

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Fig. 17. Deviation of the azimuthal radiation patterns across (a) a 20 MHz bandwidth (from 4.37 to 4.39 GHz) and (b) a 40 MHz bandwidth (from 4.36 : . to 4.40 GHz). All the patterns are normalized to that of f

= 4 38 GHz

Fig. 16. Full-wave 3D radiation patterns at several frequencies around f . :

4 38 GHz

= Fig. 18. Ferrite loop machined by a Nd-YAG laser from a ferrite slab.

at 9.4 GHz). It was observed that suppression of ferrite loss (almost achieved with a single crystal YIG) leads to a quasi perfect (omnidirectional) gain of 0 dBi. The radiation efficiency evaluated by full-wave simulation is 30%. This value is improved to 50% for the loss-less case. The limitation of the efficiency is attributed to two factors. One is the imperfect feeding structure, where part of the power is circulating around the loop is re-injected toward the source. This issue could be solved by using a reflection-type isolator in the feed. The other one is the energy dissipation in evanescent and lossy mode excitation in the reverse direction. In the lossless case, the waveguide would exhibit a purely reactive behavior and therefore consume no power in the reverse direction. However, a practical waveguide exhibits some inherent loss reducing its performance. The azimuthal dissymmetry of the radiation patterns (Figs. 13, 15) is due to the presence of the the feeding structure. It could be reduced by a less invasive feeding mechanism, such as a back slot coupling excitation. , When the frequency is different from the transition point the ideal configuration shown in Fig. 2 is not perfectly realized. The evaluation of the deviation of the radiation patterns versus frequency is of interest to assess the potential of the antenna in practical applications. Fig. 16 shows the full-wave 3D radiation patterns at several frequencies around . The deviation of the azimuthal radiation patterns across a 20 MHz bandwidth (from 4.37 to 4.39 GHz) and 40 MHz bandwidth (from 4.36 to 4.40 GHz) are shown in Fig. 17. All of the patterns are normalized that of . The observed value of 1.77 dB radiation deviation across a 20 MHz bandwidth is acceptable for IEEE 802.11a (5 GHz) wireless communication systems if is designed for that frequency band. Fig. 18 shows the ferrite loop, which was used in the forthcoming experiments. This structure was machined by a Nd-YAG laser with millisecond pulses from a ferrite slab. The intense

Fig. 19. Prototype corresponding to structure of Fig. 8 and using the loop of Fig. 18. (a) Top view. (b) Bottom view.

heat generated by this laser in the ferrite transfers its energy to the lattice phonon and may even crush the slab into pieces. We were able to obtain a loop of acceptable quality, despite a small crack on one side (not visible in Fig. 18). It was later verified that the use of a Ti-sapphire femto-second laser could machine perfect structure from a ferrite slab, thanks to the fact that its super short pulses do not create sufficient heat to stress the structure. However, the imperfect loop machined by Nd-YAG was judged sufficient for the experimental proof-of-concept of the antenna. Fig. 19 shows the antenna prototype corresponding to structure of Fig. 8 and using the loop of Fig. 18. The structure is biased by a NdFeB permanent magnet placed at the bottom (matching network side) of the structure with a 1 cm foam separation layer. To ensure uniformity of the bias field in the loop ferrite structure, an NdFeB permanent magnet much larger than the antenna size is used: the dimensions of the magnet are while the dimensions of the antenna are within 12.6 12.6 mm. The intensity of the magnetic field at the magnet surface is of 0.35 T. The whole experimental setup including the magnet with the antenna is shown in Fig. 20. The full-wave

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Fig. 22. Measured elevation radiation pattern at 4.38 GHz for 

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= 0 and 90

.

Fig. 20. Complete measurement setup including the NdFeB permanent magnet with the antenna.

Fig. 23. Measured azimuthal radiation pattern corresponding to Fig. 22 in the . plane 

= 90

Fig. 21. Comparison of the reflection coefficients for the experimental (solid lines) and full-wave (dotted lines) results.

and measured reflection coefficients are plotted in Fig. 21, which shows excellent matching in excess of 10 dB at the operation frequency of 4.38 GHz. Figs. 22 and 23 show the measured elevation and azimuthal radiation patterns, respectively. These patterns are in good agreement with the full-wave prediction of Figs. 14 and 15 despite the relatively rudimentary (hand-made) prototype. The observed oscillations on the patterns are most likely due to the imperfectness of the prototype, including ferrite surface roughness and poor (manual) copper tape metallization. However, as predicted in Fig. 15, the two fundamental properties of the antenna—omnidirectional monopole patterns and low cross polarization—are clearly verified. V. CONCLUSION A novel low-profile leaky-wave electric monopole loop regime of a ferrite-loaded open waveantenna using the guide has been presented. This structure is obtained by bending the straight waveguide structure reported in [6] into a loop configuration. Despite its closed configuration, this structure supports a traveling-wave mode due to its non-reciprocity. Moreover, this mode exhibits an infinite wavelength in to the regime achieved at the transition frequency between the

backward and forward bands. As a consequence, when the loop is electrically small, an electric monopole field distribution is obtained, leading to an electric monopole antenna. The structure has been analyzed theoretically and shown to admit, for a given loop width, different groups of discrete modes. The modes within each of group converge to the unique mode of the straight waveguide when the radius of the loop becomes very large, as a consequence of the decreased curvature of the structure. A complete antenna, with a wire feed and a stub matching section, has been designed and demonstrated both by full-wave and experimental results. The omnidirectional azimuthal gain of the antenna is 5 dBi with a co-to-cross polarization discrimination in excess of 20 dB. Performance improvements are certainly possible by using a lower-loss ferrite for higher gain and an improved feeding structure for higher symmetry. It is interesting to note that removal of the ground plane would lead to a unique electric dipole loop antenna. ACKNOWLEDGMENT The authors would like to acknowledge Murata Manufacturing (Japan) for their generous donation of ferrite materials and Ansoft Corporation for their generous donation of HFSS software licenses. REFERENCES [1] C. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York: Wiley, 2005. [2] C. Caloz and T. Itoh, Electromagnetic Metamaterials, Transmission Line Theory and Microwave Applications. New York: Wiley, 2005.

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[3] J. Lee and J. Lee, “Zeroth order resonance loop antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 994–997, Mar. 2007. [4] C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antennas Propag. Mag., vol. 50, no. 5, pp. 25–39, Oct. 2008. [5] A. Oliner and D. Jackson, Antenna Engineering Handbook, J. Volakis, Ed., 4th ed. New York: McGraw-Hill, 2005. [6] T. Kodera and C. Caloz, “Uniform ferrite-loaded open waveguide structure with CRLH response and its application to a novel backfire-to-endfire leaky-wave antenna,” IEEE Trans. Microwave Theory Tech., vol. 57, no. 4, pt. 1, pp. 784–795, Apr. 2009. [7] H. Nguyen, S. Abielmona, and C. Caloz, “Highly efficient leaky-wave antenna array using a power-recycling series feeding network,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 441–444, Jan. 2009. [8] H. V. Nguyen, S. Abielmona, A. Parsa, and C. Caloz, “Novel power recycling schemes for enhanced radiation efficiency in leaky-wave antennas,” in Proc. Asia Pacific Microwave Conf., Dec. 2009, pp. 2006–2009. [9] H. V. Nguyen, A. Parsa, and C. Caloz, “Power-recycling feedback system for maximization of leaky-wave antennas radiation efficiency,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 7, pp. 1641–1650, Jul. 2009. [10] V. Dmitriyev and L. Davis, “Nonreciprocal devices using ferrite ring resonators,” Microw., Antennas Propag., IEE Proc. H, vol. 139, no. 3, pp. 257–263, Jun. 1992. [11] J. Helszajn, W. Nisbet, and J. Sharp, “Mode charts of gyromagnetic planar ring resonators,” Electron. Lett., vol. 23, no. 24, pp. 1290–1291, Jan. 1987. [12] R. Harrington, Time-Harmonic Electromagnetic Fields, 2nd ed. New York: IEEE Press Series on Electromagnetic Wave Theory, 2001. [13] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2004. [14] B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnetics. New York: McGraw-Hill, 1962. [15] P. Baccarelli, C. D. Nallo, F. Frezza, A. Galli, and P. Lampariello, “Attractive features of leaky-wave antennas based on ferrite-loaded open waveguides,” in Proc. IEEE AP-S Int. Symp., 1997, vol. 2, pp. 1442–1445.

Toshiro Kodera (M’01) received the B.E., M.E. and Dr. Eng. degrees from Kyoto Institute of Technology, Kyoto Japan, in 1996, 1998, and 2001, respectively. He developed some numerical program and devices using ferrite media. In 2001 he joined the Faculty of Engineering, Osaka Institute of Technology, as a Lecturer. In 2005, he joined Wave Engineering Laboratories, ATR International, Kyoto Japan as a Visiting Researcher, where, in 2006 he became a Researcher. In ATR, he engaged in R&D of GaAs MMICs for 802.15.3c Gbps wireless LAN system and microwave power amplifier. In 2008, he joined Department of Electrical Engineering, École Polytechnique of Montréal, where he developed several novel radiative structures using magnetic material inspired by metamaterial concept. In 2010, he joined Yamaguchi University where he is now Associate Professor. His current research is on the microwave devices utilizing magnetic material including nano structure.

Christophe Caloz (F’10) received the Diplôme d’Ingénieur en Électricité and the Ph.D. degree from École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, in 1995 and 2000, respectively. From 2001 to 2004, he was a Postdoctoral Research Engineer at the Microwave Electronics Laboratory of University of California at Los Angeles (UCLA). In June 2004, he joined École Polytechnique of Montréal, where he is now a Full Professor, a member of the Poly-Grames Microwave Research Center, and the holder of a Canada Research Chair (CRC). He has authored and coauthored over 360 technical conference, letter and journal papers, three books and eight book chapters, and he holds several patents. His research interests include all fields of theoretical, computational and technological electromagnetics engineering, with strong emphasis on emergent and multidisciplinary topics. Dr. Caloz is a Member of the Microwave Theory and Techniques Society (MTT-S) Technical Committees MTT-15 (Microwave Field Theory) and MTT-25 (RF Nanotechnology), a Speaker of the MTT-15 Speaker Bureau, and the Chair of the Commission D (Electronics and Photonics) of the Canadian Union de Radio Science Internationale (URSI). He is a member of the Editorial Board of the International Journal of Numerical Modelling (IJNM), International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE), International Journal of Antennas and Propagation (IJAP), and of the journal “Metamaterials” of the Metamorphose Network of Excellence. He received the UCLA Chancellor’s Award for Postdoctoral Research in 2004 and the MTT-S Outstanding Young Engineer Award in 2007. He is an IEEE Fellow.

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Ultrawideband Hemispherical Helical Antennas Hamad Waled Alsawaha, Student Member, IEEE, and Ahmad Safaai-Jazi, Senior Member, IEEE

Abstract—A compact ultrawideband antenna with hemispherical helical geometry is proposed and investigated both theoretically and experimentally. The radiating element of this antenna consists of a tapered metallic strip that assumes a non-conformal orientation relative to the hemispherical surface in order to yield maximum bandwidth. The number of turns is about four and half with a constant spacing between them. The antenna is fed by a coaxial cable with the inner conductor connected to the radiating strip through a matching section, and its outer conductor connected to a ground plane. Radiation properties of the proposed hemispherical helical antenna, including far-field patterns, axial ratio, directivity, input impedance and voltage standing wave ratio (VSWR), are studied numerically and evaluated experimentally. Simulation and measured results are in good agreement. This design provides a maximum directivity of 9 1 dB, a (relative to a 50 reference impedance), and nearly equal E- and H-plane far-field patterns with high degree of axial symmetry over a bandwidth of more than 50%. Also, over a bandwidth of about 24% the axial ratio remains below 3 dB. The compact size and ultrawideband performance of this antenna make it advantageous for high speed wireless communication systems and avionics.

VSWR 2



Index Terms—Hemispherical helical antennas, low profile antennas, spherical helix, ultrawideband antennas.

I. INTRODUCTION

T

HE spherical helical antenna, first introduced at Virginia Polytechnic Institute and State University, Blacksburg (Virginia Tech), offers some unique advantages, including low profile, compact size, and very broad beamwidth [1], [2]. A modified version of this antenna was later developed by truncating the spherical helix to half of its size to form hemispherical helix that is mechanically more stable. The investigation of the hemispherical helical antenna showed that this geometry, with an optimal number of turns, provides a broader beamwdith over a wider bandwidth and essentially the same directivity as that of the full spherical helix [3]. The radiating elements in both spherical and hemispherical helical antennas in the above studies are thin wires wound over spherical surfaces. These antennas are capable of providing circular polarization, but only over narrow bandwidths. However, much wider bandwidths are required to meet the increasing demands for high data rate information transmission in wireless communication systems. A number of circularly-polarized ultrawideband antenna designs have been reported in recent years [4], [5]. These antennas, while offering more bandwidth, Manuscript received October 24, 2009; revised February 13, 2010; accepted April 03, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The authors are with Virginia Tech Antenna Group (VTAG), Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0111 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055806

are larger in size and provide several dBs less gain than the hemispherical helix introduced here. Thus, in situations where compact geometry, small size, and light weight are of interest, particularly in aerospace and mobile communication systems, the hemispherical helix would be more desirable. The study presented in this paper is aimed at exploring new designs for hemispherical helical antennas which can provide ultrawideband radiation characteristics. Attention is focused on introducing new structural designs for the radiating element as well as the feed part of the hemispherical helix in order to increase the overall bandwidth of the antenna to the ultra wideband frequency range. The overall bandwidth here is considered as the frequency range over which axial ratio (AR) is 3 dB, VSWR is 2, and nearly constant gain and half-power beamwidth are maintained simultaneously. Two modifications over the basic wire design proposed in [3] are introduced, including the replacement of the wire radiating element with a tapered metal strip and the inclusion of a doubly tapered impedance matching section, tapered in both width and height (from the ground plane), that connects the radiating element of the antenna to the feeding coaxial cable. A comprehensive investigation of the proposed hemispherical helical antenna is carried out numerically as well as experimentally. Radiation characteristics of the antenna including far-field patterns, polarization, input impedance, and directivity are studied. Simulation results are obtained using the commercial software FEKO, suite 5.3 [6]. A prototype of the antenna is constructed and its far-field patterns, gain, VSWR, and axial ratio are measured. In the remaining parts of this paper, design of the radiating element is addressed first. Then, matching of the antenna to a 50 reference impedance is studied. Fabrication of a prototype of the proposed hemispherical helix, measurement of its radiation properties, and comparison of measured and simulation results are discussed. II. DESIGN OF RADIATING ELEMENT In a first attempt, the wire radiating element in the basic hemispherical helical antenna design proposed in [3] is replaced with a metal strip. The metal strip may conform to the hemispherical surface or might assume a non-conformal configuration. The numerical analysis of the hemispherical helix with strip radiating element conforming to its surface has indicated that, compared to the basic wire design in [3], very little increase in the bandwidth over which the axial ratio is less than 3 dB can be achieved [7]. On the other hand, hemispherical helices with non-conformal radiating elements as shown in Fig. 1 prove to be much more promising. For non-conformal strip geometries, it is assumed that the outer edge of the strip resides on the surface of the hemisphere, so that the volume of the hemispherical helix remains

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Fig. 1. Geometry of a 4.5-turn hemispherical helical antenna with nonconformal strip radiating element.

Fig. 2. Variations of axial ratio (AR) versus frequency for hemispherical helical antennas with non-conformal constant-width strip radiating elements and tilt and . Both helices have 4.5 turns, angles , strip width , and are side fed by 5 mm short radius vertical wires.

9=0 ( ) a = 20 mm

9 = 45 (. . . . . .) w = 2 mm

unchanged and equal to that with the wire radiating element presented in [3]. The geometry of the outer edge of the strip in a spherical coordinate system can be described by the following equations: (1) (2) where is the radius of the hemispherical surface encompassing the antenna, and N is the number of turns of the radiating element. To properly describe the orientation of the non-conformal radiating element we introduce a new parameter referred to as ‘tilt and defined as the angle between the ground plane angle’ (xy-plane) and a line lying on the strip and perpendicular to its edges. The tilt angle for the geometry shown in Fig. 1 is 45 . The tilt angle may be regarded as a new design parameter that can be adjusted for optimum performance; namely, the widest bandwidth over which the requirements for axial ratio, VSWR, gain, and half-power beamwidth as stated above are met. The investigation of hemispherical helices with wire radiation elements presented in [3] has shown that the optimal number of turns for achieving maximum axial ratio bandwidth is about four and half. Our extensive investigation of hemispherical helices with strip radiating elements revealed that the same number of turns provides the widest bandwidth over which the axial ratio is 3 dB. Thus, for all hemispherical helical designs discussed here the number of turns is considered to be 4.5. Variations of axial ratio versus frequency for hemispherical helices with metal strip radiating elements of tilt angles 0 and 45 are depicted in Fig. 2. For both helices the outer radius is and the width of the strip radiating ele. Also, they are side fed by 5 mm vertical ment is short wires as shown in Fig. 1 and infinite ground planes are assumed in their simulations. The results in Fig. 2 clearly indicate that non-conformal radiating element designs indeed increase the axial ratio bandwidth significantly. For example, the design over a bandwidth of with a tilt angle of 0 provides about 18%. Furthermore, comparison of axial ratio results for

Fig. 3. Hemispherical helical antenna with tapered strip radiating element and . tilt angle

9=0

tilt angles of 0 and 45 in Fig. 2 shows that a left shift in the bandwidth associated with the larger tilt angles occurs. Thus, the tilt angle might be used as a means of fine tuning the frequency range without changing other antenna parameters. Next, we consider tapering the strip radiating element, expecting to further increase the bandwidth. It is well known that replacing the wire with a tapered radiating element, as in the case of bowtie antenna, increases the antenna bandwidth [7]. Based on this observation, it is expected that tapering the radiating element shall provide improvements in both the axial ratio bandwidth as well as the bandwidth over which VSWR is 2 (impedance matching is discussed in the next section). Tapering the width of the strip from 1 mm at the feed point to 4 mm at the top of the hemisphere resulted in little change in the axial ratio bandwidth of the conformal design, but for the case of the non-conformal design tapering increased the bandwidth to 20%. Fig. 3 shows the geometry of a hemispherical helix with non-conformal tapered strip radiating element with . Numerical results for the axial ratio of the tilt angle non-conformal designs with tapered strip with tilt angles 0 and 45 are shown in Fig. 4. Comparison of results in Figs. 2 and 4

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Fig. 4. Variations of axial ratio (AR) versus frequency for hemispherical helical antennas with tapered radiating elements (initial width = 1 mm, nal width = 4 mm) and tilt angles 9 = 0 ( ) and 9 = 45 (. . . . . .). Both helices have 4.5 turns, confined to a hemisphere of radius a = 20 mm, and are side fed by 5 mm short vertical wires.

indicates that in addition to improving the axial ratio bandwidth, tapering also causes a right shift in the mid-band frequency. This frequency shift is opposite of that due to increasing the tilt angle. Since nonzero tilt angles, as noted in Fig. 4, result in only a small frequency shift, in the remaining parts of this paper only designs will be considered. with tilt angle III. IMPEDANCE MATCHING In the hemispherical helix design introduced in [3] a horizontal wire parallel to the ground plane is used to connect the feed vertical wire at the center of the hemisphere to the helical radiating wire structure. This horizontal feed wire has some impact on the radiation characteristics of the antenna. Among these characteristics the input impedance is the one affected most, while the directivity, axial ratio, and radiation pattern remain largely unchanged. The feed wire and the ground plane together form a transmission line which can transform the input impedance of the hemispherical helix. On the other hand, if the horizontal feed wire is sufficiently close to the ground plane, due to its opposite image current in the ground plane, it will have negligible impact on radiated fields and thus very little impact on axial ratio and directivity. One may take advantage of the above mentioned impedance transformation property and try to match the antenna to the signal cable which is considered to have a characteristic impedance of, say, 50 . However, it is desirable that the impedance matching occurs over a wide bandwidth. Tapered transmission lines [8] are known to be capable of wideband impedance transformation. Accordingly, the horizontal wire that connects the feed to the helical wire in the antenna design presented in [3] is replaced with a tapered microstrip line that acts as an impedance matching section. This technique was first implemented on a helical antenna by Kraus [9]. The design of the matching section in this work evolved into its final form after several steps. In a first attempt, we considered an impedance matching section that was a linearly tapered triangular shape microstrip starting from the center of the base of the hemisphere and extending to the radiating element of the

Fig. 5. Geometry of a 4.5-turn hemispherical helical antenna with tapered radiating element (initial width = 1 mm, nal width = 4 mm) and tilt angle 9 = 0 .The helix is confined to a hemisphere of radius a = 20 mm and is side-fed by a nonlinearly tapered matching section. (a) 3-D view, (b) front view, (c) bottom view.

hemispherical helical antenna [10]. This matching section is, in fact, a doubly tapered transmission line acting as a wideband impedance transformer. However, simulation results for the antenna of Fig. 3 incorporating such matching section indicated that, while the VSWR bandwidth is improved significantly, the axial ratio bandwidth is decreased slightly. Since the linearly tapered matching section does not yield a satisfactory increase in the axial ratio bandwidth, we considered a tapered matching section that is closer to the ground plane and

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Fig. 6. Variations of directivity versus frequency at several angular directions for the hemispherical helical antenna of Fig. 5. Also shown are normalized power ) and yz-plane (. . . . . .) at six different frequencies. patterns in the xz-plane (

its start point is at the side instead of the center of hemispherical helix. One way to implement this modification is to vary the height of the matching section nonlinearly rather than linearly. This way, a major part of the matching section will remain closer to the ground plane. Furthermore, instead of varying the width of matching section linearly, which leads to abrupt discontinuity or sharp angles at the point of connection to the helix, the width is varied such that connection to the radiating element occurs smoothly, as shown in Fig. 5(a) and (b). The width of this matching section is very small (approximated as zero in simulations) at the start point and is equal to the starting width of the tapered radiating element at the end point. The median curve, that divides the width of the matching section in half, lies on the hemispherical surface as shown in Fig. 5(c). The height of the tapered matching section is 1 mm at the start point and 5 mm at the end point; see Fig. 5(b). A number of different functions were tried for height variations as well as the width of the matching section, and found out that an exponentially varying height and a sinusoidally varying width provide a satisfactory matching performance. By adjusting the exponent of the exponential function, this design permits a major part of the matching section to be very close to the ground plane, thus influencing the radiated fields negligibly. The hemispherical helical antenna shown in Fig. 5, with its radiating element and impedance matching section now designed for best VSWR and axial ratio performance, was analyzed extensively. The simulation results for directivity and beamwidth of this antenna are presented in the remaining part of this section, while simulation results for axial ratio, VSWR, gain, and far-field patterns are presented and compared with their measured counterparts in the next section. Fig. 6 shows variations

of directivity versus frequency along several angular directions. In calculating these results a finite ground plane, equal in size to that used in the constructed prototype (discussed in the next section), is used. It is noted that the maximum directivity at is about 9 dB with about 1 dB fluctuations over a bandwidth of more than 50% (2.2 GHz–3.7 GHz). For and the directivity remains relatively constant at about 8.5 dB and 7 dB, respectively, over even a wider bandwidth. The power patterns at six different frequencies in xz- and yz-planes are also presented in Fig. 6. Closer examination of these patterns indicates that each has a broad main beam with a high degree of axial symmetry as manifested by nearly overlapping xz- and yzplane patterns. Fig. 7 illustrates variations of the antenna beamwdith in two different plane cuts (xz- and yz- planes) with frequency. The half power beamwidth (HPBW), which measures a 3 dB drop from the maximum directivity, varies between 50 to 80 . For such a broad radiation beam, it is also worth examining the 10-dB beamwdith. This beamwidth remains nearly con. It is worth stant over the entire frequency range, mentioning that, as observed in Fig. 7, there is only about 5 difference between the beamwidths of two different cuts. This property emphasizes the fact that the three-dimensional radiation pattern of the antenna is highly symmetric about the z-axis. IV. COMPARISON BETWEEN MEASURED AND SIMULATED RESULTS A prototype of the design shown in Fig. 5, which yields the maximum overall bandwidth, was fabricated and measured. Measurement results for far-field patterns, axial ratio, VSWR,

ALSAWAHA AND SAFAAI-JAZI: ULTRAWIDEBAND HEMISPHERICAL HELICAL ANTENNAS

Fig. 7. Variations of 3-dB and 10-dB beamwidths versus frequency for the hemispherical helical antenna of Fig. 5.

Fig. 8. View of the constructed prototype of the hemispherical helical antenna. The design parameters are the same as those in Fig. 5.

and directivity are presented and compared with the corresponding simulation results in this section. In calculating the simulation results a finite size PEC circular ground plane, with the same diameter as that used in the constructed prototype, is used. Since the metallic strip used as the radiating element is very thin, a support structure, on which the helix will be mounted, is required. Balsa wood is chosen for this purpose because of its light weight and can be shaped easily. The dielectric properties of Balsa wood do not have considerable impacts on the antenna radiation characteristics. It has a relative permittivity of 1.2 [11]. The constructed prototype is shown in Fig. 8. The assembly of helical strip on support structure and the matching element is mounted on a steel pan which serves as ground plane. Finally, a 50 SMA coaxial connector is attached to the ground plane at the starting point of the matching section. The inner and outer conductors of the SMA connector are soldered to the matching section and the ground plane, respectively. After the prototype was carefully constructed, its radiation characteristics were measured using the facility of the Virginia Tech Antenna Laboratory.

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Fig. 9. Comparison of simulated and measured axial ratios for the hemispherical helices shown in Fig. 5 and Fig. 8.

Fig. 10. Comparison of simulated and measured VSWRs for the hemispherical helices shown in Fig. 5 and Fig. 8.

Fig. 9 compares variations of the simulated and measured axial ratios versus frequency for the constructed prototype. There is generally good agreement between the predicted and measured axial ratio results. The measured 3-dB axial ratio is about 24% relative to a mid-band bandwidth frequency of 3.35 GHz, while the corresponding simulated bandwidth is about 20% relative to a mid-band frequency of 3.26 GHz. Furthermore, it is noted that the measured AR values are lower than the simulated ones over essentially the entire bandwidth. This might be attributed to the inevitable differences that occur between the actual shape of the radiating element of the constructed prototype and the simulated shape used in calculating its radiation characteristics. The simulated and measured results for the VSWR of the prototype antenna are presented in Fig. 10. Again, good agreement between the measured and calculated results is noted. The meais about 50% relative sured VSWR bandwidth to a mid-band frequency of 4.05 GHz, while the corresponding simulated bandwidth is more than 46% relative to a mid-band frequency of 3.86 GHz. Such a large VSWR bandwidth clearly

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Fig. 11. Comparison of simulated and measured directivities for the hemispherical helices shown in Fig. 5 and Fig. 8.

axial ratio and VSWR is that for both the measured bandwidths are larger than the corresponding predicted ones. In other words, the performance of the constructed prototype antenna is slightly better than expected. Variations of the simulated maximum directivity and the measured gain versus frequency are illustrated in Fig. 11. As noted, the agreement between the measured and simulated results is very good, with the measured values slightly smaller than predicted ones at most frequencies. This is partly because the conductor loss has been ignored in calculating the directivity results. Far-field patterns of the prototype antenna were also examined for a large number of frequencies. As an example, Fig. 12 illustrates measured and simulated normalized power patterns in the xz- and yz- planes at a frequency of 3.40 GHz. A closer examination of these patterns emphasizes good agreements between measured and simulated results and also ascertains the axial symmetry of the main beam. V. CONCLUSION A new hemispherical helical antenna capable of providing over a bandwidth of more about 9 dB gain and over than 50% and circular polarization a bandwidth of 24% was introduced and investigated. Compared with a wire hemispherical helix, the new design incorporates two major modifications, including the replacement of the wire radiating element with a tapered metallic strip which assumes a non-conformal configuration relative to the hemispherical surface and a matching section consisting of a nonlinearly doubly tapered strip transmission line. A comprehensive numerical analysis of the proposed antenna was carried out. Radiation characteristics including far-field patterns, directivity, axial ratio, VSWR, and beamwidth were calculated over wide frequency ranges. A prototype of the antenna was constructed and measured. Generally, there is good agreement between the simulated and measured radiation characteristics. The low profile, compact geometry, and desirable radiation characteristics over a wide frequency range are important attributes of this antenna which make it an attractive candidate for avionic and UWB communication applications. REFERENCES

Fig. 12. Simulated and measured normalized radiation patterns for the hemi) and measured (. . . . . .). spherical helices of Figs. 5 and 8, simulated (

indicates that the proposed antenna is a very efficient and compact UWB radiator. An interesting observation regarding the

[1] J. C. Cardoso and A. Safaai-Jazi, “Spherical helical antenna with circular polarization over a broad beam,” IEE Electron. Lett., vol. 29, pp. 325–326, 1993. [2] A. Safaai-Jazi and J. C. Cardoso, “Radiation characteristics of a spherical helical antenna,” IEEE Proc. Microw. Antennas Propag., vol. 143, pp. 7–12, 1996. [3] E. Weeratumanoon and A. Safaai-Jazi, “Truncated spherical helical antennas,” Electroni. Lett., vol. 36, pp. 607–609, 2000. [4] T. W. Hertel and G. S. Smith, “The conical spiral antenna over the ground,” IEEE Trans. Antennas Propag., vol. 50, pp. 1668–1675, 2002. [5] S.-G. Mao, J.-C. Yeh, and S.-L. Chen, “Ultrawideband circularly polarized spiral antenna using integrated balun with application to time-domain target detection,” IEEE Trans. Antennas Propag., vol. 57, pp. 1914–1920, 2009. [6] “FEKO User’s Manual Suite 5.3,” Jul. 2007. [7] W. L. Stutzman and G. Z. Thiele, Antenna Theory and Design, 2nd ed. New York: Wiley, 1998. [8] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005. [9] J. D. Kraus and R. J. Marhefka, Antennas, 3rd ed. New York: McGraw-Hill, 2002.

ALSAWAHA AND SAFAAI-JAZI: ULTRAWIDEBAND HEMISPHERICAL HELICAL ANTENNAS

[10] H. W. Alsawaha, “New designs for hemispherical helical antennas,” M.Sci. thesis, Bradley Dept. Elect. Computer Eng., Virginia Polytechnic Institute and State Univ., Blacksburg, 2008. [11] C. Jung-Hwan, M. Jung-Ick, and P. Seong-Ook, “Measurement of the modulated scattering microwave fields using dual-phase lock-in amplifier,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 340–343, 2004.

Hamad Waled Alsawaha (S’06) was born in Kuwait City, Kuwait, on May 16, 1980. He received the B.E. degree from Kwuait University, in 2003, and the M.S. degree from Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, in 2008, both in electrical engineering. He is currently working toward the Ph.D. degree at Virginia Tech. He has been a member of the Virginia Tech Antenna Group (VTAG) since 2007. His research interests include electromagnetic wave propagation, antennas, and their applications to UWB communication systems.

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Ahmad Safaai-Jazi (SM’86) received the B.Sc. degree from Sharif University of Technology, Iran, in 1971, the M.A.Sc. degree from the University of British Columbia, Canada, in 1974, and the Ph.D. degree (with distinction) from McGill University, Canada, in 1978, all in electrical engineering. From 1978 to 1984, he was an Assistant Professor with the Division of Electrical and Computer Engineering, Isfahan University of Technology. In 1984, he returned to the Department of Electrical Engineering at McGill University, where he conducted extensive research on propagation of ultrasound waves in optical fibers. He joined the Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, in 1986, where he is a Full Professor. His research interests include antennas, UWB propagation, wireless and optical communications, and guided-wave optics. He is the author or coauthor of more than 130 refereed journal papers and conference publications. He has also contributed two book chapters and holds four patents. Dr. Safaai-Jazi was the co-recipient of the Wheeler Award for the Best Application Paper from the IEEE Antennas and Propagation Society, 1995. He received the Dean’s Award for Excellence in Teaching from the College of Engineering at Virginia Tech in 2002.

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Transmitter and Receiver Isolation by Concentric Antenna Structure Won-Gyu Lim, Han-Lim Lee, and Jong-Won Yu, Member, IEEE

Abstract—A concentric antenna structure with transmitter (Tx) and receiver (Rx) isolation characteristic is proposed. For both cases that Tx and Rx use the same frequency or different frequencies, Tx leakage can be removed by the proposed antenna structure. The field couplings between antennas are mathematically presented and Tx leakage cancellation is theoretically examined. To theoretically verify the case that Tx and Rx use the same frequency, a set of 16 inverted-F antennas and a set of 4 inverted-F antennas are designed for Tx and Rx, respectively. For the case that Tx and Rx use different frequencies, a set of 4 monopole antennas is designed for Tx and Rx each. The measured result shows the Tx/Rx isolation characteristic of more than 50 dB. Index Terms—Bi-static antenna, concentric antenna, isolator, RF front-end, Tx leakage canceller.

I. INTRODUCTION Fig. 1. Geometry of the proposed concentric antenna to eliminate Tx leakage.

I

SOLATION between transmitter (Tx) and receiver (Rx) is one of the most important issues in communication systems where Tx signal unintentionally flowing into Rx is called Tx leakage. Since Tx leakage may have higher values than received signal and makes active components saturated, it degrades the overall quality of the received signal. Tx leakage generated in the front-end of mobile communications is related to antenna and thus the method to remove Tx leakage varies according to the types of antennas used. If a single antenna is used in the communication front-end, Tx/Rx isolation is determined by reflection and isolation performance of the antenna [1]–[3]. For a single antenna, isolators such as a switch in time division duplex (TDD) and a duplexer in frequency division duplex (FDD) are used to separate Tx and Rx. A circulator, directional coupler or quadrature hybrid can also be used for the system having no distinction between time and frequency. When a dual antenna system is used, Tx leakage depends on the amount of field coupling between antennas [4], [5]. In this case, the field coupling can be reduced by increasing the distance between antennas or differentiating the beam directions of each antenna. There is also a method to cancel coupled Tx leakage partially or totally through Tx/Rx antenna feed circuit

Manuscript received October 08, 2009; revised February 04, 2010; accepted March 31, 2010. Date of publication July 19, 2010; date of current version October 06, 2010. W.-G. Lim is with the Korea Aerospace Research Institute (KARI), Daejeon, South Korea (e-mail: [email protected]; [email protected]). H.-L. Lee and J.-W. Yu are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055801

[6]. That is, the Tx leakage is cancelled by making the signals coupled and flowed into a receiver have the opposite phases. This method is mainly used for the case that Tx/Rx frequencies are the same. This paper presents Tx/Rx isolation for the case of dual antenna. Using the Tx/Rx antenna structure with the center points in -axis as shown in Fig. 1, a novel method to provide excellent isolation can be achieved regardless of whether the Tx/Rx frequencies are the same or not. Tx leakage component in a receiver is theoretically derived and a fundamental structure to fully remove the Tx leakage is introduced. Experiments on Tx/Rx antennas having the same center frequency and different center frequencies are deployed to verify the proposed structure. II. TX/RX ISOLATION THEORY OF CONCENTRIC ANTENNA A. Isolation for Identical Tx/Rx Frequency Fig. 1 shows the proposed dual antenna structure having the center points in -axis. Each of the Tx and Rx antennas has antenna feed points and keeps the same distances and are radii of the two antennas, and on the circles. indicates the distance between the Tx and Rx antennas (for theoretical development in this paper, these values are assumed to be arbitrary while is set to 0 for experiment). The Tx and Rx antennas have the same center frequency. Fig. 2 represents the equivalent circuit of the proposed dual antenna and shows the process of Tx leakage flowing into Rx. A Tx signal is delivered to an antenna by the Tx feed circuit . has a delay of having a delay element, . The signal entered to Tx antenna is then radiated in the air while some of the signal is coupled to Rx

0018-926X/$26.00 © 2010 IEEE

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Fig. 2. Block diagram of the proposed Tx leakage cancellation method for identical Tx/Rx center frequency.

antenna. At this moment, Tx leakage is generated. The Rx feed , sends the Tx circuit having a delay element, leakage to a receiver. has a delay of . With the reference to Fig. 2, the input and output relationship between Tx/Rx signals and Tx/Rx circuit feed signals are given in the form:

(1) where and denote the phases of the signal entered to the Tx and indicate and Rx feed circuit, respectively. the reflected (incident) signals from (to) Tx antenna and Rx anand tenna, respectively ( is the antenna port number). describe the reflected signals from Tx feed circuit and Rx feed and describe the incident sigcircuit, respectively while nals to Tx feed circuit and Rx feed circuit. As shown in (1), the Tx and Rx feed circuits divide a corresponding input signal into signals having equal magnitude and phase difference of or combine signals into one. The Tx/Rx antenna can be represented by matrix as follows:

Fig. 3. Equivalent mutual coupling matrix factor for the case of n

= 4.

is shown in Fig. 3, The proposed antenna structure for where the distances between antenna feed points are identically configured. If Tx antennas are symmetrically placed from the center of the circle and Rx antennas also keep a symmetric structure, 4 cases having identical Tx-to-Rx transmission characteristics can be classified as shown in Fig. 3(a)–(d). Similarly, between Tx and Rx antenna the mutual coupling matrix can be represented as (3), shown at the bottom of the page. Tx is found as follows by substituting (2) into (1): leakage

(4) is a component of the matrix and Tx leakage where is affected by the coupling of Tx/Rx antenna. When decomposing (4) with two shaded areas in (3), the following can be found:

(2) is a matrix and T denotes a transpose of the where matrices. and indicate incident signals to Tx antenna and Rx is antenna, respectively. If a matched load condition assumed in the receiver, of the Rx feed circuit becomes 0 and . the coupling related to Tx leakage depends on

(5) where . The first term and second term in the brace correspond to the right-shaded area and left-shaded area of (3), re-

(3)

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Fig. 4. Block diagram of the proposed Tx leakage cancellation method for different Tx/Rx frequency.

spectively. By substituting the equivalent relation of (4) into (5), Tx leakage is found as follows:

Fig. 5. Geometry of the generalized proposed concentric antenna structure to eliminate Tx leakage.

C. Tx/Rx Isolation of the Generalized Concentric Antenna

(6) From (6), it is observed that Tx leakage becomes 0. That is, the proposed antenna structure for identical Tx and Rx center frequencies can fully remove Tx leakage regardless of the number of arbitrary antenna feed points, , or the distance between Tx and Rx center points or the radius of Tx/Rx antenna concentric placement.

The antenna structure proposed in Fig. 1 removes Tx leakage when Tx antennas symmetrically placed on Tx circle and Rx antennas symmetrically placed on Rx circle are concentric. This structure can be generalized into arbitrary Tx/Rx circles having the same center point. For Fig. 5, both number of Tx circles and number of Rx circles are composed of number of feeding sets and concentric in -axis. Here, and are arbitrary numbers. The center distance between each circles along -axis and the radii of the circles are also arbitrary. Tx leakage of the antennas located on the arbitrary th Tx circle to the antennas located on the arbitrary th Rx circle is removed by the equations: (6) for identical Tx/Rx frequency and (7) for different Tx/Rx frequencies. That is, Tx and Rx circles of Fig. 5 have Tx leakage presented as follows and all components of the matrix are 0:

B. Isolation for Different Tx/Rx Frequency Fig. 4 shows the proposed structure to remove Tx leakage for the case that Tx and Rx frequencies are different. The antenna structure is similar with Fig. 1 that Tx and Rx are concentric. The difference from the case that Tx and Rx frequencies are the same is the Tx feed circuit which is now changed to a -way power divider delivering signals with equal magnitude and phase to the output. By taking into account the changed Tx feed circuit, Tx leakage can be calculated as follows:

(7) where denotes the constant delay of Tx feed circuit. As seen from (7), Tx leakage is also removed for the case that Tx and Rx center frequencies are different. The proposed method is different from the generally used methods such as keeping Tx/Rx distances far from each other and placing a metal to block the field coupling between antennas. Theoretically, (6) and (7) represent that Tx leakage is removed regardless of the distance between Tx and Rx antennas.

.. .

.. .

.. .

..

.

.. .

.. .

denotes the field coupling of where th Rx circle.

(8)

th Tx circle to

III. ANTENNA DESIGN A. Antenna Structure for Identical Tx/Rx Frequency The proposed Tx/Rx antenna structure has a set of antennas placed equidistantly and circumferentially from a concentric point [7]. As shown in Fig. 6, the Rx antenna on a circle and the Tx antennas on 4 circles , , , ) are concentric. Assuming the ground size to be infinite, the radiation efficiency according to Tx/Rx antenna distances is simulated and circle is the results are shown in Table I. When the radius of greater than or equal to 95 mm, Tx/Rx antenna has the radiation efficiency of more than 70% (the antenna is designed under this condition). The radius of Rx antenna circle is and the radii of Tx antenna circles are , , . The set of antennas in blue

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Fig. 6. Schemetic of the designed Tx leakage cancellation method for identical Tx/Rx center frequency.

TABLE I SIMULATED RADIATION EFFICIENCY ACCORDING TO THE DISTANCE BETWEEN TX AND RX

Fig. 7. Block diagram of the designed Tx leakage cancellation method when 16 inverted-F antennas are used for Tx and 4 inverted-F antennas are used for Rx.

on circle and the set of antennas in brown on circle (with . reference to Fig. 6) have the same radius At 900 MHz center frequency, Tx and Rx antennas have radiation efficiencies of 77% and 71%, respectively. Fig. 7 represents the equivalent schematic and feed circuits of Fig. 6. The four phases supplied to antennas are generated by power diand circles, 4 inverted-F anviders and delay lines. On tennas have relative phases of 0 , 90 , 180 and 270 . Fig. 8 , 2, 3, 4) represents the Tx leakage of the antennas on ( circle. The field coupling from the circle to the antennas on 4 inverted-F antennas on circle to the 4 inverted-F antennas on circle in Fig. 8(a)–(d) shows the following relation:

Fig. 8. Simulation results for the field coupling from Tx antennas to Rx antennas: (a) from T circle to R circle, (b) from T circle to R circle, (c) from T circle to R circle, (d) from T circle to R circle.

(9) The equivalent relationship between matrix components indicated in Fig. 3 remains the same. Also, the highest coupling is observed at the operating frequency, , of Tx/Rx antenna. Fig. 9 shows the simulation result for the amount of coupling from Tx to Rx antenna port and Tx leakage to Rx through Rx feed circuit (indicated in Fig. 7). At the center frequency, about 19% of Tx signal is coupled to Rx antenna. That is, Tx leakage correis detected at Rx antenna port. The signal sponding to at the Rx antenna port is the vector summation in Rx feed circuit and it is observed that Tx leakage is removed at the Rx port. Also, the proposed structure shows Tx/Rx isolation of more than in the frequency band of (15%).

Fig. 9. Simulation results for the amount of coupled Tx leakage versus isolated Tx leakage.

B. Antenna Structure for Different Tx/Rx Frequency For the proposed structure (Fig. 6) having identical Tx/Rx frequency, Tx leakage more than is removed up to the

frequency band of . The case that Tx and Rx frequencies are different is investigated by setting the difference between

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Fig. 10. Schemetic of the designed Tx leakage cancellation method for different Tx/Rx center frequency.

Fig. 12. Simulation results for the amount of coupled Tx leakage vs. isolated Tx leakage.

TABLE II SIMULATED RADIATION EFFICIENCY ACCORDING TO THE DISTANCE BETWEEN TX AND RX

Fig. 11. Simulation results for the field coupling from Tx antennas to Rx antennas.

each frequencies to be more than . Fig. 10 shows the antenna structure that Tx and Rx operate at different frequencies. The set of Tx antennas and the set of Rx antennas are concentric. The Tx antenna consists of four antenna sets having the same and Rx antenna consists of four anphases in the radius of tenna sets having relative phase difference of 90 in the radius . The simulation results for the radiation efficiency acof cording to distances are shown in Table II when the difference between the center frequencies of Tx and Rx antennas is fixed to . The designed antenna (shown in Fig. 10) has radius of and radiation efficiency 20 mm for , radius of 2.6 mm for of more than 90%. Fig. 11 shows the simulation result for the mutual coupling of , and denote the center frequencies Tx/Rx antenna.

Fig. 13. (a) Fabricated antenna structure, (b) measured and simulated Tx/Rx isolation results for the case that Tx/Rx frequencies are identical.

of Tx antenna, Rx antenna and the middle frequency between represents the coupling Tx and Rx antennas, respectively. circle and th antenna on coefficient of th antenna on circle. It is noted that the equivalent relation between matrix components considering Fig. 3 remains the same. Similar to the identical frequency case, Fig. 11 shows the simulation results for the summation of absolute amounts of coupling from Tx to Rx antenna port and Tx leakage to Rx port through Rx feed circuit. Tx signal corresponding to the center frequency of Rx antenna within becomes a leakage to Rx antenna. Tx components incident to Rx antenna port are vector-summated by Rx feed circuit and the result appears at Rx port. It is noted that Tx leakage is removed at the Rx port.

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Fig. 15. (a) Fabricated antenna structure, (b) measured and simulated Tx/Rx isolation results for the case that Tx/Rx frequencies are different.

B. Fabricated Antenna for Different Tx/Rx Antenna

Fig. 14. Measured and simulated radiation pattern for identical Tx/Rx center frequency: (a) z -x plane for Tx, (b) y -z plane for Tx, (c) x-y plane for Tx, (d) z -x plane for Rx, (e) y -z plane for Rx, (f) x-y plane for Rx.

IV. EXPERIMENTAL RESULTS A. Fabricated Antenna for Identical Tx/Rx Frequency For the case that Tx and Rx frequencies are identical, the designed antenna structure to remove Tx leakage is implemented with Tx/Rx feed circuits and inverted-F antennas on 250 mm 250 mm ground plane using Fr-4 board (Fig. 13(a)). The Tx and Rx antennas have the center frequency of 930 MHz. The height is 15 mm and the radius for antenna placement is implemented with the dimensions of the case: of Table I. Fig. 13(b) shows the simulated and measured isolations of the Tx/Rx antenna and the input/output reflection coefficients. It is noted that Tx leakage is removed around the center frequency. Fig. 14 describes the simulated and measured radiation characteristics of the fabricated antenna. The Tx/Rx antenna has a main beam pattern in -axis. Fig. 14(a)–(c) shows the measured results of the Tx antenna with respect to ( , ) . The Tx antenna has a gain of and with respect to about 5.5 dBi. Fig. 14(d)–(f) shows the measured results of the , ) and with respect Rx antenna with respect to to . Rx antenna has a gain of about 0 dBi in -axis.

For the case that Tx and Rx frequencies are different, the designed antenna structure to remove Tx leakage is implemented with Tx/Rx feed circuits and inverted-F antennas on 100 mm 100 mm ground plane (Fig. 15(a)). Tx and Rx antennas have the center frequencies at 1350 MHz and 1850 MHz, respecand tively. The radii of antennas are as in described in Table II. Fig. 15(b) shows the input/ output reflection coefficients, and the simulated and measured isolation of Tx/Rx antenna. It is noted that Tx leakage is removed around the center frequency of Rx antenna. Fig. 16 describes the simulated and measured radiation characteristic of the fabricated antenna. Fig. 16(a)–(c) shows the measured re, ) and sults of the Tx antenna with respect to ( . The Tx antenna has a gain of about with respect to . Fig. 16(d)–(f) shows the 0.5 dBi and a main beam at , measured results of the Rx antenna with respect to ( ) and with respect to . The Rx antenna has a gain of 0.3 dBi and a main beam at . V. CONCLUSIONS Tx leakage cancellation for the case using separate Tx and Rx antennas was examined by the proposed concentric Tx/Rx antenna. Based on the theoretical analysis illustrated in this paper, Tx leakage was removed regardless of the distance between Tx and Rx antenna. Tx/Rx isolation characteristic was achieved for both cases that Tx and Rx use the same frequency and different frequencies. In order to verify the proposed method, two types of antenna were designed. Using 4 sets of Tx antenna (each set consists of 4 inverted-F antennas) located in different distances from

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[3] J. Ho Jung, J. H. Kim, S. M. Kim, and K. C. Lee, “New circulator structure with high isolation for time division duplexing radio system,” in Proc. IEEE Vehicular Technology Conf., Sep. 2005, vol. 4, pp. 2766–2769. [4] H.-W. Son, J.-N. Lee, and G.-Y. Choi, “Design of compact RFID reader antenna with high transmit/receive isolation,” Microw. Opt. Technol. Lett., vol. 48, no. 12, pp. 2478–2481, Dec. 2006. [5] K. Kirose, N. Saito, and H. Nakano, “A cavity backed spiral antenna with an unbalanced feed system-analysis using FD-TD and moment methods,” Electron. Commun. Jpn., vol. 88, no. 1, pt. 1, pp. 55–63, Sep. 2005. [6] W.-G. Lim, W.-I. Son, K. S. Oh, W.-K. Kim, and J.-W. Yu, “Compact integrated antenna with circulator for UHF RFID system,” IEEE Antenna Wireless Propag. Lett., vol. 7, pp. 673–675, Feb. 2009. [7] W. I. Son, W. G. Lim, M. Q. Lee, S. B. Min, and J. W. Yu, “Printed square quadrifilar spiral antenna for UHF RFID reader,” in Proc. IEEE Antenna and Propagation Int. Symp., Jun. , pp. 305–308.

Won-Gyu Lim received the B.S. degree in electrical engineering from Kyungpook National University, Daegu, South Korea, in 2002, and the M.S. and Ph.D. degrees in electrical engineering from KAIST, Daejeon, South Korea, in 2004 and 2008, respectively. During his M.S. degree studies, he focused on the packaging for microwave circuit on compound semiconductor substrate. For his Ph.D. studies, he emphasized RFID system and UWB systems. He is currently is with Korea Aerospace Research Institute (KARI). His research interests are wireless transmitter/receiver front-end isolation, multilayer EMI/EMC analysis and small antenna.

Han-Lim Lee received the BASc. degree in electronics engineering from Simon Fraser University, British Columbia, Canada, in 2008 and is working toward the M.S. degree at KAIST, Daejeon, South Korea. His research interests are microwave/millimeter wave circuit, RF hybrid/wireless communication systems, RFID and antenna.

Fig. 16. Measured and simulated radiation pattern for different Tx/Rx center frequency: (a) z -x plane for Tx, (b) y -z plane for Tx, (c) x-y plane for Tx, (d) z -x plane for Rx, (e) y -z plane for Rx, (f) x-y plane for Rx.

the center point and 1 set of Rx antenna, the case that Tx and Rx frequencies were the same was verified. 4 monopole antennas were used for Tx and Rx each to verify the case that Tx and Rx frequencies were different. The measured results showed an excellent agreement with the simulation and theoretical analysis. REFERENCES [1] W.-G. Lim, S.-Y. Park, W.-I. Son, M.-Q. Lee, and J.-W. Tu, “RFID reader front-end having robust Tx leakage canceller for load variation,” IEEE Trans. Microw. Theory Tech., vol. 57, pp. 1348–1355, May 2009. [2] S. L. Karode and V. F. Fusco, “Feedforward embedding circulator enhancement in transmit/receive applications,” IEEE Microw. Wireless Compon. Lett., vol. 8, pp. 33–34, Jan. 1998.

Jong-Won Yu received the B.S., M.S., and Ph.D. degrees in electrical engineering from KAIST, Daejeon, South Korea, in 1992, 1994 and 1998, respectively. From 1995 to 2000, he worked at Samsung Electronics. He also served Wide Tecom Head and Telson from 2000 to 2001 and from 2001 to 2004, respectively. Since 2004, he has been an Assistant Professor of electrical engineering at KAIST, where he is currently an Associate Professor. His research interests emphasize microwave/millimeter wave circuit (MMIC, hybrid), wireless communication system and RFID/USN.

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True-Time-Delay Beamforming With a Rotman-Lens for Ultrawideband Antenna Systems Andreas Lambrecht, Student Member, IEEE, Stefan Beer, and Thomas Zwick, Senior Member, IEEE

Index Terms—Antenna arrays, beam steering, phased arrays, Rotman-lens, true time delay, ultrawideband (UWB).

loss of the ports is given and a phase center approximation of the tapered microstrip ports, too. The design of the dummy port region and the optimization of the whole Rotman-lens is done numerically, whereas the initial model is based on the analytical design approach. The couplings of the feeding lines and the interaction of all ports together are not straightforward to implement into the analytical system model. Based on the developed design procedure, a demonstrator for the FCC UWB range from 3.1 GHz to 10.6 GHz is developed. The Rotman-lens prototype is presented here.

I. INTRODUCTION

II. ANALYTICAL SYSTEM DESIGN APPROACH

Abstract—An analytical design model for Rotman-lenses is presented, with an accurate treatment of the return loss of the ports and their phase centers. The detailed design equations allow a very fast prototyping prior to numerical simulations for the final adjustment. Based on the presented model a prototype has been built for the true-time-delay phasing of an antenna array for the ultrawideband frequency range from 3.1 GHz – 10.6 GHz. Measurement results of the prototype together with an antenna array for transient systems are presented in the time and frequency domain.

F

OR beam-steering of ultrashort transient voltage pulses on transmit over a large angle width true time delay concepts for phasing of the antenna elements are favourable. Examples of currently used RF beamformers are the Rotman-lens [1], the Luneberg Lens [2], the Blass matrix [3], and switched RF lines [4]. In [5] an overview of these technologies is given. The main advantage of the Rotman- and the Luneberg lens are the low number of switching elements needed, compared to the Blass matrix and switched line concepts. The Rotman-lens can feed a linear array, is capable of handling high voltages and is more compact than the Luneberg lens. Consequently it is attractive for RF-systems needing wide angle scanning and wide bandwidth. This may also find applications in automotive radar systems [6], [7]. The ray-optical model of the Rotman-lens is modified with a treatment of the port structures similar to [8]. The input and output ports and the dummy region of the parallel-plate region are the main restrictions to the usable bandwidth [9]. Although the Rotman-lens is inherently broadband many publications use it narrowband. Additionally to the treatment in the spectral domain, the lens is investigated in the time domain, too, which was, as far as the authors know, only ever done in one paper [10]. There the dispersion of a Rotman-lens was characterized between 2 GHz and 20 GHz. This investigation is complemented here for all beam ports and the careful analysis of different measures of quality. An analytical approach to predict the return Manuscript received July 10, 2009; revised January 25, 2010; accepted March 19, 2010. Date of publication June 14, 2010; date of current version October 06, 2010. This work was supported by the German Research Council (Deutsche Forschungsgemeinschaft, DFG). A. Lambrecht was with the Institut für Hochfrequenztechnik und Elektronik, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany. He is now with EADS, Ulm 89081, Germany (e-mail: [email protected]). S. Beer and T. Zwick are with the Institut für Hochfrequenztechnik und Elektronik, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2052558

A. Transient Modeling of Antennas and Beamsteering Since an antenna is a linear-time-invariant (LTI) system, it can be fully described by the directional antenna transfer function in the frequency or in the time domain [11]. An antenna can be described in the time domain with its impulse response , where is characterized by transmission measurements of two identical antennas. Generally the same is true for every other system with multiple ports. The complete details on the signal processing for the measurement of the antennas transfer function can be found in [12]. For an ideal passive power divider with adjustable true time , ( : number of antennas) the ideal delays . The animpulse responses are tenna array is fed with a single transient pulse, ideally modeled as a dirac function. The superposition of the time delayed transient pulses in transmit mode at a far-field point gives the electrical field-strength of

(1)

with point,

the speed of light, and the distance to the far-field and are elevation and azimuth angle, respectively. is the applied voltage to the antenna in transmit mode is the (i.e., Gaussian distributed transient voltage pulse), free space impedance, and is the antenna input impedance. , The sum describes the idealized time domain array factor which in the real world is realized by the Rotman-lens. For the complete system the transfer functions are measured or simulated in the frequency domain. The measurement frequency range is from 400 MHz to 20 GHz. A proper calibration

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Fig. 2. Geometrical parameters of elliptical beam port arc.

, and , similar to [13]. Following the procedure of equating the path lengths one gets the and the connecting line array port positions to the linear array : lengths

quantities are defined:

Fig. 1. Geometrical parameters of Rotman-lens.

is used in order to eliminate dispersive effects of the connecting cables. The measured data is complemented by zero padding in the frequency domain. This leads to a fine interpolation of the system’s transient response with an interpolated time resolution . Since only the positive frequencies are used for of the transformation, it results in a time-discrete analytical signal. The frequency response of the complete system is synthesized from the directional frequency responses of the antenna (Tx) and the frequency response of the beamformer (BF), where the last one is directly connected with the time domain array factor . The following equation is valid for each beamport : (2)

(5) (6a) (6b) on the beamport (BP) arc The three focal points lead to a phase error of zero, but to realize more beam directions input ports on a connecting line are needed. This connecting line . With can be realized by an elliptical focal arc as the eccentricity of the ellipse and one gets for the coordinates for the beam ports

The difference of the time delays between the antenna elements , ideally , leads to a beam scanning angle of (3)

(7a) (7b) with the expansion factor

one obtains

(4) as the element spacing in the array, and . Thus, is an additional length in the feeding network used for the realization of the true-time-delay. The differences of the time delays between the elements of the arrays should all be the same for the desired steering angle. with

Case 1:

B. Design Model for the Rotman-Lens The variables used in the following can be found in Figs. 1 and 2. In this section, the relevant equations used in the analytical design approach are reviewed. 1) Coordinates of Array and Beam Ports: The lens is etched on a substrate with permittivity , for the microstrip lines this . The procedure to obtain the coordinates of changes to the array ports (AP) and the lengths of the connecting lines to the linear antenna array by equating the path lengths in the lens can be found in many publications [1], [13], [14]. [14], The expansion factor is defined by where the desired beam-steered radiation angle is . The following definitions are made: , , , . With these two normalized

Case 2:

Based on this, one can obtain the additional line lengths for the beam squint realized by the Rotman-lens (8) 2) Radiation Characteristic of Microstrip Ports: The parallel plate region can be seen as a 2-D free space, since only TEMwaves can propagate. Hence the ports can be described as 2-D can antennas with a 1-D radiating edge. Hence the pattern

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Fig. 3. Port modeling. Fig. 5. Comparison system model versus simulation versus measurements: transmission.

creates a power density in the direction , with the substrate, of

the height of

(13) The port is orthogonal to the parallel plate region and can radiate . The radiated power can be into the angle range of expressed as follows, with :

Fig. 4. Prototype lens, left: beamports (BP), right: arrayports (AP).

be calculated as follows, with being the width of the edge of the AP or BP and is the current distribution along the edge of the microstrip port (9)

(14) and compared to the whole input The difference of power gives the reflection coefficient at this port

For microstrip ports, one can assume a constant current distribution along the edge, as long as the -mode is dominant, so

(15)

(10)

5) Phase Center Adjustment for Microstrip Tapers: With the length of the taper and the width of the feeding microstrip can be approximated line, the phase center of the port by

For the case of the microstrip realization, the dimensions of the port are restricted by the occurrence of the first hybrid mode at the maximum frequency , leading to the maximum dimension of (11) the quasi-TEM mode is deteriorated. A i.e., if careful compromise has to be chosen for -lines, especially , because the possible line width is restricted. for increasing 3) Transmission Loss Modeling: Following [14] the 2-D propagation can be modelled with a modified Friis formula, with the distance between the phase centers of the radiating edge ports and the wavelength (see Fig. 5) (12) 4) Return Loss Modeling: By calculating the radiated power from the edge antennas and known input power one can estimate the reflection coefficient at the ports. Due to reciprocity this is also true for the receiving array ports. The input power

(16) The equation for the phase center location is derived by using tangential lines onto the exponential curvature (Fig. 3). Since the phase center is frequency variant, this is a compromise, which leads to a better solution in the final prototype. C. Measures of Quality There are two important effects to be distinguished: first, the ability of a system to effectively transmit power and secondly, the distorting influence on the waveform to be transmitted. The standard deviation of the group delay is a measure for the linearity of a phase of the system. It is defined as (17) describes the mean value of the group delays, and the lower and upper frequency limit.

are

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Fig. 6. Comparison system model versus simulation versus measurements: reflection.

The maximum value of the absolute value of the envelope of the impulse response is important for the achievable signal strength at the target. When measuring the envelope of the impulse response of the array the polarization has to be considered

Fig. 7. Measured transmission amplitude BP1 to all arrayports.

(18) A measure for the linear distortion of the antenna is the envelope width, which is defined as the full width at half maximum (FWHM) of the magnitude of the responses envelope

(19) is the impulse broadening when the input is a dirac distribution and is the minimum width over time of the signal incident in the measuring point. is defined as the time until the The duration of the ringing envelope has fallen from the peak value below a certain lower bound, e.g., a fraction of the main peak (20) The value of is taken from literature [12] as 0.22. An integral parameter describing the dispersive properties of of the real a system in the time domain is the delay spread valued transient response. The delay spread is calculated from the power related

with

(21) (22)

III. UWB SYSTEM: NUMERICAL RESULTS AND MEASUREMENT A. Design of Dummy Port Region The sidewall contours, connecting the beamport arc with the arrayport arc, have a triangular layout, see Fig. 4. These contours are terminated with dummyports, which absorb the incident power. This leads to certain restrictions for a good sidewall design. The dimensions of these dummyports are restricted by the same rules as the beam-/arrayports. To achieve an optimal angle of the dummyport to the incident wave it should point to the middle of the beam-/arrayport contour.

Fig. 8. Measured transmission phase BP1 to all arrayports.

B. Results The Rotman-lens, see Fig. 4 is realized on RT/duroid 6010 LM, with a permittivity of 10.6 and a thickness of 1.27 mm. The dimensions of the prototype are 19.5 cm 15.2 cm. In Fig. 5 the transmission from the outermost right beamport (BP1) to the corresponding arrayport (AP1) is shown. This is the worst case result for the transmission, because the other beamports BP2 – BP5 have a better performance. The beamports BP6-BP9 are performing the same way, since the structure is symmetrical. Numerical simulation with CST Microwave Studio and measurement with a vector network analyzer confirm the validity of the analytical model. The difference of 1 – 2 dB between the analytical result and the others is due to the missing of metal, dielectric and matching losses. In Fig. 6 the analytical model predicts correctly the lower frequency limit using (15), but it is a best case approximation, since the reflection does not decay completely due to internal reflections. Simulation and measurement coincide well in the whole range from 3.1 to 10.6 GHz. For the measurements only four out of seven beamports are used, because the validation system has only four antennas. The phases of the transmission are nearly perfectly linear, for the transmission phase from BP1 to the arrayports see Fig. 8 and for the transmission phase from BP5 to the arrayports see Fig. 13. The amplitudes have a low variance over the whole frequency range, for the transmission amplitude from BP1 to the arrayports see Fig. 7 and for the transmission amplitude from BP5 to the arrayports see Fig. 12. The realized prototype system consists of the Rotman-lens and a 4 1 Vivaldi antenna array (Fig. 9). The

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Fig. 12. Measured transmission amplitude BP5 to all arrayports.

Fig. 9. Prototype system with a 4

2 1 antenna array.

Fig. 13. Measured transmission phase BP5 to all arrayports.

Fig. 10. Measured frequency domain pattern of antenna array fed with BP1 of the Rotman-lens.

Fig. 14. Power distribution in Rotman Lens for feeding BP 5 from numerical data.

Fig. 11. Normalized time domain pattern of antenna array fed with BP1 of the Rotman-lens.

measurement results for the pattern in the frequency domain and in the time domain are shown in Fig. 10 and in Fig. 11. A detailed investigation of the power distribution inside the Rotman-lens shows how much power is transfered to the arrayports and the dummyports, how much is coupled to the other beamports and what amount is lost due to the loss mechanisms (conductivity, dielectric, higher modes, radiation). This investigation was done from 0 Hz 10.6 GHz. The values are achieved by comparing different numerical simulations. In Fig. 14 the power distribution is shown, when feeding BP5. The Fig. 15 gives the corresponding results when feeding BP1. Efficiency is degraded

Fig. 15. Power distribution in Rotman Lens for feeding BP 1 from numerical data.

for the outer beamport BP1 compared to BP5. From 7 GHz onward radiation takes place due to the occurrence of a higher order

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Fig. 16. Measured maximum amplitude of transient output.

Fig. 19. Measured ringing of transient output.

Fig. 20. Measured delay spread of transient output. Fig. 17. Measured standard deviation of group delay of transient output.

antennas themselves which is of the order of 0.1 ns – 0.5 ns, compare [12]. The measured values are even better than the simulated ones. This is due to the numerical accuracy chosen for reasonable simulation time. Thus the system will not add significantly to the impulse distortion, despite the attenuation. IV. CONCLUSION

Fig. 18. Measured FWHM of transient output.

mode. The radiation at lower frequencies (around 1 GHz) is due to the Rotman-lens reacting like a patch antenna. The coupling to the neighboring beamports reduces nearly completely with increasing frequency and is the main limiting factor for efficiency at lower frequencies. For BP1 the power transferred to the dummyports and the parasitic radiation at higher frequencies is significantly higher. The losses due to metal conductivity and substrate losses are nearly the same for both beamports. In Fig. 16, the maximum amplitude (18) at the arrayports for each beamport is given. The measured values are slightly less than the simulated ones, indicating higher losses than considered during simulation. The standard deviation of the group delay proofs the very good phase performance, see Fig. 17. The FWHM (19), the ringing (20), and the delay spread (21) of the output transients at the arrayports are shown in Figs. 18–20. These quantities show a very good performance, respectively to the ringing of the

A set of design equations for the development of a Rotman-lens prior to numerical optimization is given. The results confirmed the analytical design approach. The lens was implemented in an antenna array, with good pattern results. A detailed analysis of the transient impulse distorting effects gave promising results regarding an undistorted transmission of a transient voltage pulse through the lens. The concept of the Rotman lens is far from being lossles, especially for wide frequency ranges, but it comprises instantaneous bandwidth with parallel operability of the beams. The final system will use all seven beamports, thus improving the performance of the system further. The Rotman-lens is nontheless an attractive solution for transient beam-steering, e.g., in through-the-wall radars with Gaussian like transient shapes of the signal. Thus this beamformer is a premium choice for realizing multibeam antennas for the frequency range of 3.1 to 10.6 GHz. REFERENCES [1] W. Rotman and R. Turner, “Wide-angle microwave lens for line source applications,” IEEE Trans. Antennas Propag., vol. 11, no. 6, pp. 623–632, Nov. 1963. [2] Mathematical Theory of Optics, R. K. Luneburg, Ed. Cambridge, U.K.: Cambridge Univ. Press, 1965.

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[3] J. Blass, “Multidirectional antenna – A new approach to stacked beams,” in IRE Int. Convention Record, Mar. 1960, vol. 8, pp. 48–50. [4] J. Schoebel, J. Scheuer, R. Caspary, J. Schmitz, and M. Jung, “A true-time-delay phase shifter system for ultra-wideband applications,” presented at the German Microwave Conf., Hamburg, Germany, Mar. 10–12, 2008. [5] P. S. Hall and S. J. Vetterlein, “Review of radio frequency beamforming techniques for scanned and multiple beam antennas,” Inst. Elect. Eng. Proc. H, Microwaves Antennas Propag., vol. 137, no. 5, pp. 293–303, Oct. 1990. [6] J. Hall, H. Hansen, and D. Abbot, “Rotman lens for mm-wavelengths,” Proc. Smart Structures, Devices, and Systems, SPIE, vol. 4935, pp. 215–221, 2002. [7] J.-G. Lee, J.-H. Lee, and H.-S. Tae, “Design of a nonradiative dielectric Rotman lens in the millimeter wave frequency,” in IEEE MTT-S Int. Microwave Symp. Digest, 2001, vol. 1, pp. 551–554. [8] M. S. Smith and A. K. S. Fong, “Amplitude performance of Ruze and Rotman lenses,” Radio Electron. Eng., vol. 53, no. 9, pp. 329–336, Sep. 1983. [9] E. O. Rausch and A. F. Peterson, “Rotman lens design issues,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 3–8, 2005, vol. 2B, pp. 35–38. [10] V. K. Tripp, J. E. Tehan, and C. W. White, “Characterization of the dispersion of a Rotman lens,” in Proc. Antennas and Propagation Society Int. Symp., Jun. 26–30, 1989, vol. 2, pp. 667–670. [11] C. Baum and E. Farr, “Impulse radiating antennas,” in Proc. on Ultra-wideband, short-pulse electromagnetics; WRI Int. Conf., Brooklyn, NY, Oct. 8–10, 1992, pp. 139–147. [12] W. Soergel and W. Wiesbeck, “Influence of antennas on the ultrawideband transmission,” EURASIP J. Appl. Signal Processing, Special Issue UWB - State of the Art, pp. 296–305, Mar. 2005. [13] T. Katagi, S. Mano, and S. Sato, “An improved design method of Rotman lens antennas,” IEEE Trans. Antennas Propag., vol. 32, no. 5, pp. 524–527, May 1984. [14] P. S. Simon, “Analysis and synthesis of Rotman lenses,” presented at the 22nd AIAA Int. Communications Satellite Systems Conf., Monterey, CA, 2004, paper 2004-3196.

Andreas Lambrecht (S’09) received the Dipl.-Ing. (M.S.E.E.) degree from the Universität Karlsruhe (TH), Germany, in 2006, and the Dr.-Ing. (Ph.D.E.E.) degree from the Karlsruhe Institute of Technology (KIT), Germany, in 2010. From 2006 to 2010, he was a Research Assistant at the Institut für Hochfrequenztechnik und Elektronik (IHE), KIT. His research topics include ultrawideband arrays, true-time-delay beam steering techniques, HPEM systems, arrays for HPEM radiation, material measurements, microwave techniques. In 2010, he joined EADS, Ulm. Mr. Lambrecht received the 2006 EADS ARGUS Award for his diploma thesis.

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Stefan Beer received the Dipl.-Ing. degree from the Universitaet Karlsruhe (TH), Germany, in 2009. In 2009, he joined the Institut für Hochfrequenztechnik und Elektronik (IHE), as a Research Assistant and is currently working toward the Dr.-Ing. degree. His current research topics include the design of integrated millimeter-wave antennas, probe based antenna measurements and millimeter-wave packaging. He also has experience in designing ultrawideband antennas and in the analysis of ultrawideband truetime delay beamformers.

Thomas Zwick (S’95–M’00–SM’06) received the Dipl.-Ing. (M.S.E.E.) and the Dr.-Ing. (Ph.D.E.E.) degrees from the Universität Karlsruhe (TH), Germany, in 1994 and 1999, respectively. From 1994 to 2001, he was a Research Assistant at the Institut für Hochfrequenztechnik und Elektronik (IHE), TH. In February 2001, he joined the IBM T. J. Watson Research Center, Yorktown Heights, NY, as a Research Staff Member. From October 2004 to September 2007, he was with Siemens AG, Lindau, Germany. During this period he managed the RF development team for automotive radars. In October 2007, he was appointed Full Professor at the Universität Karlsruhe (TH), Germany, where he is the Director of the IHE. His research topics include wave propagation, stochastic channel modeling, channel measurement techniques, material measurements, microwave techniques, millimeter wave antenna design, wireless communication and radar system design. He participated as an expert in the European COST231 Evolution of Land Mobile Radio (Including Personal) Communications and COST259 Wireless Flexible Personalized Communications. He is the author or coauthor of over 80 technical papers and over 10 patents. Prof. Zwick received the Best Paper Award at the International Symposium on Spread Spectrum Techn. and Appl. ISSSTA 1998. In 2005, he received the Lewis Award for Outstanding Paper at the IEEE International Solid State Circuits Conference. Since 2008, he has been the President of the Institute for Microwaves and Antennas (IMA). He served as a lecturer for Wave Propagation for the Carl Cranz Series for Scientific Education.

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Frequency Agile Switched Beam Antenna Array System Javier R. De Luis, Student Member, IEEE, and Franco De Flaviis, Senior Member, IEEE

Abstract—A dual frequency reconfigurable antenna array and its associated phase shifter are presented. The system is capable of operation at two independent frequencies (4.7 GHz and 7.5 GHz) while switching between four different radiation patterns pointing at different spatial locations for each frequency. The reconfigurability is achieved by using high performance PIN diodes acting as microwave switches. Index Terms—Beam switched antenna, frequency agile antenna, multibeam antennas, reconfigurable antennas, smart antennas, switched system.

I. INTRODUCTION

I

NCREASING the angular resolution of antenna systems enhances the performance of the wireless mobile communication link. A phased array comprising of several elements and a control algorithm provides virtually unlimited control over azimuth and elevation scan angles at high speeds. This capability, however, comes with a high complexity and cost associated with a large number of elements, each having a digital phasing network with switching semiconductor or electromechanical elements. The high complexity and cost become more severe if a wideband (or multiple band) phased array is needed. Thus, as of today the use of phased array antennas is nearly limited to sophisticated military and space systems. However, many applications do not require a full scan capability involving the complexity of a phased array. In these cases, a simpler system, such as a reconfigurable switched beam antenna array, can be used. Reconfigurable antennas have received much attention in the past years playing an important role in the design of smart and adaptive systems [1]. Recent improvements in the performance of switching technologies such as RFMEMS and solid state switches, integrated in antennas and microwave circuits, have proved to be useful for a wide range of applications [2]–[10], including switched beam arrays. 3D structures, based on circular array configurations having a single active antenna surrounded by several loaded parasitics have been proposed to provide endfire beam switching for single [11] and dual frequency applications [12]. A multi-layered three beam system using triangular

Manuscript received November 23, 2009; revised February 17, 2010; accepted March 31, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by the Balsells fellowship program and under the California-Catalonia Innovation Program 2009. The authors are with the Electrical Engineering and Computer Science department, University of California at Irvine, Irvine, CA 92697 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055813

Fig. 1. (a) Omnidirectional antenna system with single frequency operation showing interference between aircrafts. (b) Switched beam dual frequency antenna system proposed in this paper.

microstrip antennas has been presented in [13]. Other multibeam approaches using Butler matrix [14] or phase switched solutions [15] can be found in the literature. However, the systems in [13]–[15] are limited to operation at a single frequency band. In this paper, a novel dual frequency reconfigurable microstrip antenna array for next generation telemetry application with beam switching capability at 4.7 GHz and 7.5 GHz is presented. In contrast to [12], only one of the two operating bands remains active at each time, rejecting the non-active frequency without the need of an external diplexer. Furthermore, the system is a true planar implementation printed in a two layer PCB, reducing the complexity, space and total cost of the overall system. The dual frequency switching capability will provide a better usage of the spectrum resources and will also allow several nearby systems to communicate at high speed with the ground station simultaneously. A system comprising of omnidirectional antennas and single frequency operation as shown in Fig. 1(a) may create harmful interferences between aircraft. However, a system deployed as shown in Fig. 1(b), with two different frequencies and switched beam capability, minimizes the jamming risk and enables a fully functional system. Based on this concept, the system proposed in this paper is composed of a dual frequency (DF) reconfigurable antenna and a single

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2

Fig. 2. Complete system diagram formed by a 4 1 dual frequency phased array and two different types of switched line phase shifters (Type 1, Type 2).

feed dual frequency reconfigurable switched line phase shifter. Two different features, frequency and beam maximum position, can be selected by introducing PIN diodes in the system. Progressive phases between antennas of can be chosen to generate a switched beam with beam maxima and degrees pointing to at 4.7 GHz and 7.5 GHz respectively. The proposed design is shown in Fig. 2 and can be divided in two blocks: a DF reconfigurable microstrip antenna and DF switched line phase shifter explained in Sections II and III, respectively. Section IV presents system simulations and measurement results. The input and output of each block is matched to a system impedance of 50 . The modular design approach of each sub-block allows for a simple design process flow that can be reproduced for any desired frequency ratio. II. DESIGN OF THE SINGLE ANTENNA ELEMENT The antenna element used in the 4 1 linear array of Fig. 2 is shown in detail in Fig. 3. The design consists of a printed patch , over a 1.575 mm thick Duroid RT-5880 substrate with at and 34 copper thickness, backed with a metallic ground plane. A small inner rectangular patch (IP) is partially surrounded by a larger U-shaped outer patch (OP). Both patches are connected through three high frequency GaAs PIN diodes (model MA4AGCFCP910, vendor M/A-COM) acting as microwave switches with equivalent circuit model shown in Fig. 4. The element values of the equivalent lumped circuit model were found in order to match a single diode measurement mounted in a microstrip line test fixture over the frequency range from 4 GHz to 8 GHz. The device was characterized and the test fixture was de-embedded from the measurement, using a TRL calibration procedure. The antenna is fed by a microstrip line connected to the middle point of the IP left edge. In the fundamental mode, the left and right edges of the antenna are responsible for the radiation phenomena and the RF current resonates along the -axis (Fig. 3). The diodes are placed on the right edge of the inner patch to allow the currents to flow to the OP when required.

Fig. 3. The antenna element consists of a dual microstrip patch topology con: ,  nected through PIN diodes, printed over RT duroid 5880 ("r : , 35  copper thickness, 1.575 mm dielectric thickness) with dimen: : , , sions: : , : : , , , , : ,g , : . Diode and : : : .

= 2 2 tan = 0 0009 m Lhf = 11 1 mm Whf = 12 15 mm Llf = 19 5 mm Wlf = 21 mm = 0 5 mm gd = 0 3 mm iip = 2 mm iop = 0 34 mm Wdc = 0 2 mm length = 0 7 mm width = 0 3 mm pad gap = 0 3 mm

Rs = 1 52

Fig. 4. Diode circuit model for the “On” and “Off” states. : , : and , . Maximum insertion loss @4–8 GHz 0.4 dB. Minimum isolation @4–8 GHz 11 dB.

Ls = 0 25 nH

Co = 47 fF Rp = 10 K

The patches are separated by a distance of 0.5 mm, which is sufficient to leave space for the diode placement and to provide physical isolation. When the diodes are in the “Off” state (or reverse bias), the RF signal flows mainly in the inner patch while the outer patch acts as a parasitic element and the antenna operates in the high frequency mode (7.5 GHz). Similarly, when the diodes are switched to the “On” state (forward bias), the current flows in both patches resulting in an increase of the effective area of the antenna and the system operates in the low frequency mode (4.7 GHz). The sizes of both patches have been optimized simultaneously for resonance at both frequency bands, as explained in Sections II-A–C. The DC current path returning to ground necessary for the diode biasing, is designed using a high impedance 4.7 GHz quarter wave microstrip line ending in a pad connected to the antenna ground plane. The short circuit at the pad is transformed into an open circuit seen from the OP edge, making the ground path section transparent to the RF signal. It was observed that the effect of the quarterwave biasing line on the 7.5 GHz resonance, when the diodes are in the “Off” state, was negligible due to the small RF current flowing in the OP as shown in Fig. 5. When the diodes are in the “On” state, the effect of the biasing line on the radiation pattern at 4.7 GHz produced a decrease of 0.4 dB on the antenna gain.

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Fig. 6. Resonant frequency change with increasing gap between IP and OP for cases: (a) Ideal “On”: Metal connection between patches (low frequency), (b) Ideal “Off”: No connection between patches (high frequency), (c) Real “On”: Diodes between patches in “On” state (low frequency), (d) Real “Off”: Diodes between patches in “Off” state (high frequency). Fig. 5. Scalar current distribution over the antenna at 4.7 GHz for the cases: (a) Ideal “On” with metallic connections, (b) Ideal “Off” with no connections, (c) Real “On” with diode connections and (d) Real “Off” with diode connections. Center of figure, standard rectangular patch resonating at 4.7 GHz.

Due to the different topology for both operation states, the edge resonance resistance seen from the feeding point has different values depending on the operative frequency band. Therefore, a dual frequency matching network based on a dual stub configuration was designed. A. Gap Size Effect on the Resonant Frequencies As a first step in the design of any dual-frequency dual-patch antenna, it is important to understand the effects of the gap size between the IP and OP on the high and low resonant frequencies. In order to analyze this effect, two stand alone rectangular patches were designed to resonate at 4.7 GHz ( at 4.7 GHz, ) and 7.5 GHz ( at 7.5 GHz, ) respectively according to [16]–[18]. Both independent patches were then combined together into the same space leaving a 0.5 mm gap between them as shown in Fig. 5(a)–(d). Depending on the interconnection and operating state, four different cases of the current distribution in the antenna were plotted. The antenna current distribution gives a qualitative idea of the level of IP/OP isolation in the diode “Off” state and the current ability to flow from IP to OP in the “On” state. Fig. 5 shows the low frequency operation mode with three metallic connections (ideal “On” state) and their absence (ideal “Off” state), respectively. On the other hand, Fig. 5 shows the high frequency operation using the forward biased diode S-parameters (real “On” state) and reverse biased diode parameters (real “Off” state) respectively. It is observed that the current distributions of the antenna at low frequency for the “ideal” and “real” “On” states are very similar to the standard reference patch in the center of the figure. Fig. 5(b) gives an idea on how much current is induced in the OP only due to the proximity coupling effect when both patches are physically isolated. In addition to this coupling, Fig. 5(d) produces

an additional current leakage near the diode connections due to the imperfect isolation of the diode in “Off” state. Intuitively, it is expected that the biggest deviation in frequency value from the stand alone patch case would be produced for the case of Fig. 5(d) because the IP current distribution presents more differences with respect to the stand alone patch. To verify this statement and to study the gap size effect on the antenna resonant frequency, a numerical experiment using full wave analysis was performed. The gap size was increased from 0.05 mm to 2 mm in 0.05 mm steps, while the associated frequency change was monitored as shown in Fig. 6. The difference, between “ideal” (a) and “real” (c) “On” states curves for low frequency is clearly less significant than the difference between (b) and (d) “Off” state (high frequency) cases, which matches with the current distribution comparison. The average low frequency in Fig. 6 is 4.6 GHz which is only 1% lower than the stand alone patch case. This decrease is due to the slightly longer current path forced by the connections between both patches. When the gap distance gets larger, curve (b) shows no difference in frequency with respect to the 7.5 GHz stand alone patch due to the negligible proximity coupling between IP and OP. However, this effect is not seen on curve (d) due to existing signal leakage produced by the diode in the “Off” state. Similarly, when the gap distance is small, both curves (b) and (d) show a decrease in the resonant frequency produced by the effective increase on antenna size created by the IP/OP mutual coupling. B. Optimum Number of Diodes Analysis In this study we found that the number of diodes integrated in the antenna affects the radiation efficiency and system losses. In addition, a higher number of devices will require more DC power consumption and will have higher noise generation. Therefore, a minimum number of diodes for good operation at both frequencies must be found. One to five diodes were integrated in the antenna as shown in Fig. 7 and considered in the following analysis. Fig. 8 shows the behavior of the imaginary

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Fig. 7. Scalar current distribution over antenna different number of diodes in “On” state: 1 diode (a) first resonance, (b) second resonance; 2 diodes (c) first resonance, (d) second resonance; 3 diodes (e) first resonance, (f) second resonance and 5 diodes (j) first resonance, (k) second resonance.

Fig. 8. Imaginary part of the antenna edge impedance versus frequency showing first and second resonances for the cases of (a) 1 diode “On,” (b) 3 diodes “On,” (c) 3 diodes “On.”

part of the input impedance with frequency when one, two and three diodes are switched “On” forcing the antenna to operate at low frequency. The fundamental resonant modes for these are observed at 4.48 GHz, 4.59 GHz and 4.6 GHz respectively. When a smaller number of diodes is used, frequency is lowered due to the increased current path enforced by the topology. Even more important is the higher frequency resonances observed at 8.5 GHz, 8.73 GHz and 9.2 GHz for the one, two and three diode cases respectively. In order to study all the mentioned resonances, the average current distribution over the patch operating in “On” state is shown in Fig. 7. The low frequency fundamental modes and higher frequency resonances are shown in the top and bottom rows, respectively. An additional case of 5 diodes is also introduced (Fig. 7(j)–(k)) to show that further increase in diode number does not create a significant difference in the patch current distribution. This is because the two diodes placed horizontally in the non radiating edges do not contribute to the vertical current flow in the patch. The fundamental mode current flow and the value of the associated resonant frequency are clearly closer to the stand alone patch case of Fig. 5 when three diodes are used.

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Fig. 9. Normalized radiation pattern (dB) corresponding to the second resonance of cases (a) 1 diode “On,” (b) 3 diodes “On,” (c) 3 diodes “On.”

The radiation patterns corresponding to the higher frequency resonances for all considered cases are shown in Fig. 9. The pattern for the second resonance (at 8.5 GHz), corresponding to 1-diode case has a radiation behavior typical of a fundamental mode. This is due to the fact that the current is very confined to the diode region when crossing to the OP, thus the IP remains resonant resulting in a dual frequency antenna (instead of a switchable one). One can actually perceive that behavior on the current plot of Fig. 7(b). However, when three diodes are used, the second resonance appears considerably higher at ) and the radiation pattern shows a null 9.2 GHz (around in the normal direction typical of a higher order mode of the radiating OP (instead of a fundamental one). In this case, the IP is not contributing to radiation and switching becomes effective. The two diode pattern case of Fig. 9(b) is in between the two previous cases. If only one diode was chosen for the design, its 8.5 GHz resonance could create an undesirable spurious radiating mode close to the target high frequency of 7.5 GHz. This could potentially harm the receiver if filtering circuitry is not implemented. Therefore, three diodes will be used for the design to avoid requiring an external diplexer while guaranteeing a switchable behavior with good operation at 4.7 GHz. C. Radiation Efficiency Analysis Each diode can be considered like a passive device that introduces losses into the system. The loss sources are mainly ohmic loss (insertion loss) due to the device itself and mismatch loss due to the diode placement within the antenna. The transducer gain expression [19] takes into account both loss sources and can be expressed as (1) and in this cases, refer the power deWhere is the livered to the outer patch and inner patch respectively. reflection coefficient looking towards the IP from the diode posis the reflection coefficient looking to the positive terminal. itive terminal of the diode from the IP. is the reflection coefficient looking to the OP from the diode negative terminal, is the reflection coefficient looking towards the IP from the negrepresents the transmission ative terminal of the diode and coefficient in 50 environment. Dealing with passive devices, the maximum achievable transducer gain is 0 dB. Assuming

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Fig. 10. Radiation efficiency as a function of IP and OP slit lengths. The dynamic range between worst and best results is found to be 35%.

ohmic losses, the optimum transducer gain for a specific device can be achieved satisfying simultaneous matching conditions and , which minimize (SMC) imposed by the mismatch loss. This is an important condition as long and it can be related to the radiation efficiency of the antenna. The input impedance seen at the edge of the IP and OP where the terminals of the diode are connected can be considerably different. In order to find impedances that satisfy the SMC condition, slits are introduced in the IP and OP sections as shown in Fig. 3. The length of both slits is changed independently from 0 mm (no slit) to 5.3 mm (half of the IP length). The associated radiation efficiency of the antenna is evaluated using Full Wave (Zeland IE3D MoM base software) simulations for each different pair of slit length for a total of 2500 combinations. The radiation efficiency versus slit length information is then extracted from the simulation data and plotted in Fig. 10. Efficiency varies from 40% to 74% between the two extreme cases. The efficiency improvement due to the proper slit length has an associated gain increase from 5.88 dB to 6.48 dB. This result demonstrates the importance of finding SMC for any diode placed within the antenna. In a second study, the slits for the center and corner diodes were optimized independently, but no improvement above the mentioned 74% efficiency was observed by using this approach. III. THE DUAL FREQUENCY PHASE SHIFTER A microwave phase shifter is a two port network that provides a specific phase delay to a signal travelling from its input to output ports at a given frequency. Different design approaches and topologies are widely found in the literature such us loaded line phase shifters [20], quadrature reflection phase shifters [21] or Schiffman phase shifter [22]. However, when identical phase delays are required at two different frequencies, a solution based on a non-tunable passive circuit is not available or its complexity increases [23] allowing only a very specific combination of phase delays. In general, switchable solutions are required for dual frequency operation, but the complexity and fabrication challenges may play an important role as in some reported solutions [24]. In this paper, a novel reconfigurable switched

line phase shifter using commercial PIN diodes is presented following a simple design procedure. In a switched line phase shifter, the RF input signal can travel between different transmission paths before arriving to the output resulting in different phase delays. In this system, four paths or branches can be selected as shown in Fig. 11. The phase shifters (PS) for antennas #1 and #4 (Type 1 PS) shown in Fig. 2 are identical, and can provide a switchable phase delay degrees at 4.7 GHz and 7.5 GHz simultaneof ously. Similarly, the PS connected to antennas #2 and #3 (Type degrees at both 2 PS) can provide a delay of frequencies. The different combinations of states in PS #1 to #4 shown in Table I can produce four different progressive phases , that have between antennas of associated radiation beam maxima at degrees for 4.7 GHz and degrees for 7.5 GHz. The design consists of four passive sub-circuits that are designed separately and cascaded to form the complete system. Each of these will be explained in Sections IV and V: the input/output matching networks, the SP4T distribution blocks, the dual frequency phase delay section and the dual frequency DC bias network. The design shown in Fig. 11 is fully planar using microstrip transmission lines printed on a , at RT Duroid 6006 substrate ( , 35 copper thickness and 0.635 mm board thickness). The same PIN diode model used for the antenna element was also used for the phase shifter design. In reality, due to the non perfect isolation of the diode, the non active “Off” branches always contribute with some degree of shunt parasitic reactance, deteriorating return loss conditions in the input and output ports of the phase shifter. Therefore, dual frequency input and output matching networks based on stepped impedance transmission lines are connected to the input and output SP4T ports (Fig. 12), to provide good reflection coeffifor all eight possible combicient characteristics nations (four phases and two frequencies). Both, matching networks and distribution sections must be identical for both types of phase shifter to avoid any phase perturbation. A. The Dual Frequency Phase Delay Section The DF phase delay section of Fig. 13 is the fundamental building block of the phase shifter. It consists of a transmission line loaded with two shunt opened stubs that are able to provide the same phase delay at two different frequencies simultaneously. In [25], a dual frequency delay network is presented and the closed form expressions are obtained for the particular case of 90 degrees phase delay. However, phases different from 90 degrees make the mathematical approach unfeasible so an optimization tool is required. The length (Lstub), width (Wstub) and separation (L_line) between stubs are set as optimization variables in a microwave circuit simulator (Applied Wave Research, AWR) to achieve each specific phase delay goal. IE3D was used afterwards to perform full wave analysis and for fine tuning of the phase shifter purposes. The simulation result for the case of zero degree phase delay at 4.7 GHz and 7.5 GHz with magnitude at both frequencies of the reflection coefficient below is shown in Fig. 14. The same approach is used for the rest of the targeted phases with similar results.

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Fig. 11. Complete type 1 phase shifter with phase delays from top to bottom branch of f0;45;90;135g degrees. TABLE I PHASE DELAYS FOR EACH ANTENNA PHASE SHIFTER

Fig. 13. Dual frequency phase delay section with dual stub configuration and showing optimization variables.

Fig. 12. Zoom into the phase shifter input region that contains the DF matching network, input SP4T distribution section and equal length launches. The output SP4T distribution section is also included in the figure to show the direction of the diodes.

B. The Dual Frequency DC Bias Biasing current is required to control the diode states for switching the circuit “On” and “Off.” Surface mount components may be used for this purpose. However, when operating at high frequencies, catalog rated values may change and issues may arise with component SRF. In these cases a printed approach is preferred such as the DC Bias circuit shown in Fig. 15, which does not require surface mount components. Two quarter wavelength radial open stubs are connected in shunt to a microstrip line. Each of them transforms the open

Fig. 14. Simulation results from a zero degree phase delay section. Input/Output magnitude of the reflection coefficient in dB (left axis) and Input/Output phase delay in degrees (right axis). Both frequencies remain matched while providing the desired zero degrees phase delays.

circuit in point A into a short circuit seen from point B for each frequency. The network after point B is a dual frequency quarterwave transformer that transforms the short circuit in B’ to open circuit in C for 4.7 GHz and 7.5 GHz. Thus, the network seen from point C becomes transparent to the RF signal flow. Point D is used as a DC-current input pad that allows the diodes to be biased in the circuit. Anything surrounding the DC pad should not affect RF performance if both radial stubs are properly designed. Each diode requires a current of 10 mA in forward mode (“On”), while zero volts are used in reverse mode (“Off”). If the RF signal needs to travel through the upper

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Fig. 15. Dual frequency DC bias formed by two quarter wavelength stubs, a dual frequency quarterwave section and a DC input pad to bias the diodes. Fig. 17. Simulated versus measured antenna magnitude of the reflection coefficient. For simplicity, the cases “On” and “Off” corresponding to operation at 4.7 GHz and 7.5 GHz are superimposed in the same graph. TABLE II SIMULATED AND MEASURED ANTENNA CENTER FREQUENCIES

Fig. 16. Fabricated single element antenna and zoom into the diode region. The devices are attached using silver epoxy to avoid overheating.

branch of Fig. 11, 0 V are applied to the pads DC#1,3,4,5,6 while a voltage that provides 10 mA would be applied to DC pad #2. IV. SIMULATIONS AND MEASUREMENT RESULTS The single element antenna, with optimized slit lengths and patch sizes was fabricated as shown in Fig. 16. The three PIN diodes were placed by hand using conductive epoxy to avoid overheating the device. For testing purposes, the bias current was introduced through an external coaxial bias T. However, a printed dual frequency bias could be designed for the same purpose. The magnitude of the reflection coefficient comparison between simulated (IE3D) and measured results is in very good agreement as shown in Fig. 17, where both frequency bands present good . For convenience,the “On” and “Off” cases matching are superimposed in the same graph. Table II shows the target versus simulated and measured frequencies. In all cases, return loss below 10 dB was achieved for the target frequency. The measured radiation patterns in both principal planes for the single antenna element at 4.7 GHz and 7.5 GHz are shown in Fig. 18(a)–(d), respectively. Peak gains of 5 dB and 5.4 dB are obtained for 4.7 GHz and 7.5 GHz respectively. Efficiency values of 68% for the low band and 79% for the high band were obtained. Four identical antenna elements were used to form the 4 1 linear array. Deciding the distance between elements in a dual frequency array is not a simple task and requires a tradeoff analysis of array gain and side lobe level (e.g., in order to obtain such gain at 4.7 GHz, an increase in SLL must be accepted at

Fig. 18. Measured radiation patterns for the single element antenna operating in the (a),(b) “On” and (c),(d) “Off” cases. (a) and (c) show the E -plane while (b) and (d) show the H -plane. Solid and dashed line represents co-pol and cross-pol components respectively. The measured efficiency is 79% and 68% with associated gains of 5.4 dB and 5 dB at 7.5 GHz and 4.7 GHz, respectively.

7.5 GHz). For this specific application a high system gain greater than 10 dB was required. Therefore a separation distance between antennas of 30 mm (0.47 at 4.7 GHz, 0.75 at 7.5 GHz) was chosen. With this distance, the simulated array gains are greater than 10 dB at both frequencies, despite of an increased side lobe level at 7.5 GHz. In addition, the distance avoids input

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TABLE III SIMULATED AND MEASURED PHASE DELAYS AT 4.7 GHz

TABLE IV SIMULATED AND MEASURED PHASE DELAYS AT 7.5 GHz

Fig. 19. Fabricated 4

2 1 switched beam antenna array and phase shifters.

impedance distortion due to mutual coupling, keeping good return loss conditions in the antennas. Phase shifter Types 1 and 2 were fabricated and arranged resembling Fig. 2, and all the possible phase combinations were measured to evaluate their performance. Tables III and IV show the simulated and measured progressive phase values between antennas compared to the ideal target values for 4.7 GHz and 7.5 GHz respectively. All measured values remain close to the simulations and target phases with an average phase error with respect to target values of 4.43 degrees at 4.7 GHz and 4.63 degrees at 7.5 GHz. Magnitude of the reflection coefficient conwere measured for the input and output ditions below phase shifter ports when all eight paths were activated at both frequencies. Average insertion loss of 2.8 dB including connectors and diode losses was measured. The final system is shown in Fig. 19. The four input ports of the phase shifters shown in Fig. 19 were connected to a 4-way SMA power divider to converge to a single input port. The DC control lines for the PIN diodes were externally controlled by a

Fig. 20. Simulated with IE3D infinite ground (solid) and measured (dashed) normalized E-total radiation patterns for 4.7 GHz (lf) and 7.5 GHz (hf) corresponding to different progressive phases between antennas (a) lf,  : (b) lf,  , : (c) lf, , : ; (e)  (d) lf,  : hf,  : (f) hf,  : (g) hf,  : (h) hf,  : .

= 0135sim-gain = 10 8dB = +135 sim-gain = 10 5 dB = 0135sim-gain = 14 8dB = +135 sim-gain = 12 7 dB

= 045 sim-gain = 12 5dB = +45 sim-gain = 13 7 dB = 045sim-gain = 14 7dB = +45 sim-gain = 15 2 dB

Labjack U3 device [26] connected through a USB interface to a PC, were custom software was designed. The system radiation patterns were measured in the UCI far field anechoic chamber. The simulated (solid line) and measured (dashed line) normalized radiation patterns for the array system with each one of the progressive phases between antennas provided by the phase shifter is shown in Fig. 20. A total of four switchable beams at each frequency can be obtained by using the proposed phase shifter, with the measured patterns resulting reasonably close to the simulated ones. The simulated patterns were obtained using Zeland IE3D

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with infinite ground plane, which explains the absence of the back radiation that is present in measurements. On the other hand, in order to simulate all different patterns the antenna array without phase shifter was fed directly with the theoretical phases. Therefore, the simulated absolute peak gain is on average 3.1 dB higher than the measured one, which corresponds to the phase shifter and inter-stage connector average losses. V. CONCLUSION In this paper, a dual frequency reconfigurable switched beam antenna array with phase shifter using PIN diodes for telemetry applications was presented. A dual patch approach was used to design the single antenna element, while a switched line topology was chosen for the phase shifter. The system is capable of switching the beam to four selectable space positions at two different frequencies with 1.6:1 ratio. ACKNOWLEDGMENT The authors would like to acknowledge the support given by Zeland Software (IE3D), Applied Wave Research (AWR), and Rogers Corporation. They would also like to express their gratitude for the kindly patronage and guidance given by the California-Catalonia Innovation program and Balsells program, and to Prof. R. H. Rangel and Prof. N. G. Alexopoulos for their continuous support and valuable input. REFERENCES [1] C. A. Balanis and P. I. Ioannides, Introduction to Smart Antennas. Seattle, WA: Morgan and Claypool, 2007, ch. Synthesis Lectures on Antennas #5. [2] S. Liu, M. Lee, C. W. Jung, G. P. Li, and F. De Flaviis, “A frequency-reconfigurable circularly polarized patch antenna by integrating MEMS switches,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2005, pp. 413–416. [3] J. R. De Luis and F. De Flaviis, “A reconfigurable dual frequency switched beam antenna array and phase shifter using PIN diodes,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2009, pp. 1–4. [4] C. W. Jung and F. De Flaviis, “Reconfigurable multi-beam spiral antenna with RF-MEMS capacitive series switches fabricated on rigid substrates,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2005, pp. 421–424. [5] S. V. Shynu, C. K. Gijo Augustin, P. Aanandan, P. Mohanan, and K. Vasudevan, “A reconfigurable dual-frequency slot-loaded microstrip antenna controlled by PIN diodes,” Microw. Opt. Technol. Lett., vol. 44, no. 4, pp. 374–376, Feb. 2005. [6] A. Daryoush, K. Bontzos, and P. Herczfeld, “Optically tuned patch antenna for phased array applications,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 1986, pp. 361–364. [7] F. Yang and Y. Rahmat-Samii, “Patch antenna with switchable slot (PASS): Dual frequency operation,” Microw. Opt. Technol. Lett., vol. 31, no. 3, pp. 165–168, Sep. 2001. [8] F. Yang and Y. Rahmat-Samii, “Switchable dual-band circularly polarized patch antenna with single feed,” Electron. Lett., vol. 37, no. 16, pp. 1002–1003, Aug. 2001. [9] J. Costantine, C. G. Christodoulou, C. T. Abdallah, and S. E. Barbin, “Optimization and complexity reduction of switch-reconfigured antennas using graph models,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1072–1075, Sep. 2009. [10] L. M. Feldner, C. D. Nordquist, and C. G. Christodoulou, “RF MEMS reconfigurable triangular patch antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2005, pp. 388–391. [11] M. D. Migliore, D. Pinchera, and F. Schettino, “A simple and robust adaptive parasitic antenna,” IEEE Trans. Antennas Propag., vol. 53, pp. 3262–3272, Oct. 2005. [12] R. W. Schlub, “Practical realization of switched and adaptive parasitic monopole radiating structures,” Ph.D. dissertation, Griffith University, Brisbane, Queensland, Australia, 2004.

[13] Basari, M. F. Purnomo, K. Saito, M. Takahashi, and K. Ito, “Simple switched-beam array antenna system for mobile satellite communications,” IEICE Trans. Commun., vol. E92-B, no. 12, pp. 3861–3868, Dec. 2009. [14] C. H. Tseng, C. J. Chen, and T. H. Chu, “A low-cost 60-GHz switchedbeam patch antenna,” IEEE Antennas Wireless Propag. Lette., vol. 7, pp. 432–435, Dec. 2008. [15] M. Barba, J. E. Page, and J. A. Encinar, “Planar C-band antenna with electronically controllable switched beams,” Int. J. Antennas Propag., vol. 2009, pp. 1–7, Oct. 2008. [16] D. R. Jackson and N. G. Alexopoulos, “Simple approximate formulas for input resistance, bandwidth, and efficiency of a resonant rectangular patch,” IEEE Trans. Antennas Propag., vol. 39, pp. 407–410, Mar. 1991. [17] R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip Antenna Design Handbook. Norwood, MA: Artech House, 2001, pp. 265–269. [18] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York: Wiley, 2005, pp. 819–820. [19] G. Gonzalez, Microwave Transistor Amplifiers Analysis and Design, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1997, pp. 183–184. [20] H. A. Atwater, “Circuit design of the loaded-line phase shifter,” IEEE Trans. Microw. Theory Tech., vol. 33, pp. 626–634, Jul. 1985. [21] D. M. Klymyshyn, S. Kumar, and A. Mohammadi, “Linear reflection phase shifter with optimised varactor gamma,” Electron. Lett., vol. 33, no. 12, pp. 1054–1055, Jun. 1997. [22] B. M. Schiffman, “A new class of broad-band microwave 90-degree phase shifters,” IRE Trans. Microw. Theory Tech., vol. 6, no. 2, pp. 232–237, Apr. 1958. [23] C. Collado, A. Grau, and F. De Flaviis, “Dual-band butler matrix for WLAN systems,” in Proc. IEEE Microwave Theory and Techniques society Int. Symp., 2005, pp. 2247–2250. [24] K. Tang, Y. Wu, Q. Wu, H. Wang, H. Zhu, and L. Li, “A novel dualfrequency RF MEMS phase shifter,” in Proc. Asia-Pacific Symp. on Electromagnetic Compatibility APEMC, 2008, pp. 750–753. [25] K. K. M. Cheng and F. L. Wong, “A novel approach to the design and implementation of a dual-band compact planar 90 branch-line coupler,” IEEE Trans. Microw. Theory Tech., vol. 52, pp. 2458–2463, Nov. 2004. [26] Labjack: Measurements and Automation Simplified 2009 [Online]. Available: http://labjack.com/

Javier R. De Luis (S’08) was born in Elche, Spain, in 1982. He received the Telecommunication Engineer degree from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain and the M.S. degree from the University of California at Irvine, in 2008, where he is currently working toward the Ph.D. degree. His current research is focused on reconfigurable antennas for tunable RF handset front end and smart systems.

Franco De Flaviis (SM’08) was born in Teramo, Italy, in 1963. He received the Laurea degree in electronics engineering from the University of Ancona (Italy) in 1990 and the M.S. and Ph.D. degrees in electrical engineering from the University of California at Los Angeles (UCLA), in 1994 and 1997, respectively. In 1991, he was an Engineer at Alcatel, as a researcher specialized in the area of microwave mixer design. In 1992, he was Visiting Researcher at UCLA, working on low intermodulation mixers. He is currently a Professor with the Department of Electrical Engineering and Computer Science, University of California, Irvine. He has authored and coauthored over 100 papers in refereed journals and conference proceedings, filed several international patents and authored one book and three book chapters. His research interests include the development of microelectromechanical systems (MEMS) for RF applications fabricated on unconventional substrates, such as printed circuit board and microwave laminates with particular emphasis on reconfigurable antenna systems. He is also active in the research field of highly integrated packaging for RF and wireless applications. Prof. Flaviis is a member of the URSI Commission B.

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A Measurement System for the Complex Far-Field of Physically Large Antenna Arrays Under Noisy Conditions Utilizing the Equivalent Electric Current Method Tore Lindgren, Member, IEEE, Jonas Ekman, Member, IEEE, and Staffan Backén, Student Member, IEEE

Abstract—Precipitation in the form of snow or rain could severely degrade the performance of large antenna arrays, in particular if knowledge about the beam shape and pointing direction in absolute numbers is necessary. In this paper, a method of estimating the far-field of each individual antenna element using the equivalent electric current approach is presented. Both a least squares estimator and a Kalman filter was used to solve the resulting system of equation and their performance was compared. Simulation results shows that the estimated far-field for one antenna element is very accurate if there is no noise on the signal. During noisier conditions the Kalman filter gives less noisy results while the systematic errors are slightly larger compared to the least squares estimator. Index Terms—Antenna arrays, antenna measurements.

I. INTRODUCTION

ARGE antenna arrays operating in an arctic environment may have their performance significantly degraded as the properties of the individual antenna elements change due to weather effects, in particular snowfall [1]. This could cause severe problems in applications where knowledge about the gain and pointing direction of the main beam of the antenna array is needed in absolute numbers. Good knowledge about each antenna element’s active radiation pattern can significantly improve the performance of the beamforming process. In this paper, the method of calculating equivalent electric current in order to estimate the far-field of the antennas [2], [3] is used and adapted to the situation when the antenna array consists of relatively simple antenna elements (e.g., dipoles or yagi-antennas) but where the whole antenna array is physically and electrically large. The antenna array considered here is the planned EISCAT 3D incoherent scatter radar [4] which is an upgrade of the existing tristatic EISCAT UHF radar in northern Scandinavia. The radar system will consist of one main transmit/receive site and multiple receive-only sites at 90–280 km from the main site. Each

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Manuscript received March 13, 2009; revised February 10, 2010; accepted April 12, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The authors are with the Department of Computer Science and Electrical Engineering, Luleå University of Technology, Luleå 97187, Sweden (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055780

site will have an antenna array consisting of up to 16000 antenna elements operating in the 210–240 MHz frequency band. Since the ESCAT 3D radar will be a multistatic radar with a very narrow beam, the pointing direction of the beam must be known with good accuracy. The maximum allowed timing error between any two antenna elements in the array has been found to be 160 ps [5]. The timing error is composed of jitter in the local oscillator and in the analog-to-digital converter (ADC), errors introduced in the clock distribution system, and changes in the phase of the far-field pattern of the antenna elements [6]. Since the radar will operate continuously the far-field of the antenna elements will need to be measured regularly. In particular, the measurement system must be able to detect any changes in the phase characteristics. The following limiting factors of the measurement system have been identified due to the specific application of ionospheric radar and the physical and electrical size of the system as follows. • The signals of interest to the users of the system are very weak [7]. The probes used for the measurements should therefore be located outside the field-of-view of the antenna array, as much as possible, in order to minimize interference and diffraction effects that could degrade the performance of the radar. • The beamforming of the antenna array will be digital with each antenna element sampled individually. With up to 16000 antenna elements it is essential to keep the calculations to a minimum. Thus, the number of probes used for the measurements should be minimized. • It is not practical to place the probes in the far-field of the antenna array due to the physical size of the antenna. Hence, the calibration system must be able to accurately calculate the phase of the far-field with probes located at varying distances from the antenna elements. As a result of the limiting factors mentioned above some of the conventional measurement methods may not be suitable for the type of antenna system considered in this paper. The far-field pattern of large aperture antennas is traditionally measured using radio sources such as quasars or distant galaxies [8]. With this approach it is possible to get accurate gain and phase characteristics of the antenna. The typical signal strength is, however, too low to be detected by the individual antenna elements in an array. A similar technique, described in [9], uses spacecrafts as receivers for signals transmitted by the Antenna Under Test (AUT). To apply this to the EISCAT 3D radar, the

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spacecraft would need to act as the transmitter since not all sites will have transmitting capabilities. A major drawback of such a calibration system is that it is unlikely that a spacecraft can be dedicated to the radar and the calibration of the radar would therefore rely on signals transmitted for other purposes (signals-of-opportunity). Although the measurement methods discussed above are not suitable as a sole calibration system they could be a valuable complement to a measurement system located in the near-field of the antenna array. The far-field of an antenna can also be estimated using measurements in the near-field. This typically requires more calculations than the far-field techniques. A comprehensive overview of near-field measurement techniques can be found in [10]. The methods described assume that all measured points lie on the same surface (planar, cylindrical, or spherical) within a fraction of a wavelength. This may not be possible to achieve in the EISCAT 3D radar due to the size of the system. The method described in this paper is instead based on the equivalent current approach where the antenna is replaced by an equivalent electric and/or magnetic current, which is described in [2]. A clear advantage is that the equivalent current approach is less sensitive to non-ideal probe positions than traditional near-field measurement methods. It was shown in [11] that the current measurement technique gives accurate results when estimating the far-field of large antenna arrays. In this paper a measurement system using the equivalent electric current method for the EISCAT 3D antenna array is described. To reduce the size of the matrices in the system of equations to be solved, the near-field of each antenna element is measured separately, with the probes used as test transmitters. This is possible since all elements will have a separate front-end. The method presented has been simulated in order to illustrate the performance of the approach.

are constants and are suitable basis functions. where The near-field integral (1) can then be written as (4) with (5) and (6) The field measured at the point will be affected by the polarization properties of the probes. Therefore, (7) where is the complex conjugate of the polarization vector which describes the polarization of the probe. B. The Difference Between Two Antennas in an Antenna Array The expressions above relates the current distribution, described using basis functions, to the electric near-field. Of interest when implementing the beamforming algorithms is the difference in the amplitude and phase of the far-field between two antenna elements, as this will affect the direction of the beam. Also, calculating this difference enables the use of signals-of-opportunity, where the absolute phase is not known, in the estimation process. One antenna element could then be temperature controlled (i.e., free from snow) and used as a reference element. To calculate the difference in the measured electric field between two antenna elements, and , (7) is rewritten as

II. THE MEASURED ELECTRIC FIELD

(8)

A. The Electric Field Due to the Current Distribution on One Antenna The electric field at a point in the near-field of an antenna consisting of a perfect electric conductor can be calculated using (1)

Assuming that is the reference element and constants can be written as

is the AUT, the (9)

where is the difference between possible to write (8) as

and

. This makes it

with (10) (2) where is the angular frequency, and is the permeability and the permittivity of the medium, is the surface of the antenna, is the electric current at the point on the antenna, and . The current distribution can be expanded using basis functions according to

. If the reference element is aswhere sumed to be unaffected by the environment the term will be known, with and given by (5). The only remaining unknowns are then the constants . Using matrix notation (10) is rewritten as (11) where

(3)

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III. ESTIMATING THE CURRENT DISTRIBUTION ON THE ANTENNAS There are several methods that can be used to solve the system of equations in (11). In the original work on the equivalent current technique by Petre and Sarkar [2] the conjugate gradient (CG) method was used [13]. This method results in a stable solution and it is also computationally efficient, which is crucial when large matrices are involved, such as when aperture antennas are analyzed. In the present paper the antenna elements in the array are crossed Yagi-antennas, which can be modeled using a relatively low number of unknowns. Thus, the number of probes used for the measurements are kept to a minimum, which also reduces the computational load. The performance of the measurement system is in this paper evaluated under noisy conditions. The observed field strength is modeled as (13) where is the measurement noise. The noise is assumed to be zero mean and uncorrelated between measurements. The coefof the reference element are assumed to be known a ficients priori since the reference element is unaffected by the environment. Including this into the equation does therefore not affect the properties of the noise. The system of equations can thus be written as (14) with (15) Two fundamentally different approaches of estimating are classical and Bayesian estimation [14]. In the classical approach, the parameters of interest are assumed to be deterministic but unknown. The Bayesian approach, on the other hand, assumes some prior knowledge about the properties of the parameters, such as the dynamic behavior of the system. This knowledge can then be used to improve the estimation accuracy. As a classical method the least squares (LS) estimator is chosen and the Kalman filter is chosen as a Bayesian estimator. Both these methods are used extensively in the fields of signal processing and control theory and are under certain conditions optimal (see the following two subsections). It should however be noted that other methods may be more suitable in an implementation where the properties of the signal are affected by hardware. The errors in the estimated radiation pattern of the antenna elements will consist of a random- and a systematic part. For a small antenna array the random part could be expected to have the largest influence on the beam-forming if the systematic error is small. This might not be the case for large antenna arrays, where the random error will be averaged over the array. In this case a small systematic error could have more severe consequences on the beam-forming than random variations over the array. This need to be kept in mind when comparing the results from the LS estimator with the Kalman filter.

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A. Least Squares In the LS estimator the square error is minimized using (16) is the hermitian transpose of . The error in will where be zero on average if the measurements are composed of a deterministic signal and a zero mean noise, see chapter 6 in [14]. Further, it is the optimal minimum variance unbiased (MVU) estimator if the error is white Gaussian noise (with zero mean). This is assumed in the simulations presented in the next section but will not be the case in an implementation where the white noise is filtered and thus no longer completely white. Other estimators may be more appropriate, the LS estimator is nevertheless usable even in this case since the error will be zero on average. B. Kalman Filter Since it is necessary to measure the performance of each antenna element continuously, a Kalman filter can be used to reduce the errors induced by noisy measurements, see chapter 13 in [14]. The Kalman filter has the advantage over the LS estimator that it works even if the noise are non-stationary since the properties of the noise is estimated in the filter. Also, the Kalman filter is a sequential minimum mean square error (MMSE) estimator. Thus, it may not provide a better estimate of the parameters in all cases but, given a system such as described in this paper, the estimate will be better on average. The performance of the AUT relative to the reference antenna at a given time, , is in this case described by the state (17) where are the coefficients describing the difference in the current distribution on the AUT and the reference antenna eleare the time derivatives of ment at time , from (14), and these coefficients. Assuming that there is a linear change of between time and , the state is related to the prewith vious state (18) where (19) where is the identity matrix and and . Further, the observations, related to the state using

is the time between at time can then be

(20) where

is the noise and (21)

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is given by (5). The Kalman filter uses the previous and the observations to estimate the current . This is given by (22)

where the Kalman gain is given by (23), shown at the and are the meabottom of the page, where surement- and system covariance matrices. If these matrices are known or are possible to estimate they are used in the Kalman filter to weight the signals. The system covariance matrix contains the information about the uncertainties affecting the es. In an implementation of the measurement timated state system this would, in addition to errors induced in the updating process, also include error sources related to the AUT. These could be both electrical, such as crosstalk and limited accuracy in the timing between front-ends, and mechanical, such as variations in the position of the AUT (e.g., due to wind). , includes informaThe observation covariance matrix, tion about uncertainties related to the observations. These are mainly caused by error sources affecting the probes. Apart from electrical and mechanical errors similar to the ones affecting the AUT, the main error is the thermal noise on the observation. There may also be a polarization error due to non-ideal probes. All the uncertainties discussed here will have the effect of an amplitude and phase-shift of the far-field pattern of the AUT. The estimated far-field pattern of a given antenna element will thus include all other error sources in addition to the effect of snowfall, which is the desired information. IV. NUMERICAL RESULTS In this section, the performance of an implementation of the method described above is assessed using numerical data from simulations using the Numerical Electromagnetics Code (NEC) [15]. In the simulations, all effects due to mutual coupling between nearby antenna elements (e.g., scan blindness) have been neglected. These effects can be taken into account when performing the measurements by using additional basis functions and careful selection of the probes used at a given time. A. System Setup The antenna array considered here is similar to the EISCAT 3D test array outside Kiruna, Sweden. The antenna elements are the same crossed Yagi-antennas as the ones analyzed in [1], these are also shown in Fig. 1. The measurement system is designed with the final radar system in mind, where the size of the antenna array could be on the order of 125-by-125 m. The antenna elements are mounted at

Fig. 1. The proposed measurement system setup (b) and antenna element (a). The number of antenna elements plotted is in the figure for clarity lower than in the actual system. The probes are in (b) denoted by ‘’.

an elevation angle of 55 and also rotated 45 with respect to its own axis. It should be noted that this system setup is only considered in order to define realistic locations of the probes. It is thus not designed with any array parameters in mind. The probes are assumed to be mounted on 150 m high towers which are spread out around the antenna array. Although there may be some interference due to reflections from the towers this effect should be small since there are no towers in the main beam direction of the antenna array. There are three probes on each tower at 50, 100, and 150 m. This setup is shown in Fig. 1. Fig. 2 shows the simulated far-field (solid line) of one antenna element and the far-field estimated using the equivalent electric current method (dotted line) for the element in the middle of the array in Fig. 1. This element is assumed to be used as the reference element and it is therefore temperature controlled and free from snow. The reference is the results from a full NEC-2 simulation while, for the estimation, only the field calculated at the probe locations was used. It can be seen that the results for both the amplitude and phase is accurate for the considered element. Only the results for one polarization is shown (the lower of the ones shown in Fig. 1(a). Since the position of the probes relative to the AUT will be different for different antenna elements in the array, the error introduced in Fig. 2 could vary between the antenna elements.

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Fig. 2. The far-field amplitude (a) and phase (b) for the antenna element in middle of the array. The probes are assumed to be noise free.

Fig. 3. The estimated far-field amplitude (a) and phase (b) error in the direction of the main beam of the antenna elements for different positions in the array.

This is shown in Fig. 3 for the main beam direction of the antenna element (along the y-axis with an elevation angle of 55 . The error is significantly larger in the middle, front part of the array. This is because there are no probes in the main beam of these antenna elements. This numerical error can be reduced when implementing the methods as discussed previously (e.g., using the CG method to solve the normal equations if the LS estimator is used). The main interest here is however to compare the LS estimator with the Kalman filter during noisy conditions. For this reason, the simulations are performed for an element in the middle of the array where the numerical errors in this case is small.

desired. Also, numerous other models have been proposed, see, e.g., [17]–[19]. Since the parameters affecting the permittivity of snow can vary quickly with changing conditions it is not feasible to take every possible situation into account. Instead a worst case scenario was adopted. The water content is assumed to be 10% by volume and the density 600 kg/m . This corresponds to wet and heavy snow, which have the highest effect on the antennas. The real part of the relative permittivity is in this case 3.7 while the imaginary part is 0.037. The snow-covered antenna elements are modeled in the same way as insulated wire antennas. This can be done by a modification of the impedance matrix as described in [20].

B. Modeling Snow Covered Wire Antennas The snow is in this paper modeled as a dielectric medium with a complex permittivity using a model proposed in [16]. The relative permittivity, , of snow is in the model given by (24) where is the relative density of snow compared to water, and is the water content by volume. For the imaginary part, only the frequency and water content is of importance, and this is given by (25) The imaginary part of the relative permittivity is often neglected but is included here for completeness. This model is empirical and only the effects of the density of dry snow, the water content, and the frequency are included even though the effect of both temperature and pollutants may be added to the model if

C. Performance During Snowfall Under Noise-Free Conditions Simulations were done for a test case to compare the performance of the LS estimator method with the Kalman filter approach. These simulations were done for one antenna element located in the left, back corner of the antenna array shown in Fig. 1 with the antenna element located in the middle of the array used as the reference element. In this simulation the signal is assumed to be noise-free. The Kalman filter here gave identical results as the LS estimator. The test scenario that is used both here, in the noise-free, and in the noisy conditions considered later is shown in Fig. 4. At , there is the onset of the snowfall. The thickness time of the snow layer increases gradually until there is a 0.5 mm thick layer of snow covering the antennas. In the simulations, the snow covers all wires of the antennas completely. This will most often not be the case in reality where the snow will cover only the top part of the wires. The simulation is divided into

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Fig. 4. The thickness of the snow layer as a function of time. The letters (a) to (d) denotes periods used for the analysis of the results.

Fig. 6. The complex error induced due to the noise on the test signal (" denotes the error). The four subplots (a) to (d) refer to the corresponding periods shown in Fig. 4. The units are V/m.

Fig. 5. Performance of the least squares solution during snowfall.

TABLE I ACCURACY OF THE ESTIMATED ELECTRIC FIELD DIFFERENCE

four different periods in order to evaluate the performance of the measurement system under different dynamic situations. In Fig. 5 the amplitude of the difference between reference antenna’s and the AUT’s far-field is shown. The estimated farfield difference here gives accurate results. There is a small bias during the period when there is a constant snow layer on the antennas. This bias is, however, well within the acceptable limits of the system. D. Performance During Snowfall Under Noisy Conditions In the simulations considered here, the signal transmitted by the probes is assumed to be noisy. The noise is assumed to have a gaussian distribution with zero mean. Since the amplitude of the signal received by the antenna elements will depend on both the distance to the probes and the antenna gain in the direction of the probes the signal-to-noise ratio (SNR) will vary significantly between different probes. The SNR for the AUT considered in dB to over 32 dB for difthis section varies from less than ferent probes. To evaluate the performance, the complex error in the estimated far-field was calculated. This can be seen in Fig. 6 where four different periods were analyzed. The letters (a), (b), (c), and (d) in the Fig. 6 are denoted by the same letters in Fig. 4. To create these plots, a total of 100 simulations were used. The numerical values are also shown in Table I, where the amplitude of the mean error is used to compare the systematic errors induced by the estimation methods. The standard deviation is used as a metric of the precision. During all periods, the Kalman filter gives more precise results than the least squares solution, which is the result of the averaging over time. During period (c) and (d)

there is, however, a larger systematic error in the Kalman filter while the bias in the LS estimator is slightly larger in period (b). The difference in the amplitude of the mean error is statistically significant for periods (b), (c), and (d). V. CONCLUSION The measurement system presented in this paper uses the equivalent electric current method to estimate the far-field of an antenna element in a physically large antenna array. The method enables continuous measurements of the performance of all antenna elements in array without the need of locating any probes in the main beam of the antenna array. Also, using the difference of the far-field between two elements it is possible to incorporate far-field sources, which may be at an unknown distance from the antenna element, into the system of equations and retain the phase information. The simulations shows that, in some cases, the Kalman filter gives a larger systematic error than the LS estimator. The LS estimator might therefore be more suitable in spite of its larger random variations. On the other hand, a Kalman filter can be used to estimate the state of the antenna array in more general terms, where calibration systems further

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down the receiver chain may be included. This is a topic that should be studied further. Although only one test scenario has been considered the results from the simulations shows that by using this type of measurement system, it is possible to improve the knowledge about the far-field characteristics of an antenna. This means that the availability of the radar system and the quality of the scientific measurements can be increased. REFERENCES [1] T. Lindgren and J. Ekman, “Performance of a Yagi antenna during snowfall,” presented at the Int. Symp. on Antennas and Propagation, Taipei, Taiwan, Oct. 2008. [2] P. Petre and T. K. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE Trans. Antennas Propag., vol. 40, no. 11, pp. 1348–1356, Nov. 1992. [3] T. K. Sarkar and A. Taaghol, “Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current approach,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 566–573, Mar. 1999. [4] G. Wannberg, EISCAT 3D Design Specification Document EISCAT Scientific Association, Tech. Rep., 2005 [Online]. Available: https://e7. eiscat.se/groups/EISCAT_3D_info/P_S_D_7.pdf [5] G. Stenberg, J. Borg, J. Johansson, and G. Wannberg, “Simulation of post-ADC digital beam-forming for large area radar receiver arrays,” presented at the IEEE Int. RF and Microwave Conf. Putra Jaya, Malaysia, Sep. 12–14, 2006. [6] G. Stenberg, T. Lindgren, and J. Johansson, “A picosecond accuracy timing system based on L1-only GNSS receivers for a large aperture array radar,” presented at the ION GNSS Savannah, GA, Sep. 16–19, 2008. [7] G. Wannberg, I. Wolf, L.-G. Vanhainen, K. Koskenniemi, J. Röttger, M. Postila, J. Markkanen, R. Jacobsen, A. Stenberg, R. Larsen, S. Eliassen, S. Heck, and A. Huuskonen, “The ElSCAT svalbard radar: A case study in modern incoherent scatter radar system design,” Radio Sci., vol. 32, no. 6, pp. 2283–2307, Nov.–Dec. 1997. [8] P. G. Smith, “Measurement of the complete far-field pattern of large antennas by radio-star sources,” IEEE Trans. Antennas Propag., vol. 14, no. 1, pp. 6–16, Jan. 1966. [9] P. Talaga, “The measurement of a large antenna using a spacecraft as a receiver,” IEEE Trans. Antennas Propag., vol. 38, no. 6, pp. 883–888, Jun. 1990. [10] A. D. Yaghian, “An overview of near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. 34, no. 1, Jan. 1986. [11] D. C. Law, J. R. Khorrami, W. B. Sessions, and M. K. Shanahan, “Radiation patterns of a large UHF phased-array antenna: A comparison of measurements using satellite repeaters and patterns derived from measurements of antenna current distributions,” IEEE Antennas Propag. Mag., vol. 39, no. 5, pp. 88–93, Oct. 1997. [12] P.-S. Kildal, Foundations of Antennas, a Unified Approach. Lund, Sweden: Studentlitteratur, 2000. [13] T. K. Sarkar and E. Arvas, “On a class of finite step iterative methods (conjugate directions) for the solution of an operator equation arising in electromagnetics,” IEEE Trans. Antennas Propag., vol. 33, no. 10, Oct. 1985.

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[14] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice Hall, 1993. [15] G. Burke and A. Poggio, Numerical Electromagnetics Code Method of Moments Lawrence Livermore Nat. Lab., Livermore, CA, Rep. UCID18834, 1981. [16] M. Tiuri, A. Sihvola, E. Nyfors, and M. Hallikaiken, “The complex dielectric constant of snow at microwave frequencies,” IEEE J. Oceanic Eng., vol. 9, no. 5, pp. 377–382, Dec. 1984. [17] D. A. Boyarskii and V. V. Tikhonov, “Microwave effective permittivity model of media of dielectric particles and applications to dry and wet snow,” in Proc. Int. Geoscience and Remote Sensing Symp., Aug. 1994, vol. 4, pp. 2065–2067. [18] C. Mätzler, “Microwave permittivity of dry snow,” IEEE Trans. Geosci. Remote Sens., vol. 34, no. 2, pp. 573–581, Mar. 1996. [19] A. D. Frolov and Y. Y. Macheret, “On dielectric properties of dry and wet snow,” Hydrol. Process., vol. 13, pp. 1755–1760, 1999. [20] J. H. Richmond and E. H. Newman, “Dielectric coated wire antennas,” Radio Sci., vol. 11, no. 1, pp. 13–20, Jan. 1976. Tore Lindgren (M’09) received the Ph.D. degree in electrical engineering from Luleå University of Technology, Luleå, Sweden, in 2009. He is currently working as a Researcher at Luleå University of Technology. His research interest is in antenna array measurement techniques for radar and GNSS receiver applications, and simulation of radiofrequency identification systems.

Jonas Ekman (M’06) was born in Boden, Sweden, in 1972. He received the Ph.D. degree in electrical engineering from Luleå University of Technology, Luleå, Sweden, in 2003. From 2003 to 2007, he was working as a Researcher at Luleå University of Technology. During 2005 and 2006, he did his Postdoctoral research in full-wave, time domain, PEEC modeling at the EMC Laboratory, University of L’Aquila, Italy. In 2008, he was appointed Associate Professor at Luleå University of Technology. His research interests are in computational electromagnetics, in particular, the use of the PEEC method for realistic electromagnetic modeling.

Staffan Backén (S’10) received the M.Sc. degree in electrical engineering from Luleå University of Technology, Luleå, Sweden, in 2004 and the Licentiate degree in electrical engineering from Luleå University of Technology, Luleå, in 2007, where he is currently working toward the Ph.D. degree. His research focus is on GNSS array processing and receiver architecture.

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Scalar and Tensor Holographic Artificial Impedance Surfaces Bryan H. Fong, Joseph S. Colburn, Member, IEEE, John J. Ottusch, John L. Visher, and Daniel F. Sievenpiper, Fellow, IEEE

Abstract—We have developed a method for controlling electromagnetic surface wave propagation and radiation from complex metallic shapes. The object is covered with an artificial impedance surface that is implemented as an array of sub-wavelength metallic patches on a grounded dielectric substrate. We pattern the effective impedance over the surface by varying the size of the metallic patches. Using a holographic technique, we design the surface to scatter a known input wave into a desired output wave. Furthermore, by varying the shape of the patches we can create anisotropic surfaces with tensor impedance properties that provide control over polarization. As an example, we demonstrate a tensor impedance surface that produces circularly polarized radiation from a linearly polarized source. Index Terms—Antennas, artificial materials, electromagnetic scattering, holographic gratings, impedance sheets, surface impedance, surface waves. Fig. 1. Holographic leaky wave antenna concept. Surface waves (undulating arrows) are excited on an artificial impedance surface, and are scattered by variations in the surface impedance to produce the desired radiation (straight arrows).

I. INTRODUCTION

A

common challenge for antenna designers is the integration of antennas onto complex metallic shapes while maintaining the desired radiation characteristics. The antenna excites currents in nearby conductors, and these currents can contribute to unexpected artifacts in the radiation pattern, including shadowing, nulls, cross-polarization, or radiation in undesired directions. We have developed a method to control the radiation from surface currents on metallic bodies. Our method is based on an artificial impedance surface consisting of a grounded dielectric layer covered with a pattern of conductive patches. The patches are small compared to the electromagnetic wavelength, and we can describe their scattering properties in terms of their effective surface impedance. By varying the size and shape of the patches, we can control the surface impedance as a function of position and direction. The surface impedance is patterned over the body of interest using a holographic technique [1] in which we calculate the interference pattern between the currents generated by a source and the fields associated with the desired radiation, and design

Manuscript received November 13, 2009; revised February 12, 2010; accepted April 08, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by the United States Air Force Office of Scientific Research under Contract FA9550-06-C-0021, The Boeing Company, General Motors, and in part by Raytheon. B. H. Fong, J. S. Colburn, J. J. Ottusch and J. L. Visher are with HRL Laboratories, LLC, Malibu, CA 90265 USA. D. F. Sievenpiper was with HRL Laboratories, LLC, Malibu, CA 90265 USA. He is now with the University of California, San Diego, La Jolla, CA 920930407 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055812

the spatial profile of the surface impedance to match this interference pattern. Currents from the source are scattered by the modulated surface impedance to produce the desired radiation pattern. The source can be a small feed or a plane wave impinging on the body. The concept of the holographic artificial impedance surface was first introduced in several recent conference papers [2]–[4]. In this work, we expand on the previous publications to provide a detailed explanation of how to design and characterize holographic artificial impedance surfaces, including the equations necessary for others to reproduce our results. We also provide the first measured data for a tensor impedance surface, and demonstrate that such a surface can generate a high gain beam with circularly polarized radiation from a small linearly polarized feed. Holographic artificial impedance surfaces are based on three established concepts: leaky waves on modulated impedance surfaces, artificial impedance surfaces, and holographic antennas, illustrated in Fig. 1. The foundation for this work is Oliner’s comprehensive analysis of leaky waves on modulated impedance surfaces [5] in which he describes how the propagation and radiation of leaky waves are controlled by the magnitude, modulation depth, and period of the surface impedance. We have implemented Oliner’s impedance boundary using an artificial impedance surface, and extended the modulation concept using a two-dimensional holographic patterning method. We have also expanded this idea to include anisotropic or tensor impedance surfaces which can control polarization.

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wave modes, but not so far as to introduce other propagating modes in the free space volume. A practical height for the top wall is about 1/2 wavelength. We then solve for the eigenmode that satisfies these boundary conditions to obtain the properties of the surface wave, and extract the effective surface impedance. The effective surface impedance can be most easily understood as the ratio of the electric to magnetic fields near the surface, averaged over the unit cell. For transverse magnetic direction, the effective surwaves (TM) propagating in the face impedance is (1)

Fig. 2. Unit cell used for scalar impedance surface. A square metal patch sits atop a dielectric layer with a PEC backing. The gap between square patches determines the effective scalar impedance.

II. ARTIFICIAL IMPEDANCE SURFACES Various kinds of artificial impedance surfaces have been studied in the past, such as pin-bed structures [6], and high-impedance surfaces [7]. Recently, other kinds of two-dimensional metamaterials have been designed to provide other exotic properties such as left-handed materials [8]–[10]. Because the metallic patterns in these structures are small compared to the wavelength of interest, they can be described as effective media. This permits the analysis of their scattering properties using only the macroscopic effective surface impedance instead of the detailed local geometry, enabling us to model electrically large structures based on these artificial materials using practical computing hardware. Our artificial impedance surface is based on a square lattice of sub-wavelength conductive patches on a grounded dielectric substrate. Because the structure is quasi-periodic, we simulate its effective surface impedance using a single unit cell with periodic boundary conditions, as shown in Fig. 2. This approach assumes that the structure is uniform, but the results can be used to design a non-uniform surface if its properties are slowly varying. For this analysis, we have used both the commercial Ansoft HFSS code, and our proprietary FastScat code [11] that has been adapted to include periodic boundary conditions. The HFSS code is an FEM-based solver, while the FastScat code is a high-order Nyström discretization frequency domain integral equation solver. The surface impedance can be calculated using one of three methods, which are described below. Using HFSS, we apply periodic boundary conditions to the four vertical walls of the simulation volume. The bottom wall is an electric conductor, and the top wall is a radiation boundary. The simulation also includes a block of dielectric, and a single square conductive patch. The top wall must be sufficiently far from the structure not to affect calculation of bound surface

Throughout the paper TM modes have magnetic field transverse to the direction of surface wave propagation; transverse electric (TE) modes similarly have electric field transverse to the direction of propagation. Since both fields decay away from the surface at the same rate, this ratio is independent of the position above the surface. However the integration should be done at a height that it is sufficiently far from the surface that near-field effects are negligible, yet also sufficiently far from the absorbing boundary condition that it does not affect the calculation of the surface impedance. The midpoint of the simulation volume is a good practical choice. The surface impedance can also be calculated using another method that does not require integration of the fields. For a bound surface wave, the refractive index seen by the wave is the ratio of the speed of light in free space to the phase velocity , where is of the wave along the surface, the surface wave wave vector. Using HFSS, we can calculate an eigenfrequency for a given phase difference across a unit cell of length . Since the phase difference across the unit cell , the index is is related to the wave vector through completely determined. Alternatively, using our FastScat code we can solve for the surface wave wave vector at frequency , again determining the index . With bound surface waves , where the having the functional dependence subscript denotes quantities in the X-Y plane, Maxwell’s equations and the impedance boundary condition require that the surface impedance for TM modes is related to the decay constant of the fields away from the surface by [12] (2) where and is the impedance of free space. Notice that the ratio gives the modulus of the surface impedance normalized to the free space impedance; this observation will be important in the analysis of tensor impedance surfaces. The along the surface to wave equation relates the wave vector the decay constant (3) giving the relationship between the effective refractive index and the effective surface impedance (4)

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This function is inverted to give the gap size as a function of desired impedance value. After characterizing the surface impedance for a range of gap sizes, we then build an artificial impedance surface in which the gap and the effective impedance vary as a function of position using our holographic patterning technique. III. HOLOGRAPHIC PATTERNING

Fig. 3. Plot of impedance Z versus gap g at 17 GHz for the unit cell shown in Fig. 1. Points show the result of HFSS calculations, and the line is given by (5). The unit cell has a lattice constant of 3 mm, and the dielectric is 1.57 mm thick with dielectric constant 2.2. For this range of impedance between 161j

and 234j , X and M in (8) take the values 197.5 and 36.5 , respectively.

Originally demonstrated at optical frequencies, holography involves producing an interference pattern using two waves, and then using the interference pattern to scatter one wave to produce the other. In optical holography, one wave is produced by scattering a laser beam from the object to be imaged, and the other is a reference plane wave from the same laser. The interference pattern formed by these two waves is recorded on photographic film. When the reference wave illuminates the developed film, it is scattered by the recorded interference pattern to produce a copy of the original object wave. For a reference wave and an object wave , the interference pattern contains a term proportional to . When the interference pattern is illuminated by the reference wave, this term gives (6)

This expression is used for transverse magnetic waves, and a similar expression can be derived for TE waves. This method produces the same result as integrating the fields, but is much simpler because it only requires calculating the surface wave frequency or wave vector. A third method involves calculating the reflection coefficient of the surface, and extracting the sheet impedance of the metallic layer. The sheet impedance, and the properties of the dielectric layer are then used to calculate the effective surface impedance [13]. As a general rule, higher impedance values are obtained with a narrower gap, a higher dielectric constant substrate, a thicker substrate, or a larger period. However, one must take care to ensure that the period does not exceed the effective medium limit, and that the substrate is sufficiently thin to suppress higher order modes. As an example, we have analyzed a structure based on 1.57 mm thick Rogers Duroid 5880, with a dielectric constant of 2.2, and a lattice constant of 3 mm. For this analysis, we arbitrarily choose a phase difference of 72 degrees per cell for the periodic boundary conditions. The choice of phase difference determines the location on the dispersion curve, and affects the frequency of operation. By sweeping the size of the gaps between the metal patches, we determined the impedance as a function of the gap width using HFSS. For gaps ranging from 1 mm to 0.2 mm, the effective impedance varied from 161j to 234j ohms at 17 GHz, plotted in Fig. 3. Using the HFSS computed impedance data, we computed a least-squares fit to the impedance as a function of the inverse gap: (5) Here the impedance is in ohms and the gap

is in mm.

i.e., a copy of the original object wave [1]. Microwave holograms [14], [15] are created using a similar concept. A source antenna produces the reference wave, which can be a surface wave, [16] and the desired radiation pattern corresponds to the object wave. The microwave hologram is built as a collection of scatterers that correspond to the interference pattern produced by these two waves, and it scatters the reference wave to produce the object wave. In our approach, we implement the hologram as a two-dimensional modulated artificial impedance surface. The surface impedance is designed by generating the intercorresponding to the curference pattern between a wave rents generated by the antenna and a wave corresponding to the radiation pattern we would like to generate. For a given on the impedance radiation pattern, the method requires surface, i.e., the near field values of the radiating wave evaluated on the impedance surface. The fidelity of the holographically reproduced far field is of course limited by the size of the impedance surface. As a simple example, assume that we would like to use a small monopole antenna located at the origin on a surface occupying the X-Y plane. The currents generated by this antenna can be approximated as a cylindrical wave (7) where corresponds to the effective index seen by the surface currents, and is the radial distance from the antenna. Assume that we would like this antenna to produce a narrow pencil beam in the X-Z plane. The fields on the in a particular direction surface associated with this plane wave radiation pattern are represented by (8)

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Fig. 4. Scalar impedance pattern formed from holographic interference (8). Impedance values are shown on a linear scale, with light areas corresponding to high impedance. The impedance pattern scatters the surface wave from a point source into a plane wave 60 from the normal.

where is an arbitrary phase offset that sets the impedance value at the origin. We define the surface impedance as the interference pattern between these two waves

Fig. 5. A portion of the holographic pattern of Fig. 3 implemented using square patches. High impedance regions have small gaps (and are darker). Gaps vary between 0.2 mm and 1 mm, i.e., the full range of gaps and impedances shown in Fig. 2. The unit cell size is fixed at 3 mm. At 17 GHz the free space wavelength is 17 mm.

(9) where is an arbitrary real average value, is the real modulation depth, and is a point on the impedance surface in the X-Y plane. As in the optical holography case, the radiated (object) wave results from the scattering of the surface wave from . The holothe modulated impedance, given by gram pattern shown in Fig. 4 is the interference pattern formed by a point source and a plane wave propagating 60 degrees from normal. In practice, we typically set to the average impedance value for the geometry of choice, and we set to span the entire available impedance range. However, since the radiation rate is proportional to , it can be adjusted depending on the size of the surface to obtain a desired aperture profile. The holographic pattern shown in Fig. 4 is combined with the impedance data shown in Fig. 3 to determine the required gap as a function of position. We combined (5) with (9) to produce a function describing the gap size versus position on the surface, and used this to produce a file describing a pattern of squares. These squares were then printed as metal patches on a 25.4 cm (Y-dimension) by 40.64 cm (X-dimension) 1.6 mm thick (Z-dimension) Duroid 5880 printed circuit board. A small section of the pattern of metal squares is shown in Fig. 5. The surface was fed by a 3 mm long monopole antenna inserted from the back, located at the focus of the ellipses shown in Fig. 4. Currents generated by the monopole are scattered by the holographic impedance surface to produce a narrow pencil beam with a measured gain of about 20 dBi, as at shown in Fig. 6, which plots the measured polarized radiation pattern in the X-Z plane. For comparison, a monopole on a similar smooth metal surface produces the expected low-gain pattern, shown on the same plot. Thus, the holographic impedance surface has scattered the surface wave to produce the desired plane wave, with little of the feed energy being directly radiated into free space. This example involves TM waves propagating on an inductive impedance surface to produce vertically polarized radiation. A similar result can be obtained with TE waves on a capacitive

Fig. 6. Measured radiation patterns from monopole antenna placed above the holographic impedance surface (black) and a smooth metal surface (grey). The inclination angle is defined from the impedance surface normal, i.e., the Z-axis, in the X-Z plane.

impedance surface to produce horizontally polarized radiation. To generate arbitrary polarization from any source requires a method of controlling the coupling between currents in any direction and fields in any direction. This requires an anisotropic surface, which can be described using an impedance tensor. IV. ARTIFICIAL TENSOR IMPEDANCE SURFACES In the scalar impedance case the relationship between impedance and geometrical parameter (metal patch size) is mapped out by determining the bound mode wave number for each value of gap . In the tensor impedance case, the relationship between tensor impedance components and geometrical

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parameters is similarly determined. We derive an equation relating tensor impedance components to surface wave properties, and we use simulations or measurements to determine the relationship between geometrical patterning and surface wave parameters. For a uniform surface in the X-Y plane, bound surface waves decay into the surrounding space, and have spatial dependence ; again, subscripts refer to in-plane quantities. For scalar impedance surfaces, we divide surface waves into TM, and TE modes, where transversality is with respect to the surface wave propagation direction . For TM waves, using the spatial dependence above, the fields are given is assumed throughout) [12] by (a time dependence of (10.a) (10.b) For TE waves, the fields are given by (11.a) (11.b) The term is the transverse wave vector for a wave traveling across the surface at angle (12) From the wave equation, the terms in the wave vector are re. Note that the exponentially lated through decaying dependence in the direction results in a negative sign before . On a tensor impedance surface, the surface wave modes are in general neither pure TM nor TE, but rather a hybrid. The electric and magnetic fields can thus be written as (13) and (14) The tensor impedance boundary condition on the is written as

surface

(15) for the tangential components of and , and the current . Energy conservation requires that be anti-Hermitian, and reciprocity requires that be pure imaginary [17]. (Notice that this requires .) Substituting the combined TM and TE fields (13) and (14) into the boundary condition (15) gives

(16)

For a given impedance tensor and propagation direction ratio. The ratio is we can solve for and the

,

(17) where the minus sign corresponds to a TM-like mode, and the plus sign to a TE-like mode. For TM-like modes, the impedance tensor has purely inductive eigenvalues and requires that the negative sign be taken in (17) to produce bound surface waves. Similarly, for TE-like modes, the impedance tensor’s purely capacitive eigenvalues requires the plus sign be taken in (17) for bound surface waves. In the scalar impedance case the ratio (2) gives the -normalized scalar impedance, which is independent of surface wave propagation direction. In the tensor ratio as giving an efimpedance case we can interpret the fective normalized scalar impedance that is propagation direction dependent. However, this interpretation does not extend to the surface electric field and current since they are not generally aligned. Equation (17) also implies that the mode phase velocity is direction dependent, as is generally true in an anisotropic medium. Equation (17) expresses the effective normalized scalar impedance as a function of tensor impedance components and propagation direction. To determine the impedance tensor components using (17) requires computing or measuring the effective normalized scalar impedance at three different propa, gation directions and then solving for the three unknowns , and from the three nonlinear constraint equations. If data from more than three different propagation directions is available, a least squares solution to the constraint equations derived from (17) can be used. The periodic array of metal squares described in the scalar case can be extended to produce a tensor impedance surface by adding a slice through each square, and varying the width and angle of the slice, as illustrated in Fig. 7. For a surface with given lattice constant, dielectric thickness, and dielectric con, stant the three independent terms in the impedance tensor, , and , are controlled by the three degrees of freedom, , , and , the width of the gap between the squares. To understand how the sliced square produces a tensor impedance surface, consider an electric field in the X-direction applied to the square with a diagonal slice. Some portion of the currents in the square will tend to run along the direction of the slice. Thus, the slice provides coupling between currents in the X- and Y-directions and fields in the X- and Y-directions. Fig. 8 plots the effective normalized scalar impedance as a function of surface wave propagation direction , for the patch geometry shown in Fig. 7, at a frequency of 10 GHz. The ef, fective normalized scalar impedance is computed at 90 , 105 , 120 and 150 using FastScat, and the associated

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of a given geometry’s impedance tensor components; one need never compute the scalar impedance as a function of propagation direction to construct the tensor hologram.) For a given metal patch geometry, we are able to compute the associated impedance tensor using this characterization method. Implementing a given tensor impedance hologram requires the solution of the inverse problem: for the given tensor impedance determine the corresponding metal patch geometry. We solve the inverse problem by constructing a database of metal patch geometries and associated impedance tensors, and then numerically invert this mapping. V. TENSOR SURFACE HOLOGRAPHIC PATTERNING

Fig. 7. Unit cells used for tensor impedance surface. A square metal patch with slice sits atop a dielectric layer with a PEC backing. The gap g between squares, the width of the slice g , and the angle  of the slice control the three independent tensor components.

The tensor impedance hologram generalizes the scalar impedance hologram described above. In the tensor impedance case, the surface electric field and surface current are related through (18) The tensor impedance function is constructed from the outer product of the expected vector surface current and the desired outgoing electric field vector. The outer product generalizes the simple multiplicative scalar pattern described in (9). In the scalar case, the desired radiated wave term results from the scattering of the surface wave from the modulated impedance pattern. In the tensor case, we desire a resulting from the radiated vector wave term scattering of a vector surface wave from a modulated tensor impedance pattern. A modulated tensor impedance proportional creates this radiated vector wave term when to . (The dagger illuminated by the reference surface wave gives the Hermitian conjugate.) Thus, we have (19)

Fig. 8. Plot of effective scalar impedance as a function of surface wave propa, gation direction  for the sliced patch unit cells shown in Fig. 6 (g : : ), at a frequency of 10 GHz. Points show results of g , FastScat computations at propagation angles of 60 , 90 , 105 , 120 , and 150 . Least squares solution of (12) gives the continuous curve, with tensor components Z : ,Z : : , and Z . Principal axes : and  : , with principal values of 384.2j and are at  161.2j , respectively.

= 0 2 mm

= 0 2 mm

= 60

= 184 1j

= 71 3

= 67 6j

= 018 7

= 361 4j

Satisfying energy conservation requires subtracting from its Hermitian conjugate, while satisfying reciprocity requires keeping only the pure imaginary terms. The resulting impedance tensor that satisfies the energy conservation and reciprocity requirements is then



(20) impedance tensor components , , and are computed from a least squares solution to (17). With the computed tensor components, (17) can again be used to plot the effective normalized scalar impedance as a function of . Instead of the three , , and , the anti-Hermitian impedance components tensor may alternatively be described by its principal axes and angle of the principal axes. The principal axes and angles of the impedance tensor are also shown in Fig. 8. (Note that the construction of the tensor hologram requires only the determination

In order to demonstrate an antenna that cannot be implemented with a scalar impedance surface, we designed a tensor impedance surface to scatter currents from a linearly polarized source to produce circularly polarized radiation. For a feed, we used a WR90 (X-band) waveguide, mounted centered along the Y-edge of a 25.4 cm (Y-dimension) by 40.64 cm (X-dimension) surface. We calculated the interference pattern between currents from a point source at the waveguide aperture and a left hand

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. Substituting (21) and (22) into (20), we where compute the impedance tensor as a function of position (shown in Fig. 9)

where

(23)

Fig. 9. Tensor impedance components formed from holographic interference. Impedance values are shown on a linear scale, with light areas corresponding to high impedance. The tensor impedance pattern scatters the surface wave from a vertically polarized point source into a circularly polarized plane wave 45 from the normal. The functional dependences are given in (15).

circularly polarized plane wave at . Taking the origin of the coordinate system to be at the point source location and the impedance surface to lie in the X-Y plane, the desired outgoing circularly polarized plane wave has electric field given by (21) where by the point source is given by

. The surface current excited

(22)

, is the average reactance, and Here is the reactance modulation depth. The impedance tensor given in (23) is the desired functional form for a surface scattering a linearly polarized excitation into a circularly polarized output. To implement this functional form for the tensor impedance we used the sliced-patch geometry pattern on a 1.27 mm thick Rogers 3010 microwave substrate, with a lattice constant of 3 mm. Using FastScat, we simulated at 10 GHz a range of varying from 0.2 mm to 1 mm and geometries with and 0.2 to 0.5 mm, respectively. The slice angle varied over 360 degrees. With these parameters the FastScat computed major principal axis impedance varied from 116j to 384j Ohms, and the minor principal axis impedance varied from 115j to 161j Ohms. The angle of the principal axes varied over 360 degrees. For the Rogers 3010 substrate, a scaling function was required in order to match the FastScat (or HFSS) computed impedance values to near-field measured impedance values. The reason behind the need for a scaling function to match the measured values to the simulated values, which agree among multiple simulation methods, or to analytical solutions, [18] is still unknown. We constructed a set of uniform scalar impedance surfaces and determined their surface impedances using near-field measurements and FastScat computations. The scaling function is the function mapping FastScat computed impedance values to measured impedance values. The scaling function varied from 1 at low impedance values to 1.37 at high impedance values. The scaled values corresponding to the FastScat computed values given above are 121j to 509j Ohms for the major axis impedance, and 119j to 185j Ohms for the minor axis impedance. Fig. 10 shows the principal axes’ scaled impedances for a range of gaps and two slice widths and . The functional form in (23) requires that the major principal and while the axis impedance vary between minor principal axis impedance vary between and . From Fig. 10 we see that it is impossible for the sliced gap structure to satisfy the principal axes’ impedance values required by (23). For the sliced gap structure, the principal axes’ accessible impedance ranges never overlap. Because we

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Fig. 12. A portion of the tensor holographic impedance surface of Fig. 8 and (15) implemented using sliced gap unit cells. Fig. 10. Tensor impedance components along a one-dimensional slice on the impedance surface. Solid lines are impedance components realized using sliced gap patches; dashed lines are desired impedance components specified by (15). Thick black lines are the Z component, grey lines are Z , and thin black lines are Z (= Z ).

Fig. 13. Measured radiation patterns from monopole antenna placed above the tensor holographic impedance surface. Left-hand circularly polarized radiation is shown in black; right-hand circularly polarized radiation is in grey. Fig. 11. Major and minor principal axes impedances as a function of gap size for two slice gap widths g , at slice angle  = 90 . Black lines are major axis impedances; grey lines are minor axis impedances. Solid lines are for g = 0:2 mm; dashed lines are for g = 0:5 mm.

are unable to match both principal values of (23) simultaneously, we have chosen to match only the major axis impedance and the angle of the principal axes; the matching can be accomplished by fixing the slice width at 0.2 mm while varying only the gap and the slice angle. Fig. 11 shows the impedance tensor components demanded by (23) and the constructed impedance tensor components matching only the angle and major axis impedance, along a diagonal on the surface. The constructed components match the general trend of the required components. For the structure that we built, the average reac, and the reactance modulation depth tance .

We fabricated and measured the tensor impedance surface pattern shown in Fig. 9, implemented using our sliced patch geometry, for which a small section is shown in Fig. 12. Note that the angle of the slice rotates with each unit cell with increasing distance from the source, which is at the center of the left edge of the figure. Fig. 13 contains plots of the measured far field patterns, in terms of left hand and right hand circular polarization components. The measure left hand circular polarization beam peak gain was 21.8 dB at 38 degrees from the surface normal. At the left hand circular polarization beam peak the right hand circular polarization component was down by 19.6 dB. This measured far field data shows good agreement in terms of polarization purity compared to the computed pattern for the ideal tensor impedance surface, shown in a previous publication [4] but the beam peak is off by 7 degrees. An error in beam angle may be caused by an error in

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the calculated phase velocity of the surface wave. This may be due to the fact that the impedance of the tensor surface was not directly measured using the near field technique to determine the scaling function, but instead we applied a scaling function obtained from scalar surfaces. VI. CONCLUSION We have demonstrated that we can build both scalar and tensor artificial impedance surfaces, and that these can be used to design conformal antennas with a range of important properties. Scalar surfaces can be implemented using a simple lattice of square metal patches with variable gap width, while tensor surfaces can be implemented by introducing a slice through the patches having a variable angle. The scalar surface can be designed to scatter a given surface wave into a desired far-field radiation pattern, and the tensor surface extends this concept to provide polarization control. We have used this concept to design and build a surface that can generate a circularly polarized plane wave from a linearly polarized source. It is possible that this work can be extended to include a wide range of alternative geometries that may provide even greater control over the values of the impedance tensor to enable greater accuracy of designs and more control over the radiation pattern. REFERENCES [1] P. Hariharan, Optical Holography: Principles, Techniques and Applications. Cambridge, U.K.: Cambridge Univ. Press, 1996, pp. 11–13. [2] D. Sievenpiper, J. Colburn, B. Fong, J. Ottusch, and J. Visher, “Holographic artificial impedance surfaces for conformal antennas,” presented at the IEEE Antennas and Propagation Symposium Digest, Washington DC, Jul. 5, 2005. [3] J. Colburn, D. Sievenpiper, B. Fong, J. Ottusch, and P. Herz, “Advances in artificial impedance surface conformal antennas,” in IEEE Antennas and Propagation Symposium Digest, Washington DC, Jun. 9, 2007, pp. 3820–3823. [4] B. Fong, J. Colburn, P. Herz, J. Ottusch, D. Sievenpiper, and J. Visher, “Polarization controlling holographic artificial impedance surfaces,” in IEEE Antennas and Propagation Symposium Digest, Washington DC, Jun. 9, 2007, pp. 3824–3827. [5] A. Oliner and A. Hessel, “Guided waves on sinusoidally-modulated reactance surfaces,” IRE Trans. Antennas Propag., vol. 7, no. 5, pp. 201–208, Dec. 1959. [6] R. King, D. Thiel, and K. Park, “The synthesis of surface reactance using an artificial dielectric,” IEEE Trans. Antennas Propag., vol. 31, no. 3, pp. 471–476, May 1983. [7] D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2059–2074, Nov. 1999. [8] C. Caloz and T. Itoh, “Transmission line approach of Left-Handed (LH) materials and microstrip implementation of an artificial LH transmission line,” IEEE Trans. Antennas Propag., vol. 52, pp. 1159–1166, May 2004. [9] C. Caloz, S. Lim, C. A. Allen, and T. Itoh, “Leakage phenomena from negative refractive index structures,” in Proc. URSI Int. Symp. on Electromagnetic Theory, Pisa, Italy, May 2004, pp. 156–158. [10] C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antennas Propag. Mag., vol. 50, no. 5, pp. 25–39, Oct. 2008. [11] L. F. Canino, J. J. Ottusch, M. A. Stalzer, J. L. Visher, and S. M. Wandzura, “Numerical solution of the Helmholtz Equation in 2D and 3D using a high-order nystrom discretization,” J. Comput. Phys., vol. 146, pp. 627–663, 1998.

[12] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 2nd ed. New York: Wiley, 1984. [13] A. M. Patel and A. Grbic, “A printed leaky wave antenna with a sinusoidally modulated surface reactance,” presented at the IEEE Antennas and Propagation Symp., Charleston, SC, Jun. 1–5, 2009. [14] R. Dooley, “X-band holography,” Proc. IEEE, vol. 53, no. 11, pp. 1733–1735, Nov. 1965. [15] W. Kock, “Microwave holography,” Microwaves, vol. 7, no. 11, pp. 46–54, Nov. 1968. [16] P. Checcacci, V. Russo, and A. Scheggi, “Holographic antennas,” IEEE Trans. Antennas Propag., vol. 18, no. 6, pp. 811–813, Nov. 1970. [17] D. J. Hoppe and Y. Rahmat-Samii, Impedance Boundary Conditions in Electromagnetics. Washington, DC: Taylor & Francis, 1995, pp. 135–137. [18] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and accurate analytical model of planar grids and high-impedance surfaces comprising metal strips or patches,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1624–1632, Jun. 2008.

Bryan H. Fong received the B.S. degree in physics from Yale University, New Haven, CT, in 1993 and the M.A. and Ph.D. degrees in plasma physics from Princeton University, Princeton, NJ, in 1995 and 1999, respectively. He is a Senior Research Staff Physicist in the Computational Physics Department, HRL Laboratories, LLC, Malibu, CA. He joined HRL Laboratories in 2001, following postdoctoral research at the National Center for Atmospheric Research. His research has been in nonlinear instabilities in laboratory and astrophysical plasmas, computational electromagnetics, high-order numerical methods, and quantum information. Dr. Fong is a member of the American Physical Society.

Joseph S. Colburn (M’92) received the B.S. degree from the University of Washington, Seattle, in 1992, and the M.S. and Ph.D. degrees from University of California, Los Angeles, in 1994 and 1998 respectively, all in electrical engineering. Since 1998, he has been with HRL Laboratories, Malibu, CA, where he is currently the Antenna Department Manager in the Applied Electromagnetics Laboratory. At HRL he has worked on millimeter and microwave antennas and circuits for aerospace and automotive applications. From 1995 to 1997, he was with the TRW Space and Electronics Group where he was involved with the design and measurement of satellite antennas. He has over 25 papers published in technical publications and was issued 10 patents.

John J. Ottusch was born in Landstuhl, Germany, in 1955. He received the S.B. degree in physics form the Massachusetts Institute of Technology, Cambridge, and the Ph.D. degree in physics from the University of California, Berkeley, in 1977 and 1985, respectively. In 1985, he joined Hughes Research Laboratories (now HRL Laboratories), Malibu, CA. Until 1994, his research involved experimental investigations of nonlinear optics, including Raman and Brillouin scattering and optical phase conjugation. Since that time he has focused primarily on developing fast, high-order algorithms and software for electromagnetic modeling.

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John L. Visher received the B.A. degree from the University of California in Santa Cruz, in 1979 and the M.A. degree from Columbia University, New York, in 1981, both in physics. For over 23 years he has worked in HRL Laboratories, Malibu, CA, and its previous corporate incarnations. He has written more code for more projects than he can or cares to remember. His most recent code models EM in the time domain with novel high order stable and explicit techniques.

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Daniel F. Sievenpiper (M’94–SM’04–F’09) received the B.S. and Ph.D. degrees in electrical engineering from the University of California, Los Angeles, in 1994 and 1999, respectively. Since 2010, he has been a Professor at the University of California, San Diego, where his research focuses on antennas and electromagnetic structures. Previously, he was the Director of the Applied Electromagnetics Laboratory at HRL Laboratories in Malibu, CA, where his research included artificial impedance surfaces, conformal antennas, tunable and wearable antennas, and beam steering methods. He has more than 60 issued patents and published more than 50 technical publications. Dr. Sievenpiper was awarded the URSI Issac Koga Gold Medal in 2008, and in 2009 he was named as a Fellow of the IEEE. In 2010, he was elected to the Antennas and Propagation Society Administrative Committee.

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Reflections From Multiple Surfaces Without Edges William B. Gordon, Senior Member, IEEE

Abstract—An algorithm is presented for calculating the positions of the specular points that appear when a collection of reflecting surfaces is illuminated by an external source. The set of specular points is represented as the fixed point of a certain mapping, and this fixed point is calculated by the method of successive approximations (MSA). The MSA is an iterative technique which is essentially different from a search or shooting and bouncing ray technique. The latter require much larger numbers of functional of reflecting surfaces evaluations, especially when the number is greater than unity. A search technique requires a number of function evaluations that varies exponentially with , whereas the number of function evaluations required by the MSA varies linearly with . Index Terms—Electromagnetic reflection, geometrical optics, optical reflection, reflection.

I. INTRODUCTION A. The General Problem E are concerned with multiple reflections from PEC surfaces embedded in a homogeneous medium. For this case the Principle of Least Time reduces to Fermat’s Principle, according to which light will follow a path of minimum length [1], [2]. This gives rise to the following general problem. , Problem: Given an ordered set of surfaces and two points and external to these surfaces, find the points which minimize the path length function

W

(1) subject to the constraints (2) In applications is the location of the source and is the obare called the servation point. The solutions specular points. As stated, the problem is bistatic in the sense that and can be distinct. The corresponding monostatic problem is the case for which . The surfaces need not be distinct; however, when we assume that no two , for successive surfaces in the list are identical: . This study was motivated by a program for the calculation of ship radar cross section. In this application the concern is to calculate the scattered field produced by the bouncing ray

Manuscript received April 29, 2009; revised January 25, 2010; accepted March 30, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The author is with the Naval Research Laboratory Division, GTEC Inc., Crofton, MD 21114 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055785

, where the surfaces correspond to various ship components. Once the specular points have been located, it is assumed that geometrical optics will be used to calculate the field scattered from one bounce to the next. Throughout our discussion we make the following assumptions. have boundary edges; (a) None of the surfaces in the list (b) None of the surfaces in the list have singular points at which the surface normal is not defined; (c) For each surface in the list and for each point external to there exists a unique point which minimizes . Departing from our the distance function usual custom of representing points in 3-space with boldface lower case letters, this point will be denoted by the . Hence is the point on which is symbol closest to , and (3) (d) For every surface in the list an effective method exists . For our purposes it does for the calculation of not matter whether is given by a closed form formula or an algorithm whose rate of convergence is sufficiently rapid to be practical. Surfaces satisfying (a)–(d) include ellipsoids and infinitely extended flat planes. One could also include paraboloids in the approved list, provided that the points and are “external” in the sense that they lie on the convex side of the parabolic surface. Condition (b) excludes conical surfaces. Condition (c) excludes non-convex surfaces such as tori. is the specular point for the monostatic The point problem involving the single surface and source . Methods for the calculation of will be briefly reviewed in Section II-A. B. Nature of the Solution In the following sections it will be shown how the solution to the general problem can be expressed as a fixed point of a certain mapping, and how this fixed point can be calculated by the method of successive approximations (MSA). into itself. Recall that a Let be a mapping of a space fixed point of is a solution to the equation (4) The MSA consists of locating the position of a fixed point by taking the limiting value of a sequence defined according to the iteration (5) where is an initial guess. (The MSA is commonly discussed in the context of establishing the existence of solutions to sys-

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tems of differential or integral equations. See, e.g., [3]–[7].) For the problem of bistatic reflection from a single surface, (the case ), it turns out that the specular point is a fixed point for the function , where

(6) For the case , is a more complicated func, with , tion of the N-tuple . In both the cases and the MSA reduces to a series of calculations of for various surfaces and points . We emphasize that the MSA is essentially different from a is search technique. In a straight forward search the space sampled at a large number of closely spaced points, and an approximate solution to the problem is obtained by selecting a sample point which most closely satisfies the required criteria. Thus to find the specular point for reflection from a single surface, in the absence of symmetry the problem would require sample a two-dimensional search on approximately points, where is the sample interval and is the area of that portion of the surface which is visible from both and . This number can be very large since must be small enough to insure that the estimate for is accurate to within a small fraction of a wavelength (so that the relative phases of all the bounce components can be accurately calculated). For the general problem of minimizing the path length function for surfaces, each with areas whose sizes are comparable to , one would need to calculate the path length at approximately N-tuples of sample points. On the other hand, we have found that the number of iterations of (5) that are required to achieve a given degree of accuracy by means of the MSA technique does not increase to any significant extent when is increased. It follows that the number of function evaluations required for the MSA technique varies linearly with , rather than exponentially with . For these reasons we believe that an iterative technique based on the MSA, when feasible, is to be preferred over a search technique. The accuracy requirements for the MSA are discussed in the Appendix. A shooting and bouncing ray approach to the general problem can be considered as a type of search. The problem of reflection from a single surface would require the launching of a large a much number of rays, and as pointed out in [8], when more densely packed bundle of launched rays would be required due to the fact that the divergence between two neighboring rays doubles with each bounce. The construction of a practical search technique would require a reduction of the areas to be searched or the cases to be examined, or better, that special features of a problem be used to reduce the dimension of the problem. Thus for example, in studying propagation paths for communication systems in an urban environment, it is desirable to formulate the problem as a 2D problem rather than a 3D problem [9]–[11]. Recently an MSA approach has been successfully applied to solve the general problem for a collection of straight lines, the lines being the edges of wedges [8], [12]. In this case the MSA takes the form of a multi-dimensional generalization of the Newton Algorithm. The path function for this problem is

convex, and this convexity implies that the solution is unique, and that the Newton Algorithm will always converge to a solution. However, in a problem involving ellipsoidal surfaces, the , underlying space of N-tuples , is more topologically complex, which leads to questions regarding the uniqueness of solutions and the convergence of the algorithm. These mathematical issues are discussed in the Appendix. II. REFLECTIONS FROM A SINGLE SURFACE A. Monostatic Reflections Since our solution to the general problem reduces to calcufor a single surface , a brief account of the lations of latter problem is now in order. For the problem of monostatic scatter from a single surface , we are given a point external to and we seek to find a point which minimizes the path function , subject . It is well known and easily established to the constraint that the stationary points of the function are points at which the vector is normal to the surface, i.e., the solutions to (7) denotes the outward pointing unit normal to at . where For a closed convex surface the solution set to (7) will contain is an absolute maximum and one one solution for which is an absolute minimum, the latter by definition for which . being We now assume that is defined by the equation , where is a smooth differentiable function. Then since the grais the outward pointing normal to the surface dient , the specular points are the points for which (8) for some positive number . In principle one could solve (8) to obtain as a function of and : . Substituting , we get the equation this result into the equation , which, in principle, can then be solved to . With this solution obtain as a multi-valued function of in hand, it is then an easy matter to determine which t-value . provides a minimum for It is well known that if is an ellipsoid the problem of calreduces to the solution of a polynomial equaculating tion [13, p. 119]. This result can be established by the procedure just outlined. For an ellipsoid centered at the origin we have , where is a symmetric matrix with pos, itive eigenvalues. Equation (8) then becomes from which we get (9) Substituting this expression into the equation the result

we get

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Using a rectangular coordinate system in which ized, this last equation can be written as

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is diagonal-

and Setting defined by (14) we have

, for the function

(10) are the eigenvalues of and where are the coordinates of . Clearing fractions, (10) becomes a polynomial equation in , which is of degree 6 if the eigenvalues are all distinct. However, we note that the left hand side of (10) is positive when because is assumed to be external to , whereas the left hand side decreases monotonically to ( 1) as . Hence there will be a unique positive value of which satisfies (10). Denoting this value of by , the quantity can be found by using Newton’s algorithm (11) where denotes the left hand side of (10). Due to the convexity of the convergence of to is rapid, and the sois obtained by setting in (9). lution Returning now to the case of a general surface defined by , it is easily established that when is the equation sufficiently close to we have the first order approximation (12) From (12) it follows that a first order approximation for the minto is given by imum distance from a point

where we have set (16) Hence, for

, (15) reduces to (17)

Comparing this result with (7), we see that if the quantities and were somehow known (which they are not), the solution . Otherwise to our bistatic problem would be given by and satisfy put, suppose that a point and the quantities the following three identities:

Then is the specular point that minimizes the right hand side of (14). This suggests the following algorithm: Initialization: The initial estimates for and , denoted by and respectively, are taken to be the minimum distances from and to , i.e.,

(13) We often use the approximation (13) as a sanity check to verify does in fact that a calculated position of a pecular point place the point on its surface, i.e., we require that

(18a) The initial estimate for the specular point , is then given by

, denoted by

(18b) Updating: Thereafter, for

B. Bistatic Reflections For the problem of bistatic reflection from a single surface we are given two distinct points and external to , and we wish to find the point which minimizes the path function

, we set

(18c) and

(14) (18d) defined for points in space, For a general function denote the restriction of to a surface . The stationary let are those points for which the tangential points of components of all vanish; hence the stationary points for which are those points (15)

which, This algorithm generates a series of points if it converges, will converge to the specular point. To obtain a practical stopping criterion, we recall that at a specular point the incident ray and the reflected ray must satisfy Snell’s reflection law according to which the angles of incidence

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and reflection are equal. These angles are calculated according to

(19) In practice, after each iteration we calculate the angles of incidence and reflection, and when these angles differ by a small preset amount, say 0.1 , the process is terminated, and the last is accepted as the solution. This criterion calculated value of is justified in the Appendix, where it is shown to be suitable for ship sized complex radar targets. We also impose a maximum on the allowed number of iterations. If the stopping criterion is not achieved within the allowed maximum, the process is terminated. Empirically, we find that this only happens when blockages occur. For more details see Section IV. III. REFLECTIONS FROM MULTIPLE SURFACES We now consider the general problem discussed in Section I, . The algorithm used to obtain a solution to this with problem is a modification of the technique used in Section II-B to solve the bistatic problem for a single surface. Setting

Initialization: For the initial estimates , we first set

Then, if , we let denote the integral part of , we set and for increasing in the range

while for decreasing we set

in the range

Updating: Thereafter, for

,

,

, we set

and (see equation at the bottom of the following page). (Note that there is no stipulation for the values and ). As in Section II-B, the iteration is continued until the angles of incidence and reflection differ by less than 0.1 at each bounce. IV. EXAMPLES A. Reflections From Spheres

and letting (1), we have

denote the path function

We now let denote the specular points, or estimates thereof. As in the case of bistatic scatter from a single surface, the equations above suggest the following algorithm.

We recall that the iterative algorithm described in Section III is a series of successive approximations where at each step we obtain an estimate for the positions of all the specular points. At each of these estimated positions we calculate the angles of incidence and reflection, and the process is stopped when the angles of incidence and reflection differ by less than 0.1 degree, at which point the last calculated values of the specular points are accepted as the solution. The process is also terminated if the stopping criterion is not attained after a certain preset number of iterations, usually taken to be 15. Empirically, we find that the stopping criterion is not attained when a physically meaningful solution does not exist due to the existence of blockages. In such cases the angles of incidence and reflection at one or more of the specular points are complementary rather than equal. In these cases the algorithm does in fact converge, but the stopping cri” is not terion “ satisfied. In general we find that the speed of convergence is only weakly dependent on , but more strongly dependent on the angles of incidence. Specifically, convergence is slower when one or more of the angles of incidence is close to 90 ; in other words, when the configuration is close to one involving a blockage. In the examples shown below the points and and the surfaces are rather close, but we have found that the rate of convergence is faster when the complex of points and surfaces is more spread out. This is to be expected since when a point is very distant from a surface , the distances undergo only a very small fractional change as varies over , so that the distances , , and can be expected to converge more rapidly to their limiting values.

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Fig. 1. Schematic view of specular reflections from three spheres. Solution obtained with four iterations.

Fig. 2. Schematic view of the same three spheres shown in Fig.1, but with the positions of p and q moved to produce a blockage.

Our first set of examples involves spheres, and for a sphere with center and radius we have

Fig. 3. Schematic view of specular reflections from a configuration of five spheres showing another type of blockage not detected by the algorithm.

Fig. 4. Monostatic specular reflections from a “ladder” of ten spheres.

Fig. 1 shows an application of the algorithm to a simple configuration of three spheres with different radii. For graphical ease the points , and the sphere centers are all assumed to be co-planar. The large open circles represent the spheres and the small filled in circles represent the calculated specular points.

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Fig. 5. (a)–(c) Examples of bistatic specular reflections from a ladder of ten spheres whose positions have been perturbed by a random number generator.

In this case only four iterations were required to obtain the solution, as indicted in the figure. In Fig. 2 everything is the same except that we have moved the point so as to achieve a blockage. We have written because the stopping criterion is not attained since the angles of incidence and reflection at the third sphere are complementary rather than equal. However, the algorithm does produce the solution depicted in the figure with only four iterations, and it would be easy to program a modified stopping criterion that would include this possibility. A blockage of another type is shown in Fig. 3. Here the last sphere in the sequence (the one in the center, labeled #5) blocks the path between the second and third spheres; however, the algorithm doesn’t sense this, and has no difficulty in obtaining a solution with 5 iterations. In Fig. 4 we show a “ladder” of ten spheres. The source and observation point are co-located, and only three iterations were required to obtain a solution, as indicated in the figure. In Fig. 5(a)–(c) a random number generator was used to perturb the positions of the sphere centers in the x-direcand are co-located, and in tion. In Fig. 5(a) the points Fig. 5(b) and (c) the source point is moved off to the left, so as to obtain a larger angle of incidence at the first sphere. As these angles of incidence are increased, a larger number of iterations is required to obtain a solution. In Fig. 5(a)–(c) these angles of incidence are 31.2 , 70.2 , and 84.8 , respectively. We have found that when the number of spheres is increased from 10 to 20, the number of iterations required for a solution remains approximately the same. When the number of spheres in increased beyond 20 we sometimes find that the required number of iterations actually decreases. The algorithm can be used to calculate multipath effects, such as that shown in Fig. 6. Here we consider bouncing rays that take a path that hits the first sphere twice; viz., the path . The solution is obtained by telling the algorithm that there are three spheres, and then giving the third sphere the same parameters as the first. In all the examples thus far shown, the points , and the sphere centers are all coplanar. This was done for ease of graphical presentation, and the coplanar condition is not at all required

Fig. 6. Specular reflections from two spheres showing a multipath effect.

Fig. 7. 3D view of specular reflections from a configuration of four spheres whose parameters are shown in Table–I.

for the algorithm’s rapid convergence. In fact, the algorithm has been tested on non-coplanar complexes of spheres whose sizes and positions are set by using a random number generator. An example (not using a random number generator) is shown in Fig. 7. The input parameters for Fig. 7 are shown in Table I. Table II shows how the algorithm converges to the solution for this given

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TABLE I INPUT PARAMETERS FOR FIG. 7

, defined according to

, and let the quantity

be

(20) so that

set of input data. The column labeled S# shows the sphere identifying number, and for each iteration the table shows the xyz-coordinates of the estimated positions of the specular points as well as the angles of incidence and reflection calculated according to (19). The “Initial Configuration” shows the xyz-coordinates of the initial estimates dedata shown under the heading scribed in Secton III. The “Fifth (FINAL) Iteration” are the final estimates for the specular point positions. We recall that the iteration is terminated when the angles of incidence and reflection differ by less than 0.1 at each of the spheres.

The expression for the scattered field will contain the phase factor , where is the wave number. Therefore an acin the escurate calculation of phase requires that the error timate of satisfies the inequality , where is the radar wavelength, and the parameter is given some preset . Let denote the unsigned difference bevalue, usually tween the angles of incidence and reflection. In Section II-B we , where is given discussed a stopping criterion a preset value. Using the fact that the surface of in a neighborhood of is approximated by its tangent plane, one can establish the second order approximation

Hence the condition where

is satisfied if

, (21)

Setting limit on

equal to a given a preset value imposes an upper ; viz, from (21) we get the condition (22)

B. Other Surface Types The algorithm has also been tested on collections of ellipsoids with arbitrary positions and orientations, and it has been found that the algorithm works as well for ellipsoids as it does for spheres. In our final two examples we consider problems in which a flat planar reflecting surface is involved. Two examples are shown in Fig. 8(a) and (b). In these examples we consider the , optical path where the plane is taken to be the plane . In Fig. 8(a) , while in Fig. 8(b) the positions of the point and the two ellipsoids are unchanged while the point has been made to be coincident with , so that the configuration is monostatic rather than bistatic. Although there were no surprises, we did find that the number of iterations required to achieve convergence was usually larger than that for a similar configuration of three ellipsoids—roughly by a factor of 2. APPENDIX MATHEMATICAL APPENDIX A. Accuracy Requirements In the following analysis we first suppose that we are dealing with a complex target . Suppose that is one of the links in the chain . Let be the specular point and set ,

The preset value was , and setting in (22) we get the condition , which for an X-band radar amounts to the requirement that . But , so that for a complex frigate sized target we are guaranteed that , which is well within the 2.7 km limit prescribed by condition (22), with and . Consider now the case of a simple target . Since , the stopping criterion would not be sufficiently stringent if the source and observation point were both more than 5.4 km distant from the target. Hence it can happen that the stopping criterion for a simple target can be more stringent than that for a complex target of which the simple target forms a part. B. Convergence This section is devoted to open questions concerning the uniqueness of solutions and the convergence of the algorithm. Uniqueness. An extended Fermat principle states that light will move along paths for which the path length function is stationary, but not necessarily a minimum [1]. For the general problem involving closed convex reflecting surfaces it can stationary paths. (This is be asserted that there are at least a standard result from Morse critical point theory which follows from the fact that the Euler characteristic of the topological product of topological spheres is equal to [14].) There

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TABLE II CONVERGENCE OF THE ALGORITHM IN THE FIG. 7 COMPLEX

must be at least one path at which the path length is minimized, one at which it is maximized, and according to Morse Theory there necessarily must be other paths at which the path length is neither a local minimum nor a local maximum, these being the so-called saddle solutions [14]. Our algorithm is not capable of generating non-minimizing stationary paths. However, based on empirical studies of the one and two sphere cases, we conjecture that for complexes consisting entirely of closed convex bodies these non-minimizing stationary paths always involve blockages, and hence are of no interest in the context of geometrical optics. If this conjecture turns out to be is false, then non-minimizing stationary paths could be as important as the minimizing paths.

The amplitude of the component of the scattered field produced by each one of the specular points is a function of the surface curvature at that point, while the critical point type (maximum, minimum or saddle) only affects the phase [15]. Convergence. Returning for the moment to the general fixed with its MSA solution (5), when point problem there is more than one fixed point then there necessarily must be some initial “guesses” for which the sequence (5) does not converge. The reasons are topological, and will not be further discussed, except to say that the set of all initial guesses for which the MSA does not converge is often a fractal object with a very complicated structure. (This is the case, e.g., for Newton’s of algorithm applied to locate the zeros of a polynomial

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Fig. 8. Schematic view of specular reflections from a configuration of two ellipsoids and a flat planar surface. (a) A bistatic example. (b) A monostatic example.

degree three or higher in the complex plane. See, e.g., the color plates contained in [16].) Applying this general observation to our specific problem, from what has just been said we know that there is more than one fixed point, and this raises the question as to whether the initialization schemes used in Section II-B (for ) and Section III (for ) guarantee the convergence of the algorithm. Empirically, we find that our algorithm always converges unless there are blockages, but we have no mathematical proof that this is generally the case. REFERENCES [1] R. P. Feynman, Lectures on Physics. Reading, MA: Addison-Wesley, 1963. [2] J. L. Shaeffer, “Physics and overview of electromagnetic scattering,” in Radar Cross Section, E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Eds. Dedham, MA: Artech House, 1985, ch. 3. [3] F. Riesz and B. Sz.-Nagy, Functional Analysis. New York: Frederick Ungar, 1955. [4] G. F. Simmons, Introduction to Topology and Modern Analysis. New York: McGraw-Hill, 1963. [5] J. Dieudonne, Foundations of Modern Analysis. New York: Academic Press, 1960. [6] H. A. Antosiewicz, “Fixed point theorems and ordinary differential equations,” in MAA Studies in Mathematics Vol. 14. Washington, DC: The Mathematical Association of America, 1977. [7] C. Goffman, “Preliminaries to functional analysis,” in MAA Studies in Mathematics Vol. 1. Washington, DC: The Mathematical Association of America, 1962. [8] G. Carluccio and M. Albani, “An efficient ray tracing algorithm for multiple straight wedge diffraction,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3534–3542, Nov. 2008.

[9] A. S. Alvar, A. Ghorbani, and H. Amindavar, “A novel hybrid approach to ray tracing acceleration based on pre-processing and bounding volumes,” Progr. Electromagn. Res., vol. PIER 82, pp. 19–32, 2008. [10] K. Rizk, J.-F. Wagen, and E. Gardiol, “Two-dimensional ray-tracing modeling for propagation prediction in microcellular environments,” IEEE Trans. Veh. Technol., vol. 46, no. 2, pp. 508–518, 1997. [11] G. Liang and H. L. Bertoni, “A new approach to 3-D ray tracing for propagation prediction in cities,” IEEE Trans. Antennas Propag., vol. 46, no. 6, pp. 853–863, June 1998. [12] P. Bagnerini, A. Buffa, and A. Cangiani, “A fast algorithm for determining the propagation path of multiple diffracted waves,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1416–1422, May 2007. [13] E. F. Knott, “High frequency RCS prediction techniques,” in Radar Cross Section, E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Eds. Dedham, MA: Artech House, 1985, ch. 5. [14] J. Milnor, Morse Theory. Princeton, NJ: Princeton U. Press, 1963. [15] W. B. Gordon, “High frequency approximations to the physical optics scattering integral,” IEEE Trans. Antennas Propag., vol. 43, no. 3, pp. 427–432, March 1994. [16] M. R. Schroeder, Number Theory in Science and Communication, 2nd ed. Berlin: Springer-Verlag, 1986. William B. Gordon (SM’87) was born in Washington, DC. He received the B.S. and M.S. degrees in mathematics from George Washington University, in 1959 and 1960, respectively, and the Ph.D. degree in mathematics from The Johns Hopkins University, Baltimore, MD, in 1968. He retired from the Radar Division, Naval Research Laboratory, Washington, DC, after 32 years of federal service. He currently works as an independent researcher at the Naval Research Laboratory Division, GTEC, Inc., Crofton, MD. He is the author of more than 30 papers on the application of differential geometry to problems in dynamical systems, signal processing, and electromagnetic scattering. Dr. Gordon is a member of the American Mathematical Society and the Mathematical Association of America.

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On the Convergence of the Eigencurrent Expansion Method Applied to Linear Embedding via Green’s Operators (LEGO) Vito Lancellotti, Senior Member, IEEE, Bastiaan P. de Hon, and Anton G. Tijhuis, Member, IEEE

Abstract—The scattering from a large complex structure comprised of many objects may be efficiently tackled by embedding each object within a bounded domain (brick) which is described through a scattering operator. Upon electromagnetically combining the scattering operators we arrive at an equation which 1 of the structure: involves the total inverse scattering operator We call this procedure linear embedding via Green’s operators (LEGO). To solve the relevant equation we then employ the eigencurrent expansion method (EEM)—essentially the method of moments with a set of basis and test functions that are approxi1 (termed eigencurrents). We mations to the eigenfunctions of have investigated the convergence of the EEM applied to LEGO in cases when all the bricks are identical. Our findings lead us to formulate a simple and practical criterion for controlling the error of the computed solution a priori. Index Terms—Boundary integral equations, composite structures, domain decomposition method, eigencurrent expansion method, equivalence principle, method of moments (MoM).

I. INTRODUCTION

L

INEAR embedding via Green’s operators (LEGO) is a domain decomposition method (DDM) particularly suited for solving the scattering and radiation of electromagnetic (EM) waves from (both 2-D and 3-D) large composite structures comprised of many bodies [1]–[4]. In the LEGO concept, similarly to other DDMs (e.g., [5]–[9]), we tear apart the structure into its constituent elements and we embed them within bounded (possibly interconnected) sub-domains, dubbed bricks. After describing each brick by means of a scattering operator, we capture the underlying physics of the whole original structure (i.e., the multiple scattering among the objects) by means of the total inverse scattering operator [2], [4], which can be written analytically in a formal fashion. has to be solved for the (equivaThe equation involving by a suitable numerical method. To lent) scattered currents this purpose, a direct application of the method of moments Manuscript received November 13, 2009; revised March 24, 2010; accepted April 09, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported by the Postdoctoral Fund TU/e project no. 36/363450 and performed in the framework of the MEMPHIS project (http:// www.smartmix-memphis.nl). The authors are with the Department of Electrical Engineering, Technical University of Eindhoven, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055804

(MoM) [10] with some kind of sub-domain basis functions, such as the Rao-Wilton-Glisson (RWG) [11] div-conforming vector elements, may not be advisable when a large 3-D problem is addressed. In fact, the matrix of the resulting algebraic system becomes too big to be stored and formally inverted using a direct solver, even for moderate numbers of bricks. In order to handle larger EM problems, in [2, Section IV], we proposed the eigencurrent expansion method (EEM), as an alternative to the bare MoM. In short, the EEM consists of using a set (termed eigencurrents) of approximated eigenfunctions of to represent . To obtain the scattering operators of the bricks ) we actually resort to the MoM. Hence, the EEM (and hence is practically implemented as a basis change from the underlying set of RWG functions. In [2] we also demonstrated that the eigencurrents can be separated into two types, i.e., coupled and uncoupled. This observation enabled us to reduce the system matrix (in the basis of the eigencurrents) to block-diagonal form with just two blocks. The first one—associated with the coupled eigencurrents and fully populated—is ordinarily quite small as compared to the size of the whole matrix. Conversely, the other block—associated with the uncoupled eigencurrents and possibly huge—is diagonal and therefore it is effortlessly stored and (formally) inverted. No clear transition exists between the two types of eigencurrents: Hence one may expect the computed solution to depend on the relative dimension of the sets of coupled and uncoupled eigencurrents. Concerning this, we numerically demonstrated in [2, Section V] that the EEM does converge, when the number of coupled eigencurrents is gradually increased (at the expense of the number of uncoupled eigencurrents). We also argued that a rule for dividing the eigencurrents is hard to state, for many factors come into play. However, in an attempt to shed light on the convergence properties of the EEM applied to LEGO, we have worked out two purposeful case studies in the notable instance when the structure involves identical objects. Upon analyzing the eigenvalues of the scattering operator, we are indeed able to formulate a quite general criterion for determining the number of coupled eigencurrents a priori for a desired level of approximation. To be more specific, in Section III we first investigate how the size and the shape and the EM properties of the body enclosed in a brick affect the spectrum of the relevant scattering operator. Secondly, we relate the error committed on the scat(calculated through the EEM) to the eigentered currents values of the scattering operators. It turns out that there is a simple relationship (practically linear) between the error and the

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eigenvalue pertinent to the highest-order coupled eigencurrent. What’s more, this relationship appears virtually independent of the bricks’ content and only weakly dependent on the frequency and the distance of the LEGO bricks. Then, in Section IV we discuss our findings and draw up a practical rule for estimating the error of the computed solution or, conversely, for choosing the number of coupled eigencurrents required to attain a desired level of accuracy. We mention that a convergence study was conducted for the synthetic-functions approach [7]. However, [7] does not provide an a-priori criterion for truncating the basis of the synthetic functions. To the best of our knowledge, the quest for a criterion for controlling the error has not been attempted so far for other DDMs. II. OVERVIEW OF LEGO/EEM AND NOTATION In what follows we conform to the assumptions and definitions adopted in our previous work on LEGO and EEM [2]. To reiterate briefly, we deal with a 3-D structure comprised of PEC or penetrable objects (immersed in a homogeneous background medium) which we embed in as many LEGO bricks , . We employ the MoM to compute the scattering operator of and the transfer operator from to , . To this purpose, we model both the surface of and the boundary with 3-D surface the object inside and RWG functriangular meshes on which we define tions, respectively. The algebraic counterparts of and are then [2, Section IV]

is rank-deficient whenever , as the MoM matrix in (1) has size . We will go back to this point in Section IV-A. For the time being, we solely observe that , “contributed” by a the number of coupled eigencurrents or, stated another way, the brick, cannot exceed , if they eigencurrents corresponding to the null space of exist, are uncoupled. Keeping this in mind, it is evident that inverting (3) through the EEM and as many coupled eigencurrents as possible is equivalent to solving (3) with the original basis of RWG functions1. In fact, in this instance the EEM boils down to a mere basis change with no order reduction (see [2, Section IV]). For this reason we assume the latter solution as the . reference for our comparisons, and we denote it by Then, as a measure of the global accuracy of the solution computed with the EEM, we define the relative error on the scattered current coefficients as (cf. [7, Eq. (33)]) (5) where denote the vector 2-norm [13] in the space spanned . To assess the pointwise accuracy of , by the rows of we introduce the local deviation with respect to the reference solution (cf. [7, Eq. (34)]), namely,

(6) where

(and analogously

) are defined in (4).

(1) III. TWO CASE STUDIES

(2) where the relevant operators are listed in [2, Tables I–III]. Finally, the algebraic equation to be solved is: (3) where the system matrix has rank . The diagonal blocks are the (formal) inverse of (1), whilst the off-diagonal of , conblocks are the negative of (2). The column vectors tain the expansion coefficient of the scattered and incident cur, viz., rent densities defined on either side of

(4) where a normalization factor is included in , with being the intrinsic impedance of the background medium. To solve (3) with the EEM, we form a basis of by juxtaposing the eigenapproximated eigencurrents of vectors of [2]. Thereby, vanishes everywhere but on the th brick, and two distinct eigencurrents with are orthogonal with respect to the standard inner product. Moreover, it is worth noticing that the scattering matrix

The structures we considered are: A) an aggregate of perfect electric conducting (PEC) cubes and B) an aggregate of penetrable cylinders; in both cases the incident field where was the plane wave with the wavelength in vacuum. Below we first describe the case studies. The discussion is postponed to Section IV for the sake of clarity. A. Scattering From PEC Cubes With Varying Size The PEC cubes (edge ) are arranged in a regular rectanplane. gular pattern (lattice constant ) parallel to the In accordance with LEGO, we enclosed the cubes within touching cubic bricks (edge ). We considered five realizations of the cubes with increasing edge length, . Additionally, we alnamely, lowed for three different electrical sizes of the bricks, viz., . For the sake of clarity, in Fig. 1 we report the case as well as the corresponding LEGO bricks. Also shown are 1To verify our assumption, for the 4-cube case study discussed in Section III-A we have computed the scattered current coefficients through the baseline MoM as follows. By solving an EFIE [12] over the cubes (with 4 RWG functions) we have obtained the current density coefficients [ ]. Hence, = [ ] [ ]([ ]) , with 2 functions we have computed over each brick’s surface. The local deviation [defined similarly to (6)] between the LEGO/EEM solution with = minf2 g and results no larger than 10 in all of the instances we considered.

q

P

N

P q N ;N

N

q

q

N

LANCELLOTTI et al.: ON THE CONVERGENCE OF THE EIGENCURRENT EXPANSION METHOD APPLIED TO LEGO

Fig. 1. Case study: (left) four PEC cubes arranged in a rectangular lattice (the realization a=d bricks each one enclosing a cube.

[

]

Fig. 2. Eigenvalues of S pertinent to the four cubic bricks shown in Fig. 1 as a function of the index p for d= : and for increasing values of a=d. The circles  mark the eigenvalues whose index p N  N . Inset: vs. a=d and cartoon of a cube inside its LEGO first four eigenvalues of S brick.

()

[

= 0 33357

]

=

2

the 3-D surface triangular meshes that are support to the underlying set of RWG basis functions employed in conjunction with MoM. In order to keep the mesh density at a constant value, over each cube’s surface we adjusted the . Consequently, we number of facets while increasing to , whereas the number of set was fixed at . With RWG functions on these choices the average (electric) length of the edges in the and is and mesh on , respectively. (1) To begin with, we obtained the scattering operators through the MoM: As a matter of fact, since the bricks are identical, we had to do the calculations for just one brick. Then, for we computed the eigenvalues of . each pair ) are plotted in Figs. 2–4 as a function of The results (i.e., the eigenvalue index and for the three values of . The , whilst the insets show parameter of the lines is the ratio . Finally, the circle on each the first four eigenvalues versus . curve tags the eigenvalue

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= 0:7 is shown) and (right) LEGO model with as many cubic

Fig. 3. Same as Fig. 2 for d=

= 0:66713.

Fig. 4. Same as Fig. 2 for d=

= 1:0007.

Secondly, we repeatedly solved (3) with the EEM, while retaining an increasingly larger number of coupled eigenper brick. To be specific, was set upon currents

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N

N

FOR THE

FOR THE

TABLE II 4-CUBE CASE STUDY AND d=

TABLE III 4-CUBE CASE STUDY AND d=

= 0:66713

= 1:0007

Fig. 5. 2-norm error  of the equivalent scattered current coefficients [q ] relevant to the four cubic bricks shown in Fig. 1 as a function of the eigenvalue pertinent to the last coupled eigencurrent and for increasing values of a=d.

N

FOR THE

TABLE I 4-CUBE CASE STUDY AND d=

= 0:33357

Fig. 6. 2-norm error  of the equivalent scattered current coefficients [q ] relevant to the four cubic bricks shown in Fig. 1 as a function of the eigenvalue pertinent to the last coupled eigencurrent and for increasing values of w=d. Inset: cartoon of the cubic bricks and geometrical quantities.

counting all the eigencurrents whose eigenvalues satisfy , : The relevant values and ) are listed in Tables I–III. (versus the threshold Afterwards, from the knowledge of we calculated the corresponding 2-norm error with (5). In doing so, we obtained . the sets of curves plotted in Fig. 5 for the three values of The quantity on the horizontal axis (note the reversed scale) is , i.e., the magnitude of the eigenvalue corresponding to . Once again the last coupled eigencurrent contributed by . the parameter of the lines is Thirdly, to probe the effect of the lattice constant, we repeated , and the test above for

, and we calculated . The selected values of can be retrieved from Table III as well. The results are plotted , where the parameter is now . in Fig. 6 versus Eventually, for all of the cases above we calculated the local deviation of from (6). As an example, in Fig. 7 we report for the realization , and —which corresponds to and . It can be noted that the largest value taken on by is and that the accuracy of the scattered current den[2] is basically the same. sities and B. Scattering From Dielectric Cylinders With Varying Permittivity The dielectric cylinders (radius and height ) are arranged plane. We emin a regular triangular pattern parallel to the bricks which are hexagbedded the cylinders within onal right prisms (edge and height .) To investigate the effect

LANCELLOTTI et al.: ON THE CONVERGENCE OF THE EIGENCURRENT EXPANSION METHOD APPLIED TO LEGO

Fig. 7. Local deviation from the reference solution of (top) normalized J and (bottom) normalized M relevant to the four cubic bricks shown in Fig. 1 for : , d= : and j j  j j . the case a=d

=04

= 1 0007

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Fig. 8. Case study: (top) three dielectric cylinders arranged in a triangular lattice and (bottom) LEGO model with as many hexagonal bricks each one enclosing a cylinder.

10

of the material properties, we considered five different realizations of the cylinders with different relative permittivity, namely, . Instead, we held the relative size of cylinders and bricks fixed and the brick’s size constant ( .) Shown in Fig. 8 are the cylinders and the bricks along with their 3-D surface triangular patching. In particular, the number of RWG functions defined on a brick’s boundary and on a cylinder’s surface is and . Accordingly, the average (electric) length of the edges in the mesh is on and on . As in the previous case study, we computed the scattering operator of one brick. Calculation of the induced current densities on the embedded cylinder was carried out through the PMCHWT set of integral equations [12]. The spectrum of is plotted in Fig. 9 versus ; the parameter of the curves is , whereas the inset displays the first four eigenvalues as a function of . Upon repeatedly inverting (3) by applying the EEM with (see Table IV for the more and more coupled eigencurrents relevant values), we computed and subsequently from (5). The results are presented in Fig. 10 as a function of for and the five values of permittivity selected above.

[

]

Fig. 9. Normalized eigenvalues of S pertinent to the three hexagonal bricks shown in Fig. 8 as a function of the index p for increasing values of " . Insets: vs. " and cartoon of a cylinder inside its LEGO first four eigenvalues of S brick.

[

]

IV. DISCUSSION OF RESULTS A. Spectrum of From Figs. 2–4 and 9 we observe the following. increases as the 1) The decay rate of the spectrum of of the body enclosed in is reduced relative size

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TABLE V CONDITION NUMBERS OF [P

]

AND [X

]

FOR THE

4-CUBE CASE STUDY

Fig. 10. 2-norm error  of the equivalent scattered current coefficients [q ] relevant to the three hexagonal bricks shown in Fig. 8 as a function of the eigenvalue pertinent to the last coupled eigencurrent for l= = 0:13343 and increasing values of " .

N

2)

3)

4)

5)

FOR THE

TABLE IV 3-CYLINDER CASE STUDY

or, from another perspective, the boundary of the brick is moved farther away from the surface of the object. The bigger the object with respect to the enclosing brick, . Apparthe larger the amplitude of the th eigenvalue, ently, this applies to most of the spectrum, but it may not be true for the very first eigenvalues (as confirmed by the inset in Fig. 3) which correspond to the strongest reflections in . view of the physical meaning of It is worthwhile noticing at this point that, as long as the medium embedded within a brick is passive, the eigencannot exceed unity (in magnitude), based values of on physical grounds. Numerical experiments (not reported ’s do approach unity when the enhere) show that the closed object fills the brick almost totally. depends only weakly on the EM The spectrum of properties of the embedded object, with the most evident variations regarding the first eigenvalues (inset of Fig. 9.) does not seem to play The electric size of the bricks a major role, at least in the present numerical experiments. At any rate, for a given geometry, increasing the frequency causes the whole spectrum to drift toward higher values of . , the spectrum of exhibits In cases when an abrupt discontinuity occurring for : From that index on the eigenvalues are theoretically zero and in practice close to or below the threshold of numerical noise for double-precision floating-point operations. However, we emphasize that we are not in trouble with the calculation , even though this entails . of

In fact, as detailed in [2, Section IV], on the one hand, we as such, but to apply the EEM we never compute just invert the eigenvalues corresponding to the coupled eigencurrents, whereas, on the other hand, we need not invert the eigenvalues of the uncoupled eigencurrents (which may be null.) Therefore the stability of the EEM is , as annot endangered, provided , has ticipated in Section II. Only, when is meaningless. to be interpreted formally, because One may also argue, in the light of Figs. 2–4 and 9, that the is very high. In general, spectral condition number of nor [see (1)] are to be blamed for though, neither . To back up our statement, the overall ill-conditioning of and for the 4-cube case in Table V we list study of Section III-A. In practice, happens to be more or less well-conditioned depending on the kind of integral and on the mesh density. Thereby, equations posed on and (and hence the calculation of the inversion of ) is not an issue. is mostly affected by the propAs a matter of fact, and , which relate currents on agation matrices to fields on and viceversa [2, Table III]. These matrices are intrinsically rank-deficient in that the number of basis functions and need not be the same. Rather, on and have to be adjusted independently, according to the commonly accepted meshing criterion (i.e., mesh edge ). Nevertheless, even though and coinand could be smaller than cided, the effective rank of , since the entries of and can be proven to decay to zero when the distance between and is inand become creased. Accordingly, the columns of almost linearly dependent, and a singular value decomposition and reveals that the higher-order singular values of are null or nearly so. B. Relative Error on the Computed Currents From Figs. 5, 6 and 10 we observe the following. 1) The relative error decreases linearly (in a logarithmic , viz., the amplitude of the eigenvalue perscale) with taining to the highest-order coupled eigencurrent. Notice , relathat the actual number of coupled eigencurrents , may be different for different tive to a given value of

LANCELLOTTI et al.: ON THE CONVERGENCE OF THE EIGENCURRENT EXPANSION METHOD APPLIED TO LEGO

geometries and frequencies, as suggested by Figs. 2–4 and 9. appears virtually independent of 2) The relative error the size and the EM properties of the objects and only weakly affected by the frequency. In practice, when increases, diminishes, which shows as a leftward drift of the error curves. improves when the 3) For given frequency and geometry, bricks are located farther away. This effect, however, may , since is not not be explained by the behavior of affected by the relative position of the bricks in space. The latter, though, enters through the transfer operators (2). Based on established multipole expansion methods [14], to become smaller and [15], we expect the entries of smaller when and are set farther and farther away. Consequently, the eigencurrents are more and more attento . This means uated while propagating from is that ever more eigencurrents become uncoupled, if increased. Therefore, holding (and hence ) fixed must yield more accurate results, as confirmed by Fig. 6. C. Number of Coupled Eigencurrents The diagrams of Figs. 5, 6 and 10 can be used in two complementary ways. First of all, entering the plots with , we can determine the accuracy of the solution to (3) obtained coupled eigencurrents. through the EEM applied with Alternatively, and perhaps more fruitfully, entering the plots , whence we determine , i.e., with , we can read off the number of coupled eigencurrents necessary for attaining the desired accuracy. Now, since the error curves in Figs. 5, 6 and 10 are practi(recall the horizontal scale cally straight lines with slope is reversed), by direct inspection we are able to fit the pairs by means of the mathematical linear mapping (7) (which constitutes the vertical intercept) represents where when . the value of From the preceding discussion, it is clear that in (7) ought to be determined specifically for the problem under consideration. Nonetheless, our findings also tell us that is only weakly , , and the EM properties of the emdependent on , bedded object. In the light of Figs. 5, 6 and 10, we conservatively , which should be fine for reasonable values of the relset evant parameters. Thus, for the sake of argument, assuming we , then from (7) we want to solve (3) with an error . From the knowledge of we know that deduce by simply counting the eigenvalues that satisfy the . condition V. CONCLUSIONS Upon investigating two case studies, we have addressed the convergence of the EEM applied to LEGO. In particular, by analyzing the effects that geometrical and EM parameters as well as the frequency have on the eigenvalues of a brick’s scattering operator, we have empirically found that there exists a simple

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relationship between the error on the current and the amplitude of the eigenvalue corresponding to the highest-order coupled eigencurrent. We have translated this law into a compact approximate mathematical formula, which can be practically implemented in a numerical code for choosing the required number of coupled eigencurrents a priori. ACKNOWLEDGMENT The authors would like to thank the Reviewers for their constructive comments which helped improve the manuscript. REFERENCES [1] A. M. van de Water, B. P. de Hon, M. C. van Beurden, A. G. Tijhuis, and P. de Maagt, “Linear embedding via Green’s operators: A modeling technique for finite electromagnetic band-gap structures,” Phys. Rev. E, vol. 72, no. 5, pp. 1–11, Nov. 2005. [2] V. Lancellotti, B. P. de Hon, and A. G. Tijhuis, “An eigencurrent approach to the analysis of electrically large 3-D structures using linear embedding via Green’s operators,” IEEE Trans. Antennas Propag., vol. 57, no. 11, pp. 3575–3585, Nov. 2009. [3] V. Lancellotti, B. P. de Hon, and A. G. Tijhuis, “A total inverse scattering operator formulation for solving large structures with LEGO,” in Proc. Int. Conf. on Electromagnetics in Advanced Applications, ICEAA’09, Sep. 2009, pp. 335–338. [4] V. Lancellotti, B. P. de Hon, and A. G. Tijhuis, “Sensitivity analysis of 3-D composite structures through linear embedding via Green’s operators,” Progr. Electromagn. Res., vol. 100, pp. 309–325, Jan. 2010. [5] J. Yeo, V. V. S. Prakash, and R. Mittra, “Efficient analysis of a class of microstrip antennas using the characteristic basis function method (CBFM),” Microw. Opt. Technol. Lett., vol. 39, no. 6, pp. 456–464, 2003. [6] M. K. Li and W. C. Chew, “Wave-field interaction with complex structures using equivalence principle algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 130–138, Jan. 2007. [7] L. Matekovitz, V. A. Laza, and G. Vecchi, “Analysis of large complex structures with the synthetic-functions approach,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2509–2521, Sep. 2007. [8] P. Ylä-Oijala and M. Taskinen, “Electromagnetic scattering by large and complex structures with surface equivalence principle algorithm,” Waves Random Complex Media, vol. 19, no. 1, pp. 105–125, Feb. 2009. [9] G. Xiao, J.-F. Mao, and B. Yuan, “A generalized surface integral equation formulation for analysis of complex electromagnetic systems,” IEEE Trans. Antennas Propag., vol. 57, no. 3, pp. 701–710, Mar. 2009. [10] R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968. [11] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, May 1982. [12] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. Piscataway, NJ: IEEE Press, 1998. [13] G. H. Golub and C. F. van Loan, Matrix Computations. Baltimore, MD: Johns Hopkins Univ. Press, 1996. [14] N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic problems,” IEEE Trans. Antennas Propag., vol. 40, no. 7, pp. 634–641, Jun. 1992. [15] J. D. Jackson, Classical Electrodynamics, 3rd ed. Chichester, U.K.: Wiley, 1999. Vito Lancellotti (M’04–SM’09) was born in Torino, Italy, in 1968. He received the Laurea (M.Sc.) degree (with honors) in electrical engineering and the Ph.D. degree in electronics and communications both from Politecnico di Torino, Italy, in 1995 and 1999, respectively. In early 1999, he joined the Telecom Italia Lab, Torino, as a Senior Researcher and was involved in projects concerning TCP/IP and ATM networks. In June 2000, he became Senior Researcher at the Milan-based subsidiary of Corning (now Avanex), where he worked on broadband electro-optic lithium niobate modulators and optical waveguides. From 2002 to 2008, he served as a Research Fellow and Lecturer in the Department of Electrical Engineering, Politecnico di Torino, where he substantially contributed to the development of the TOPICA code,

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devised for analysis and design of plasma facing antennas for magnetically controlled nuclear fusion. In 2005, he was appointed Visiting Scientist at Massachusetts Institute of Technology, Cambridge, and in 2007 conducted research at the Max-Planck-Institut fur Plasmaphysik, Garching, Germany. As of April 2008, he has been a Research Fellow and Lecturer with the Faculty of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands. His current research interests mainly concern efficient techniques for the computational modeling of electromagnetic fields in large structures.

Bastiaan P. de Hon was born in 1966 in Amstelveen, The Netherlands. He received the M.Sc. and Ph.D. degrees (both cum laude) in electrical engineering from the Delft University of Technology, Delft, The Netherlands, in 1991 and 1996, respectively. Since 1996, he has been with the Electromagnetics Group, Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands. From 1996 until 2000, he was on a fellowship awarded by the Royal Netherlands Academy of Arts and Sciences and, from 2000, he was a Lecturer. He has been a summer student at CERN, Geneva (Switzerland) and at Schlumberger Cambridge Research, Cambridge, U.K., and a Visiting Scientist at the University of Tel Aviv, Tel Aviv, Israel, and the University of Glasgow, UK. His research interests include theoretical and numerical aspects of electromagnetic, acoustic and elastic wave phenomena.

Dr. de Hon received the Steven Hoogendijk Award in 1998 for his Ph.D. thesis.

Anton G. Tijhuis (M’88) was born in Oosterhout N.B., The Netherlands, in 1952. He received the M.Sc. degree in theoretical physics from Utrecht University, Utrecht, The Netherlands, in 1976, and the Ph.D. degree (cum laude) from the Delft University of Technology, Delft, The Netherlands, in 1987. From 1976 to 1986 and 1986 to 1993, he was an Assistant and Associate Professor with the Laboratory of Electromagnetic Research, Faculty of Electrical Engineering, Delft University of Technology. In 1993, he became a Full Professor of electromagnetics with the Faculty of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands. He has been a Visiting Scientist with the University of Colorado at Boulder, the University of Granada, Granada, Spain, the University of Tel Aviv, Tel Aviv, Israel, and with McDonnell Douglas Research Laboratories, St. Louis, MO. Since 1996, he has been a Consultant with TNO Defence, Security, and Safety, The Hague, The Netherlands. His research interests are the analytical, numerical, and physical aspects of the theory of electromagnetic waves. In particular, he is involved with efficient techniques for the computational modeling of electromagnetic fields and their application to detection and synthesis problems from several areas of electrical engineering.

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Unified Time- and Frequency-Domain Approach for Accurate Modeling of Electromagnetic Radiation Processes in Ultrawideband Antennas Diego Caratelli and Alexander Yarovoy, Senior Member, IEEE

Abstract—A singularity-expansion-method-based methodology is proposed for the accurate time- and frequency-domain analysis and modeling of wave radiation processes in ultrawideband antennas. By means of this approach, the transient electromagnetic field distribution in the Fraunhofer region is presented in analytical closed form as the superposition of outgoing propagating nonuniform spherical waves. The time dependence of the wave amplitudes is determined by the resonant phenomena occurring in the structure. Analytical expressions for the antenna gain and effective height are derived. The major novelties of the presented formulation lie in a dedicated two-step vector fitting procedure for a minimal pole/residue spherical harmonic expansion of the time-domain equivalent electric and magnetic currents excited on a suitable spherical Huygens surface enclosing the antenna under analysis, as well as in the introduction of a new class of incomplete spherical Bessel functions useful to describe transient wave phenomena in truncated structures. The proposed approach is validated by application to an ultrawideband resistively loaded bow-tie antenna for ground-penetrating radar applications. Index Terms—Incomplete spherical Bessel functions, singularity expansion method (SEM), spherical harmonic expansion, time-domain effective height and gain, transient wave radiation, ultrawideband (UWB) antennas.

I. INTRODUCTION

T

HE characterization of ultrawideband (UWB) antenna radiation processes in the far-field region is performed by the numerical computation of radiation integrals involving the free-space dyadic Green’s functions [1]. This is usually carried out in the frequency domain, and the spatial distribution of the electromagnetic field is typically presented by a set of radiation patterns at different frequencies. However, such an approach does not provide an integral physical insight into the mechanisms which are responsible for the electromagnetic behavior of the structure, and requires large computational resources and storage of a large amount of data (think about a series of three-dimensional radiation patterns over a large frequency band). Some attempts have been made in the recent past to come to more compact semi-analytical representations of Manuscript received October 05, 2009; revised March 16, 2010; accepted March 28, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This research has been partly carried out in the framework of the EU-sponsored project ORFEUS, and the STARS program funded by the Dutch government. The authors are with Delft University of Technology, International Research Centre for Telecommunications and Radar, Delft 2628 CD, Netherlands (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055800

the electromagnetic field distribution excited by complex UWB antennas [2]. In this paper, a completely different approach, generalizing and expanding the model in [3], is proposed. The radiated field is presented directly in the time domain as the superposition of outgoing propagating non-uniform spherical waves related to the complex resonant processes occurring in the structure. To this end, any time-domain integral-equation or finite-difference technique [4], [5] can be adopted to carry out the full-wave analysis within a volume surrounding the antenna, and to determine on-the-fly, in step with the numerical simulation, a spherical harmonic expansion [6] of the equivalent electric and magnetic currents excited on a suitable Huygens surface enclosing the radiating structure [7]. Then, a pole/residue representation [8], [9] of the aforementioned currents is derived by a specially developed time-domain vector fitting procedure [10]–[12]. This allows for the convenient evaluation of the electromagnetic field distribution in the Fraunhofer region by applying a modified singularity expansion method (SEM), and making proper use of the theory of advanced special functions for mathematical physics [13]. As known, special functions are frequently adopted in different fields, such as acoustics, optics, and electromagnetics, to express exact or approximate analytical solutions of complex physical problems [13]. Recently, incomplete Hankel and modified Bessel functions have been introduced in [14]–[17] to determine, in analytical closed form, the spatial distribution of the electromagnetic field associated with progressive and evanescent wave contributions excited in truncated cylindrical structures. In this paper, the new class of the incomplete modified spherical Bessel functions of arbitrary order, useful to describe electromagnetic wave phenomena concerning spherical structures, is presented. The properties of the considered class of functions, with particular attention to the relevant governing differential equation and recurrence formulae as well as asymptotic expansions for small and large arguments, are investigated and discussed thoroughly. The general theory is then applied to derive a simple analytical expression of the transient electromagnetic field distribution excited by a general antenna in the relevant Fraunhofer region, using the said SEM-based approach. As a result, antenna far-field parameters, such as gain and effective height [18]–[21], can be easily evaluated in closed form in both the time and frequency domains, so providing a deep and meaningful physical insight into wave radiation processes. The paper is organized as follows. Section II describes the singularity expansion method for modeling of transient radiation

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with (2) to evaluate the aforementioned convolution integrals. As discussed in [19], such approximation can be performed once the time-domain Fraunhofer condition is verified (3)

Fig. 1. Antenna enclosed by a spherical Huygens surface S . The radiating structure is assumed to be connected to a uniform transmission line excited by a real voltage generator v (t) with internal resistance R . The reference system adopted to express the field quantities is also shown.

being the pulse-length of the antenna excitation signal. So, under the considered assumption, the transient radiative electromagnetic field excited by the antenna outside the surface is found to be, after some algebra, as the following vector slant-stack transform (SST) of the surface equivalent current densities:

(4) phenomena. The spherical wave expansion coefficients are analytically evaluated in Section III. Section IV details a dedicated time-domain vector fitting procedure for field-related quantities. In Section V closed form expressions of the antenna gain and effective height are derived in terms of the newly introduced incomplete modified spherical Bessel functions. The application of the proposed methodology to the analysis of an ultrawideband resistively loaded bow-tie antenna for ground-penetrating radar (GPR) applications is then presented in Section VI. The concluding remarks are summarized in Section VII.

(5) where , and is the spherical-wave delayed time. The radiation integral appearing in (4) can be conveniently handled by applying a two-step data-fitting procedure aimed to the expansion of the equivalent currents in terms of time-variant vector spherical harmonic functions with proper , the aforementioned currents coefficients. At any time are expressed as the linear combination of orthonormalized surface harmonics [23], defined as (6)

II. POLE/RESIDUE MODELING OF TRANSIENT ELECTROMAGNETIC RADIATION PROCESSES Let us consider an antenna operating in free space, and enhaving radius (see Fig. 1). closed by a spherical surface in the three-dimensional space is the Naturally induced by useful to express the physical polar coordinate system quantities of interest. As known from theory [22], the spatial distribution of the can be readily radiated electromagnetic field for determined by convolving the equivalent current densities , excited along the Huygens sphere with the radiative component of the dyadic Green’s functions, and

with , where , , and . As usual, the quantities and denote the observation and is the speed of light, and is source points respectively, the free-space wave impedance. Being primarily interested in the far-field antenna characteristics, we may use the following approximation [7]:

(1)

being the associated Legendre function of the first kind of degree and order [24]. In particular, it can be shown that the following identity holds [23]: (7) where the superscript denotes complex conjugation. Hence, one can readily obtain (8) where the vector coefficients

,

are computed as

(9) The series expansion (8) is exact as long as goes to infinity. Truncation errors will arise when limiting the sum over to a finite discrete angular bandwidth . To mitigate such problem, according to the theory of the optimal interpolation of radiated

CARATELLI AND YAROVOY: UNIFIED TIME- AND FREQUENCY-DOMAIN APPROACH FOR ACCURATE MODELING

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electromagnetic fields over a sphere [25], [26], the parameter is selected to be as (10)

being the usual ceiling function [24]. In (10) the excess bandwidth factor is used to control the approximation error which [25]. It is worth decreases more than exponentially with noting that (10) becomes identical to the frequency-domain cri, which terion provided in [26] upon the substitution is actually the upper cut-off frequency of the excitation pulse. A dedicated procedure to compute expansion integrals in (9) has been developed, and analytical details are investigated thoroughly in Section IV. Provided that the considered antenna is excited by a finiteduration signal, the current expansion coefficients in (8) can be represented using the following SEM-based expression [8]

(11) where and , are the complex poles and vector residues, respectively, of the exponential terms accounting for the damped natural resonant processes is the usual Heaviside occurring in the structure. In (11) unit-step distribution. From the SEM theory it is known that the entire functions , are needed to accurately describe the earlytime behavior of the antenna [8], [9]. However, as discussed in [20], [21], such terms are usually neglected, and their contribution efficiently taken into account by means of a modified pole/residue representation. So, substituting (8) and (11) into (4)-(5) yields

Fig. 2. Angular domain (# ; ' ; #; '; =t ) of the equivalent currents, excited on the Huygens surface S , contributing, at the spherical-wave delayed time  = t r=c , to the radiated electromagnetic field excited at the observation point r; #; ' .

0 f

As it can be easily recognized, for (see Fig. 2). Using the Laplace integral representation of the Heaviside distribution (15) and exploiting the following projection property of the surface harmonics [27]:

(16) being the canonical modified spherical Bessel function of the first kind and order [24], the radiation integrals in (12) can be evaluated as

(17)

(12) where

where

g

is the infinitesimal solid angle, and

(13) (18)

In (12) denotes the angular domain contributing, at the normalized of the equivalent currents on time , to the radiated electromagnetic field excited at the ob, namely servation point

having introduced the incomplete modified spherical Bessel function of order

(14)

(19)

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To the best knowledge of the authors at the time of writing, this class of special functions for electromagnetics and mathematical physics has not yet been presented in the scientific literature. In particular, it is worth noting that the following identity holds: (20) where the complementary distribution (21) accounting for pseudo-diffraction phenomena, gives a non-zero contribution if the end point is not equal to unity. The analytical properties of the considered class of functions, with particular attention to the relevant governing differential equation and recurrence formulae, as well as the asymptotic expansions for small and large arguments, are derived and discussed thoroughly in Appendix A. Upon combining (17) and (18) with (12), and setting for shortness (22) the time-domain electromagnetic field radiated by the antenna in the Fraunhofer region is obtained as the following superposition of outgoing propagating non-uniform spherical waves attenuating along with the radial distance and time according to the real part of complex poles

M

Fig. 3. Three-dimensional view of the computational mesh adopted to evaluate the spatial distribution of the electromagnetic field within the Huygens sphere S enclosing the antenna under analysis.

Let us assume that the electromagnetic field is known at any discrete node . Under such hypothesis, after setting for shortness , the field distribution excited along can be conveniently computed, by using a local inverse distance weighting (IDW) interpolation technique [28], as

(25)

(23) Finally, using (18) and the shifting properties of the unilateral Laplace transform, the complex frequency-domain equivalent representation immediately reads

the summation being carried out over the triplets relevant to the field samples located in the neighborhood of the , and where deobservation point notes the adopted weighting function. In this way, the Huygens can be currents on readily determined using the tangential component of . According to the theory of the optimal interpolation of radiated fields over a sphere [26], the following cardinal series (CS) representation holds:

(24) where is the free-space complex wave-number. Adopting (23) and (24) one can conveniently determine the antenna characteristics and far-field response to any arbitrary excitation [18]–[21]. III. ANALYTICAL EVALUATION OF SPHERICAL EXPANSION COEFFICIENTS OF SURFACE EQUIVALENT CURRENTS In the presented modeling approach, the electromagnetic field enclosing the antenna distribution over the Huygens surface under analysis is supposed to be evaluated by means of a fullwave time-domain procedure, such as FEMTD, FDTD, FVTD, FIT, MRTD [4], [5], based on a suitable three-dimensional space (see Fig. 3). lattice

(26) with (27)

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and where a trivial reasoning about the vector transformation in spherical coordinates leads to the conclusion that

In (26) the expansion orders by (10) and

and

(28) are given, respectively,

(29) so that the sampling elevation and azimuth angles are evaluated as (30) (31) and . for each As outlined in [25], the CS representation (26) is exact as far as is strictly band-limited to . Anyway, aliasing errors due to the unavoidable violation of such assumption can be controlled by proper selection of the excess bandwidth factor [25], [26]. Finally, the general time-variant spherical harmonic expanrelevant to sion coefficient the surface equivalent currents can be obtained upon combining (26), (27) and (28) with (9), so yielding after some algebra

Fig. 4. Asymptotic computational burden of the developed IDW interpolation procedure carried out in step with the time-domain full-wave simulation aimed to the evaluation of the equivalent currents excited along the Huygens sphere enclosing the antenna under analysis.

It is worth noting that the symbolic computation of expansion , can be conveniently carried out before coefficients the time marching scheme starts; hence, only a minor correction is required in the core of the adopted electromagnetic field prediction algorithm in order to implement (32), resulting in a floating negligible extra computational burden of point operations (FPOs) at each time iteration, as it can be inferred from Fig. 4. IV. TIME-DOMAIN VECTOR FITTING PROCEDURE

(32)

Let us now focus the attention on the modified pole/residue representation of the general time-variant spherical harmonic relevant expansion coefficient to the surface equivalent currents excited on

where (37)

(33)

(34) with

being the Kronecker’s delta and (35)

. Due to the vectorial nature where , the matrix pencil method [29], [30], or of the residues the modified Prony’s algorithms [31], [32] can not be applied by complex exponential functo perform the fitting of tions as in [20], [21]. So, to overcome this limitation, a dedicated non-conventional time-domain vector fitting (TD-VF) procedure has been specifically developed. Such procedure, originally introduced in [10]–[12] to extract frequency-independent equivalent circuits of multiport interconnect structures, is aimed to the derivation of a rational approximation to the Laplace transform , that is of (38)

(36) , denoting the Chebyshev polynomials of first and second kind, respectively, and degree [13].

To this end, a scalar weight function (39)

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with assigned initial poles , and unknown residues is introduced. In addition, let us assume that the following approximation holds: (40) Since the right-hand-side term of (40) features the same complex poles as the weight function, a cancellation beof and the poles of tween the zeros must occur. This condition provides a way to estimate by solving in a least squares sense, for the unknown , the following time-domain equation: residues

is carried out, Once the evaluation of the residues , the poles can be determined by enforcing are computed using a suitable algowhere the zeros rithm, such as the Collins-Krandick technique [33]. The described method, known as pole relocation, avoids the use of ill-conditioned non-linear least squares methods for the direct fitting of (37)–(38). In particular, the pole relocation can as starting poles be iterated using the estimated poles of the scalar weight function at the next iteration. The resulting procedure is stable by construction, but might be non passive because of the least squares solution of (41), which does not guarantee passivity a priori [11], [12]. So, to avoid a possible physical inconsistency of the model, the passivity must be checked and imposed by enforcing at each iteration (46) for all indices , , . The convergence of the described procedure is usually obtained in a few iterations (less than ten), and verified by the following stopping criterion:

(41) where

(47)

, and

(42) is the transient waveform resulting from the inverse Laplace transform of the – th partial fraction term appearing on the left-hand side of expansion (40). The convolution integral in (42) can be accurately approximated by means of the quadratic recursive equation

(43)

where is a suitable convergence parameter (selected to be equal to in the developed numerical procedure). are known, the vector Finally, once the poles can be conveniently determined by solving residues (37) in a least squares sense. As pointed out in [3], the set of poles and residues to be used for the evaluation of the radiated electromagnetic field distribution in the time and frequency domains by means of (23) and (24) respectively, can be thinned according to the level of the energy indicators defined as follows:

with weighting coefficients (48) (44)

Summation in (23), (24) will be therefore restricted to those poles among all the whose energy satisfies the condition , where and is a given threshold. In order to achieve an adequate numerical accuracy, has been in the developed algorithm. set to V. ANTENNA EFFECTIVE HEIGHT AND GAIN

(45) denoting the time step adopted in the full-wave numerical procedure to compute the spatial distribution of the electromagenclosing the netic field excited within the Huygens surface antenna under analysis.

Let us assume that the antenna under analysis is driven by a having internal resistance . matched voltage generator Then, according to [19], the time-domain electromagnetic field radiated by an antenna in the Fraunhofer region can be expressed as the convolution integral of the propagating input current wave and the effective at the antenna terminals , namely height in transmit mode (49)

CARATELLI AND YAROVOY: UNIFIED TIME- AND FREQUENCY-DOMAIN APPROACH FOR ACCURATE MODELING

Therefore, under the assumption of impulse excitation , one easily obtains

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are where the vector residues easily determined by solving (57) in a least squares sense. In this way, it follows that:

(50) (58) being the effective height of the antenna in receive mode, and where the superscript is used to denote the field . So, quantities due to the impulse current wave substituting (4) into (50) yields

and

(59)

(51)

. The time-domain equivalent representation for of (57) is clearly given by (60)

where (52) The closed form evaluation of the antenna effective height is achieved by introducing the spherical harmonic representation of the impulse equivalent current densities

the Dirac-pulse term accounting for the instantaneous effect of the driving voltage source on the angular distribution of equivalent currents excited on the Huygens surface. By combining (53) and (60) with (51), the approximate expression of the antenna effective height is found to be, after some algebra

(53) So, by enforcing (8), and using the orthonormality property [23]

(61) ,

(54)

where (62)

the integral (52) can be conveniently rewritten as follows: (55) where denotes the general time-variant spherical harmonic vector coefficient of the im. pulse current By the convolution theorem, the (55) may be Laplace-transformed to (56) being comthe complex natural resonant frequencies puted by means of the TD-VF numerical procedure described at any in Section IV. Provided that point in the complex – plane, and as , the following SEM-based fitting model can be introduced: (57)

. It should be noticed that the first for each integral term appearing on the right-hand side of expansion only in the normalized time in(61) contributes to . Hence, adopting the following Fourier series terval representations: (63)

(64)

relevant to the shifted Dirac pulse, and the Legendre polynomial of order [24] respectively, it is straightforward to show that

(65)

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In (64) denotes the canonical spherical Bessel function of the first kind and order [24]. Finally, using (17)–(18) to compute the second integral term appearing on the right-hand side of (61), the effective height of the antenna in receive mode can be evaluated in closed analytical form, without any limitation involving the time or the observation spatial direction , as

equation (A28). Using (20) and the shifting properties of the unilateral Laplace transform, it can be shown that

(70) Therefore, the complex frequency-domain equivalent representation of the antenna effective height is easily derived as follows:

(71) (66)

where is the surface area of the Huygens sphere . The relevant effective height in transmit mode can be obtained by differentiating (66) in the sense of distributions. So, making use of the following identity:

(67)

readily yields

It is to be pointed out that the impulse and truncated polynomial terms in the SEM expansion (68) describe the instantaneous effect of the driving voltage on the radiated electric field, whereas the damped wave contributions are related to the complex resonant processes occurring in the structure. In particular, the incomplete modified spherical Bessel functions account for the very early transient when only the portion of the surface corresponding to the solid angle intersected by the half space gives contribution to the radiated electromagnetic field value excited at the observation point . In the limit each term in (68) approaches to a time-independent canonical modified spherical Bessel function-related quantity, meaning that the observation point starts to collect wave contributions from the whole Huygens sphere. Finally, as discussed in [18], [19], the time-domain antenna gain is defined by simply extending the expression of the corresponding frequency-domain parameter (72) denoting the field polarization direction, and the usual energy ( ) norm. So, taking advantage of (49) one obtains, after some manipulations (73) where the continuous auto-correlation operator

(68)

(74) has been introduced, and

with (69)

(75) being the first-order derivative of Legendre polynomial of order [24], and where the quantities are evaluated by the

is the so-called gain correlator [19], which is an intrinsic antenna characteristic describing the relevant radiation properties

CARATELLI AND YAROVOY: UNIFIED TIME- AND FREQUENCY-DOMAIN APPROACH FOR ACCURATE MODELING

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in the Fraunhofer region. After setting for shortness and , it follows from the (66) that:

(76)

Fig. 5. Magnitude of the cross-correlation function V (s; z ) of orders n = 3 and  = 2 with s = 1 + j . The agreement between the exact integral representation and the truncated series expansion of order 10 is very good.

0

where the following cross-correlation functions have been considered ( ):

(77)

(78)

(79) The integrals appearing in (77)–(79) can be easily expressed in terms of the incomplete spherical Bessel functions described in Appendix A. For sake of brevity, we here point out one par, by ticular case only. Under the assumption that making use of the Fourier series expansion (64), it is not difficult to reduce (78) to this form

Fig. 6. Three-dimensional view of a low-frequency ultrawideband resistively loaded bow-tie antenna for ground-penetrating radar applications. Antenna characteristics: l = 40 cm, 2 = 75 , h = 1 mm. The local coordinate system adopted to express the field quantities is shown.

with (82)

(80) Then, adopting the Whittaker’s integral representation (19) and applying Fubini’s theorem yields

denoting the incomplete spherical Bessel function of the first kind and order , whose application to the analytical solution of advanced electromagnetic problems will be the subject of future papers. Finally, upon combining (81) with (80), a rapidly convergent in the aforespecified variation series expansion for range of the parameter is obtained. As it can be noticed in Fig. 5, only a few terms in (80) are to be retained in order to guarantee a compact but very accurate representation. Cross-correlation integrals (77)–(79) for any value of can be evaluated analytically in a similar way, as shown in Appendix B. VI. AN EXAMPLE OF UWB ANTENNA MODELING

(81)

The considered approach has been validated by application to an UWB resistively loaded bow-tie antenna for GPR applications [34]. A sketch of the structure geometry is shown in Fig. 6. As it appears, the antenna features two circularly ended flairs,

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with half angle , whose electrical conductivity is a function of the radial distance from the central delta gap. and internal resisHere, a voltage source of amplitude tance is employed to excite the radiating element. To properly enlarge the antenna bandwidth, thus reducing the late-time ringing due to the spurious multiple reflections between the open ends and the feed point, the resistive loading has been assumed to have the following Wu-King-like distribution [34] (83) where is the flairs length, and is the electrical conductivity value at the antenna end sections. appearing in (83) has been determined by The parameter equating the static resistance of the flairs (84) to the characteristic impedance of the bow-tie antenna regarded as an infinitely long coplanar fin transmission line [35] (85) denoting the complete elliptic integral of the first kind [24] with moduli (86) (87) By doing so, after some algebra, it follows that (88) where is the negative branch of the Lambert function the thickness of the metal flairs. As a [24], and has been consequence, the electrical conductivity parameter found to be equal to about 40 S/m. The near-field full-wave analysis of the considered radiating structure has been performed by means of a locally conformal finite-difference time-domain (FDTD) scheme [36] useful to model dielectric, metal, and radiation losses avoiding the staircase approximation of the geometry, and providing in this way a clear advantage in terms of accuracy over the use of the conventional FDTD algorithm, as well as unstructured and/or stretched space lattices potentially suffering from significant numerical dispersion and instability [5]. The antenna has been meshed on a UPML-backed [37] uni, form cubic grid with spatial increment where is the free-space wavelength at the upper cut-off frequency of the excitation voltage signal, which is the Gaussian pulse defined by (89)

Fig. 7. Frequency-domain behavior of the antenna input reflection coefficient. The radiating structure, featuring a nearly flat input impedance around Z , is well matched to the feeding line starting from the frequency f .

= '

200

130 MHz where

, and (90)

The source pulse is coupled into the finite-difference equations used to update the time-domain electric field distribution within the delta gap [36]. As it appears in Fig. 7, the antenna with the specified resistive loading is well-matched to the feeding line to . in the frequency band from This excellent performance is useful to meet demanding specifications of modern GPR applications both in terms of fractional bandwidth, and low-frequency operation. Using the developed SEM-based approach, the time-variant of spherical harmonic expansion vector coefficients the surface equivalent currents excited on the Huygens sphere with radius have been computed on-the-fly in step with the numerical FDTD simulation, and then fitted to the modified pole/residue expansion (37) with order , as predicted by (10) with . Hence, the proposed non-uniform spherical-wave representation (23), , has been applied to with residue energy threshold evaluate the transient radiative electromagnetic field distribution without any limitation involving the time. In doing outside so, the pole/residue expansion order has been selected heuristically in such a way as to guarantee an adequate degree of accuracy in the modeling of the natural resonant processes occurring in the structure. The electromagnetic field behavior in the frequency domain can be directly computed by means of . (24) with As it appears in Fig. 8, the agreement with the numerical results obtained using the full-wave locally conformal FDTD technique is pretty good. But, it is worth stressing that the computational times and memory usage required to carry out the pole/residue spherical harmonic expansion are negligible in comparison with those relevant to the FDTD numerical simulation for computing the far-field distribution. The residue energy distribution and the location of dominant antenna poles in the complex frequency domain are shown in Fig. 9. The pair of conjugate poles with the smallest damping can be readily noticed. The coefficient value

CARATELLI AND YAROVOY: UNIFIED TIME- AND FREQUENCY-DOMAIN APPROACH FOR ACCURATE MODELING

Fig. 8. Transient (a) and frequency-domain (b) behavior of the co-polarized electric field component excited by the resistively loaded bow-tie antenna at the observation point r R ; 90 ; 0 (R = 1:1 m), as computed by a locally conformal FDTD technique and the proposed pole/residue model with energy threshold  = 10 .

f

g

relevant spherical harmonic mode having orders and is characterized by the resonant frequency , where the antenna response peaks (see Fig. 8) and the electrical . As it can be easily inferred, the length of each flair is about pair of dominant poles, together with the one lying on the real – axis, primarily determines the late-time characteristics of the radiating structure, whereas the poles with the larger damping coefficients account for the very early transient. The space-time behavior of the electromagnetic field radiated by the antenna has been thoroughly investigated to gain a physical insight into the transient emission phenomena responsible for the structure characteristics. Shown in Fig. 10 is the spatial distribution of the – component of the electric field excited – plane of the antenna ( – plane) at different along the times. In this way, one can observe different wave-fronts propaoutside the Huygens surface . gating in the air region related to the radiation process In particular, the wavefront occurring at the driving point is nearly a sphere meaning that, , the radio wave contribuat any observation point tion from the feed arrives at the same time

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Fig. 9. Residue energy distribution (a) and location in the complex frequency = 0:1) in the SEM-based repreplane (b) of dominant poles ( sentation of the electromagnetic field radiated by the resistively loaded bow-tie antenna shown in Fig. 6.

E

E



resulting in two symmetrical wave-fronts and . The time of arrival of such radio signals changes with the azimuth angle of observation as (92) The diffracted field then propagates back to the feed point, generating secondary undesired emission/diffraction phenomena. This process repeats, but results in a negligible ringing effect thanks to the resistive loading of the flairs. The frequency-domain distribution of the radiative compohas been computed nent of the electromagnetic field for via the SEM-based expansion (24). As it appears in Fig. 11, the spatial decay electric field magnitude exhibits the typical . Fiwith global maximum along the end-fire direction nally, using the analytically based approach outlined in the pre(see Fig. 12) vious section, the antenna effective height have and, hence, the time-domain gain in transmit mode been evaluated under the hypothesis of Gaussian pulse excitation (89) implying

(91) The interaction of this wave with the open ends of the antenna is responsible for the excitation of a diffracted field contribution

(93)

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Fig. 10. Spatial distribution of the y – component of the electric field excited along the E – plane of the resistively loaded bow-tie antenna at the spher(a) and  : (b). The radiation ical-wave delayed time  process from the feed, as well as the diffraction phenomena arising from the field interaction with the flair ends can be noticed. Maximum observation radius: R : .

= 5 ns

= 6 5 ns

= 15m

Fig. 12. Snapshot of the co-polarized component of the time-domain effective r;  along the E – plane of the resistively loaded height in receive mode bow-tie antenna at the delayed time  (a) and  (b). The : : spherical wave excited at the feed point can be noticed. Maximum observation : . radius: R

(^ )

= 3 5 ns

= 4 5 ns

= 15m

As expected from the theory, the antenna radiation pattern, not shown here for sake of brevity, features a stretched donut– plane and like shape with almost uniform level along the deep nulls along the bow-tie axis. It has been found that the , since half the power peak value of the gain is accepted by the antenna is dissipated along the flairs. VII. CONCLUSIONS

Fig. 11. Magnitude of the y – component of the radiative electric field excited along the E – plane of the resistively loaded bow-tie antenna at the fundamental resonant frequency f . The typical end-fire radiation process can : . be noticed. Maximum observation radius: R

= 200 MHz

= 15m

with (94) denoting the usual error function [24].

A novel SEM-based approach has been developed for the accurate time- and frequency-domain characterization of electromagnetic radiation processes in ultrawideband antennas. By the use of a dedicated two-step vector fitting procedure, a suitable pole/residue harmonic expansion of the transient electromagnetic field distribution in the Fraunhofer region has been derived. In this way, the radiated field has been expressed in analytic way as the superposition of outgoing propagating non-uniform spherical waves attenuating along with the radial distance and time according to the real part of the poles accounting for the complex resonant processes occurring in the structure. To allow the closed form evaluation of the antenna far-field parameters, such as gain and effective height, a new class of special functions has been introduced. This class, formed by the incomplete modified spherical Bessel functions, can be usefully employed to describe wave phenomena concerning truncated spherical structures. The main analytical characteristics, as well as the small and large argument asymptotic approximations of the considered functions have been derived and discussed. In particular, the differential and recurrence equations

CARATELLI AND YAROVOY: UNIFIED TIME- AND FREQUENCY-DOMAIN APPROACH FOR ACCURATE MODELING

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respectively. So, applying (A1) to (19) yields: (A3) By using the following recurrence relations for the Legendre polynomials [40]:

(A4) (A5) it may be easily verified that: Fig. 13. Distribution of the incomplete modified spherical Bessel function i (; w ) of order n = 8 for w = 3=4.

of the incomplete spherical Bessel functions present additional terms with respect to those relevant to the classical spherical Bessel function theory. By using the suggested methodology, one can gain a meaningful insight into the physical mechanisms which are responsible for the electromagnetic behavior of complex radiating structures. Such information can be usefully exploited to optimize the performance of UWB antennas for a wide variety of applications. Furthermore, the suggested approach allows for a significant reduction of the computational resources. The considered model has been validated by application to a resistively loaded bow-tie antenna. The transient response of the structure evaluated using the minimal pole/residue model has been found to be in good agreement with the reference solution obtained by means of a full-wave locally conformal FDTD technique. The proposed approach can be applied in a straightforward manner to electromagnetic scattering problems as well.

(A6) From this identity it is straightforward to show that: (A7) Finally, the right-hand side of (A7) is expressed in terms of partial derivatives of the incomplete function with respect to the parameter resulting in the homogeneous spherical Bessel-like differential equation: (A8) Recurrence Equations: The incomplete modified spherical Bessel functions satisfy the following recurrence equations:

(A9)

APPENDIX A In this appendix we turn to the consideration of the new and important class of incomplete modified spherical Bessel functions (see Fig. 13), stemming from the analysis of time-domain wave radiation phenomena in arbitrary three-dimensional electromagnetic structures. From the integral representation (19) in Whittaker form [38], , [39], it can be easily seen that, in the limit of coincides with , explaining why the terminology of calling an incomplete spherical modified Bessel function is pertinent. Actually, there is a strong interrelationship between these classes of special functions, the analytical properties of the former paralleling and generalizing those of the latter. Differential Equation: In order to derive the differential equation defining the incomplete modified spherical Bessel functions, let us firstly introduce the Bessel and Legendre operators:

(A10) Since both formulas are obtained in the same way, only (A9) is derived below for sake of brevity. Using the integral representation in Whittaker form (19), one obtains [40]

(A11)

(A1)

The last term in the right-hand side of (A11) is readily evaluated as follows:

(A2)

(A12)

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Furthermore, carrying out integration by parts and making use of (A8) yields, after some mathematical manipulations

(A13) Thus, substituting equations (A12) and (A13) into equation (A11) results in the recurrence relation (A9). It is not difficult to see that formulas (A9) and (A10) collapse to the well-known recurrence equations for canonical modified spherical Bessel functions of the first kind [13], [24] as the parameter approaches to unity. Asymptotic Expansion for Large Arguments: Since the asymptotic expansion of the classical modified spherical Bessel functions for large arguments is already available in the literature [41], in view of the representation (20) we can confine our attention to the analysis of the behavior of the complementary as . To this end, using the distribution following sum formula for the Legendre polynomials [42]: (A14) one can rewrite (21) in the convenient form (A15)

Fig. 14. Behavior of the incomplete modified spherical Bessel function of order n and parameter w = for real (a) and imaginary (b) values of the argument.

=2

=34

where (A16) denoting the usual incomplete gamma function [13]. By virtue of the identity

Therefore, an asymptotic expansion of the incomplete function for large values of , uniformly with respect to , can be derived by retaining the leading terms of the polynomials in (A20). As a result, we obtain that

(A17) holding for integer values of the parameter , we have

(A21) (A18)

with

as the polynomial of degree

given by (A19)

It follows from combining (A18) with (A15) that:

(A20)

where a proper use has been made of the sum formula (A14), of as well as of the symmetry property the Legendre polynomials [40]. As it can be noticed in Fig. 14, the asymptotic expression , and (A21) provides an accurate approximation of , already for reasonably small values of the hence argument . Furthermore, it is worth noting that, in the electromagnetic field theory concerning the radiation and scattering from truncated structures, the right-hand side of (A21) may be regarded as the superposition of a uniform spherical wave and the diffracted field contribution arising from the source truncations.

CARATELLI AND YAROVOY: UNIFIED TIME- AND FREQUENCY-DOMAIN APPROACH FOR ACCURATE MODELING

Asymptotic Expansion For Small Arguments: For arbitrary value of the parameter , the expansion of the exponential term in (19) gives

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and accordingly, after some mathematical manipulations

(A28)

(A22) where where, according to (A14)

(A29) denotes the confluent Lommel function of orders [41].

and

[39],

(A23) APPENDIX B being the generalized incomplete beta function [13]. A tedious calculation, the details can be of which we omit, shows that the coefficients expressed in terms of the generalized hypergeometric functions [13] as follows:

In this appendix, the theory of the incomplete spherical Bessel functions is applied to the analytical evaluation of the cross-correlation integrals (77)–(79) useful to derive the explicit expression of the antenna gain in time domain. , by using the Fourier Under the assumption that series expansion (64) of the Legendre polynomial of order and the Whittaker’s integral representation (82) of the incomplete spherical Bessel function of first kind and order , one readily obtains

(A24)

Hence, taking advantage of some advanced properties of the associated Legendre functions of the first kind, among which the useful identity [41]

(B1)

(A25) one can conveniently derive a suitable elementary function-based representation of the leading term in (A22) dein the limit of scribing the asymptotic behavior of uniformly with respect to

(A26)

In a similar way, it is straightforward to show that

(B2)

for . A trivial application of Fubini’s theorem leads to the following general identity involving the arbitrary parameters and

Finally, using (A22) we can also determine the value of the . The incomplete modified spherical Bessel functions for equation (A23) implies

(A27)

(B3)

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which can be used to define the auxiliary function of the two real variables

easily yields

(B9) It is to be pointed out that a suitable quadrature rule, such as the Gauss-Kronrod formula [43], can be conveniently adopted for the accurate numerical approximation of the definite integral appearing on the right-hand side of term only (B9), and giving a non-zero contribution to . for (B4) REFERENCES As a result, upon denoting for shortness and , the integral (78) can be evaluated in closed form as follows:

(B5) Finally, we turn to the consideration of the cross-correlation . It is not difficult to reduce (79) to this function form

(B6)

with and

, , . Therefore, introducing the quantities (B7)

(B8)

[1] H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach. New York: McGraw-Hill, 1985. [2] C. Roblin and A. Sibille, “Ultra compressed parametric modeling of UWB antenna measurements using symmetries,” presented at the Proc. URSI General Assembly, Chicago, Aug. 10–16, 2008. [3] G. Marrocco and M. Ciattaglia, “Ultrawide-band modeling of transient radiation from aperture antennas,” IEEE Trans. Antennas Propag., vol. 52, pp. 2341–2347, Sep. 2004. [4] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, 2nd ed. New York: Wiley-IEEE Press, 1997. [5] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [6] J. E. Hansen et al., “Spherical near-field antenna measurements,” in IEE Electromagnetic Waves, ser. 26. London, U.K.: Peter Peregrinus, 1988. [7] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed. New York: Wiley, 1997. [8] C. E. Baum, “The singularity expansion method,” in Transient Electromagnetic Fields, L. B. Felsen, Ed. Berlin: Springer-Verlag, 1976, pp. 129–179. [9] C. E. Baum, “On the singularity expansion method for the solution of electromagnetic interaction problems,” AWFL Interaction Note, vol. 88, Dec. 1981. [10] S. Grivet-Talocia, F. Canavero, I. Maio, and I. Stievano, “Reduced-order macromodeling of complex multiport interconnects,” presented at the URSI General Assembly, Maastricht, Belgium, Aug. 19–23, 2002. [11] S. Grivet-Talocia, “Generation of passive macromodels from transient port responses,” in Proc. IEEE Topical Meeting on Electrical Performance of Electronic Packaging, Princeton, NJ, Oct. 27–29, 2003, pp. 287–290. [12] S. Grivet-Talocia, “Package macromodeling via time-domain vector fitting,” IEEE Microwave Wireless Comp. Lett., vol. 13, pp. 472–474, Nov. 2003. [13] G. E. Andrews, R. Askey, and R. Roy, Special Functions. Cambridge, U.K.: Cambridge Univ. Press, 2001. [14] R. Cicchetti and A. Faraone, “Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, pp. 3373–3389, Dec. 2004. [15] R. Cicchetti and A. Faraone, “Exact closed-form expression of the electromagnetic field excited by pulse-shaped and triangular line currents,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1706–1716, Jun. 2008. [16] D. S. Jones, “Incomplete Bessel functions I,” in Proc. Edinburgh Math. Soc. (Series 2), 2007, vol. 50, pp. 173–183. [17] D. S. Jones, “Incomplete Bessel functions II – Asymptotic expansion for large argument,” in Proc. Edinburgh Math. Soc. (Series 2), 2007, vol. 50, pp. 711–723. [18] E. G. Farr and C. E. Baum, “Extending the definition of antenna gain and radiation pattern into the time domain,” Sensor and Simulation Notes, Nov. 1992. [19] A. Shlivinski, E. Heyman, and R. Kastner, “Antenna characterization in the time domain,” IEEE Trans. Antennas Propag., vol. 45, pp. 1140–1149, Jul. 1997.

CARATELLI AND YAROVOY: UNIFIED TIME- AND FREQUENCY-DOMAIN APPROACH FOR ACCURATE MODELING

[20] G. Marrocco and M. Ciattaglia, “Approximate calculation of time-domain effective height for aperture antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 1054–1061, Mar. 2005. [21] S. Licul and W. A. Davis, “Unified frequency and time-domain antenna modeling and characterization,” IEEE Trans. Antennas Propag., vol. 53, pp. 2882–2888, Sep. 2005. [22] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. New York: Prentice-Hall/IEEE, 1994. [23] T. M. MacRobert and I. N. Sneddon, Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications, 3rd ed. Oxford, U.K.: Pergamon Press, 1967. [24] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. New York: Academic Press, 2007. [25] O. M. Bucci and G. Franceschetti, “On the spatial bandwidth of scattered fields,” IEEE Trans. Antennas Propag., vol. 35, pp. 1445–1455, Dec. 1987. [26] O. M. Bucci, C. Gennarelli, and C. Savarese, “Optimal interpolation of radiated fields over a sphere,” IEEE Trans. Antennas Propag., vol. 39, pp. 1633–1643, Nov. 1991. [27] J. A. Stratton, Electromagnetic Theory. Piscataway, NJ: IEEE Press, 2007. [28] D. Shepard, “A two-dimensional interpolation function for irregularly-spaced data,” in Proc. 23rd ACM National Conf., Aug. 1968, pp. 517–524. [29] Y. Hua and T. K. Sarkar, “Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp. 814–824, May 1990. [30] T. K. Sarkar and O. Pereira, “Using the Matrix Pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag., vol. 37, no. 1, pp. 48–55, 1995. [31] M. L. Van Blaricum and R. Mittra, “A technique for extracting the poles and residues of a system directly from its transient response,” IEEE Trans. Antennas Propag., vol. 23, pp. 777–781, Nov. 1975. [32] A. Poggio, M. Van Blaricum, E. Miller, and R. Mittra, “Evaluation of a processing technique for transient data,” IEEE Trans. Antennas Propag., vol. 26, pp. 165–173, Jan. 1978. [33] G. E. Collins and W. Krandick, “An efficient algorithm for infallible polynomial complex root isolation,” in Proc. ISSAC, Berkeley, CA, Jul. 27–29, 1992, pp. 189–194. [34] D. Caratelli, A. Yarovoy, and L. P. Ligthart, “Full-wave analysis of cavity-backed resistively loaded bow-tie antennas for GPR applications,” in Proc. Eur. Radar Conf., Amsterdam, The Netherlands, Oct. 27–31, 2008, pp. 204–207. [35] R. L. Carrel, “The characteristic impedance of two infinite cones of arbitrary cross section,” IRE Trans. Antennas Propag., vol. 6, no. 2, pp. 197–201, Apr. 1958. [36] D. Caratelli and R. Cicchetti, “A full-wave analysis of interdigital capacitors for planar integrated circuits,” IEEE Trans. Magn., vol. 39, no. 3, pp. 1598–1601, May 2003. [37] S. D. Gedney, “An anisotropic PML absorbing media for FDTD simulation of fields in lossy dispersive media,” Electromagn., vol. 16, pp. 399–415, 1996. [38] M. M. Agrest and M. S. Maksimov, Theory of Incomplete Cylinder Functions and their Applications. Berlin, Germany: Springer-Verlag, 1971. [39] G. N. Watson, A Treatise on the Theory of Bessel Functions. Cambridge, U.K.: Cambridge Univ. Press, 1962. [40] R. A. Askey, Orthogonal Polynomials and Special Functions. Philadelphia, PA: SIAM, 1975, vol. 21, Regional Confe. Series in Applied Mathematics.

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[41] Y. Luke, The Special Functions and Their Approximations. San Diego, CA: Academic Press, 1969. [42] W. Koepf, Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998. [43] D. Calvetti, G. H. Golub, W. B. Gragg, and L. Reichel, “Computation of Gauss-Kronrod quadrature rules,” Math. Comput., vol. 69, no. 231, pp. 1035–1052, Jul. 2000.

Diego Caratelli was born in Latina, Italy, on May 2, 1975. He received the Laurea (summa cum laude) and Ph.D. degrees in electronic engineering from “La Sapienza” University of Rome, Italy, in 2000 and 2004, respectively. In 2005, he joined the Department of Electronic Engineering, “La Sapienza” University of Rome, as a Contract Researcher. Since 2007, he has been with the International Research Centre for Telecommunications and Radar (IRCTR), Delft University of Technology, The Netherlands, as a Senior Researcher. His main research activities include the design, analysis and experimental verification of printed microwave and millimeter-wave passive devices and wideband antennas for satellite, WLAN and GPR applications, the development of analytically based numerical techniques devoted to the modeling of electromagnetic field propagation and diffraction processes, as well as the analysis of EMC/EMI problems in sensitive electronic equipment. Dr. Caratelli is a member of the Italian Electromagnetic Society (SIEm).

Alexander G. Yarovoy (M’96–SM’04) received the Diploma in radiophysics and electronics (with honors), and the Candidate Phys. and Math. Sci. (Ph.D. equivalent) and the Doctor Phys. and Math. Sci. (D.Sc. equivalent) degrees in radiophysics from Kharkov State University, Ukraine, in 1984, 1987, and 1994, respectively. In 1987, he joined the Department of Radiophysics, Kharkov State University, as a Researcher and became a Professor in 1997. From September 1994 through 1996, he was with Technical University of Ilmenau, Germany, as a Visiting Researcher. Since 1999, he has been with the International Research Centre for Telecommunications-Transmission and Radar (IRCTR), Delft University of Technology, The Netherlands, where, since 2009, he has held the Chair in Microwave Technology and Systems for Radar. His main research interests are in ultrawideband (UWB) microwave technology and its applications (in particular, UWB radars) and applied electromagnetics (in particular, UWB antennas). Prof. Yarovoy is the recipient of a 1996 International Union of Radio Science (URSI) “Young Scientists Award” and co-recipient of the 2001 European Microwave Week Radar Award for the paper that “best advances the state-of-the-art in radar technology.” He served as the Chair of the 5th European Radar Conference (EuRAD’08), Amsterdam, The Netherlands and was Co-Chairman and Technical Program Committee Chair of the Tenth International Conference on Ground Penetrating Radar (GPR2004), Delft.

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Enhanced A-EFIE With Perturbation Method Zhi-Guo Qian, Member, IEEE, and Weng Cho Chew, Fellow, IEEE

Abstract—The recently developed augmented electric field integral equation (A-EFIE) remedies the well-known low-frequency breakdown. However, it loses accuracy of current when applied to certain low-frequency applications like plane wave scattering. This paper addresses the low-frequency inaccuracy problem, and proposes a perturbation method as a remedy. A-EFIE with perturbation solves the same A-EFIE matrix system with updated right hand side vectors repeatedly to obtain the current and charge on different frequency orders as a perturbation series. This method does not require a search for the loop and tree basis, but the loop and tree currents can also be recovered. Numerical experiments validate the novel method. Index Terms—Augmented electric field integral equation (A-EFIE), electromagnetic scattering, loop-tree, low frequency, perturbation.

I. INTRODUCTION HE electric field integral equation (EFIE) with Rao-Wilton-Glisson (RWG) basis functions [1] has been a prevailing integral equation method over the last two decades for various types of electromagnetic problems, such as scattering, antenna, and RF circuit [2]–[4]. However, it only works well at mid-frequencies because of the well-known low-frequency breakdown [5]. Various remedies have been proposed for a stable full-wave formulation in the low-frequency regime [6]–[20]. A widely used numerical method is the loop-tree or loop-star decomposition [6]–[13]. It achieves a quasi Helmholtz decomposition (also known as Hodge decomposition) by searching for the loop-tree or loop-star basis. These bases separate the contributions from the vector potential and the scalar potential in the impedance matrix, so that the contribution from the vector potential will not be swamped by that from the scalar potential at low frequencies. However, it is tricky to search for loop basis in a complex structure, as encountered in a real-world chip package. In recent years, the multiresolution basis has been introduced to control the conditioning and improve the convergence of EFIE at low frequencies [14], [15]. The self-regularizing property of EFIE

T

Manuscript received August 26, 2009; revised February 22, 2010; accepted March 25, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported by grants SRC 2006-KJ-1401 and AFOSR F9550-04-1-0326. Z. G. Qian was with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. He is now with Apache Design Solutions, Inc., San Jose, CA 95134 USA. W. C. Chew is with the The University of Hong Kong, Pokfulam, Hong Kong, on leave from the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055795

has also been exploited for preconditioning by leveraging the Calderòn identity [16]–[18]. The resulting system is shown to be stable at low frequencies. Some other methods have used more unknowns to expand EFIE for better conditioning and stability in the low-frequency regime [19]–[25]. Among the various approaches, the recently developed augmented electric field integral equation (A-EFIE) is one of the simplest methods to remedy the low-frequency breakdown. A-EFIE is developed based on the saddle point form EFIE, which includes charge as an additional unknown to expand the mixed potential form EFIE [26], [27]. Only recently, the low-frequency stability of the saddle point form is discovered [20]–[23]. First, including charge avoids the imbalance between the vector potential and the scalar potential in an easy manner: a much simpler remedy for the low-frequency breakdown than the loop-tree decomposition. Second, appropriate frequency scaling creates a scale invariance matrix in the low-frequency regime. Last, enforcement of charge neutrality ensures the matrix system full rank down to DC. Numerical examples have shown that the new formulation can be easily combined with fast algorithm to achieve high efficiency. It has been successfully applied to solve large-scale real world package structures with over one million surface integral equation unknowns [25]. However, we observed that A-EFIE loses accuracy of current for certain applications such as plane wave scattering. In this paper, we investigate the low-frequency inaccuracy of current based on the frequency dependence of the matrix system and its effect on the final solutions. It is shown that if the current , the vector potential is on a frequency order higher than contribution is diminished and swamped. The result is that the divergence-free current cannot be retrieved. We thus propose a perturbation method to enhance the accuracy of A-EFIE. By solving the same matrix with consecutively updated right hand side vectors, the small current can be retrieved with high accuracy. It is also shown that the loop current and tree current can be obtained from the results of A-EFIE without a need to search for loop basis. This paper is organized as follows. In Section II, A-EFIE is briefly reviewed. In Section III, the frequency dependence of several typical applications is analyzed to present the lowfrequency physics. Then, the perturbation method is proposed in Section IV, and the recovery of loop and tree currents is introduced in Section V. Numerical examples are included in Section VI to validate the method.

II. A-EFIE FORMULATION First, A-EFIE is briefly reviewed. Then the low-frequency inaccuracy of the current is illustrated out.

0018-926X/$26.00 © 2010 IEEE

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A. A-EFIE For an arbitrarily shaped PEC surface , EFIE in mixed potential form is written as

(1) where

is With our choices of the RWG and pulse basis functions, simply expressed as patch is positive part of RWG patch is negative part of RWG (9) patch is not part of RWG . It is straightforward to derive that the matrix has an interesting left null space spanned by a vector of ones. The physically meaning of the null space is that purely solenoidal (divergence-free) current does not have charge accumulation. The RWG based EFIE can thus be written as

(2) (10) is the free space Green’s function, is the wave-number in free space, is the wave impedance in free space, is the relative permeability, is the relative permittivity, is the surface is the incident electric field. To solve electric current, and the surface integral equation by the method of moments (MoM), the surface is discretized into flat triangles ( triangle pairs). Then, the surface current is expanded with basis functions defined on these elements

where the right hand side vector is defined as (11) The vector potential matrix edges

comprises interactions among (12)

(3) The most popular basis function is Rao-Glisson-Wilton (RWG)[1], which is defined on a pair of triangles. In our formulation, it is modified slightly by removing the edge length (4) otherwise. where are the area of triangles . The nodes represent the free vertexes of the triangle pair. We also define pulse basis on the th triangle with area as function

It is a full-rank matrix of dimension The scalar potential is factorized into three terms, the middle of which is formed by interactions among patches: (13) It is a full-rank matrix of dimension . Since is less than , the scalar potential matrix is rank deficient. Notice that EFIE breaks down when the rank deficient scalar potential matrix swamps the vector potential matrix at low frequencies. Combing (8) and (10), we obtain a saddle point system with both current and charge as unknowns [21], [22] (14)

otherwise.

(5)

These pulse functions can be used to expand surface charge density: (6) The relationship between current and charge is defined by the current continuity condition, which is usually written as (7) If the surface current and charge are expanded with the aforementioned basis functions, the current continuity equation can be converted into a matrix-vector form as (8) where is a vector whose th element is in (3), and is a vector whose th element is in (6). The divergence operator is represented by a sparse matrix , which is related to the incidence matrix of a graph defined on the original triangular mesh. The graph has a vertex for each triangle in the mesh, and it has an edge for each common edge in the mesh.

where is an identity matrix of dimension , and is the speed of light. Similar saddle point form of EFIE was studied more than two decades ago [26], [27], however its low-frequency advantage has not been discovered until recently [20], [22]. Additional frequency scaling has been used in (14) to reach a frequency invariance matrix system in the low-frequency , because regime. However it is still singular at DC, i.e., of the left null space of matrix . An easy remedy is to squeeze out the charge redundancy by eliminating one charge unknown for each singly, connected surface [25] (15) where we defined the backward matrix and forward matrix to map the charge unknowns back and forth. These two matrices are simple and highly sparse. Matrix is an identity matrix of reduced dimension, and vector is the reduced unknown vector of charge. This matrix system is the low-frequency stable A-EFIE. The first equation accounts for the electric field contributed from the vector potential and the scalar potential, and the second equation poses a constraint for the non-divergencefree current.

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are frequency inAt low frequencies, matrices and variant. Charge is frequency invariant. Therefore, the frequency determines the contribution of the vector podependence of is the only term containing the contribution tential, and of divergence-free current. If is on the order of , is frequency independent, so that the two terms are balanced down to DC. However, if is on the order smaller than , decreases with frequency, so that the contribution from the vector potential and divergence-free current is lost. It does not cause ill-conditioning of the matrix, but the divergence-free current is non-retrievable. The next section demonstrates the frequency dependence of the current for three different types of applications.

m

Fig. 1. Geometry of a parallel plate capacitor. Unit: 50  .

III. LOW-FREQUENCY ANALYSIS Quasi Helmholtz decomposition is a useful method because it helps to decouple the magneto- and electrostatic physics. The following analysis invokes the loop-tree decomposition. It is to be noted that the analysis in this section is to offer a physical understanding of the problem, and loop-tree decomposition is never invoked in the actual A-EFIE method and the ensuing perturbation method. For any structure, the loop-tree based EFIE is written in a matrix form as [9] (17) and tree basis function where loop basis function have been used. The sub-matrices represent the interaction among loop basis and tree basis. For example

Fig. 2. The extracted capacitance by A-EFIE.

B. Low-Frequency Inaccuracy of Current A number of numerical examples have demonstrated that A-EFIE does not have low-frequency breakdown, and it can solve complex real world structures with a large number of unknowns [24], [25]. Most of these examples are inductive at low frequencies. A-EFIE has also been applied to other types of problems, such as capacitive problem and scattering problem. The formulation is stable down to DC, and the charge solution is always correct. However, the current may lose accuracy. For instance, a parallel plate capacitor is shown in Fig. 1. A delta-gap voltage source is applied in the middle of the connecting strip. We solve for the current and charge on the surface using A-EFIE, and then extract the capacitance from the current. The capacitance shown in Fig. 2 is not consistent with the low-frequency physics: the capacitance should have been a constant. A-EFIE does not deliver correct current solution for capacitive problems at low frequencies. This can be classified as a low-frequency inaccuracy problem. It is imperative to understand the underlying reason and find a remedy. The inaccuracy can be explained by the frequency dependence of the current and the charge because their contributions exhibit different roles in the first equation of A-EFIE in (15)

(16)

(18) Due to the divergence free property of loop basis function, the other three sub-matrices only have vector potential term (19) (20) (21) The right hand side vectors are simply (22) (23) The frequency dependence of these matrices originates from the Green’s function. The scalar Green’s function is (24)

QIAN AND CHEW: ENHANCED A-EFIE WITH PERTURBATION METHOD

where by mated as

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. A low-frequency problem is characterized , so the scalar Green’s function is well approxi-

frequency decreases, so that the current loses accuracy for plane wave scattering problems at low frequencies. B. Circuit Problem: Capacitor

(25) More importantly, physics of different types of problems determines the frequency dependence of the right hand side vectors. Next, we show three different problems. A. Plane Wave Scattering This problem at low frequency involves both magneto-quasistatic and electro-quasi-static physics. The phase factor of a plane wave can be expanded in a similar manner. Thus, the frequency dependence of the matrices and vectors for a plane wave scattering problem is estimated as

(26) where the property of loop basis is used, so the corresponding excitation and sub-matrices have different frequency dependence than tree basis [9], [28]. The pair in the bracket represents the dominant frequency dependence of the real and imaginary parts, respectively. The frequency is a normalized notation of frequency because it is the small electrical size that defines a low-frequency problem, not the frequency along. Next, we need to know the frequency dependence of the loop current and the tree current. Matching the real and imaginary part of the two equations, , and . For we have four equations to deduce the value of , example, the real part of the top equation yields (27) The remaining three sets of constrains are written as

This problem involves electro-quasi-static physics. For simplicity, a circuit is usually excited by delta-gap voltage sources. At sufficiently low frequency, any circuit degenerates into a capacitor or an inductor. For a capacitor problem, there is no global loop through the excitation port, otherwise, the global loop provides a closed current return path and the problem is inductive instead. Such a property leads to the following frequency dependence of the right hand side vectors, (31) The above notation means that excitation of loop basis is exactly zero, and the excitation of tree basis is a real constant at the excitation ports. Substituting the relations into (17), the zeros of the right hand side indicate complete cancellations between the matrix vector products on the left hand side. These cancellations make the above deduction for plane wave scattering inappropriate. Therefore, we resort to a numerical example, the parallel plate capacitor in Fig. 1. The loop-tree based EFIE [9] is used to solve for the loop current and the tree current at several low frequencies. Comparing these solutions, we are able to deduce the following relationships: (32) The real part is much smaller than imaginary part because loss is negligible for PEC structures at low frequencies. The loop current and the tree current are on the same order of frequency. is on the order Substituting the relations back to (16), of . It also diminishes as frequency decreases, so that capacitive problems also experience the low-frequency inaccuracy problem. C. Circuit Problem: Inductor

(28) It is straightforward to draw the conditions satisfying all four sets

This is a magneto-quasi-static physics problem. The difference between inductor problems and capacitor problems is that there are excitations of global loop currents in the former. Specifically, the right hand side vectors are (33)

(29)

Such a difference leads to completely different physics. Similar to the capacitive problem, we use a numerical experiment, a strip loop inductor, to deduce the following relationships

(30)

(34)

It shows that the loop current is much larger than the tree current at low frequencies, therefore, the leading term of current is on the order of . Then in (16), the first term on the left hand , is of order while the second term on the leftside, hand side is of order . Hence, the first term is diminishing as

Due to the negligible loss for PEC structures at low frequencies, real part is much smaller than imaginary part. The loop current is inversely proportional to frequency, which is consistent with the physics of an inductor. Therefore, the leading order term of . Then, in (16) is on the order of as the current is

Additional numerical experiments verify the relations.

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frequency decreases. The vector potential and the scalar potential are balanced at low frequencies, so that no low-frequency inaccuracy problem exists in inductive problems.

To accommodate the matrices and excitation vector, the unknown vectors are expanded in a similar manner (49)

IV. PERTURBATION METHOD

(50)

A perturbation method is proposed to remedy the low-frequency inaccuracy in this section [28], [29]. With the expansion of Green’s function in (25), the sub-matrices in (15) can be ex, where is a typical length scale. panded. We denote Then, can be written as , where the tilde above indicates the normalization by . At low frequencies, , and the vector potential sub-matrix is expanded. Usually, second , order is enough for practical applications if (35) where (36) (37) (38) At low frequencies, matrix is expanded as

Substituting them into (15) and equating like powers of , we can obtain a series of equations. At first, matching zeroth order of gives rise to the lowest order equation: (51) We denote the matrix as , which is equal to the A-EFIE matrix in (15) in the static regime. No special treatment is needed to construct the matrix. From the zeroth order equation, we can and charge . This is solve for the zeroth order current enough for inductive problems. However, the aforementioned is zero for plane frequency dependence analysis shows that wave scattering problems. The first order equation is needed to . obtain the correct Then, matching the first order of yields the first order equation,

. Similarly the scalar potential (52) (39)

where (40) (41) (42) It is also straightforward to show that The incident plane wave can be written as

as

With the zeroth order current and charge, we can solve for the first order of them using this equation. Notice that for plane wave scattering problems, is zero and the second term on the right side can be eliminated. It is not surprising to see that the leading term of current may be the second order for certain problems. For example, capacitor problems fall into this category. To tackle them, the second order equation needs to be invoked

. (53) (43)

is the unit vector of the incident direction. The exwhere pansion at low frequencies is (44) Thus, the right hand side vector is

Once , , , and are obtained using the zeroth and first order equations, the second order of charge and current and can be solved. Notice that both terms associated with should be eliminated if the leading order of current is . With the series of equations, current and charge on different frequency orders can be solved for accurately.

(45)

V. RECOVERY OF LOOP AND TREE CURRENTS

(46)

This section shows how to get loop-tree currents without a search for topological loop basis [23]. Quasi Helmholtz decomposition indicates that current can be decomposed into and non-divergence solenoidal (divergence-free) current free current

where

(47) (48)

(54)

QIAN AND CHEW: ENHANCED A-EFIE WITH PERTURBATION METHOD

In loop-tree decomposition method, and tree basis by loop basis

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and are represented , respectively

For all types of problems, the leading order of tree current is always (60)

(55) is the th element of vector , and is the th where element of vector . As we have stated before, the search for loop basis is complicated, and matrix element evaluation related to loop basis also needs special treatment because of the potential cancellations. It is possible to recover loop and tree currents without a search for topological loop basis. Both of them can be expanded by RWG basis functions instead. Without loss of generality, we consider the recovery of both the loop current and the tree current on the th order of frequency, assuming that both coefficient vectors of loop current and tree current are expanded as (56) where both expansion vectors and are of dimension . According to the current and charge expansion in (49) and (50), and loop and tree currents on the th order are related to . This is because of the factor on the left-hand side of (49). There are two steps for the recovery. In the first step, tree current is recovered using a procedure similar to the basis rearrangement. For arbitrary mesh, the spanning tree can be easily obtained using the breadth first search. RWG basis functions belonging to the spanning tree constitute the tree basis function set. The relationship between these tree basis functions and pulse basis functions is represented by a , which is the incidence matrix of the sparse matrix spanning tree. To be specific (57) The expansion coefficient vector of the th order tree current equals to . The tree basis is a subset of RWG basis, and the projection from tree basis to RWG basis can be repre. Thus, we have sented by a matrix .

(58)

The computation bottleneck is the inverse of matrix , but it has been shown that the matrix vector product can be operations [9]. The efficient computation is effected in attributed to its sparsity pattern. As the incidence matrix of a spanning tree, the unknown at the tip of any branch is easily obtained because only one current unknown is related to the charge on the patch. Then the neighboring current unknown can be calculated in turn along the branch. Finally, the unknown at the junction is solved once unknowns in all branches are solved. In the second step, loop current is recovered by subtracting the tree current from the total current. Since all of them are expanded with the same set of RWG basis, it is equivalent to subtracting the expansion vector (59)

But the leading order of loop current is determined by the physical nature. For an inductive problem, it is simply (61) For a plane wave scattering problem, however, it becomes (62) because is zero. If both and problem, it should start from

are zero like a capacitive (63)

In this process, only RWG basis are used. No loop basis is needed. This technique was first presented in [21], and also in [23]. Recently, it also has been reported in [30]. VI. NUMERICAL EXAMPLES The application of A-EFIE to inductor problems have already been reported in the literature [22], [24], [25]. In this section, two other types of problems are presented. One is the plane wave scattering of a PEC sphere, and the other is a parallel plate capacitor. As we shown before, A-EFIE loses accuracy of the current. However, with the perturbation method, the enhanced A-EFIE successful remedies the problem. In the first example, an -polarized plane wave impinges onto direction. The sphere centers at the a PEC sphere from the origin and has a radius of 1 m. We discretize the surface into 1,568 triangular patches, equivalent to 2,352 inner edges. At very low frequencies, the far field comprises equally important contributions of two equivalent dipoles, a magnetic dipole plus an electric dipole, according to Rayleigh scattering theory. Fig. 3 shows the comparison between Mie series, A-EFIE, and A-EFIE with perturbation at 100 kHz. The frequency is not very low, and A-EFIE does not lose accuracy of current. With perturbation method, A-EFIE also gives right results. Then we did the comparison at 1 Hz, seeing Fig. 4. A-EFIE is wrong due to the low-frequency inaccuracy of current, while A-EFIE with perturbation is able to deliver correct results. The relative error of the bi-static RCS in Fig. 5 shows good accuracy of A-EFIE with perturbation. Notice that the spike is around 120 degree, which corresponds to the null of the bi-static RCS. As our prein (49) equals zero in the plane wave vious analysis shows, scattering problem. It is necessary to invoke the perturbation method for accurate current. By solving (51), (52), and (53) conand . According to the frequency secutively, we obtain is the leading dependence of loop and tree currents in (30), is the summation of second order order loop current, while loop current and leading order tree current. Their radiation pattern is shown in Fig. 6 separately. It validates that regarding far field radiation, the tree current is equivalent to an -polarized electric dipole, while the loop current is equivalent to a -polarized magnetic dipole.

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Fig. 3. Comparison of the bi-static RCS of a PEC sphere for the vertical polarization. The sphere has a radius of 1 m, and the frequency is 100 kHz.

Fig. 6. Split the bi-static RCS into two terms: the first order j order j . The frequency is 1 Hz.

and the second

Fig. 7. Geometry of a torus. Unit: 0.1 m.

Fig. 4. Comparison of the bi-static RCS of a PEC sphere for the vertical polarization. The sphere has a radius of 1 m, and the frequency is 1 Hz.

Fig. 8. Comparison of the bi-static RCS of a PEC torus for the vertical polarization. The frequency is 10 Hz.

Fig. 5. Relative error of the bi-static RCS using A-EFIE with perturbation. The reference is Mie series solution. The frequency is 1 Hz.

In the second plane wave scattering example, we replace the sphere with a PEC torus shown in Fig. 7. The radius of the tube

is 0.1 m and the distance from the center of the tube to the center of the torus is 1 m. We discretize the surface into 1,800 triangular patches, equivalent to 2,700 inner edges. The torus structure is topologically unique because it has two global loops. Fig. 8 shows the comparison of RCS between loop-tree decomposition [9], A-EFIE, and A-EFIE with perturbation at 10 Hz. Similar to the sphere case, A-EFIE loses accuracy at the low frequencies,

QIAN AND CHEW: ENHANCED A-EFIE WITH PERTURBATION METHOD

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capacitance between A-EFIE and A-EFIE with perturbation. When using A-EFIE with perturbation, the second order equa. A-EFIE loses accuracy tion in (53) is used to solve for around 100 kHz, while A-EFIE with perturbation delivers accurate results down to DC. Fig. 10 shows the current at 1 Hz using A-EFIE, and Fig. 11 demonstrates the current at 1 Hz using A-EFIE with perturbation. VII. CONCLUSION

Fig. 9. Comparison of the extracted capacitance between A-EFIE and A-EFIE with perturbation.

A-EFIE was proposed as a simple and efficient remedy for the low-frequency breakdown of EFIE. In this paper, we addressed the possible low-frequency inaccuracy problem of A-EFIE. It is related to the physics of a problem. If the contribution of the vector potential to the electric field is swamped by that of the scalar potential, the current loses accuracy. It happens to capacitive problems and scattering problems. A perturbation method has been proposed to resolve the issue. It solves the same A-EFIE matrix system with updated right hand side vectors repeatedly to obtain the current and charge on different frequency orders as a perturbation series. In addition, the loop and tree current can be recovered accordingly without a need to search the topological loop basis. REFERENCES

Fig. 10. Wrong surface current density at 1 Hz solved by A-EFIE. The plot is in dB scale. Unit of geometry: 50  . Unit of current density: A/m.

m

Fig. 11. Correct surface current density at 1 Hz solved by A-EFIE with perturbation. The plot is in dB scale. Unit of geometry: 50  . Unit of current density: A/m.

m

while A-EFIE with perturbation generates correct results as the loop-tree decomposition does. The third example is the parallel plate capacitor. A delta-gap voltage source is assigned in the middle of the strip connecting the two plates. Fig. 9 shows the comparison of the extracted

[1] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surface of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, May 1982. [2] R. F. Harrington, Field Computation by Moment Methods. New York: IEEE Press, 1993. [3] N. Morita, N. Kumagai, and J. R. Mautz, Integral Equation Methods for Electromagnetics. Boston, MA: Artech House, 1990. [4] W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Boston: Artech House, 2001. [5] Z. G. Qian and W. C. Chew, “A quantitative study of the low frequency breakdown of EFIE,” Microw. Opt. Tech. Lett., vol. 50, no. 5, pp. 1159–1162, May 2008. [6] D. R. Wilton, J. S. Lim, and S. M. Rao, “A novel technique to calculate the electromagnetic scattering by surfaces of arbitrary shape,” presented at the URSI Radio Science Meeting, Jun. 1993. [7] M. Burton and S. Kashyap, “A study of a recent, moment-method algorithm that is accurate to very low frequencies,” Appl. Computat. Electromagn. Soc. J., vol. 10, no. 3, pp. 58–68, Nov. 1995. [8] W. Wu, A. W. Glisson, and D. Kajfez, “Study of two numerical solution procedures for the electric field integral equation at low frequency,” Appl. Computat. Electromagn. Soc. J., vol. 10, no. 4, pp. 69–80, Nov. 1995. [9] J. S. Zhao and W. C. Chew, “Integral equation solution of Maxwell’s equations from zero frequency to microwave frequencies,” IEEE Trans. Antennas Propag., vol. 48, pp. 1635–1645, Oct. 2000. [10] G. Miano and F. Villone, “A surface integral formulation of Maxwell equations for topologically complex conducting domains,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4401–4414, Dec. 2005. [11] F. Ling, V. I. Okhmatovski, W. Harris, S. McCracken, and A. Dengi, “Large-scale broad-band parasitic extraction for fast layout verification of 3-D RF and mixed-signal on-chip structures,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, Jan. 2005. [12] V. I. Okhmatovski, J. D. Morsey, and A. C. Cangellaris, “Enhancement of the numerical stability of the adaptive integral method at low frequencies through a loop-charge formulation of the method-of-moments approximation,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 962–970, Mar. 2004. [13] G. Vecchi, “Loop-star decomposition of basis functions in the discretization of EFIE,” IEEE Trans. Antennas Propag., vol. 47, pp. 339–346, Feb. 1999. [14] F. P. Andriulli, F. Vipiana, and G. Vecchi, “Hierachical bases for nonhierachic 3-D triangular meshes,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2288–2297, Aug. 2008.

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[15] F. Vipiana and G. Vecchi, “A novel, symmetrical solenoidal basis for the MoM analysis of closed surfaces,” IEEE Trans. Antennas Propag., vol. 57, no. 4, Apr. 2009. [16] R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antennas Propag., vol. 52, pp. 362–372, Feb. 2004. [17] H. Contopanagos, B. Dembart, M. Epton, J. Ottusch, V. Rokhlin, J. Visher, and S. M. Wandzura, “Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering,” IEEE Trans. Antenna Propag., vol. 50, no. 12, pp. 1824–1930, Dec. 2002. [18] F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative Calderón preconditioner for three-dimensional electromagnetic scattering,” IEEE Trans. Antenna Propag., vol. 56, no. 8, pp. 2398–2412, Aug. 2008. [19] M. Taskinen and P. Ylä-Oijala, “Current and charge integral equation formulation,” IEEE Trans. Antenna Propag., vol. 54, no. 1, pp. 58–67, Jan. 2006. [20] D. Gope, A. R. Ruehli, and V. Jandhyala, “Solving low-frequency EM-CKT problems using the PEEC method,” IEEE Trans. Adv. Packag., vol. 30, pp. 313–320, May 2007. [21] Z. G. Qian and W. C. Chew, “An augmented electric field integral equation for low frequency electromagnetics analysis,” presented at the IEEE Int. Symp. on Antennas and Propagation and USNC/URSI National Radio Science Meeting, San Diego, CA, Jul. 2008. [22] Z. G. Qian and W. C. Chew, “An augmented EFIE for high speed interconnect analysis,” Microw. Opt. Tech. Lett., vol. 50, no. 10, pp. 2658–2662, Oct. 2008. [23] Z. G. Qian and W. C. Chew, “Full-Wave packaging modeling using fast broadband surface integral equation solver,” University of Illinois, 2008, Res. Rep. CCEML 4-08. [24] Z. G. Qian and W. C. Chew, “Packaging modeling using fast broadband surface integral equation method,” in Proc. IEEE 17th Topical Meeting on Elect. Perf. of Electronic Packag., Oct. 2008, pp. 347–350. [25] Z.-G. Qian and W. C. Chew, “Fast full-wave surface integral equation solver for multiscale structure modeling,” IEEE Trans. Antennas Propag., vol. 57, pp. 3594–3601, Nov. 2009. [26] A. Bendali, “Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. I. The continuous problem,” Math. Comp., vol. 43, no. 167, pp. 29–46, 1984. [27] A. Bendali, “Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. II. The discrete problem,” Math. Comp., vol. 43, no. 167, pp. 47–68, 1984. [28] Y. Zhang, T. J. Cui, W. C. Chew, and J.-S. Zhao, “Magnetic field integral equation at very low frequencies,” IEEE Trans. Antennas Propag., vol. 51, pp. 1864–1871, Aug. 2003. [29] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. New York: Springer, 1999. [30] F. P. Andriulli, S. Campione, and G. Vecchi, “A stable fast solver for quasi-helmholtz decomposition methods,” presented at the IEEE Int. Symp. on Antennas and Propagation and USNC/URSI National Radio Science Meeting, Charleston, SC, Jun. 2009.

Zhi-Guo Qian (S’07–M’09) received the B.S. and M.S. degrees in electrical engineering from Southeast University, Nanjing, China, in 2001 and 2004, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois, Urbana, in 2009. From 2004 to 2009, he was a Research Assistant with the Center for Computational Electromagnetics and the Electromagnetics Lab at the University of Illinois. In 2009, he joined Apache Design Solutions, San Jose, CA. His research interests are various computational electromagnetics methods and their application to power integrity and signal integrity analysis for package, System-in-Packages and board designs. Dr. Qian was the recipient of the INTEL Best Student Paper Award presented at the IEEE 17th Topical Meeting on Electrical Performance of Electronic Packaging (EPEP) in 2008.

Weng Cho Chew (S’79–M’80–SM’86–F’93) received the B.S. degree in 1976, both the M.S. and Engineer’s degrees in 1978, and the Ph.D. degree in 1980, all in electrical engineering from the Massachusetts Institute of Technology, Cambridge. He is serving as the Dean of Engineering at The University of Hong Kong. Previously, he was a Professor and the Director of the Center for Computational Electromagnetics and the Electromagnetics Laboratory at the University of Illinois, Urbana. Before joining the University of Illinois, he was a Department Manager and a Program Leader at Schlumberger-Doll Research. His research interests are in the areas of waves in inhomogeneous media for various sensing applications, integrated circuits, microstrip antenna applications, and fast algorithms for solving wave scattering and radiation problems. He is the originator several fast algorithms for solving electromagnetics scattering and inverse problems. He has led a research group that has developed parallel codes that solve dense matrix systems with tens of millions of unknowns for the first time for integral equations of scattering. He has authored a book entitled Waves and Fields in Inhomogeneous Media, coauthored a book entitled Fast and Efficient Methods in Computational Electromagnetics, and authored/coauthored over 300 journal publications, over 400 conference publications and over ten book chapters. Dr. Chew is a Fellow of IEEE, OSA, IOP, Electromagnetics Academy, Hong Kong Institute of Engineers (HKIE), and was an NSF Presidential Young Investigator (USA). He received the Schelkunoff Best Paper Award from the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE Graduate Teaching Award, UIUC Campus Wide Teaching Award, and IBM Faculty Awards. He was a Founder Professor of the College of Engineering, and previously, a Y.T. Lo Endowed Chair Professor in the Department of Electrical and Computer Engineering at the University of Illinois. From 2005 to 2007, he served as an IEEE Distinguished Lecturer. He served as the Cheng Tsang Man Visiting Professor at Nanyang Technological University in Singapore in 2006. In 2002, ISI Citation elected him to the category of Most-Highly Cited Authors (top 0.5%). In 2008, he was elected by IEEE AP Society to receive the Chen-To Tai Distinguished Educator Award. He is currently the Editor-in-Chief of JEMWA/PIER journals, and on the Board of Directors of Applied Science Technology Research Institute, Hong Kong. He served on the IEEE Adcom for Antennas and Propagation Society as well as Geoscience and Remote Sensing Society.

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Exponentially Converging Nystrom Methods in Scattering From Infinite Curved Smooth Strips— Part 1: TM-Case John L. Tsalamengas

Abstract—Low-order subdomain basis methods of moments provide little help when highly accurate electromagnetic computations are required. High order modeling, on the other hand, can fill the need for enhanced accuracy but at the expense of greater implementation cost. To supplement such methods, this paper presents Nyström techniques, both easily implemented and highly accurate, relevant to TM scattering by arbitrarily shaped smooth infinite curved strips. The analysis takes full account of both the singular nature of the kernels and the singularities of the solution at the edges; as a result, the proposed solutions are exponentially converging. In addition, by eliminating inner product integrals, closed form analytical expressions are obtained for all matrix elements; thus our algorithms have very low implementation and computational cost. Detailed numerical examples and case studies amply demonstrate the efficiency, stability, and extremely high accuracy of the algorithms. These algorithms apply uniformly from electrically small to electrically large conducting screens. With only slight modifications the present analysis can be also used to obtain exponentially converging Galerkin solutions. Index Terms—Electromagnetic scattering, Galerkin method, integral equations, Nystrom method, strip scatterers.

I. INTRODUCTION IFFRACTION from screens is of importance in studying reflector antennas and wing-like scatterers, frequency-selective surface reflectors, remote sensing etc. The relevant boundary value problems are most conveniently formulated via integral equations (IE) which are usually solved by methods of moments (MoM) such as Galerkin method and its collocation variants [1]–[6]. Low-order subdomain MoM are simple and reasonably accurate for ordinary engineering applications. Despite their simple formulation, fast computer implementation, and low operation count, however, low-order subdomain MoM lack the extra accuracy of exponentially convergent methods. The main difficulty in obtaining MoM that are both highly accurate and fast concerns the presence of several singular integrals that require specialized treatment for efficient computation. High order modelling, on the other hand, definitely fills the

D

Manuscript received June 18, 2009; revised April 01, 2010; accepted April 09, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The author is with the Department of Electrical and Computer Engineering, National Technical University of Athens, GR-15773 Zografou, Athens, Greece (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055788

need for enhanced accuracy, but at the expense of greater implementation cost (see, e.g., [7]–[12]). The highly accurate method of analytical reguralization (MAR) [13] is a sophisticated Galerkin technique, based on singular integral equation methods, suitable for treating scattering by curved PEC strips. The method of dual series equations, leading to the so-called Riemann-Hilbert Problem, is another powerful analytical technique (originally developed in [14] and extensively covered in [15]) which was very successfully applied to analyze scattering by slotted circular cylinders [14]–[17]; later, this technique was extended to a certain class of 2D curved smooth strips [18], [19]. Exponentially converging Galerkin algorithms pertinent to oblique scattering by a flat strip have been developed in [20], [21] based on singular integral equation (SIE) methods. SIE methods have been, also, successfully applied in scattering by a slotted circular cylinder [22]. For a certain class of 2D problems, Nyström method (NM) has been advocated as an alternative solution method which is much more profitable than MoM [23], [24]. Firstly, NM converges exponentially. Moreover, by eliminating inner product integrals, NM offers computational speed-up over a traditional MoM or Galerkin formulation. The central idea behind NM is very simple as discussed in [23]–[25]: approximate the integral by a proper quadrature rule, which involves evaluation of the integrand at a finite number of quadrature points, and then satisfy the resulting equation at those quadrature points. The success of the method depends crucially on finding an appropriate quadrature rule [23], [24]. When the kernel is singular, one has to extract the singularity and, then, to use two different quadrature rules, one for the singular part and another for the analytic part, with common quadrature points [23], [24], [26]. Variants of the Nyström method can be found, for instance, in [7]–[12], [26]–[30]. Working in the framework of the NM of [23], [24] this paper primarily concerns the efficient (exponentially converging) solution of several integral equations relevant to TM scattering by infinite smooth curved strips. The special and definitely simpler case of a flat strip has been recently treated in the context of the NM in [29]. Like [29], all proposed solutions follow. a) Take full account of both the singular nature of the kernels and the singularities of the solution at the edges. This advantage is a requirement of every exponentially converging Nyström or MoM implementation. In this context, proper decomposition of each of the kernel functions into a singular part and an analytic part is very crucial as pointed out at pertinent points in Sections II and III.

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direction normal to the z axis with its electric field parallel to . The total electric field at is z-directed the axis (1)

Fig. 1. Cross-sectional view of a curved smooth strip excited by a plane wave.

b) Eliminate inner product integrals. c) Yield closed form analytic expressions for all matrix elements; this stands in contrast to MoM whose matrix elements require in general the numerical evaluation of integrals [1], [2], [4]. Advantages a)–c) are important characteristics that greatly add to the elegancy and efficiency of our Nyström algorithms. MoM have been studied for years, and the studies have led to computer programs capable of providing results with sufficient accuracy for ordinary engineering applications. The aforementioned studies, and the ensuing computer programs, are general in the sense that they apply to a variety of geometries. The analysis herein only concerns the specific 2D geometries mentioned above. But, as a result of the specialized nature of our analysis, we obtain results of very high accuracy using smaller-than-usual matrices. Our lower computational costs may be useful, for example, in analysis and design problems for multi-element strip scatterers. Furthermore, our results can be used as benchmark tests for general-purpose computer programs. The present study, therefore, is not intended to compete with analyses leading to general-purpose computer programs, and can in fact test and complement such programs. The paper is organized as follows: In Section II, TM scattering of a plane wave at normal incidence is formulated in terms of a first-kind SIE. The relevant SIE is then solved by two exponentially converging Nyström methods as outlined in Sections III–IV. The first method is a direct solution of the initial logarithmically singular IE, which closely resembles the global Nyström method developed in [29] for scattering by a flat strip. The second method, a radically improved (exponentially converging) variant of the powerful indirect technique outlined in [27], recasts the initial IE to a Cauchy singular integral equation solvable by the NM; the Cauchy-type integrals encountered are computed by suitably selected quadratures. Detailed numerical examples and case studies presented in Section V amply demonstrate the efficiency, stability, and remarkable accuracy of the proposed algorithms. Moreover, with only slight modifications the present analysis enables one to obtain exponentially converging Galerkin solutions in the way outlined in Section VI. Both our Nyström and Galerkin algorithms are found to apply uniformly from electrically small to electrically large strips. In the following, the assumed time dependence is suppressed. II. GENERAL FORMULATION Fig. 1 shows an infinitely long, perfectly conducting curved smooth strip in free space with edges parallel to the z axis. The primary excitation is an incident plane wave propagating in a

, and is the contour of the where conductor. The first term in the right side in (1) is the incident electric field

The second term is the scattered field, that is the field due to the induced surface current density , expressed as an integral involving the Green function (2) Integral Equation: Assume that has the parametric , satrepresentation for all . In the isfying following, source points on will be designated by . , (1) By enforcing the boundary condition gives the logarithmically singular first-kind integral equation (3) with unknown

and kernel (4)

where (5) (6) in (3) is identified with For simplicity in notation, . In conformity with the edge condition, will be sought in the form (7) which only incorporates the leading-order edge singularity. (The advisability of using high order representations for the edge singularity to enhance accuracy is discussed in [31] in connection with high order MoM solutions of integral equations.) The electric field integral equation (3) has been the starting point of many past investigations, see, e.g., [1], [2], [5], [6]. Our twofold aim in revisiting this IE here is 1) to present a global, direct, easily implemented, and exponentially converging Nyström solution; and 2) to show how the efficiency of the solution may depend crucially on the details of the implementation (Section III). In addition, a drastically improved variant of the powerful indirect method proposed in [27] for the solution of (3) will be presented (Section IV). We will show, in particular,

TSALAMENGAS: EXPONENTIALLY CONVERGING NYSTROM METHODS IN SCATTERING–PART 1: TM-CASE

that an exponential convergence can be attained by the indirect method, too; a crucial requirement for this is the suitable decomposition of the kernel function into a singular part and an analytic part. Finally, with only slight modifications, the present analysis will be also applied to obtain exponentially converging Galerkin solutions (Section VI).

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can most efficiently be evaluated via the Gauss-Chebyshev quadrature rule [32, Eq. (25.4.38)] (14)

where the nodes coincide with the zeros of the first kind , that is Chebyshev polynomial

III. DIRECT SOLUTION BY THE NYSTROM METHOD In view of [32, Eq. (9.1.89)]

(15)

(8)

is the Bessel function of (C denotes the Euler’s constant and is confined order ), it is clear that the singularity of . Because of this, we make use of to the term the decomposition (9) where sions

and

Because of the analyticity of , the error of the quadrature (14) decays exponentially with increasing [32]. Consesettles down to its final value for very small values quently, of . is concerned, following the outline in [29] we As far as let (16) and make use of the finite Chebyshev expansion

assume the closed form expres(17) (10)

The weights [29], [33]

can be found from the interpolation formula

(18) (11a) Using (14) and (17) together with the basic result [29, Eq. (55)] (11b) Equation (11b) is readily obtained from (8). Observe that the singularity of the kernel function is confined to the logarithmic term in (9), that is and are analytic. As will be seen later on, the analyticity and is of critical significance in so far as obtaining of an exponential convergence rate is concerned. Substitute (7) and (9) into (3). In the resulting expressions, one then encounters the singular integral

(19) (12) gives (20)

where (12) (21) and the non-singular integral

with (13)

(22)

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Finally, by successively satisfying (3) at , we end up with the linear algebraic system

where

(28) (23) (29)

where

(30) (24) As seen, the matrix elements assume closed form analytical expressions. Simplicity in implementation and exponential convergence are very important features of our algorithm. Remark: As stated above, the analyticity of and is a crucial requirement in obtaining an exponentially converging algorithm; that is, mere continuity is insufficient. To shed light on this point further, consider the decomposition similar to (9)

with and denoting the derivatives with reand . spect to of The IE (27) has to be accompanied by the condition (see [27, Eq. (8)])

(31)

(25) where where (26a)

(32)

(26b) often used in practice. In view of (8), the decomposition (25) is quite a legitimate choice, because is continuous and finite everywhere, taking the value when . Nevertheless, is not analytic: its derivatives of second order and beyond are singular . Because the rate of convergence of the quadrature rule at (14) depends crucially on the smoothness of the integrand (according to error estimates for the remainder term given in [32]), Nyström methods applied to (3) using (25) will inevitably yield a mediocre convergence rate rather than exponential; convergence issues will be discussed later on in Section V. Of course, it is also possible to use quadrature rules different from (14) and (20) obtained from other approximations for the integrand (see, e.g., [34]). However, the application of (14) and (20) is most advantageous because of their simplicity and high approximation order.

Equation (31) results after multiplying (3) by , integrating from to , and interchanging the order of integrations with respect to and . : Observe that Computation of ; hence, the integrand in (28) is continuous and bounded everywhere. Motivated by this remark, one may be tempted as a regular integral via a Gauss-Chebyshev to compute rule similar to (14). Although permissible, this is by no means an advisable choice in so far as obtaining an exponential convergence rate is concerned: the derivatives with respect to of of order 2 and beyond become singular . Because the rate of convergence of a quadrature at rule depends crucially on the smoothness of the integrand, such a choice would inevitably yield a mediocre convergence rate rather than exponential. The best practice, therefore, is to treat as a singular integral of the type , see equation (12); this yields

IV. INDIRECT SOLUTION BY THE NYSTROM METHOD An indirect solution method is to recast (3) into a Cauchy IE solvable by the exponentially converging NM. Following the outline in [27], we differentiate (3) with respect to and use (9) to obtain the Cauchy singular integral equation (27)

(33)

Computation of : Because of the analyticity of the integral defining , see (29), is of the type and can thus be most efficiently computed by the standard GaussChebyshev quadrature formula.

TSALAMENGAS: EXPONENTIALLY CONVERGING NYSTROM METHODS IN SCATTERING–PART 1: TM-CASE

Computation of : The Cauchy-type integral encountered in (30) can be computed with the help of the quadrature [35]

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TABLE I

jJ j

=0 = 1 V=m; kw = ; = 45

2

AT THE CENTER x OF A FLAT STRIP OF WIDTH w AS OBTAINED BY THE DIRECT AND INDIRECT NM FOR INCREASING L WHEN E

(34) where is the Chebyshev polynomial of the second kind. , where The selection (35) are the zeros of term quadrature rule

, enables one to eliminate the perplexing from (34) and, thus, to obtain the

(36) Most advantageously, in (36) only values of at the nodes are involved. : By using the decomposition (9), the inComputation of tegral in (32) can be expressed as the sum of a logarithmically and a regular integral of the singular integral of the type , see (12), (13); these integrals have to be computed type as outlined in Section III. The final result is (37) where has been defined in (21). Finally, with the help of (37) the left side of (31) can be evaluated as a regular integral via the Gauss-Chebyshev rule. The right side of (31) can be evaluated in the same way. Discretization of (27) and (31): By successively satisfying both , we obtain their (27) and (31) at discrete counterparts1

(38a) (38b) respectively, where

(39)

in place of (38a), and a similar equation in place of (38b); (40) is identical with the first (11) of [27]. The implications on the rate of convergence from selecting (25),(40) rather than (9),(38) are discussed below. V. CONVERGENCE—COMPARISON BETWEEN THE DIRECT AND INDIRECT NYSTROM METHODS enables one to compute any quantity Knowledge of of practical interest. For instance, the total electric field and the induced current density are given by

(41a) (41b)

[Note: (41a) results from (1) whereas (41b) may be obtained from expansions similar to (17)–(18) for the function ]. As a first validation test, Table I—referring to a flat strip of , the amplitude of the induced surface width —shows of the strip, as obtained current density at the center both by the direct and the indirect NM for increasing when . In this special case of a flat strip, the problem is exactly solvable by separation of variables in elliptic coordinates in terms is of Mathieu functions. The exact value of

Remark: If (25) had been selected instead of (9), one would have obtained as in [27] the expression

(40) 1The reason for the choice t with (34) and (36).

= t~

has been explained above in connection

(42)

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0

j

Fig. 2. log (j[E (a; 0) E (a; 0)]=E (a; 0) ) versus L for a circular strip of radius a that extents from  =  to  = 2  . The parameter values are = 0 ; E = 1;  = 10 ; ka = 1. (Solid line: direct solution by the NM, dashed line: indirect solution by the NM).

0

j

0

j

j

0

j

j

0

j

Fig. 3. log ( [E (a; 0) E (a; 0)]=E (a; 0) ) versus L for a circular strip of radius a that extents from  =  to  = 2  . The parameter values are = 0 ; E = 1;  = 10 ; ka = 10. (Solid line: direct solution by the NM, dashed line: indirect solution by the NM).

0

where (43) (44a) (44b)

In (42), (44), and are the even and odd Mathieu and functions of the first kind of order whereas are solutions of the modified Mathieu equation and denote the derivatives with respect to ) [37], [38]. As seen, the convergence of our Nystrom methods is exponential. The final results coincide within 15 decimals with the exact value obtained via (42). Fig. 2 refers to a circular strip of radius , shown in the inset, which extents from to ; the pa. For rameter values are increasing , the number of points of the Gauss-Chebyshev rules used in evaluating the matrix elements, we show the logarithm (base 10) of the relative error . Apparently, the convergence is exponential and essentially the same for the direct and the indirect from (41) using eimethods. (Note: Computation of ther (23) or (38) shows that settles down to its final , for ; this value, asymptotic value can be treated as exact value.) For the same configuration as in Fig. 2, Fig. 3 shows versus when and . Again, the convergence rate is exponential both for the direct and the indirect NM. As noted above, the role of the decomposition (9) in obtaining an exponential convergence is crucial. This is corroborated by the plots in Fig. 4 and Fig. 5 which pertain to the same configuration as in Fig. 2. Firstly, Fig. 4 compares to one another the direct Nystrom solution based on (9), as implemented by

Fig. 4. log ( [E (a; 0) E (a; 0)]=E (a; 0) ) versus L for the circular strip shown in the inset, as obtained by the direct method using the decompositions (9) and (25), respectively. The parameter values are = 0 ; E = 1;  = 10 ; ka = 1.

Fig. 5. log ( [E (a; 0) E (a; 0)]=E (a; 0) ) versus L for the circular strip shown in the inset, as obtained by the indirect method using the decompositions (9) and (25), respectively. The parameter values are = 0 ; E = 1;  = 10 ; ka = 1.

(23)–(24), and the direct Nystrom solution based on (25); in the latter case, the solution is again implemented by (23) but the matrix elements are now given by (45) instead of (24). Apparently, the slight change from (9) to (25) has a striking impact on the rate of convergence. (The interpretation offered by these plots may suggest that, for the result based on (25), the error drops exponentially at the beginning.

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TABLE II

E (a; 0)

AS OBTAINED 1) BY THE DIRECT NM AND BY THE INDIRECT NM FOR INCREASING L. 2) BY THE GM FOR INCREASING j

j

N

(E = 1; = 0 ;  = 5 )

=0; = 1. (Solid

Fig. 6. Condition numbers of the system matrices versus L when E . The circular strip shown in the inset has  and ka line: Direct NM, dashed line: Indirect NM).

=1

=5

However, after the overall error is reduced to a low level by increasing the order of the quadrature rule, an error that was minor becomes the dominant part in the overall error.) Next, Fig. 5 shows that these remarks apply equally well to the indirect solution method as implemented by using (9), (38) and (25), (40) , respectively. Remark: It appears that there in no obvious difference between the results plotted in Fig. 4 and those in Fig. 5. This observation may be interpreted as a confirmation of the equivalence between the direct and indirect solution methods. For a circular strip, Fig. 6 compares to one another the condition numbers of the system matrices in the direct NM and in . the indirect NM when As seen, the condition number in the former method is smaller. However, for this range of both methods lead to reasonable values of the condition number. It is stressed that larger values of (which would lead to larger and possibly problematic condition numbers) are not necessary in any of the two methods considered here: Because of the rapid convergence, excellent accuracy (within 15 significant digits) has already been obtained as will be shown in Table II. with VI. GALERKIN SOLUTION

using the standard L-point Gauss-Chebyshev rule yields the Galerkin linear algebraic system (48) In (48)

We make use of the expansion (46) (49) and substitute from (46) into (3). All integrals encounterd may be computed via the same L-point Gauss-Chebyshev quadratures used by the Nystrom method as outlined in Section III ( is selected as high as needed to assure any prescribed accuracy). This yields (47) where is given by (24). Finally, multiplying both sides of (47) by and integrating from

to

(50) given by (24), based on the decomposition (9). with [Note: The use of (45), based on the (deficient) decomposition (25), instead of (24) is not recommended for reasons explained in the preceding Section V; this point is further discussed below]. As seen from (50), filling the matrix of the Galerkin calls of the function , system (48) requires just like the NM. Convergence of the Galerkin Method: The convergence of , the number the GM depends on two parameters, namely

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TABLE III

jE ; j FOR A NON-CONVEX KITE-SHAPED STRIP AS OBTAINED 1) BY (0 0)

THE

Fig. 7.

L

log

(j[

E a; 0 E (

0)

a; =E N

(

0)]

, solid curve, and 2) versus L for parameter values are = 0 ; E = 1;  = = 40

j

0

10

DIRECT NM FOR INCREASING L AND 2) BY THE GM FOR INCREASING

N

E

(

= 1

;

= 0

;

= 10 )

a; j : 1) versus N for , dotted curve. The ; ka . (

0) )

= 40 = 1

j

Fig. 8. log ( [E (a; 0) E (a; 0)]=E (a; 0) ), (a) versus N for several values of L and (b) versus L for N = 40. The curve labeled “Galerkin exponential” has been obtained via (48)–(50) using (24), based on (9). The curves labeled “Galerkin deficient” have been obtained via (48)–(50) using (45), based on (25). The parameter values are = 1;  = = 0 ; E 10 ; ka = 1.

of the basis functions used in (46), and , the number of points of the Gauss-Chebyshev rules used in (49)–(50). The convergence of the Galerkin algorithm implemented by given by (24), based on the de(48) and (50) [with composition (9)] is illustrated in Fig. 7 for a circular strip of radius , shown in the inset, when . As seen, the converge is exponential both versus and and simultaneously, versus . Thus, using suffice to assure an error close to , i.e., within the round off errors of the computer. To emphasize the vital role of the decomposition (9) in attaining such an exponential convergence, Fig. 8 compares to one

another: 1) the GM using (24), based on the decomposition (9), and 2) the GM using (45), based on the deficient decomposition (25); the parameter values are the same as in Fig. 7. It is evident is essenfrom Fig. 8(a) that the convergence rate versus tially the same for both implementations; that is, in both cases the number of basis functions needed in order to assure any specified relative error is essentially the same. In contrast, the characteristics of their convergence versus are quite dissimilar. Thus, in suffice to get a relative error close to . This case 1, stands in contrast with case 2, where the values of needed to attain specified error levels are much greater than in case 1. For , or instance, in order to attain an error less than , one has to select , or 2000, respectively. In particular, as Fig. 8(b) reveals, error levels close to are unattainable in the context of the decomposition (25). Comparison Between Nystom and Galerkin Methods: For a , Table II shows circular strip of radius as obtained 1) by the exponentially converging direct and indirect Nystrom methods for increasing , the number of points needed to evaluate the Nystrom matrix elements using an -point Gauss-Chebyshev rule, and 2) by the Galekin method for increasing , the number of basis functions used in (46). The Galerkin matrix elements have been computed using (50) together with (24). As seen, all three methods yield extremely accurate results (coinciding within 15 decimals) from electrically small to electrically large strips even though the aperture is very narrow. Moreover, our results are in very good agree, our ment with those of [5] and [38]. [Note: For as obtained by the Galerkin computations shown that method settles down to its final value, 0.103180835843027 and, simultaneously, (2.13602908532912), when . The results in Table II have been computed in (50)]. using

TSALAMENGAS: EXPONENTIALLY CONVERGING NYSTROM METHODS IN SCATTERING–PART 1: TM-CASE

The differentiability of the contour of the strip plays a role similar to the proper choice of the kernel decomposition (the smoother the better; however, a double smoothness guarantees only a couple of correct digits). Table III refers to a nonconvex kite-shaped strip, shown in the inset, whose boundary has the parametric representation . For , and this table shows as obtained 1) by the direct Nystrom method for increasing , and 2) by the . The Galerkin matrix eleGalekin method for increasing ments have been computed from (50) using (24). As seen, both methods are stable and extremely accurate, with their final results coinciding within 15 significant digits. VII. CONCLUSION The Nyström method properly implemented provides exponentially converging solutions to the integral equation of TM scattering by infinite smooth perfectly conducting strips. Proper decomposition of the kernels into a singular part and an analytic part is a crucial requirement in so far as obtaining such exponentially converging solutions is concerned. Sophisticated computation of the integrals encountered, with due regard to both the singular nature of the kernels and the singularities of the solutions at the edges, yields simple closed form analytic expressions for all matrix elements, thus greatly adding to the efficiency and simplicity of the algorithms. With only slight modifications, the present analysis enables one to obtain exponentially converging Galerkin solutions, too. All our Nyström and Galerkin algorithms apply uniformly with remarkable accuracy from electrically small to electrically large conducting screens. The proposed solutions can be used as a basis for treating more complicated structures involving smooth curved strips in conjunction with material inclusions. Such applications will be presented in future works. The proposed method is restricted to smooth strips. Thus, for instance, contours with corners cannot be treated by the proposed algorithms, at least in their present form. ACKNOWLEDGMENT The author is grateful to the anonymous Reviewers for their constructive criticism and valuable comments and suggestions. Those critical comments helped to substantially improve the quality of this work. REFERENCES [1] J. D. Shumpert and C. M. Butler, “Penetration through slots in conducting cylinders—Part 1: TE case,” IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1612–1621, Nov. 1998. [2] J. D. Shumpert and C. M. Butler, “Penetration through slots in conducting cylinders—Part 2: TM case,” IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1612–1621, Nov. 1998. [3] A. Frenkel, “External modes of two-dimensional thin scatterers,” IEE Proc., vol. 130, pp. 209–214, 1983, Pt.H. [4] A. G. Tyzhnenko and Y. N. Ryeznik, “Convergent Galerkin MoM solution for 2-D H-scattering from screens,” Electromagnetics, vol. 25, pp. 329–341, 2005. [5] J. R. Mautz and R. F. Harrington, “Electromagnetic penetration into a conducting circular cylinder through a narrow slot, TE case,” J. Electromagn. Waves Applicat., vol. 2, no. 3/4, pp. 269–293, 1988. [6] J. R. Mautz and R. F. Harrington, “Electromagnetic penetration into a conducting circular cylinder through a narrow slot, TM case,” J. Electromagn. Waves Applicat., vol. 3, no. 4, pp. 303–336, 1989.

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[7] L. F. Canino, J. J. Ottusch, M. A. Stalzer, J. L. Visher, and S. M. Wandzura, “Numerical solution of the Helmholtz equation in 2D and 3D using a high-order Nyström discretization,” J. Comput. Phys., vol. 146, pp. 627–663, 1998. [8] S. D. Gedney, “On deriving a locally corrected Nyström scheme from a quadradure sampled moment method,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2402–2412, 2003. [9] A. F. Peterson, “Accuracy of currents produced by the locally-corrected Nyström method and the method of moments when used with higher-order representations,” ACES J., vol. 17, pp. 74–83, 2002. [10] A. Zhu, D. Gedney, and J. L. Visher, “A study of combined field formulations for material scattering for a locally corrected Nystrom discretization,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4111–4120, Dec. 2005. [11] M. S. Tong and W. C. Chew, “A higher-order Nyström scheme for electromagnetic scattering by arbitrarily shaped surfaces,” IEEE Antennas Wirless Propag. Lett., vol. 4, pp. 277–280, 2005. [12] M. S. Tong and W. C. Chew, “Nyström method with edge condition for electromagnetic scattering by 2D open structures,” in Proc. Progr. Electromagn. Res., PIER 62, 2006, pp. 49–68. [13] A. I. Nosich, “MAR in the wave-scattering and eigenvalue problems: Foundations and review of solutions,” IEEE Antennas Propag. Mag., vol. 42, no. 3, pp. 34–49, 1999. [14] V. N. Koshparenok and V. P. Shestopalov, “Diffraction of a plane electromagnetic wave by a circular cylinder with a longitudinal slot,” USSR J. Comput. Mathematics Mathem. Physics, vol. 11, no. 3, pp. 222–243, 1971, English translation. [15] A. I. Nosich, “Green’s function-dual series approach in wave scattering by combined resonant scatterers,” in Analytical and Numerical Methods in Electromagnetic Wave Theory, H. Hashimoto, M. Idemen, and O. A. Tretyakov, Eds. Tokyo, Japan: Science House, 1992, pp. 419–468. [16] W. A. Johnson and R. W. Ziolkowski, “The scattering of an H-polarized plane wave from an axially slotted infinite cylinder: A dual series approach,” Radio Sci., vol. 19, pp. 275–291, 1984. [17] R. W. Ziolkowski, “n-Series problems and the coupling of electromagnetic waves to apertures: A Riemann-Hilbert approach,” SIAM J. Math. Anal., vol. 16, pp. 358–378, 1985. [18] T. Oguzer, A. I. Nosich, and A. Altintas, “E-polarized beam scattering by an open cylindrical PEC strip having arbitrary “conical section” profile,” Microwav. Opt.Technol. Lett., vol. 31, no. 6, pp. 480–484, 2001. [19] T. Oguzer, A. I. Nosich, and A. Altintas, “Analysis of arbitrary conic section profile cylindrical reflector antenna, H-polarization case,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 3156–3162, 2004. [20] J. L. Tsalamengas and J. G. Fikioris, “Efficient solutions for scattering from strips and slots in the presence of a dielectric half-space: Extension to wide scatterers -Part I: Theory,” J. Appl. Phys., vol. 70, no. 3, pp. 1121–1131, Aug. 1991. [21] J. L. Tsalamengas, “Direct singular integral equation methods in scattering and propagation in strip- or slot-loaded structures,” IEEE Trans. Antennas Propag., vol. 46, no. 10, pp. 1560–1570, Oct. 1998. [22] R. W. Scharstein, M. L. Waller, and T. H. Shumpert, “Near-field and plane-wave ectromagnetic coupling into a slotted circular cylinder: Hard or TE polarization,” IEEE Trans. Electromagn. Compat., vol. 48, no. 4, pp. 714–724, Nov. 2006. [23] R. Kress, Linear Integral Equations. Berlin, Germany: Springer-Verlang, 1989. [24] R. Kress, “Numerical solution of boundary integral equations in timeharmonic electromagnetic scattering,” Electromagn., vol. 10, pp. 1–20, 1990. [25] L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations. Cambridge, U.K.: Cambridge Univ. Press, 1992. [26] Z. T. Nazarchuk, “Singular integral equations in two-dimensional diffraction problems,” in Math. Methods Electromagn. Theory, Proc. 3rd Int. School Seminal, Crimea, Ukraine, Apr. 1990. [27] A. A. Nosich and Y. V. Gandel, “Numerical analysis of quasioptical multireflector antennas in 2-D with the method of discrete singularities: E-wave case,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 399–406, Feb. 2007. [28] A. A. Nosich, Y. V. Gandel, T. Magath, and A. Altintas, “Numerical analysis and synthesis of 2-D quasioptical reflectors and beam waveguides based on an integral-equation approach with Nystrom’s discretization,” J. Opt. Society of Amer.- A, pp. 2831–2836, Sep. 2007.

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[29] J. L. Tsalamengas, “Exponentially converging Nyström’s methods for systems of singular integral equations with applications to open/closed strip- or slot-loaded 2D structures,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1549–1558, May 2006. [30] J. L. Tsalamengas, T. K. Dikaliotis, and E. C. Pitsavos, “Strip-loaded slot antennas driven by multilayered parallel-plate waveguides with optimized matching and enhanced gain: TM and TE cases,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1423–1432, May 2006. [31] M. M. Bibby, A. F. Peterson, and C. M. Coldwell, “High order representations for singular currents at corners,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2277–2287, Aug. 2008. [32] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972. [33] S. Krenk, “On the use of the interpolation polynomial for solutions of singular integral equations,” Quart. Appl. Math., vol. 33, pp. 479–483, Jan. 1975. [34] P. Kolm and V. Rokhlin, “Numerical quadratures for singular and hypersingular integrals,” Comput. Math. Applicat., vol. 41, pp. 327–352, 2001. [35] M. M. Chawla and T. R. Ramakrishnan, “Modified Gauss-Jacobi quadrature formulas for the numerical evaluation of Cauchy type singular integrals,” BIT, vol. 14, pp. 14–21, 1974.

[36] D. S. Jones, Acoustic and Electromagnetic Waves. Oxford, U.K.: Oxford University Press, 1986. [37] S. Zhang and J. M. Jin, Computation of Special Functions. New York: Wiley, 1996. [38] J. L. Tsalamengas, “Direct singular integral equation methods in scattering from strip-loaded dielectric cylinders,” JEMWA, vol. 10, no. 9, pp. 1331–1358, 1999.

John L. Tsalamengas was born in Karditsa, Greece, in 1953. He received the Diploma of Electrical and Mechanical Engineering and the Doctor’s degree in electrical engineering from the National Technical University of Athens (N.T.U.A.), Greece, in 1977 and 1983, respectively. From 1983 to 1984, he worked at the Hellenic Aerospace Academy. He then joined N.T.U.A. where he has been a Professor of electrical engineering since November 1995. His fields of interest include problems of wave propagation, radiation and scattering in presence of complex media, computational electromagnetics, and applied mathematics.

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Exponentially Converging Nystrom Methods in Scattering From Infinite Curved Smooth Strips— Part 2: TE-Case John L. Tsalamengas

Abstract—Low-order subdomain basis methods of moments provide little help when highly accurate electromagnetic computations are required. High order modeling, on the other hand, can fill the need for enhanced accuracy but at the expense of greater implementation cost. To supplement such methods, this paper presents Nyström techniques, both easily implemented and highly accurate, relevant to TE scattering by arbitrarily shaped smooth infinite curved strips. The analysis takes full account of both the singular nature of the kernels and the singularities of the solution at the edges; as a result, the proposed solutions are exponentially converging. In addition, by eliminating inner product integrals, closed form analytical expressions are obtained for all matrix elements; thus our algorithms have very low implementation and computational cost. Detailed numerical examples and case studies amply demonstrate the efficiency, stability, and extremely high accuracy of the algorithms. These algorithms apply uniformly from electrically small to electrically large conducting screens. With only slight modifications the present analysis can be also used to obtain exponentially converging Galerkin solutions. Index Terms—Electromagnetic scattering, Galerkin method, integral equations, Nystrom method, strip scatterers.

I. INTRODUCTION

P

ART 1 of this two-part paper [1] outlined exponentially converging Nystrom and Galerkin solutions of the integral equation (IE) relevant to TM scattering by infinite smooth curved strips. For a survey on alternative methods used in the past in connection with this scattering problem refer to [1]. Part 2 primarily concerns the efficient (exponentially converging) Nystrom solution of the integral-integrodifferential equation pertinent to TE scattering by infinite smooth curved strips. With only slight modifications of the analysis, exponentially converging Galerkin solutions will be also obtained. We emphasize the vital role of the proper decomposition of the kernel functions into a singular part and an analytic part in so far as obtaining such exponentially converging solutions is concerned. The paper is organized as follows: In Section II, TE scattering of a plane wave at normal incidence is formulated in

Manuscript received June 18, 2009; revised July 05, 2010 accepted July 06, 2010. Date of publication August 05, 2010; date of current version October 06, 2010. This work was supported in part by the Program of Basic Research PEBE 2007 of NTUA. The author is with the Department of Electrical and Computer Engineering, National Technical University of Athens, GR-15773, Zografou, Athens, Greece (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2063412

terms of a first-kind integral-integrodifferential equation. This integral equation is discretized in Section III in the context of the Nystrom method (NM) with the help of suitably selected quadrature rules. Several validation tests are presented in Section IV along with a detailed study of the convergence of the proposed Nystrom solution. Finally, Section V shows how, with only slight modifications of the analysis, exponentially converging Galerkin solutions can be also obtained. Detailed numerical examples and case studies amply demonstrate the efficiency, stability, and remarkable accuracy of the proposed Nystrom and Galerkin algorithms. These algorithms are found to apply uniformly from electrically small to electrically large time depenstrips. In the following, the assumed dence is suppressed. II. GENERAL FORMULATION Fig. 1 shows an infinitely long, perfectly conducting curved smooth strip in free space with edges parallel to the z axis. The primary excitation is an incident plane wave propagating in a direction normal to the z axis with its magnetic field parallel to . The total magnetic field at is z-directed the axis (1) where

is the contour of the conductor and (2)

The first term in (1) is the incident magnetic field

and the second term is the scattered magnetic field, that is the field due to the induced surface current density . The other by differenticomponents of the field can be found from ation. For instance (3) , . where Integral Equation: Assume that , sentation for all , where

has the parametric repre, satisfying

In the following, source points on will be designated by , .

0018-926X/$26.00 © 2010 IEEE

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and are In (6), for simplicity in notation, and , respectively. In conformity identified with will be sought in the form with the edge condition, (12)

III. SOLUTION BY THE NYSTROM METHOD Fig. 1. Cross-sectional view of a curved smooth strip excited by a plane wave.

By enforcing the boundary condition , , (1) gives the following first-kind singular integrodifferential equation

For Eq. (9)]

we will make use of the decomposition, see [1,

(13) where (14)

(4) having as the unknown. Following the outline in [2, Section 3.2.3 ], (4) can be recast in the following form which only involves tangential derivatives

(15) For reasons explained in [1] it is very important that the sinis confined to the logagularity of the kernel function and are rithmic term in (13) and, thus, both analytic. In the light of (13) and the first (9), a convenient decomposiis tion for

(5) (16) ( is the unit normal to at , see Fig. 1, is the unit normal to at , and are the directional derivatives along in the and directions, respectively, whereas and are the directional derivatives along in the and directions, tangential to at and , respectively). Using the parametric , , of , (5) gives representation

where (17) (18) Again, it is very important that both and in (18) are analytic. Substitute from (13) and (16) into (6). In the resulting equation, one then encounters several regular integrals of the form

(6) (19) where as well as several logarithmically singular integrals of the form (7) (8)

(9) (10) (11)

(20)

where is analytic. Moreover, we encounter the Cauchytype singular integral (21)

TSALAMENGAS: EXPONENTIALLY CONVERGING NYSTROM METHODS IN SCATTERING—PART 2: TE-CASE

and its derivative

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where (30) (22)

defines the so called finite-part or hypersingular integral

(31) Evaluation of : is a Cauchy-type integral and can be computed by using the quadrature [6]

Evaluation of : The integral is regular and can thus be computed via the standard Gauss-Chebyshev rule [3] (23) with nodes the zeros of the Chebyshev polynomial is

(32) A more convenient form for , valid for any , may be in the right side of (32) via the obtained by substituting finite Chebyshev expansion

, that (33) (24)

Owing to the presumed analyticity of the integrand, the error in (23) decays exponentially as increases [3]. Consequently, settles down to its final value for very small values of . : contains a logarithmic singularity Evaluation of and can be evaluated along the outlines in [4]. and use the finite Chebyshev Let expansion

where can be found from the following interpolation formula analogous to (26) (34) By substituting from (34) into (33) and then into the right side of (32), after some algebraic manipulations which are omitted here for brevity, one ends up with the result

(25) where [5]

can be found from the interpolation formula [4],

(35) where (36)

(26)

Remark: Alternatively,

can be evaluated from

Using (25) and the identity (27) (37)

together with the basic result

Using (31), after some algebra which is omitted here for brevity, gets the closed form expression if if

(38) (28) As seen, (35). Remark: For the selection

one obtains (29)

; that is, (37) is identical with where (39)

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are

the

zeros

of

, the perplexing term encountered in (32) disappears,

j

J j=Z

and (32) simplifies to

TABLE I

= 0 OF A FLAT STRIP OF WIDTH 2w FOR H = 1 A=m, kw =  , = 45

AT THE CENTER x INCREASING L WHEN

(40) Noticeably, (40) has the form of the conventional Gauss-Chebyshev rule [3]. Equation (40) can be, also, directly obtained from (35) with the help of (36). : The hypersingular integral can be Evaluation of evaluated by differentiating, with respect to , either (32) or (35). Omitting any details from brevity, the final result is (41) where (46) if

if Notes: 1) Rule (41) is valid for any . , 2) The selection simplifies (41) to

(42)

As seen, the matrix elements assume, most advantageously, simple closed form expressions. Note: The set of collocation points may be selected may be identified with either of arbitrarily. For instance, or . the sets

, see (39),

(43) where (44)

IV. VALIDATION AND CONVERGENCE As a first validation test, Table I—referring to a flat strip of —shows , the normalized amplitude of the width of the strip, induced surface current density at the center for increasing ; the parameter values are , , , and . In this special case of a flat strip, the problem is exactly solvable by separation of variables in elliptic coordinates in terms of Mathieu functions. The exact value of is

in conformity with [7]. Discretization of (6): By satisfying (6) at a set of collocation points , , we obtain the linear algebraic system (47)

(45) where

where (48) (49) (50) In (47), (49)–(50), and are the even and odd Mathieu functions of the first kind of order whereas and ( ,4) are solutions of the modified Mathieu equation ( and denote the derivatives

TSALAMENGAS: EXPONENTIALLY CONVERGING NYSTROM METHODS IN SCATTERING—PART 2: TE-CASE

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TABLE II

E (a; 0) =Z AND H (a; 0) FOR INCREASING L FOR TWO RESONANT VALUES OF ka (ka = 1:841184 AND ka = 5:331440) WHEN H = 1, =5 , =0 j

j

j

j

Fig. 3. Condition number of the system matrix versus L when H = 1,  = 10 , and ka = 1.

=

0 ,

Table II refers to a circular strip of radius , shown in the to . As a second inset, which extents from and validation test, this table shows (total values) for increasing and for two resonant values of ( and ) when , , . As seen, our results are extremely accurate and in excellent agreement with those of [7], [10], and [11]. Fig. 2 refers to a circular strip of radius , shown in the inset. , , and both for (a) and For (b) , this figure shows the logarithm (base 10) of the relative error versus when . Here stands for the (total value) settles down for suffinal value to which ficiently large . This asymptotic value can be treated as exact value. Apparently, the convergence rate is exponential. For a circular strip, Fig. 3 shows the condition number of the , , , . Nystrom matrix when As seen, the condition number for this range of takes reasonable values. It is stressed that larger values of (which would lead to larger and possibly problematic condition numbers) are not necessary: Because of the rapid convergence, excellent accuracy (within 15 significant digits) has already been obtained with as shown in Fig. 2(a). V. GALERKIN SOLUTION We make use of the expansion (51)

0

j

Fig. 2. log (j[H (a; 0) H (a; 0)]=H (a; 0) ) versus matrix size L for a circular strip of radius a that extents from  =  to  = 2  . The parameter values are H = 1, = 0 ,  = 10 and (a) ka = 1, (b) ka = 10.

and substitute from (51) into (6). All integrals encountered may be computed via the same -point Gauss-Chebyshev quadratures used by the Nystrom method as outlined in Section III ( is selected as high as needed to assure any prescribed accuracy). This yields

0

with respect to ) [8], [9]. As seen, the convergence of our Nystrom solution is exponential. The final result coincides within 15 decimals with the exact value obtained from (47).

(52) is specified in (46). Finally, multiplying both where sides of (52) by and integrating from to , using the standard

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Fig. 4.

H a; 0 H

a; =H L N

a; j H

N

log (j[ ( 0) ( 0)] ( 0) ): 1) versus for , dotted curve, and 2) versus for = 24, solid curve, as obtained = 1, = 0 , = by the Galerkin algorithm. The parameter values are = 1. 10 ,

L

= 40

ka

Fig. 5. Condition number of the Galerkin matrix versus = 10 , and = 0 , = 1.





ka

N

when

= 1

,

TABLE III

jH a; j AS OBTAINED 1) BY THE DIRECT NM AND BY THE INDIRECT NM (H , , FOR INCREASING L. 2) BY THE GM FOR INCREASING N )  (

0)

= 1

-point Gauss-Chebyshev rule, yields the Galerkin linear algebraic system

H

= 0

= 5

(53) In (53),

(54)

(55) given by (46). As seen, filling the matrix of the with calls of the function Galerkin system via (55) requires , just like the NM. Convergence of the Galerkin Method: The convergence of , the number the GM depends on two parameters, namely of the basis functions used in (51), and , the number of points of the Gauss-Chebyshev rules used in (54), (55). The convergence of the Galerkin algorithm is illustrated in Fig. 4 for a circular strip of radius , shown in the inset, when , , , and . As seen, the converge is and versus . Thus, using exponential both versus and simultaneously, suffice to get an error close to , i.e., within the round off errors of the computer. For a circular strip, Fig. 5 shows the condition number of the when , , , Galerkin matrix versus . As seen, the condition number for this range of and takes small values. (Larger values of are not necessary because, as shown in Fig. 4, excellent accuracy, within 15 as significant digits, has already been obtained with a result of the rapid convergence.) Comparison Between Nystom and Galerkin Methods: For a circular strip of radius ( , 50, 100), Table III shows

as obtained 1) by the exponentially converging Nystrom method for increasing , the number of points of the Gauss-Chebyshev rule used in valuating the Nystrom matrix , elements, and 2) by the Galekin method for increasing the number of basis functions used in (51). As seen, both methods yield extremely accurate results from electrically

TSALAMENGAS: EXPONENTIALLY CONVERGING NYSTROM METHODS IN SCATTERING—PART 2: TE-CASE

j

H (0; 0)

NM

TABLE IV

FOR A NON-CONVEX KITE-SHAPED STRIP AS OBTAINED AND 2) BY THE GM FOR INCREASING FOR INCREASING = 0 , = 10 ) j

L



N

1) BY THE ( = 1,

H

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only slight modifications, the present analysis enables one to obtain exponentially converging Galerkin solutions, too. All our Nyström and Galerkin algorithms apply uniformly with remarkable accuracy from electrically small to electrically large conducting screens. The proposed solutions can be used as a basis for treating more complicated structures involving smooth curved strips in conjunction with material inclusions. Such applications will be presented in future works. The proposed method is restricted to smooth strips. Thus, for instance, contours with corners cannot be treated by the proposed algorithms, at least in their present form. ACKNOWLEDGMENT The author is grateful to the anonymous Reviewers for their constructive criticism and valuable comments and suggestions. Those critical comments helped to substantially improve the quality of this work. REFERENCES

small to electrically large strips even when the aperture is very narrow. of the strip plays a The differentiability of the contour role similar to the proper choice of the kernel decomposition (the smoother the better; however, a double smoothness guarantees only a couple of correct digits). Table IV refers to a nonconvex kite-shaped strip, shown in the inset, whose boundary has the parametric representation , . For , , this table shows as obtained 1) by the and Nystrom method for increasing , and 2) by the Galekin method . As seen, both methods are stable and exfor increasing tremely accurate, with their final results coinciding within 14 significant digits. VI. CONCLUSION The Nyström method properly implemented provides exponentially converging solutions to the integral equation of TM scattering by infinite smooth perfectly conducting strips. In this connection, proper decomposition of each of the kernels into a singular part and an analytic part is a crucial requirement. Sophisticated computation of the integrals encountered, with due regard to both the singular nature of the kernels and the singularities of the solutions at the edges, yields simple closed form analytic expressions for all matrix elements, thus greatly adding to the efficiency and simplicity of the algorithms. With

[1] J. L. Tsalamengas, “Exponentially converging Nystrom methods in scattering from infinite curved smooth strips. Part 1: TM-case,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3265–3274, Oct. 2010. [2] N. Morita, N. Kumagai, and J. R. Mautz, Integral Equation Methods for Electromagnetics. Boston: Artech House, 1990. [3] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972. [4] J. L. Tsalamengas, “Exponentially converging Nyström’s methods for systems of singular integral equations with applications to open/closed strip- or slot-loaded 2D structures,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1549–1558, May 2006. [5] S. Krenk, “On the use of the interpolation polynomial for solutions of singular integral equations,” Quart. Appl. Math., vol. 33, pp. 479–483, Jan. 1975. [6] M. M. Chawla and T. R. Ramakrishnan, “Modified Gauss-Jacobi quadrature formulas for the numerical evaluation of Cauchy type singular integrals,” BIT, vol. 14, pp. 14–21, 1974. [7] A. M. Korsunsky, “Gauss-Chebyshev quadrature formulae for strongly singular integrals,” Quart. Appl. Math, vol. 56, no. 3, pp. 461–472, Sep. 1998. [8] D. S. Jones, Acoustic and Electromagnetic Waves. Oxford, U.K.: Oxford Univ. Press, 1986. [9] S. Zhang and J. M. Jin, Computation of Special Functions. New York: Wiley, 1996. [10] R. W. Scharstein, M. L. Waller, and T. H. Shumpert, “Near-field and plane-wave electromagnetic coupling into a slotted circular cylinder: Hard orTE polarization,” IEEE Trans. Electromagn. Compat., vol. 48, no. 4, pp. 714–724, Nov. 2006. [11] J. L. Tsalamengas, “Direct singular integral equation methods in scattering from strip-loaded dielectric cylinders,” JEMWA, vol. 10, no. 9, pp. 1331–1358, 1999.

John L. Tsalamengas was born in Karditsa, Greece, in 1953. He received the Diploma of Electrical and Mechanical Engineering and the Doctor’s degree in electrical engineering from the National Technical University of Athens (N.T.U.A.), Greece, in 1977 and 1983, respectively. From 1983 to 1984, he worked at the Hellenic Aerospace Academy. He then joined N.T.U.A. where he has been a Professor of electrical engineering since November 1995. His fields of interest include problems of wave propagation, radiation and scattering in presence of complex media, computational electromagnetics, and applied mathematics.

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Frequency-Dependent FDTD Simulation of the Interaction of Microwaves With Rocket-Plume Kiyoshi Kinefuchi, Ikkoh Funaki, Member, IEEE, and Takashi Abe

Abstract—The ionized exhaust plumes of solid rocket motors may interfere with RF transmission under certain flight conditions. To understand the important physical processes involved, we measured microwave attenuation and phase delay due to the exhaust plume during sea-level static firing tests for a full-scale solid propellant rocket motor. The measured data were compared with the results of a detailed simulation performed using the frequency-dependent finite-difference time-domain ((FD)2 TD) method. The numerically derived microwave attenuation was in good agreement with experimental data. The results revealed that either the line-of-sight microwave transmission through ionized plumes or the diffracted path around the exhaust plume mainly affects the received RF level, which depends on the magnitude of the plasma-RF interaction.

Fig. 1. Plume RF interaction in rocket flight. Solid rocket motor plume impedes microwave transmission.

Index Terms—FDTD, microwave, plasma, rocket.

I. INTRODUCTION XHAUST plumes from solid propellant rocket motors interfere with microwave transmission, as shown in Fig. 1, and such interference can result in failure of telecommunications, which could affect telemetry, command procedures, or the radar link between a vehicle and the ground-based antennas. This effect has been addressed in many papers [1]–[6] and is thought to be caused by the high-density plasma in the high-temperature exhaust of the motor, since solid propellant grains include low-ionization-energy elements such as sodium and potassium as impurities, which contribute significantly toward high-density plasma generation [7], [8]. Nevertheless, a satisfactory estimate of the attenuation level is still not agreed upon, though it is very necessary because exhaust plume RF interference can result in failure of launch vehicle telecommunications. With current technology, however, it is impossible to simulate the RF interference caused by the plume with simulations that take into account the details of all relevant processes. The reasons for this are as follows: (1) The plume plasma properties, plasma density, and electron collision frequency, all of which affect RF transmission, obey nonequilibrium ionization processes, which are affected by aluminum and aluminum oxide particles, alkali metal impurities, and chemical reactions

E

Manuscript received September 07, 2009; revised February 18, 2010, April 01, 2010; accepted April 02, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. K. Kinefuchi is with the Department of Aeronautics and Astronautics, University of Tokyo, Tokyo 113-8656, Japan (e-mail: [email protected]). I. Funaki and T. Abe are with the Institute Space and Astronautical Science, Japan Aerospace Exploration Agency, Kanagawa 229-8510, Japan (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055796

Fig. 2. Antenna setup in the experiment.

of the propellant. Because such processes are not well understood, a satisfactory estimate of the plasma properties using theoretical solutions is difficult. (2) During a rocket flight, the microwave transmission path includes three-dimensional propagation through the plasma as well as diffraction and reflection. The numerical calculations for such microwave transmissions, in which all effects are taken into account, are not practical because numerous calculations are required. To develop a technique for estimating the attenuation level under real flight conditions, it is necessary to understand first of all the essence of the physics underlying the interactions. For this purpose, it is appropriate to investigate the interactions between the plume and microwaves in a static firing test. This is because the essence of the interaction mechanism is included in a simple plume-RF interaction configuration. Hence, in this study, we investigated the microwave transmission behavior during sea-level static firing tests conducted for a full-scale solid rocket motor. This was done by measuring the

0018-926X/$26.00 © 2010 IEEE

KINEFUCHI et al.: FREQUENCY-DEPENDENT FDTD SIMULATION OF THE INTERACTION OF MICROWAVES

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Fig. 3. Microwave attenuations and S-band phase delay measurement system in the experiment.

microwave attenuation and phase delays at three frequencies by using a simple antenna setup. For detailed investigation of EM field distribution around the plume, frequency-dependent finitedifference time-domain ((FD) TD) method would be extremely useful; however, it has not as yet been applied to rocket plume RF interactions. Accordingly, we first describe the (FD) TD method that we used for computational analyses of microwave transmissions during rocket motor firing, in order to achieve a more detailed understanding of this process. Then, we report on the experimental and numerical results, including approximately estimated plume plasma properties. Finally, the mechanisms of plume-microwave interference during the ground static firing test are discussed with the (FD) TD simulation results.

Fig. 4. Picture of experiment during motor firing.

II. EXPERIMENTAL SETUP AND RESULTS The microwave attenuation experiments were conducted during static firing tests on a full-scale solid rocket motor [9]. Three microwave frequencies generally used in space communications were selected: S-band, 2.3 GHz; C-band, 5.6 GHz; and X-band, 8.5 GHz. Horn-type antennas were adopted for both the transmitter and receiver in order to achieve high directivity. The motor was positioned horizontally, and the antennas were placed on the motor test site facing each other across the motor plume, as shown in Fig. 2. The heights of the antennas were adjusted such that they were positioned along the center of the axis of the motor nozzle. Fig. 3 shows the system for measuring microwave attenuation and phase delay. The microwaves with each of the three frequencies were received and finally downconverted to 10 kHz and then recorded on a digital data recorder with a sampling rate of 48 kHz. The attenuation was evaluated by comparing the received microwave levels before and during the motor firing. For the phase delay measurement, the S-band signal from the oscillator was divided and its phase was periodically varied by using a phase shifter. The phase-delayed signal was combined with the signal received by the receiver antenna and the synthesized signal was recorded. The phase delay data were obtained by analyzing the waveform or hum of the recorded signal. A picture taken during the experiment is shown in Fig. 4. Fig. 5 shows plots of the received microwave voltage ratio for the three frequencies (not in the dB scale), the S-band phase

Fig. 5. Experimental results: microwave attenuations, phase delay and chamber pressure (S-band, 2.3 GHz; C-band, 5.6 GHz; X-band, 8.5 GHz).

delay, and the combustion chamber pressure (non-dimensionaland ized by the maximum chamber pressure of the motor). are the microwave voltages received before and during firing, respectively. In this paper, the term “attenuation” is defined as . It can be seen in Fig. 5 that low-frequency microwaves are highly attenuated during the initial 90 s. This frequency dependence of attenuation is in accordance with the characteristics attributable to the plane wave-slab plasma interaction theory [10], [11]. In addition, there is an apparent relationship between the

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voltage ratios and the chamber pressure because the plasma density distribution in the plume changes with decreasing chamber pressure [9], [12]. For approximately 20 s before the end of the firing (90–110 s in Fig. 5), it is observed that large attenuations and large phase shift appear and that the frequency dependence of attenuation changes.

TABLE I INPUT PLUME PLASMA PROPERTIES FOR (FD) TD CALCULATION

III. FREQUENCY-DEPENDENT FDTD The solid rocket motor exhaust plume includes plasma that causes RF interference. Although there are some efficient methods to solve the interaction between plasma and radio waves [13], we would like to clarify detailed microwave fields and propagation characteristics and to expand the developed method to the actual rocket flight configuration as a next step. Therefore, in order to analyze microwave transmission through such plasma, we have developed the (FD) TD method calculation code [14], [15]. The electric field update equation for the plasma region is expressed as (1) where

where is the electron collision frequency, is the electron is the time step size. The two plasma plasma frequency, and properties ( and ) are required for the (FD) TD simulation so that they are estimated by using the measured voltage and applying the plane wave-slab plasma interacratios tion theory [10], [11] as follows:

(2)

where is the microwave frequency, is the speed of light, and is the distance the wave travels in the plasma. Both (1) and (2) are based on Drude dispersion model. On deriving these equations, thermal particle motion is ignored to assume a cold plasma, which is to say that the thermal electron speed is much less than the speed of light. Applying this equation to the experimentally derived voltage ratios, the spatially mean plasma properties in the plume can be calculated. In order to estimate them,

Fig. 6. FDTD calculation model for rocket plume microwave interaction.

the experimentally derived voltage ratios for two microwave frequencies at least are required because variables to be found from and . We obtained voltage ratios for three (2) are two, microwave frequencies in the experiment. Therefore, the early period and the middle period values were determined by using the received voltage ratios of all the three frequencies and minimizing the error between measured and calculated voltage ratios. On the other hand, the values for the last period were calculated by using only the C-band and X-band voltage ratios since (2) is not appropriate for the S-band. The reason for this treatment will be given later. We assumed and the rocket nozzle exit diameter are approximately identical based on following two reasons: (1) Microwaves mainly take the shortest course to the receiver through the center of the plume, i.e., travel on the LOS (Line-Of-Sight). This consideration will be proved to be correct except in the last period of S- and C-band transmission as explained in the next section. (2) The plume diameter almost equals the nozzle exit diameter of the motor (1.7 m) under atmospheric conditions. In Fig. 5, the attenuation of each frequency changes dynamically with time; hence, the firing period is subdivided into three periods: the early period (10–20 s), the middle period (40–50 s) and the last period (95–105 s). The measured voltage ratios were averaged for each period before substituting to (2) in order to reduce the received noise mainly generated by the flow turbulence in the plume [7]. The estimated plasma properties in each period are summarized in Table I. The (FD) TD simulation model for microwave-rocket plume interaction is shown in Fig. 6. The simulation is two-dimensional and the shape of horn antennas and the ground asphalt are included in the model for the calculation. The plasma properties ( and ) in Table I were inputted for the plume region. The plume diameter in the calculation was the same as the nozzle exit diameter and the plasma distribution was spatially uniform, in accordance with the method used for estimating the values

KINEFUCHI et al.: FREQUENCY-DEPENDENT FDTD SIMULATION OF THE INTERACTION OF MICROWAVES

Fig. 7. Example of received signals at the receiver antenna for S-band before firing and in the early period.

shown in Table I. Although the plasma actually has a nonuniform distribution in the plume, this effect to the microwave propagation is supposed to be small. This is because, in the present configuration, microwaves travel on the LOS in almost all cases. In fact, we can expect that under this condition, the total attenuation in the plume for the nonuniform distribution case is the same as the one for the uniform distribution case. In order to confirm this expectation, we conducted (FD) TD simulation with nonuniform plasma distribution [16]. The result showed that the nonuniform effect to the EM field around the plume is negligible as expected when microwaves travel mainly on the LOS. Furthermore, afterburning layer effect due to mixing with surrounding air at the plume boundary is negligible because the antennas were located near the nozzle exit where the layer has not developed significantly [9]. The Mur second-order absorbing boundary condition is applied to the four sides of the computation space. The size of simulation region is 21 12 m, which is large enough to neglect the effect of the boundary reflection. The simulation grid is square and for the entire grid , where is the microwave wavelength. The time step is selected to satisfy the Courant criterion. The calculations were conducted for the 3 frequencies (S-, C-, and X-band) and 4 periods (before firing, early period, middle period, and last period). These 12 FDTD simulations were separately performed from the start of the microwave radiation at the transmitter through the receiving at the receiver.

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Fig. 8. Numerical S-band E distribution before firing and contour legend.

Fig. 9. Numerical S-band E distribution in the early firing period.

Fig. 10. Numerical S-band E distribution in the middle firing period.

IV. CALCULATION RESULTS An example of received signals at the receiver antenna for S-band before firing and in the early period is shown in Fig. 7. The origin of x-axis corresponds to the start of microwave radiation from the transmitter and the microwave reaches at the receiver about 48 ns. The received levels keep constant after receiving. The voltage ratios were found using these constant values. Figs. 8–11 show the calculation results for the S-band wave transmission ( distribution) before firing, in the early period, in the middle period, and in the last period, respectively. In these figures, the absorbing boundaries are not displayed for the sake of simplicity. The color distribution of these figures was adjusted to facilitate observation of interference fringes. The same color distribution is used for all the figures, therefore

Fig. 11. Numerical S-band E distribution in the last firing period.

the contour legend is shown only in Fig. 8. In Figs. 9 to 11, we can note that the radiated wave interferes with the exhaust plume located at the center of the calculation region. There are interference stripes above the plume, and between the plume

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Fig. 12. Comparison of the voltage ratio between the experimental results and the FDTD results.

Fig. 13. S-band E distribution on the transmitter-plume core-receiver line.

and the ground. These stripes are caused by diffraction around the plume and reflection from the ground asphalt. When we compare these results with the calculation results without the ground, the effect of the ground reflection on the received level is found to be almost negligible. A summary of the comparison of experimental and numerical results for all three frequencies is shown in Fig. 12. The experimental and numerical results show good agreement. Hence, the calculation accuracy is good enough to allow a discussion of the detailed processes associated with microwave transmission. The S-band electric field distributions of the early period and the last period on the line running through the transmitter, the plume core and the receiver are shown in Fig. 13. The attenuation level of the S-band when it passes through the plume is much greater in the last period than in the early period. In the last period, the microwave level slightly increases behind the plume. This level recovery is caused by the wave bypassing around the plume, which is the result of diffraction. The plume diffraction effect increases as plume attenuation gets stronger. In the last period, the microwave hardly penetrates the plume and the diffraction effect becomes stronger due to a large attenuation so that the diffraction effect dominates the level of the received signal. In the case of the early period, however, recovery by diffraction was hardly observed, as shown in Fig. 13, because the plasma penetration path is dominant.

Fig. 14. X-band E distribution on the transmitter-plume core-receiver line.

On the other hand, in the case of the X-band, there is no level increase behind the plume in the calculation results as shown in Fig. 14 so it can be assumed that the magnitude of the diffraction is smaller for the higher frequency case. This is because the higher frequency waves are associated with high directivity and low interference with plasma. The difference in the frequency dependence of attenuation in the last period (90–110 s in Fig. 5), would be caused by appearance of the diffraction path. The phase delay of the S-band gradually increased in the last period in the experiment, as shown in Fig. 5. Generally, phase of microwaves advances through plasma, while diffraction effect elongates the wave path length and causes phase delay. Therefore, the phase delay observed in the experiment was caused by the diffraction around the plume. A quantitative discussion regarding the phase shift on the basis of experimental data and (FD) TD results would be difficult because of the following two reasons: (1) The experimental data includes relatively large erdegrees). (2) The plume size, plasma distriburors (about tion, and so on were approximated in the (FD) TD simulation, as explained in Section IV. The latter approximation strongly affects the phase shift, especially in the diffraction case, because it is very sensitive to the diffraction path length around the plume. Hence, in order to discuss the propagation mode, we estimated the phase advance by applying the plane wave-slab plasma interaction theory [9], [10]. The equation expressing the phase advance through the plasma is

(3)

Fig. 15 shows the comparison of the phase advance as determined by the theory and the FDTD results. The values in the early period and the middle period agree well; hence, it can

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ACKNOWLEDGMENT This study was supported by many staff members of Japanese rocket motor testing and development. The authors would like to express their gratitude to all those who contributed to the testing. They are indebted to T. Kato, S. Tachikawa and H. Ogawa of the Institute Space and Astronautical Science for their valuable cooperation. REFERENCES

Fig. 15. Comparison of the phase advance according to the plane wave-slab plasma interaction theory and the FDTD results.

be concluded that the microwaves travel straight toward the receiver through the center of the plume in these periods. However, the values in the last period are different except for the X-band. This is because the phase delays of the S-band and the C-band in the last period were caused by not only propagation through the plume but also diffraction around it. Only the X-band shows good agreement even in the last period, which indicates that almost all of the X-band penetrated the plume, even in the last period. We used the voltage ratios of the C-band and the X-band to estimate the plasma properties shown in Table I in the previous section, because the S-band did not penetrate the plume in the last period and bypassed it. Diffraction would also affect the C-band transmission in the last period so that the plasma properties in that period would have some uncertainty. However, the uncertainty would be small because the (FD) TD results and experiment show good agreement as seen in Fig. 12. We can conclude that the estimated plasma properties shown in Table I are accurate enough for practical purposes. V. CONCLUSION For estimating plume plasma properties and understanding the essence of rocket plume RF interference, experiments involving the interaction between rocket exhaust and microwave transmission were conducted during a full-scale solid rocket firing test. Microwave transmission analysis using the (FD) TD approach was applied. The attenuation predicted by the (FD) TD calculations agrees well with the experimental results. The (FD) TD results indicated some interactions such as microwave attenuation through the plume and diffraction around the plume. Furthermore, the (FD) TD calculations revealed that when a higher plasma density is achieved in the plume, microwaves hardly penetrate the plume and bypass it, in other words, the diffraction effect becomes dominant. The diffraction effect changes the microwave path length to the receiver and causes the phase delay observed in both the experimental data and the numerical (FD) TD calculation results. In the low plasma density case, however, the penetration path becomes dominant and the plane wave-slab plasma interaction theory is useful for estimating the attenuation.

[1] T. Abe, K. Fujita, H. Ogawa, and I. Funaki, “Microwave telemetry breakdown caused by rocket plume,” in Proc. 31st AIAA Plasmadynamics and Lasers Conf., Denver, 2000, pp. 2000–2484. [2] D. E. McIver, Jr., “The radio frequency signal attenuation problem of rocket exhaust,” in Proce. NASA Conf. on Communicating Through Plasmas of Atmospheric Entry and Rocket Exhaust, 1964, pp. 167–179, Langley Research Center. [3] W. A. Wood and J. E. DeMore, “Microwave attenuation characteristics of solid propellant rocket exhaust products,” in Proc. AIAA 6th Solid Propellant Rocket Conf., WA, 1965, pp. 65–183. [4] F. A. Vicente, E. C. Taylor, and R. W. Phelps, “Analysis of flame effects on measured electromagnetic propagation date,” J. Spacecraft, vol. 4, no. 8, pp. 1069–1075, 1967. [5] F. P. Boynton and P. S. Pajasekhar, “Plume RF Interference Calculations for Space Shuttle,” 1978, NASA-CR-161099. [6] L. D. Smoot, “Causes of ionization in rocket exhausts,” J. Spacecraft, vol. 12, no. 3, pp. 179–183, 1975. [7] L. D. Smoot, “Rocket exhaust plume radar attenuation and amplitude/ phase noise,” J. Spacecraft, vol. 4, no. 6, p. 774, 1975. [8] E. L. Capener, J. Chown, J. E. Nanevicz, and L. A. Dickinson, “Studies on ionization phenomena associated with solid propellant rockets,” in Proc. Solid Propellant Rocket Conf., WA, 1965, pp. 65–182. [9] K. Kinefuchi, I. Funaki, H. Ogawa, T. Kato, S. Tachikawa, T. Shimada, and T. Abe, “Investigation of microwave attenuation by solid rocket exhausts,” in Proc. 47th AIAA Aerospace Science Meeting & Exhibit, FL, 2009, pp. 2009–1386. [10] J. Lawton and F. J. Weinberg, Electrical Aspects of Combustion. Oxford: Clarendon Press, 1969. [11] J. A. Blevins, R. A. Frederick, Jr., and H. W. Coleman, “An assessment of microwave measurement techniques in rocket exhaust applications,” in Proc. AIAA 32nd Aerospace Science Meeting & Exhibit, NV, 1994, pp. 94–0671. [12] L. D. Smoot and D. L. Underwood, “Prediction of microwave attenuation characteristics of rocket exhausts,” J. Spacecraft, vol. 3, no. 3, pp. 302–309, 1966. [13] J. R. Wait, “EM scattering from a vertical column of ionization in the earth-ionosphere,” IEEE Trans. Antennas Propag., vol. 39, no. 7, pp. 1051–1054, 1991. [14] R. J. Luebbers, S. Hunsberger, and K. S. Kunz, “A frequency- dependent finite-difference time-domain formulation for transient propagation in plasma,” IEEE Trans. Antennas Propag., vol. 39, no. 1, pp. 29–34, 1991. [15] J. L. Young, “A full finite difference time domain implementation for radio wave propagation in a plasma,” Radio Sci., vol. 29, pp. 1513–1522, 1994. [16] K. Kinefuchi, I. Funaki, T. Shimada, and T. Abe, “Numerical prediction of microwave-rocket plume interaction,” in Proc. Asian Joint Conf. on Propulsion and Power, Miyazaki, Japan, 2010, pp. 2010–144.

Kiyoshi Kinefuchi was born in Kanagawa, Japan, in 1978. He received the B.S. degree from the Tokyo Institute of Technology, Tokyo, Japan, in 2001 and the M.S. and Ph.D. degrees from the University of Tokyo, in 2003 and 2009, respectively, all in aerospace engineering. He has been an Engineer at the Space Transportation Mission Directorate, Japan Aerospace Exploration Agency, where he developed launch vehicles since 2003. His research interests include microwave propagation, plasma physics, thermo-fluid dynamics, and these applications for aerospace development.

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Ikkoh Funaki (M’00) received the B.E. degree from Kyoto University, Kyoto, Japan, in 1990 and the M.E. and Ph.D. degrees from the University of Tokyo, Tokyo, Japan, in 1992 and 1995, respectively. During 1995 and 2001, he was a part-time Lecturer and then a Research Associate with the Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Kanagawa, Japan, where he engaged himself in developing the microwave discharge ion engines. In 2001, he was a Lecturer with the University of Tsukuba, Tsukuba, Japan. In 2003, he joined the Department of Space Transportation Engineering, Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, where he is currently an Associate Professor. His current research interests are electric and other advanced spacecraft propulsion systems, plasma application in space, and space plasma physics.

Takashi Abe is a Professor with the Department of Space Transportation Engineering, Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Kanagawa, Japan.

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Two-Dimensional Microwave Imaging Based on Hybrid Scatterer Representation and Differential Evolution Abbas Semnani, Member, IEEE, Ioannis T. Rekanos, Member, IEEE, Manoochehr Kamyab, and Theseus G. Papadopoulos

Abstract—A hybrid method for solving two-dimensional inverse scattering problems is proposed. The method utilizes differential evolution as a global optimizer and is based on two alternative representations of the unknown scatterer. Initially, the scatterer properties are represented by means of truncated cosine Fourier series expansion that involves limited number of unknown expansion coefficients. Then, the reconstructed profile obtained is used as an initial estimate and the differential evolution is further applied to a scatterer representation based on pulse function expansion. In this representation, the scatterer region is subdivided by a fine grid and the scatterer properties are considered constant within each cell. When the truncated cosine Fourier expansion representation is adopted, the dimension of the solution space can be reduced and the instabilities caused by the ill-posedness of the problem are suppressed. In the second step of the hybrid method, where the pulse functions representation is considered, the scatterer reconstruction is finer and more accurate due to its quite accurate initial estimate. Numerical results show that the hybrid method results in lower reconstruction error compared to abovementioned representations. Also, the hybrid method outperforms the other two representations, even in the presence of noisy field measurements. Index Terms—Differential evolution (DE), finite difference timedomain (FDTD), inverse scattering, microwave imaging.

I. INTRODUCTION

M

ICROWAVE inverse scattering techniques have attracted significant interest over the last decades because of their numerous applications in areas such as medical imaging, nondestructive testing, and geophysical prospecting. The particular objective of electromagnetic inverse scattering is to reconstruct the spatial distributions of the electromagnetic properties of a scatterer by inverting scattered field measurements. This inverse problem is governed by two main difficulties [1].

Manuscript received July 15, 2009; revised March 09, 2010; accepted April 07, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported by the Iran Telecommunication Research Center (ITRC) under Grant T-500-19708. A. Semnani and M. Kamyab are with the Department of Electrical Engineering, K. N. Toosi University of Technology, Tehran, Iran (e-mail: [email protected]; [email protected]). I. T. Rekanos and T. G. Papadopoulos are with the Physics Division, School of Engineering, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055793

First, inverse scattering is a nonlinear problem, which is usually solved by iteratively minimizing a cost function describing the discrepancy between measured and estimated field data. Local gradient-based optimization techniques as well as global search optimization algorithms have been proposed and applied to the minimization problem. The local optimization algorithms are computationally fast, but may be trapped into local minima [2], [3], whereas the global search algorithms avoid being trapped into local minima (see [4] and references within). Although parallelization of global search algorithms and reduction of the number of problem unknowns have been proposed [5], the solution inverse scattering problems by means of such techniques is still computationally demanding. The second difficulty appearing in inverse scattering problems is the ill-posedness introduced by the compactness of the direct scattering operator; the operator that maps the electromagnetic properties of the scatterer to the scattered field. Usually, in solving inverse scattering problems, the properties inside the scatterer region are expressed as a sum of weighted pulse functions of rectangular support [6], [7]. We will refer to this representation as the pulse function expansion (PFE). The PFE results in fine resolution of the reconstructed scatterer profile, but it requires the determination of huge number of unknown parameters. Furthermore, the PFE is very sensitive to the ill-posedness effects, especially when noisy field measurements are involved. The effects of ill-posedness, such as instabilities in the solution, are usually suppressed by means of regularization [8], [9], exploitation of a priori information in the scatterer description [10]–[12], reduction of the problem unknowns [13], [14], and frequency-hopping techniques [3], [15]. However, implementing the abovementioned techniques is not straightforward at all, while the problem of the huge number of unknowns in PFE still remains. In an alternative approach, the number of the problem unknowns and consequently the ill-posedness can be suppressed by reducing the dimension of the functional space of the spatial distributions of the scatterer properties. In other words, the appropriate representation of the scatterer is of great importance in the treatment of ill-posedness. From this point of view, scatterer representations based on the expansion of a limited number of smooth functions have been proposed [16], [17]. For example, representing the unknown scatterer profile by a truncated cosine Fourier expansion (TCFE) makes the inverse problem less sensitive to the presence of noise. However, because of the limited number of expansion terms, TCFE is not accurate enough for some applications. Finally, multiscaling/mul-

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tiresolution strategies [18]–[23] and multistep approaches [24] for inverse scattering applications have been proposed to cope with the ill-posedness effects. In this paper, a two-step hybrid method for 2D scatterer reconstruction is proposed. In the first step, a coarse scatterer profile is obtained by utilizing the TCFE to describe the scatterer properties. The coefficients of the TCFE are iteratively estimated by means of the differential evolution (DE) algorithm [25], which has been previously proposed for inverse scattering applications [26]–[33]. Then, the reconstructed profile obtained by the TCFE is used in the second step as the initial estimate, while the PFE is adopted to improve the resolution of the profile. As in the first step, the DE is applied to the PFE. Actually, the present work is an extension of the hybrid method, which was originally proposed in [34], to the case of 2D scatterers. Moreover, in its present form, the method utilizes the DE algorithm, in contrast to the original one [34] where the particle swarm optimization (PSO) was applied. The DE has been selected instead of the PSO, because previous indicative results have shown that the DE outperforms PSO in terms of reconstruction accuracy [31], [35]. The description of the inverse scattering problem, the PFE and TCFE representations of the scatterer properties, as well as a brief overview of DE are presented in Section II. In Section III, the hybrid method is introduced and its advantages over the methods based on just the PFE or the TCFE are discussed. In Section IV, the reconstruction of several 2D inhomogeneous scatterers is considered and numerical results are presented. Finally, some brief conclusions follow in Section V.

II. DESCRIPTION OF THE PROBLEM We consider a 2D inhomogeneous scatterer (Fig. 1), which can be either lossless or lossy. The scatterer is infinite along the axis and it is bounded in the -plane. The relative permittivity, , and the conductivity, , of the scatterer depend only on and , whereas the surrounding medium is considered to be free space. The scatterer is illuminated by plane waves with the electric field polarized along the axis, while the time dependence of the field is of a gaussian pulse. A total number of incidences is considered, while for each incidence, the total electric field is measured at distinct observation points. The solution of the direct scattering problem is carried out by means of the FDTD method [36] applied to the discretized -plane. The objective of the inverse scattering procedure is to estiand , which are described by the parammate eters vector . Actually, is estimated by minimizing the cost function

Fig. 1. Geometrical configuration of the problem.

and

is the total variation regularization term [37],

(3) The latter can be introduced to cope with the ill-posedness. In (2), and are the estimated and the measured field, respectively, while denotes the incidence, is the observation point, is the time index, and is the total time of measurement. The integrations in (3) are carried out within the scatterer region and the impact of the regularization is tuned by the and . regularization factors A. Representations of the Scatterer Properties If the PFE representation of the scatterer properties is adopted, the scatterer region is subdivided by a rectangular grid with and subdivisions along the and axis, respectively. Then, the relative permittivity and conductivity are expressed as,

(4)

(5) respectively, where

(1) where

is the standard error term given by

(2)

if elsewhere

(6)

are the pulse functions. Thus, in the PFE representation, the describing the scatterer properties parameters vector components consists of .

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If cosine basis functions are used, the TCFE representation of the scatterer properties is derived. In particular, and are expressed as

(7) (8) where and vector

and are the dimensions of scatterer region along the axis, respectively. In the TCFE case, the parameters consists of components . Compared to the PFE, the TCFE usually involves much lower number of unknown co, because the support domain of the efficients cosine functions covers the entire scatterer region. Hence, the TCFE representation reduces the solution space dimension and results in significant suppression of the ill-posedness effects. As discussed in sensitivity analysis of the TCFE [17], the use of low number of expansion terms reduces the resolution of the scatterer reconstruction, whereas employing many terms causes oscillatory response or even divergence of the solution. Therefore, although an inverse scattering algorithm based on TCFE leads to a fast reconstruction with a substantial reduction in both ill-posedness and amount of computations, an important weakness is the imperfect accuracy.

Fig. 2. Flow chart of the proposed hybrid method.

fittest with respect to the cost function, it replaces generation, i.e., if if

B. Brief Overview of the Differential Evolution Algorithm In DE, the cost function is minimized with respect to the vector of parameters, , by updating a set of . is the population candidate solutions size and is the dimension of the solution space, which equals or in the PFE and the TCFE case, respectively. to After generating the initial population, the candidate solutions are refined by applying mutation, crossover and selection, iteratively. During mutation, for each individual solution , three , , and , which are mutually difdistinct solutions, ferent and different from , are randomly selected to generate a new solution, , i.e.,

(9) where is the mutation factor. Then, crossover results in the generation of the final offspring , according to the scheme if if

(10)

where , is a random number uniformly distributed within [0,1] and is a predefined crossover probability. It should be mentioned that has to inherit at least one component from . Finally during selection, the offspring, , competes with the initial solution candidate, , and if it is

in the next

(11)

The DE terminates when the cost function of the fittest candidate solution is lower than a predefined threshold. An alternative termination criterion is the completion of a predefined total number of DE iterations. III. HYBRID METHOD In order to increase the resolution of the reconstructed scatterer and at the same time to suppress the ill-posedness of the inverse problem, the use of a two-step hybrid method is proposed. The hybrid method benefits from the advantages of both TCFE and PFE representations of the scatterer properties. The flow chart of the method is shown in Fig. 2. A. First Step In the first step, the TCFE with small number of expansion terms is adopted. The coefficients of TCFE (parameters vector ) are estimated by means of DE. The DE is applied to the minimization of the standard error term (2). No additional regis introduced, because the smoothness of ularization term the cosine functions and the limited number of expansion terms are considered adequate to suppress the ill-posedness. The actual number of expansion terms in TCFE depends on the profile under reconstruction. In particular, when the scatterer profile presents a lot of discontinuities or it is rather oscillatory, then the number of TCFE terms used should increase. The first step iterations of the DE algorithm of the hybrid method employs

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Fig. 3. (a) Original relative permittivity profile of a lossless scatterer (first reconstruction example) and (b) best profile estimate present in the initial population of DE. Reconstructed profiles utilizing (c) the PFE representation, (d) the TCFE representation, and (e) the hybrid method.

resulting in a preliminary reconstruction of the scatterer properand in (7) ties. In particular, the expansion coefficients and (8), are estimated. It should be mentioned that for the solution of the direct scattering problem by means of the FDTD, the PFE representation of the scatterer is utilized. This requires that the TCFE coefficients are mapped to the PFE ones, i.e.,

(12) (13) Furthermore, at some positions within the scatterer region, the TCFE may give values of relative permittivity and conductivity, or . In these which are not feasible, i.e., cases, is set equal to one and equal to zero. B. Second Step To improve the resolution of the profiles derived by the first step, the PFE is adopted during the second step of the hybrid method. The DE minimizes the cost function (1), which apart from the standard error term, includes also the regularization . Now, the objective of DE is to search for the opterm timum sets of PFE coefficients, and . In the second step, the initial population of DE contains the parameters vector

Fig. 4. (a) Standard error term, F (p), of the cost function and (b) reconstruction error, e(p), versus the number of iterations when the PFE representation and the hybrid method are applied to the reconstruction of a lossless scatterer (first reconstruction example). The vertical dotted lines separate the first and the second step of the hybrid method.

, which is related to the best candidate solution found in the end of the first step. The parameters vector is by means of (12) and (13). Also, actually derived from part of the initial population consists of candidate solutions gen. In particular, 10% of the erated by small perturbations of population is generated by this method. By this technique, the DE exploits the best but still coarse reconstruction derived by means of TCFE. Furthermore, random solutions around the best one are utilized. The rest of the initial population is randomly generated to allow global search. In the second step, the DE teriterations. minates after From the presentation of the hybrid method as well as of the TCFE and PFE representations involved, it is evident that the proposed method has a lot of similarities with the multiscaling/multiresolution techniques [18]–[23]. In particular, as in the multiresolution methods, the scatterer is initially described by means of a small number of unknown coefficients, resulting in a low-resolution scatterer reconstruction. This reconstruction is further improved and its resolution becomes finer by increasing the number of unknowns. However, there is a significant difference between the present hybrid method and the multiresolution ones. In the hybrid method, the basis functions used in the two sequential representations are not of the same type, as they should be in a multiresolution approach. In the TCFE, the support domain of each cosine basis function covers the whole scatterer, whereas in the PFE, the pulse basis functions are discontinuous and the support domain of each pulse is much

SEMNANI et al.: 2-D MICROWAVE IMAGING BASED ON HYBRID SCATTERER REPRESENTATION AND DIFFERENTIAL EVOLUTION

Fig. 5. (a) Original relative permittivity profile of two distinct lossless scatterers (second reconstruction example) and (b) best profile estimate present in the initial population of DE. Reconstructed profiles utilizing (c) the PFE representation, (d) the TCFE representation, and (e) the hybrid method.

smaller than the scatterer region. The main advantage of using, in the first step, the TCFE instead of a coarse PFE representation is that no additional regularization is required because the cosine functions are already smooth. Moreover, one can theoretically achieve a high-resolution representation of the scatterer by just selecting a high-frequency cosine function, whereas in the PFE, a significantly larger number of expansion terms is required. IV. NUMERICAL RESULTS The proposed hybrid method is applied to the reconstruction of three different 2D dielectric scatterer profiles. In all three cases, the scatterer region is inhomogeneous, while the first two of them are considered lossless whereas the third one is lossy. The scatterer region (Fig. 1) for all three test cases is square , and it is illuminated by TM-polarized gaussian pulses generaround the region. The gaussian ated at four positions pulses are not modulated and their bandwidth frequency corresponds to a characteristic wavelength . For each illumination, the total electric field is measured at eight positions placed around the scatterer region. The positions of transmitters and receivers are depicted in Fig. 1. Actually, the measurements are simulated by means of the FDTD. To avoid the “perpetration of inverse crime”, a finer mesh with double density is utilized for the generation of simulated measurements compared to the

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Fig. 6. (a) Standard error term, F (p), of the cost function and (b) reconstruction error, e(p), versus the number of iterations when the PFE representation and the hybrid method are applied to the reconstruction of two distinct lossless scatterers (second reconstruction example). The vertical dotted lines separate the first and the second step of the hybrid method.

mesh used during the inverse problem solution. For the PFE representation of the scatterer properties, ten subdivisions are conresulting in 100 sidered along both and axis equal square cells. For the TCFE representation, five expansion . Concerning terms are utilized for each direction the DE implementation, the mutation factor is , whereas . The population size in the crossover probability is . the first and the second step of the hybrid method is The same number of DE iterations is adopted for both first and . For comparison, the DE is second step, i.e., also applied by considering strictly either the PFE or the TCFE representation for 600 iterations each. Hence, in all cases, the stopping criterion of DE is the completion of 600 iterations. It should be noted that, when the DE is applied to the strict PFE and TCFE representations, as well as to the hybrid method, the same initial populations are utilized. To quantify the reconstruction accuracy, the reconstruction error is defined as

(14)

where “ ” denotes the original scatterer properties. Even when the TCFE is adopted, the evaluation of the reconstruction error

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Fig. 7. (a) Original relative permittivity profile of a lossy scatterer (third reconstruction example) and (b) best profile estimate present in the initial population of DE. Reconstructed profiles utilizing (c) the PFE representation, (d) the TCFE representation, and (e) the hybrid method.

Fig. 8. (a) Original conductivity profile of a lossy scatterer (third reconstruction example) and (b) best profile estimate present in the initial population of DE. Reconstructed profiles utilizing (c) the PFE representation, (d) the TCFE representation, and (e) the hybrid method.

is based on the values of and derived by (12) and (13), respectively. In the first reconstruction example, we consider a lossless scatterer with discontinuous relative permittivity profile (Fig. 3(a)). The domain of investigation, within which the scatterer lies, is square with side equal to the characteristic wavelength , and it is identical for all the reconstruction examples studied. As shown in Fig. 3(a), the scatterer cross-section is square, while its relative permittivity presents a discontinuity and . Only the reby taking two distinct values construction of the relative permittivity is considered, because a priori knowledge that the scatterer is lossless is exploited. The original scatterer profile, the best estimate of the relative permittivity within the initial population of DE, as well as the final reconstructed profile by means of the PFE, the TCFE, and the hybrid method are shown in Fig. 3. When the regularization term (3) is introduced in the strict PFE case and the second step is set equal of the hybrid method, the regularization factor to 0.0001. The reconstructed scatterer profiles derived after 600 iterations of the DE for the cases of the PFE and the TCFE representation are shown in Fig. 3(c) and (d), respectively. The reconstruction based on the proposed hybrid method (Fig. 3(e)) involves 300 iterations utilizing TCFE (first step), followed by another 300 iterations utilizing PFE (second step). From Fig. 3, it is evident that the proposed hybrid method resulted in more accurate reconstruction compared to the strict PFE and the strict

TCFE. Also, the reconstructed scatterer obtained by the hybrid method preserved the edges of the original scatterer. Thus, the proposed method leads to high resolution reconstruction. The standard error term (2) of the cost function (1) and the reconstruction error (14) versus the number of iterations of the DE for the cases of the PFE representation and the hybrid method are presented in Fig. 4. The graphs of the standard error term and the reconstruction error corresponding to the TCFE representation are not presented. The reason is that, in the strict TCFE case, both the standard error term and the reconstruction error are identical to those of the hybrid method for the first 300 iterations (first step). Also, for the rest 300 iterations in the strict TCFE case, the values of the standard error term and reconstruction error do not change. Actually, numerical results showed that the DE converges before the 300th iteration when the TCFE is adopted. It should be mentioned that when the PFE representation is involved, i.e., during the strict PFE case as well as during the second step of the hybrid method, the standard error term of the cost function (Fig. 4(a)) is not monotonically decreasing with respect to the number of iterations. This comes in contradiction to what is expected when the DE is applied. The reason is that, when the PFE representation is adopted, the DE minimizes the whole cost function (1) and not just the standard error term (2). From Fig. 4(a), it is clear that when the DE is applied to the TCFE representation, rapid convergence of the reconstruction is achieved at about 300

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TABLE I STANDARD ERROR TERM F (p) AND RECONSTRUCTION ERROR e(p) FOR THE THREE RECONSTRUCTION EXAMPLES AFTER 600 ITERATIONS OF DE UTILIZING PFE, TCFE AND THE HYBRID METHOD

TABLE II STANDARD ERROR TERM F (p) AND RECONSTRUCTION ERROR e(p) AFTER 600 ITERATIONS OF DE BY APPLYING PFE, TCFE AND THE HYBRID METHOD TO THE FIRST RECONSTRUCTION EXAMPLE. RESULTS DERIVED IN THE CASE OF NOISY MEASUREMENTS WITH DIFFERENT SIGNAL-TO-NOISE RATIOS (SNR)

Fig. 9. (a) Standard error term, F (p), of the cost function and (b) reconstruction error, e(p), versus the number of iterations, when the PFE representation and the hybrid method are applied to the reconstruction of a lossy scatterer (third reconstruction example). The vertical dotted lines separate the first and the second step of the hybrid method.

iterations. Also, in the first 300 iterations, the reconstruction accuracy achieved using TCFE (Fig. 4(b)) is higher compared to the one obtained by means of PFE for the same number of iterations. Furthermore, after 600 iterations, the reconstruction error obtained using the hybrid method is lower compared to both the strict PFE and the strict TCFE representations. In the second reconstruction example, we consider two distinct lossless scatterers of square cross-section with relative perand (Fig. 5(a)). As in the first reconmittivities struction example, only the reconstruction of the relative permittivity is considered. The original relative permittivity profile, its best estimate within the initial population of DE, as well as the final reconstructed profiles by means of the PFE, the TCFE, and the hybrid method are shown in Fig. 5. As in the first example, is set equal to 0.0001. To derive the the regularization factor reconstructed scatterer profiles, the same total number of DE iterations is used, as in the first reconstruction example. Fig. 6 presents the standard error term of the cost function and the reconstruction error versus the number of iterations of the DE for the cases of the PFE representation and the hybrid method. From Figs. 5 and 6, it is evident that the hybrid method outperforms both the strict PFE and the strict TCFE in terms of preservation of scatterer edges and reconstruction accuracy. In the last reconstruction example, we consider a lossy scatterer of square cross-section with discontinuous relative permittivity and conductivity profiles (Figs. 7(a) and 8(a)). The relative permittivity and conductivity of the inner part of the scatterer

and , respectively, whereas the corare responding values at the outer part of the scatterer are and . It is clear that since the scatterer is lossy, the reconstruction of the conductivity profile of the scatterer region has to be considered, as well. Hence, the number of the unknown expansion coefficients in the PFE, the TCFE, and the hybrid method is two times the number of unknowns related to the previous reconstruction examples. In the strict PFE case and the second step of the hybrid method, the regularization factors and . The original relative permitare tivity profile, its best estimate within the initial population of DE, as well as the final reconstructed relative permittivity by means of the PFE, the TCFE, and the hybrid method are shown in Fig. 7. The corresponding conductivity profiles are shown in Fig. 8. The total number of DE iterations for the PFE, the TCFE, and the hybrid method is the same as in the previous reconstruction examples. Fig. 9 presents the standard error term of the cost function and the reconstruction error versus the number of iterations of the DE for the cases of the PFE representation and the hybrid method. As in the previous two reconstruction examples, it is clear from Figs. 7, 8, and 9 that the hybrid method results in lower reconstruction error and more accurate edge preservation compared to both PFE and TCFE. Table I presents the values of the standard error term and the reconstruction error obtained after 600 iterations of each method when applied to the three reconstruction examples. We note that in the second example,

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Fig. 10. Reconstructed relative permittivity profiles of a lossless scatterer (first reconstruction example) based on the PFE representation using noisy measure(noiseless case), (b) SNR = 20 dB, and (c) ments with (a) SNR = SNR = 15 dB.

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where two separated scatterers are considered, the strict PFE results in lower reconstruction error compared to the strict TCFE. This indicates that the TCFE with small number of expansion terms is not adequate enough for well separated scatterers. Concerning the computational burden of the hybrid method, it should be mentioned that it is almost totally governed by the time required for the solution of the direct scattering problems. In other words, the computational time needed for the generation of each new population in the DE is almost negligible compared to the FDTD application. However, the hybrid method happens to be much faster than the pure PFE, because it provides more accurate reconstruction in much smaller number of iterations. This is attributed to the use of the TCFE representation at the first step of the method. In particular, as shown in Figs. 4, 6, and 9, the TCFE converges much faster than the PFE during the first 100 iterations. The performance of the hybrid method in comparison with the PFE and the TCFE representations has also been investigated in the case of noisy measurements. In particular, the scatterer profile of the first reconstruction example has been considered, while the measurements of the total field have been corrupted by additive white Gaussian noise. Three different values of the signal-to-noise ratio (SNR) have been examined, i.e., noiseless measurements, and SNR equal to 20 and 15 dB. The SNR level in decibels is given by

(15) where is the variance of the Gaussian noise. The DE has been applied to the PFE and the TCFE representation of the scatterer for 600 iterations. The application of the DE in the iterations utilizing the hybrid method consisted of PFE, followed by another iterations utilizing the

Fig. 11. Reconstructed relative permittivity profiles of a lossless scatterer (first reconstruction example) based on the TCFE representation using noisy mea(noiseless case), (b) SNR = 20 dB, and (c) surements with (a) SNR = SNR = 15 dB.

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Fig. 12. Reconstructed relative permittivity profiles of a lossless scatterer (first reconstruction example) based on the hybrid method using noisy measurements with (a) SNR = (noiseless case), (b) SNR = 20 dB, and (c) SNR = 15 dB.

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TCFE. The final values of the standard error term and the reconstruction error for the three methods are presented in Table II. It is evident that the hybrid method outperforms PFE and TCFE in terms of reconstruction accuracy, even when the SNR is 15 dB. The reconstructed relative permittivity profiles when noiseless or noisy measurements are inverted are shown in Figs. 10, 11, and 12, for the cases when the PFE, the TCFE, and the hybrid method are applied, respectively. From Fig. 10, we conclude that the PFE representation is inadequate when the measurements are contaminated with noise. On the other hand, when the TCFE representation is used, the reconstructed profiles are almost identical for both noiseless and noisy

SEMNANI et al.: 2-D MICROWAVE IMAGING BASED ON HYBRID SCATTERER REPRESENTATION AND DIFFERENTIAL EVOLUTION

measurements (Fig. 11). This fact can be attributed to the significant regularization of the inverse problem because of the small number of unknowns in the TCFE representation and the smoothness of the cosine basis functions. It should be emphasized that the TCFE representation results in acceptable estimates of the scatterer profiles and low reconstruction errors, without the adoption of an additional regularization term in the cost function. Finally, as shown in Fig. 12, the hybrid method is robust in the presence of noise, while it also preserves the discontinuities of the scatterer profile. V. CONCLUSION A hybrid method for solving inverse scattering problems has been proposed. The proposed method is based on considering two alternative representations of the unknown scatterer properties, namely the truncated cosine Fourier expansion and the pulse function expansion representations, in a hybrid scheme. The truncated cosine Fourier expansion shows limited reconstruction precision for well separated scatterers, but it is less affected by the ill-posedness of the problem, even when no additional regularization term is introduced. On the other hand, the pulse function expansion results in accurate reconstruction, but the estimated scatterer profiles are very sensitive to the presence of noise. It has been shown that, with the aid of the proposed hybrid method, it is possible to exploit the advantages of both representations. Furthermore, the hybrid method conquers the limitations of the aforementioned representations and gives accurate reconstruction results where the discontinuities of the original scatterer properties are preserved. The numerical results show that, even in the presence of noisy field measurements, the hybrid method leads to more accurate scatterer reconstruction compared to the methods that involve either the truncated cosine Fourier expansion or the pulse function expansion only. REFERENCES [1] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Berlin: Springer-Verlag, 1992. [2] A. Abubakar and P. M. van den Berg, “Iterative forward and inverse algorithms based on domain integral equations for three-dimensional electric and magnetic objects,” J. Computat. Phys., vol. 195, no. 1, pp. 236–262, Mar. 2004. [3] I. T. Rekanos and T. D. Tsiboukis, “A finite element based technique for microwave imaging of two-dimensional objects,” IEEE Trans. Instrum. Meas., vol. 49, no. 2, pp. 234–239, Apr. 2000. [4] M. Pastorino, “Stochastic optimization methods applied to microwave imaging: A review,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 538–548, Mar. 2007. [5] A. Massa, D. Franceschini, G. Franceschini, M. Pastorino, M. Raffetto, and M. Donelli, “Parallel GA-based approach for microwave imaging applications,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3118–3127, Oct. 2005. [6] I. T. Rekanos, T. V. Yioultsis, and C. S. Hilas, “An inverse scattering approach based on the differential E-formulation,” IEEE Trans. Geosci. Remote Sensing, vol. 42, no. 7, pp. 1456–1461, Jul. 2004. [7] L. Li, W. Zhang, and F. Li, “Tomographic reconstruction using the distrorted Rytov iterative method with phaseless data,” IEEE Geosci. Remote Sens. Lett., vol. 5, no. 3, pp. 479–483, Jul. 2008. [8] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems. Washington, DC: Winston, 1977. [9] S. Caorsi, S. Ciaramella, G. L. Gragnani, and M. Pastorino, “On the use of regularization techniques in numerical invere-scattering solutions for microwave imaging applications,” IEEE Trans. Microwave Theory Tech., vol. 43, no. 3, pp. 632–640, Mar. 1995.

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[10] L. Souriau, B. Duchêne, D. Lesselier, and R. E. Kleinman, “Modified gradient approach to inverse scattering for binary objects in stratified media,” Inv. Probl., vol. 12, no. 4, pp. 463–481, Aug. 1996. [11] A. Fhager and M. Persson, “Using a priori data to improve the reconstruction of small objects in microwave tomography,” IEEE Trans. Microwave Theory Tech., vol. 55, no. 11, pp. 2454–2462, Nov. 2007. [12] T. A. Maniatis, K. S. Nikita, and N. M. Uzunoglu, “Two-dimensional dielectric profile reconstruction based on spectral-domain moment method and nonlinear optimization,” IEEE Trans. Microwave Theory Tech., vol. 48, no. 11, pp. 1831–1840, Nov. 2000. [13] S. Caorsi, A. Massa, M. Pastorino, and M. Donelli, “Improved microwave imaging procedure for nondestructive evaluations of two-dimensional structures,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1386–1397, Jun. 2004. [14] M. Benedetti, M. Donelli, G. Franceschini, M. Pastorino, and A. Massa, “Effective exploitation of the a priori information through a microwave imaging procedure based on the SMW for NDE/NDT applications,” IEEE Trans. Geosci. Remote Sensing, vol. 43, no. 11, pp. 2584–2592, Nov. 2005. [15] W. C. Chew and J.-H. Lin, “A frequency-hopping approach for microwave imaging of large inhomogeneous bodies,” IEEE Microwave Guided Wave Lett., vol. 5, no. 12, pp. 439–441, Dec. 1995. [16] T. Isernia, V. Pascazio, and R. Pierri, “A nonlinear estimation method in tomographic imaging,” IEEE Trans. Geosci. Remote Sensing, vol. 35, no. 4, pp. 910–923, Jul. 1997. [17] A. Semnani and M. Kamyab, “Truncated cosine Fourier series expansion method for solving 2-D inverse scattering problems,” Prog. Electromagn. Res. (PIER), vol. 81, pp. 73–97, 2008. [18] M. Donelli, G. Franceschini, A. Martini, and A. Massa, “An integrated multiscaling strategy based on a particle swarm algorithm for inverse scattering problems,” IEEE Trans. Geosci. Remote Sensing, vol. 44, no. 2, pp. 298–312, Feb. 2006. [19] E. Miller and A. Willsky, “A multiscale, statistically based inversion scheme for linearized inverse scattering problems,” IEEE Trans. Geosci. Remote Sensing, vol. 34, no. 2, pp. 346–357, Mar. 1996. [20] E. Miller, “Statistically based methods for anomaly characterization in images from observations of scattered radiation,” IEEE Trans. Image Processing, vol. 8, no. 1, pp. 92–101, Jan. 1999. [21] S. Caorsi, M. Donelli, and A. Massa, “Detection, location, and imaging of multiple scatterers by means of the iterative multiscaling method,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 4, pp. 1217–1228, Apr. 2004. [22] A. Baussard, E. L. Miller, and D. Lesselier, “Adaptive multiscale reconstruction of buried objects,” Inv. Probl., vol. 20, no. 6, pp. S1–S15, Dec. 2004. [23] M. Molinari, S. J. Cox, B. H. Blott, and G. J. Daniell, “Adaptive mesh refinement techniques for electrical impedance tomography,” Physiol. Meas., vol. 22, no. 1, pp. 91–96, Feb. 2001. [24] I. Catapano, L. Crocco, M. D’Urso, and T. Isernia, “On the effect of support estimation and of a new model in 2-D inverse scattering problems,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1895–1899, Jun. 2007. [25] R. Storn and K. Price, “Differential evolution-A simple and efficient heuristic for global optimization over continuous space,” J. Global Opt., vol. 11, no. 4, pp. 341–359, Dec. 1997. [26] K. A. Michalski, “Electromagnetic imaging of circular-cylindrical conductors and tunnels using a differential evolution algorithm,” Microwave Opt. Technol. Lett., vol. 27, no. 5, pp. 330–334, Dec. 2000. [27] K. A. Michalski, “Electromagnetic imaging of elliptical-cylindrical conductors and tunnels using a differential evolution algorithm,” Microwave Opt. Technol. Lett., vol. 28, no. 3, pp. 164–169, Feb. 2001. [28] A. Qing, “Electromagnetic inverse scattering of multiple two-dimensional perfectly conducting objects by the differential evolution strategy,” IEEE Trans. Antennas Propag., vol. 51, no. 6, pp. 1251–1262, Jun. 2003. [29] A. Qing, “Electromagnetic inverse scattering of multiple perfectly conducting cylinders by differential evolution strategy with individuals in groups (GDES),” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1223–1229, May 2004. [30] A. Qing, “Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems,” IEEE Trans. Geosci. Remote Sensing, vol. 44, no. 1, pp. 116–125, Jan. 2006. [31] I. T. Rekanos, “Shape reconstruction of a perfectly conducting scatterer using differential evolution and particle swarm optimization,” IEEE Trans. Geosci. Remote Sensing, vol. 46, no. 7, pp. 1967–1974, Jul. 2008.

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[32] A. Bréard, G. Perrusson, and D. Lesselier, “Hybrid differential evolution and retrieval of buried spheres in subsoil,” IEEE Geosci. Remote Sens. Lett., vol. 5, no. 4, pp. 788–792, Oct. 2008. [33] A. Massa, M. Pastorino, and A. Randazzo, “Reconstruction of twodimensional buried objects by a differential evolution method,” Inv. Probl., vol. 20, no. 6, pp. S135–S150, Dec. 2004. [34] A. Semnani and M. Kamyab, “An enhanced hybrid method for solving inverse scattering problems,” IEEE Trans. Magn., vol. 45, no. 3, pp. 1534–1537, Mar. 2009. [35] A. Semnani, M. Kamyab, and I. T. Rekanos, “Reconstruction of one-dimensional dielectric scatterers using differential evolution and particle swarm optimization,” IEEE Geosci. Remote Sens. Lett., vol. 6, no. 4, pp. 671–675, Oct. 2009. [36] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1995. [37] A. Abubakar and P. M. van den Berg, “Total variation as a multiplicative constraint for solving inverse problems,” IEEE Trans. Image Processing, vol. 10, no. 9, pp. 1384–1392, Sep. 2001. Abbas Semnani (S’07–M’09) was born in Sari, Iran, on September 7, 1978. He received the B.S. degree in electrical engineering from the University of Tehran, Tehran, Iran in 2000 and the M.S. (with first rank) and Ph.D. degrees in electrical engineering from K. N. Toosi University of Technology, Tehran, in 2002 and 2009, respectively. He is currently a Research and Teaching Associate at K. N. Toosi University of Technology. His research interests include inverse scattering, computational electromagnetics, optimization methods, antennas and passive microwave devices.

Ioannis T. Rekanos (S’92–A’00–M’02) was born in Thessaloniki, Greece, in 1970. He received the Diploma degree in electrical engineering, in 1993, and the Ph.D. degree in electrical and computer engineering, in 1998, both from the Aristotle University of Thessaloniki (AUTH), Greece. From 1993 to 1998, he was a Research and Teaching Assistant in the Department of Electrical and Computer Engineering at AUTH. From 2000 to 2002, he was a Senior Researcher in the Radio Laboratory at the Helsinki University of Technology,

Finland, holding a Marie Curie Post-Doctoral Fellowship. From 2002 to 2006, he was an Assistant Professor in the Department of Informatics and Communications, Technological and Educational Institute of Serres, Greece. Since 2006, he has been an Assistant Professor in the Physics Division, School of Engineering at AUTH. His current research interests include electromagnetic and acoustic wave propagation, inverse scattering, computational electromagnetics and acoustics, and digital signal processing in biomedicine. Dr. Rekanos has been a scholar of the Bodossaki Foundation, Greece. In 1995, he received the URSI, Commission B, Young Scientist Award.

Manoochehr Kamyab received the B.S. and M.S. degrees from the University of Tehran, Tehran, Iran, and the Ph.D. degree from Michigan State University, in 1982, all in electrical engineering. He is currently an Associate Professor in the Department of Electrical Engineering, K. N. Toosi University of Technology, Tehran. His research interests include the metamaterials and their applications in antenna engineering, electrically small antennas, microwave and millimeter-wave circuits, and mobile communication systems. He is leading a group of graduate students in the areas of negative-refraction metamaterials and their microwave applications, integrated antennas and components for broad-band wireless telecommunications, novel antenna beam-steering techniques, millimeter and submillimeter-wave circuits, as well as scattering and inverse scattering problems.

Theseus G. Papadopoulos was born in Thessaloniki, Greece, in 1983. He received the B.Sc. degree in physics from Aristotle University of Thessaloniki (AUTH), Greece, in 2006 and the M.Sc. degree in astrophysics from University College London (UCL), London, U.K., in 2007. He is currently working toward the Ph.D. degree at AUTH. His research interests include computational methods for wave propagation and inverse scattering problems.

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Scattering by an Infinite Dielectric Cylinder Having an Elliptic Metal Core: Asymptotic Solutions Grigorios P. Zouros and John A. Roumeliotis

Abstract—The scattering of a plane electromagnetic wave by an infinite circular dielectric cylinder, containing an elliptic metallic one is considered in this work. The electromagnetic field is expressed in terms of both circular and elliptical-cylindrical wave functions, connected with one another by well-known expansion formulas. When the solution is specialized to small , , where is values of the parameter the interfocal distance of the elliptic conductor and the length of its major axis, analytical expressions of the form (2) 2 (4) 4 6 are obtained for the scattered field and the various scattering cross-sections. Both polarizations are considered for normal incidence. Numerical results are given for various values of the parameters.

=

( ) = (0)[1 +

+

(

+ ( )]

1)

2

2

Index Terms—Electromagnetic scattering, elliptic cylinder, perturbative analysis, scattering cross sections. Fig. 1. The geometry of the scatterer.

I. INTRODUCTION CATTERING from composite bodies can be used for the detection of their internal structure i.e. the existence of nonsymmetries, inhomogeneities and also in diagnostic methodologies in biological tissues. The shapes of the boundaries severely limit the possibility for analytical solution of such problems. Various mathematical and numerical techniques are used for complicated geometries. Such geometries including one or more elliptic cylinders are treated, among others, in [1]–[18]. More analytically in [1] the electromagnetic cloaking of a conducting object with sharp wedges located within a hollow cylinder of circular or elliptical cross section is examined, by the finite element method. The scattering of plane electromagnetic waves by multilayer elliptic cylinders, with one or more layers made of metamaterials is treated in [2] and with all layers made of isorefractive materials is treated in [3]. In [4], [5] is analyzed the scattering of a plane electromagnetic wave by an infinite circular dielectric cylinder coating eccentrically an elliptic metallic [4] or dielectric cylinder [5], while in [6] an infinite elliptic dielectric cylinder coating eccentrially a circular metallic or dielectric cylinder is examined. The electromagnetic scattering by a lossy dielectric-coated elliptic cylinder is investigated in [7]. In [8] the scattering properties of an impendance elliptic cylinder

S

Manuscript received December 16, 2008; revised January 13, 2010; accepted March 19, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The authors are with the School of Electrical and Computer Engineering, National Technical University of Athens, Athens 15773, Greece (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055777

coated with a homogeneous material are investigated analytically. The computation of the specific absorption rate (SAR) inside a biological elliptic cylinder model illuminated by an incident plane wave or by a line current source is examined in [9] and [10], respectively. The elliptic model is made up of layers of different biological tissues. The radar cross section per unit length in the case of plane wave scattering by a lossy multilayer elliptic cylinder is computed in [11]. In [12], [13] the plane wave scattering by a dielectric elliptic cylinder coated with a confocal [12] or a nonconfocal [13] dielectric is investigated. The scattering of an obliquely incident wave by an elliptic conducting cylinder coated by a lossy dielectric is treated in [14] and by a multilayered elliptical lossy dielectric cylinder in [15]. The scattering by a conducting elliptic cylinder coated by a dielectric elliptical shell is presented in [16], [17] when the shell is confocal and in [18] when the shell is nonconfocal with the conducting elliptic cylinder. In the present work the scattering of an electromagnetic plane wave by an infinite circular dielectric cylinder containing a coaxial elliptic metallic cylinder is treated. The geometry of the scatterer is shown in Fig. 1. The interfocal distance of the and are the lengths of its elliptic cylinder is , while major and minor axes, respectively. The radius of the circular dielectric cylinder is . The present geometry is a perturbation of the coaxial circular one with radii and . All the materials are lossless. Both polarizations are considered for normal incidence. Using expansion formulas between elliptical and circular cylindrical wave functions [19], [20] and satisfying the boundary conditions we conclude, after some manipulation, to two infinite sets of linear nonhomogeneous equations for the electromagnetic field in region II.

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For general values of these sets can be solved only an analytical solunumerically, by truncation, but for tion is possible. After lengthy, but straightforward calculations, analytical expressions of the form are obtained for the scattered field and the scatand tering cross-sections. The expansion coefficients are given by exact, closed-form expressions, independently of , while corresponds to the coaxial circular problem . The main advantage of this expression is that it is valid for each small value of , free of Mathieu functions, while all numerical techniques should repeat the calculation, from the beginning, for and are known, each different small or large. So, once is immediately evaluated by quick “back-of-the-envelope” calculations, for each small . Certainly, this is the result of a great analytical effort which, made once, reduces dramatically the otherwise necessary computer time for the numerical evaluation of the many Mathieu functions, in the case of small . and The terms omitted in our solution are of the order of is not so severe as it may appear higher. So the restriction at first. Independent numerical solution of this same problem shows that the errors in the approximate analytical results of this paper remain low enough, even for values of up to 0.7 or corresponding to a metallic higher (maximum possible strip). Apart from the mathematical interest of its solution, the circular-elliptical combination of the present problem may increase or decrease the scattering cross sections, as compared to the ones of the concentric circular geometry. The -wave polarization is examined in Section II and the -wave polarization is examined in Section III. Finally, in Section IV numerical results are given for various values of the parameters. II.

is the Hankel function of the second kind with where the superscript (2) omitted for simplicity. The field in region II, satisfying the boundary condition at the elliptical boundary , has the expression [4]

(3) . where In (3) are the transverse elliptical cylindrical coordinates , and are the even (odd) with respect to radial Mathieu functions of the first and second kind, respecare the even (odd) angular Mathieu functively, while tions [19]. we use the expanTo satisfy the boundary conditions at sion formulas connecting the Mathieu functions with the coaxial circular cylindrical ones [20] in steps similar with those in [6], [21]. These boundary conditions are

(4) By satisfying them we express ’s

and

in terms of

’s and

-WAVE POLARIZATION

A. Solution of the Problem (5) The incident plane wave impinging normally on the has the form [20]

axis,

(6) (1) where are the polar coordinates with respect to , defines the direction of incidence with respect to , is the , , cylindrical Bessel function of the first kind, is the Neumann factor and is the wavenumber is suppressed in region I. The time dependence throughout. The scattered field is expressed as

where are the expansion coefficients for the both even or odd and Mathieu functions [19] with the cylindrical Bessel function of the second kind (Neumann function). Thus we finally obtain the following two infinite sets of linear nonhomogeneous equations for the expansion and coefficients

(7) (2)

(8)

ZOUROS AND ROUMELIOTIS: SCATTERING BY AN INFINITE DIELECTRIC CYLINDER HAVING AN ELLIPTIC METAL CORE

where

The set (7) takes the following form, up to the order

3301

:

(15)

(9)

A similar expression holds also for the set (8), with . As it is evident from (3) the subscripts of ’s (and ’s) are always nonnegative. Any quantity in (15) with negative subscripts is interpreted as zero. The same is valid also for the corresponding ’s (and ’s). Each of the sets (7) and (8) separates into two distinct subsets, one with even and the other with odd. ’s are obtained from the solution of the set (15) by Cramer’s rule, using formulas (27)–(29) from [23] for the determinant of ’s in (15) and for the determinant originating from after the substitution of its th column by the column of ’s. So

(16)

and (10)

(11) (12) (17) and , are both even or odd. In (9)–(12) we have used the and . substitutions For general values of the sets (7) and (8) can be solved only numerically by truncation, a complicated task due to the calculation of the Mathieu functions for each different . However, for small an analytical, closed-form solution can be obtained. Instead of the eccentricity is used, [22] for as well as Maclaurin series expansions in powers of 1) appearing in the process. After lengthy, each function of but straightforward calculations, one can find expansions of the form

where

(18) if is even(odd) and continues while starts with the value only with values of the same parity with . Using the expansions (13) and (14) into (16)–(18) we obtain

(19) where (13) (20) (21) (14) The analytical expressions of the expansion coefficients are given in the Appendix.

and

(22)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 10, OCTOBER 2010

The various symbols appearing in (20)–(22) are defined in (A.20)–(A.24) of the Appendix. is given by an expansion analogous to (19) Similarly

Using next expansions analogous to (28) for and , , we find from (25) that

,

(29) (23)

where

where , and are given by (20), (21) and (22), respectively, with some differences in the expressions of the various symbols, as it is explained in the Appendix.

(30)

B. The Scattered Far Field By using the asymptotic expansion for the Hankel function in (2), we obtain the scattered far field expression and next the back scattering or radar , forward and total scattering cross sections [4]

(31)

(32)

(24) (25)

Equation (29) can be set in the form

where (33) (26) A similar expression holds also for

and

, namely

The coefficients and are given in (5) and (6). If we substitute there the expansions of the various quantities for small , we obtain (34)

(27) The analytical expressions of the expansion coefficients in (27) are given in (A.25)–(A.40) of the Appendix. Analogous , and . So, , expansions are obtained for , 2, 4 is obtained from (26), by simply using and in place of and respectively, while

In (33) and (34) and correspond to the coaxial circular geometry. The correctness of our results was checked by the forward scattering theorem [25]

(35) which was verified to a very good accuracy for various values of the parameters. III.

(28) with Re representing the real part and the asterisk denoting the complex conjugate.

-WAVE POLARIZATION

The incident wave and the scattered wave are again given by (1) and (2), respectively. The field in region II, satisat is the fying the boundary condition and same as (3), with the only difference that are now replaced by their derivatives with re. In (4), , , should be replaced by spect to at , , , respectively.

ZOUROS AND ROUMELIOTIS: SCATTERING BY AN INFINITE DIELECTRIC CYLINDER HAVING AN ELLIPTIC METAL CORE

VALUES OF k 

,k Q

,g

and g

3303

TABLE I for  = :

Following steps identical with the ones for the -wave polarization we obtain again (5)–(35) with the aforementioned changes. IV. NUMERICAL RESULTS AND DISCUSSION In Table I the values of , , and appearing in (33) and (34) are given for various values of for , , and , for the - and -wave polarization. The results are symmetric about , as it is imposed by the geometry of the scatterer. In Figs. 2–4 the scattering cross-sections are plotted versus for the - and -wave polarization and for the same values of the parameters as in Table I. We give analytical results up to the order , as well as independent numerical results, obtained from the solution of the present problem, by truncating the sets (7) and (8). It is evident that the errors are low enough in each , case. The same is done in Figs. 5–7 for

= 2 54,  = = 1, a=b = 0:4 AND k b = 

,

and . The results are symmetric about . From Figs. 2–7 as well as from the Table I the higher sensito the change of is evident, as compared to that tivity of of and , in the case of -wave. It is also evident, from all figures, that the deviation of the scatterer examined here from the coaxial circular one, by making the inner metallic cylinder elliptic, increases or decreases the scattering cross-sections, depending on the values of the parameters. Inversely, this may be useful for the detection of a shape perturbation in the inner cylinder. In Figs. 8 and 9 we give the percentage errors , , and , respectively, resulting from the for comparison of the approximate results of this paper with the exact numerical results obtained independently. In each figure, , , , and for the - and -wave. The same is done in Figs. 10 and 11

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 10, OCTOBER 2010

= 2 54

Fig. 2. Back scattering cross section for  = : ,  = : and k b  . E -wave: black lines, H -wave: gray lines.

04

=

= 2 54

Fig. 3. Forward scattering cross section for  = : ,  = : and k b  . E -wave: black lines, H -wave: gray lines.

04

=

= 2 54

Fig. 4. Total scattering cross section for  = : ,  = : and k b  . E -wave: black lines, H -wave: gray lines.

04

=

= 1, a=b =

Fig. 5. Back scattering cross section for  = = 0:5,  = = 0:9, a=b = 0:4 and k b =  . E -wave: black lines, H -wave: gray lines.

= 1, a=b =

Fig. 6. Forward scattering cross section for  = : ,  =  . E -wave: black lines, H -wave: gray lines. a=b : and k b

=04

=

= 05

= 0 :9 ,

Fig. 7. Total scattering cross section for  = = 0:5,  = = 0:9, a=b = 0 = 1, a=b = :4 and k b =  . E -wave: black lines, H -wave: gray lines.

for , , and . From these figures, as well as from many other available, it is evident

that the percentage error for approximate results up to the order is much lower than the corresponding one for approximate results up to the order (especially for and for

ZOUROS AND ROUMELIOTIS: SCATTERING BY AN INFINITE DIELECTRIC CYLINDER HAVING AN ELLIPTIC METAL CORE

=1

=04

=

= 45

Fig. 11. Total scattering cross section percentage error for  = : , : , a=b  and . E -wave: black lines,  = : ,k b H -wave: gray lines.

= 2 54

Fig. 12. Back and forward scattering cross section percentage error for : ,  = , a=b . E -wave: black , k b  and  = lines, H -wave: gray lines.

Fig. 9. Total scattering cross section percentage error for  = : ,  = , a=b : ,k b  and . E -wave: black lines, H -wave: gray lines.

=1

= 04

=

= 45

Fig. 10. Back and forward scattering cross section percentage error for  = : ,  = : , a=b : ,k b  and . E -wave: black lines, H -wave: gray lines.

= 05

= 09

= 04

=

= 45

= 05

=

Fig. 8. Back and forward scattering cross section percentage error for  = , a=b : ,k b  and . E -wave: black lines, : ,  = H -wave: gray lines.

2 54

3305

= 09

= 2 54

= 04

=1

=

=1

= 45

=

= 45

the -wave) thus permitting values of up to 0.7 or higher , corresponding to a strip) with low (maximum possible and . The values of used in errors, especially for Figs. 2–7 keep the percentage error low enough in each case. , as can be seen from Our method is valid even for , Fig. 12, as well as from many other available, but for the sets (7), (8) and also (15) turn out to be ill-conditioned, due to the presence of a metallic line at the axis of the dielectric cylinder in this special case (see also [24]). versus , for the The backscattering cross-section , and the -wave is given in Fig. 13 for , and , and in Fig. 14 for , (metamaterials), and . The errors are low enough as in the previous cases. The parameters in Fig. 14 are the same with the corresponding ones in Fig. 2, with and have opposite values. the only difference that The main remark after comparison of these two figures is that versus in Fig. 14, referring the percentage variation of to metamaterials, is much smaller than in Fig. 2, referring to materials.

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 10, OCTOBER 2010

=09

Fig. 13. Back scattering cross section for  = : ,  =  . E -wave: black lines, H -wave: gray lines. : and k b

=2

06

= 02 54

= 0:9, a=b =

Fig. 14. Back scattering cross section for  = : ,  = a=b : and k b  . E -wave: black lines, H -wave: gray lines.

=04

=

= 01,

In Fig. 15 the backscattering cross-sections in dB are plotted, versus , for , , and in the case of the - and the -wave. The variations are greater for the -wave. Finally, in Fig. 16 we plot the percentage errors for , as varies, for the same values of the parameters as in Fig. 15 . Thus we can examine the accuracy of our reand for sults near the resonant peaks and troughs of Fig. 15. We plot the percentage errors up to an upper limit equal to 10, in order to be able to see details for errors less than this limit. It should for the -wave be noticed here that the peak left to and has a very high value, while the ones between , also for the -wave, have values greater than 10. Thus, near some resonant peaks or troughs the percentage errors remain low enough, but near others they become very high, especially for the -wave. To be more specific, the very high percentage errors occur at these values of where the backscattering cross-section becomes small (see Figs. 15 and 16). Nevertheless it should be noticed here that in the former cases the are also small, but much greater than absolute errors of its exact value, so that the percentage errors become too high.

= 2 54

: ,  = Fig. 15. Back scattering cross section in dB for  = . E -wave: black lines, H -wave: gray lines. a=b : and

=04

=0

= 1,

= 2 54

Fig. 16. Back scattering cross section percentage error for  = : ,  = , a=b . E -wave: black line, H -wave: gray : and line.

=1

= 04

=0

It is evident that in these cases terms of the order or higher should be kept in our analytical solution, in order to improve its validity. For much greater than the wavelength, i.e. for electrically large scatterers. the number of terms necessary for the convergence of our solution grows up rapidly and also the use of the present solution, and finally of the formulas (33) and (34), give results which are not correct. It should be noticed also that we have verified the plotted results given in [25] in every case, for all the values of the parameters used there. Also we have verified the results in [23] with and in our excellent accuracy, by setting formulas. APPENDIX The analytical expressions of the various expansion coefficients appearing in (13), (14), also in (20)–(23) and finally in (27), are found after lengthy but straightforward calculations, by using the expansions for the Mathieu functions from [22] and are given in what follows:

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A.

-Wave Polarization

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(A.16) (A.1)

(A.2)

(A.17)

(A.3) (A.4)

(A.18) (A.5) (A.6) (A.7)

represents the Bessel function or the Neuwhere , if , , 2, 4 is replaced by mann function or by respectively (if is simplified to ). We have also used the substitutions

(A.8) (A.19)

(A.9)

(A.10)

while the various ’s and ’s appearing in the former relations have the expressions given in eqs.(A6)–(A12) in the Appendix of [23]. The symbols appearing in (20)–(22) are defined as follows: (A.20) (A.21)

(A.22)

(A.23) (A.11) (A.12) In (A.1)–(A.12) we have used the substitutions (A.13) (A.14)

(A.24)

(A.15)

It should be reminded that the various subscripts must always be non-negative. Any quantity with negative subscript is interpreted as zero.

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Eqs. (A.20)–(A.24) are also used for the evaluation of in (23), with the difference that ’s are replaced by ’s and ’s by ’s. The expansion coefficients for , appearing in (27), have the expressions

B.

-Wave Polarization

Equations (A.13)–(A.16) remain the same with the exception that we replace the Bessel functions with their derivatives with respect to their arguments. Now, (A.17) should be totally changed and written as

(A.25) (A.41) and also (A.18) should be changed as (A.26)

(A.42) is the quantity inside the brackets in (A.17), is the same with , with the only difference that the Bessel functions are replaced by their derivatives with is the quantity inside the brackets in respect to , while (A.18), with the Bessel functions replaced by their derivatives with respect to . Finally, (A.19) should be simply changed by replacing with and with . where

(A.27) where (A.28)

REFERENCES (A.29) (A.30)

(A.31) (A.32)

(A.33) (A.34) while , , 2, 4 is replaced by or by . The expansion coefficients for , appearing also in (27), namely , and are again given by (A.25), (A.26) is now given by (A.28) with and (A.27), respectively, where replaced by , while (A.29)–(A.34) should be replaced by (A.35) (A.36)

(A.37) (A.38)

(A.39) (A.40)

[1] A. Nicolet, F. Zolla, and S. Guenneau, “Finite-element analysis of cylindrical invisibility cloaks of elliptical cross section,” IEEE Trans. Magn., vol. 44, pp. 1150–1153, Jun. 2008. [2] M. Pastorino, M. Raffetto, and A. Randazzo, “Interactions between electromagnetic waves and elliptically shaped metamaterials,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 165–168, 2005. [3] S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinders,” IEEE Trans. Antennas Propag., vol. 52, pp. 189–196, Jan. 2004. [4] J. A. Roumeliotis and S. P. Savaidis, “Scattering by an infinite circular dielectric cylinder coating eccentrically an elliptic metallic one,” IEEE Trans. Antennas Propag., vol. 44, pp. 757–763, May 1996. [5] S. P. Savaidis and J. A. Roumeliotis, “Scattering by an infinite circular dielectric cylinder coating eccentrically an elliptic dielectric cylinder,” IEEE Trans. Antennas Propag., vol. 52, pp. 1180–1185, May 2004. [6] S. P. Savaidis and J. A. Roumeliotis, “Scattering by an infinite elliptic dielectric cylinder coating eccentrically a circular metallic or dielectric cylinder,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 1792–1800, Oct. 1997. [7] A. K. Hamid and M. I. Hussein, “Electromagnetic scattering by a lossy dielectric-coated elliptic cylinder,” Canad. J. Phys., vol. 81, pp. 771–778, May 2003. [8] A. R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag., vol. 48, pp. 1574–1580, Oct. 2000. [9] S. Caorsi, M. Pastorino, and M. Raffetto, “Analytic SAR computation in a multilayer elliptic cylinder for bioelectromagnetic applications,” Bioelectromagnetics, vol. 20, pp. 365–371, 1999. [10] S. Caorsi, M. Pastorino, and M. Raffetto, “Analytical SAR computation in a multilayer elliptic cylinder: The near-field line-current radiation case,” Bioelectromagnetics, vol. 21, pp. 473–479, 2000. [11] S. Caorsi, M. Pastorino, and M. Raffetto, “Radar cross section per unit length of a lossy multilayer elliptic cylinder,” Microw. Opt. Technol. Lett., vol. 21, pp. 380–384, Jun. 1999. [12] A. R. Sebak, “Electromagnetic interaction with a magnetic coated elliptic cylinder,” IEEE Trans. Magn., vol. 27, pp. 3886–3889, Sep. 1991. [13] A. R. Sebak, H. A. Ragheb, and L. Shafai, “Plane wave scattering by dielectric elliptic cylinder coated with nonconfocal dielectric,” Radio Sci., vol. 29, pp. 1393–1401, Nov.–Dec. 1994. [14] C. S. Kim, “Scattering of an obliquely incident wave by a coated elliptic conducting cylinder,” J. Electromagn. Waves Applic., vol. 5, pp. 1169–1186, 1991. [15] C. S. Kim and C. Yeh, “Scattering of an obliquely incident wave by a multilayered elliptical lossy dielectric cylinder,” Radio Sci., vol. 26, pp. 1165–1176, Sep.–Oct. 1991. [16] J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci., vol. 23, pp. 1061–1066, Nov.–Dec. 1988.

ZOUROS AND ROUMELIOTIS: SCATTERING BY AN INFINITE DIELECTRIC CYLINDER HAVING AN ELLIPTIC METAL CORE

[17] H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phy., vol. 66, pp. 1115–1122, Dec. 1988. [18] H. A. Ragheb, L. Shafai, and M. Hamid, “Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag., vol. 39, pp. 218–223, Feb. 1991. [19] P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. [20] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. [21] J. A. Roumeliotis and S. P. Savaidis, “Cutoff frequencies of eccentric circular-elliptic metallic waveguides,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 2128–2138, Nov. 1994. [22] G. C. Kokkorakis and J. A. Roumeliotis, “Power series expansions for Mathieu functions with small arguments,” Math. Comput., vol. 70, no. 235, pp. 1221–1235, 2001. [23] G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics, vol. 27, pp. 159–182, 2007. [24] J. A. Roumeliotis and N. B. Kakogiannos, “Scattering from an infinite cylinder of small radius embedded into a dielectric one,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 463–470, Mar. 1994. [25] N. B. Kakogiannos and J. A. Roumeliotis, “Electromagnetic scattering from an infinite elliptic metallic cylinder coated by a circular dielectric one,” IEEE Trans. Microw. Theory Tech., vol. 38, pp. 1660–1666, Nov. 1990.

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Grigorios P. Zouros was born in the city of Lefkada, Lefkada island, Greece, in 1982. He received the Educational and Technological Electronic Engineering degree from the School of Educational and Technological Engineering of Athens, Greece, in 2003 and the Electrical and Computer Engineering degree from the National Technical University of Athens, Greece, in 2008, where he is currently working toward the Ph.D. degree. His research interests include scattering and wave propagation in electromagnetic theory.

John A. Roumeliotis was born in Corinth, Greece, in 1953. He received the Electrical Engineering and the Dr. Eng. degrees from the National Technical University of Athens (NTUA), Greece, in 1975 and 1979, respectively. From 1979 to 1981, he fulfilled his military service. Since 1982, he has been with the Electrical Engineering Department, NTUA, where he is a Professor. His research interests include scattering and wave propagation and boundary value problems in electromagnetic theory and acoustics, as well as applied mathematics.

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On the Absorption Mechanism of Ultra Thin Absorbers Alireza Kazemzadeh and Anders Karlsson

Abstract—Ultra thin absorbers are studied in detail to provide a comprehensive model for their absorption mechanism. It is shown that the transmission line (TML) approach is not able to model the absorber frequency response correctly. It results in large ) ratio is below a errors when the thickness to wavelength ( certain level. It is explained that large amplitudes of high order Floquet modes and excitation of non-transverse component of the scattered field at the absorption frequency are the reasons for the TML model inaccuracy. It is illustrated that at small ratios, the structure becomes a localized lossy resonator which exhibits absorption even in a finite size array configuration. For a resistive squared patch periodic array, the resonant frequency can be estimated fairly accurately by the cavity model of a single perfect conductor patch antenna. Index Terms—Cavity resonators, frequency selective surfaces, microwave absorbers, radar cross section.

I. INTRODUCTION

C

ONVENTIONAL absorbers like Salisbury [6], Jaumann [3], [4], and circuit analog absorbers [13] consist of dielectric layers, quarter of a wavelength thick at mid-frequency. The analysis of these absorbers is based on the dominance of a single propagating TEM mode between layers and neglect of the possible mutual interaction of the higher order evanescent modes. In the case of homogenous resistive sheet layers (Salisbury and Jaumann absorbers), the evanescent modes are not excited and for the circuit analog absorbers their effects at are negligible because of the large electrical distance ( mid-frequency) between the frequency selective surfaces [13]. This paper investigates the effect of these modes on the performance of the absorber when the distance between the layers is reduced such that an ultra thin absorber is formed. Ultra-thin absorbers are characterized by a thickness of or less at the operating frequency. Different designs of ultra thin absorbers are proposed by several authors [5], [8], [11], [12]. Although the geometry of the periodic array unit cell and the design approach are different among the papers, the common idea is to construct a high impedance ground plane by aid of a metasurface. Engheta [5] has explained that the surface impedance of the periodic arrangement (metamaterial) interacting with the

Manuscript received February 04, 2009; revised February 10, 2010; accepted March 29, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The authors are with the Department of Electrical and Information Technology, Lund University, SE-221 00 Lund, Sweden (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055779

reactive impedance of the ground plane can generate a magabove netic wall (open circuit) at a distance shorter than the ground plane. By placing a resistive sheet over this magnetic wall, power can be absorbed from the incident EM wave. The analysis of the absorber used by Engheta is based on the single mode transmission line model. Improvement in the structure thickness can be achieved by merging the resistive sheet and the periodic structure to have resistive cell elements [11]. In this paper it is shown that by decreasing the thickness of the absorber from the conventional quarter-wavelength to lower levels, the accuracy of the TML model decreases significantly ratio. Difand the model is not applicable below a certain ferent methods and approaches both in the time and frequency domain are applied to provide a comprehensive explanation for the absorption mechanism of ultra thin absorbers. At small ratios a perceptible non-transverse electric field component is produced by high order evanescent Floquet modes. This dominant component is not taken into account in an ordinary TML model. It results in a lossy standing wave pattern underneath the resistive element. The phenomena can be illustrated by the damping oscillation behavior of the scattered field in time domain. Close agreement between the resonance frequency of a single PEC patch element and absorption frequency of the corresponding periodic array ultra thin absorber verifies that the absorption mechanism is due to a lossy resonance. It is shown that for sufficiently large space between the array elements, where the cavity mode is the dominant field distribution, the absorber is able to operate even in finite size configurations. II. SHORTCOMINGS OF TRANSMISSION LINE MODEL ratios (about It is well-known that for sufficiently large ) the frequency response of the absorber can be simulated accurately by the TML model. In the model, the scattering (reflection, transmission) properties of the periodic array is modeled by its equivalent surface impedance. This approach is used widely in design of circuit analog absorbers [13]. Engheta [5], utilizes the same concept for formulating the reflection properties of the ultra thin absorbers. It is shown here that the approach ratios. is not applicable for small ultra thin absorber formed by strip gratings To proceed, a is considered, see Fig. 1. For this geometry only TM polarization exhibits absorption and the equivalent surface impedance of the grating for this polarization is a series RC network [10]. Therefore, the TML model of the absorber (if applicable) is a series RC network parallel with a short circuit transmission line, see Fig. 1(c). Since ultra thin absorbers are narrowband [8], [10], [12], accurate estimation of the absorption frequency is vital for a realistic model. One way to examine the accuracy of the TML model, is to compare the absorption frequency of the TML

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KAZEMZADEH AND KARLSSON: ON THE ABSORPTION MECHANISM OF ULTRA THIN ABSORBERS

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Fig. 1. The structure of the strip grating ultra thin absorber: (a) side view, (b) front view, (c) circuit model for TM mode. Fig. 3. The relative absorption frequency error of the TML model compared to full wave simulation, as a function of thickness to wavelength ratio. (a = 20 mm, w = 15 mm,  = 1).

the problem is -coordinate independent. Therefore, the fields for the TM mode can be expanded by Floquet modes as

(1)



Fig. 2. Comparison of the frequency response of an ultra thin absorber (d= 1=19) simulated by full-wave analysis (EM) and the transmission line model (TML). (a = 20 mm, w = 15 mm,  = 1; d = 2 mm).

model to an accurate full wave simulation. Therefore, the absorber is simulated in the frequency domain solver of CST Microwave Studio with high accuracy. The accuracy of the solver is selected to be 1e-6 and the tetrahedron mesh is refined adaptively at three different frequencies to result in a very fine mesh. mm, For an ultra thin absorber with parameters mm, mm and (see Fig. 1), the frequency responses of the TML model and the accurate full wave simulation, are compared in Fig. 2. As seen from the figure, there is a noticeable difference between the absorption frequencies calculated by the TML model and the full wave analysis. The difference increases rapidly as the thickness of the absorber is reduced. The relative absorption frequency error is shown in Fig. 3 as a function of ratio. The exponential growth of the error at ratios demonstrates clearly the deficiency of the TML small approach in analyzing ultra thin absorbers. In the following parts the reasons for the inaccuracy of the TML model are revealed and a comprehensive explanation for the absorption mechanism is provided. The strip grating absorber of the above example can be analyzed by mode matching technique. By expanding the fields in Floquet modes and investigating the behavior of the modes at the absorption frequency, valuable information about the physics of the absorption process can be obtained. Normal incidence in the direction of in Fig. 1 is considered. Therefore, excitation of any field component in the direction is referred as a non-transverse component. According to Fig. 1, the geometry of

(2)

(3) where (4) (5) In the above equations index represents the region, which can be air or the substrate, see Fig. 1. The index refers to the Floquet mode number. By imposing the boundary conditions, the unknown coefficients are calculated. The mode index refers to the propagating TEM mode which is used in the TML model. This mode contains no z-directed component at normal incidence. The excitation of a -component electric field for , is of great importance. For the convenmode indices is small in tional quarter-wavelength layer, the amplitude of comparison to the transverse component of the E-field and takes a negligible share in the reactive part of the grating impedance. ratios, where the correct But this is not the case for small modeling of the absorption mechanism requires full consideration of this dominant component. The following example illustrates the significance of the non-transverse component. and periodicity For a fixed value of strip grating width , the equivalent surface impedance of the grating is calculated [9], [10]. The calculated capacitance is referred to as . Then the transmission line model is formed using

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0

Fig. 4. The relative capacitance increase (C C =C ) as a function of thickness to wavelength ratio. (a = 40 mm, w = 27 mm,  = 1:2).

the calculated impedance of the grating and the exact physical , see Fig. 1(c). Obviously the thickness of the absorber TML model frequency response differs from the full wave ratios. Next the parameters of the simulation for small and transmission transmission line model, the capacitance , are adjusted such that it results in the same line length frequency response as the full wave analysis. Henceforth, this new circuit model is referred to as the adjusted TML model and . In the adjusted TML model the increase its capacitance as value from the grating capacitance of its capacitance level may represent the growth of the component. ratio for The relative capacitance increases as a function of a typical grating is shown in Fig. 4. The adjusted capacitance can be considered as a superposition of the grating capacitance which is formed between adjacent array elements and an component between the extra capacitance formed by the component and ground plane and the resistive element. The consequently the extra capacitance are thickness dependent. This effect is not considered in an ordinary TML model which is formed by just considering the grating capacitance as the reactive part of the equivalent impedance. Another way to illustrate the existence and significance of the component in the absorption process is to investigate the amplitudes of the higher order Floquet modes that contribute component inside and outside of to the expansion of the the absorption band, see (2). For this purpose a strip grating is considered ultra thin absorber with absorption frequency 5.53 GHz. The normalized amplitudes (to the incident wave) of high order Floquet modes (excluding mode number zero corresponding to the propagating TEM mode) at/out of the absorption frequency band are shown in Fig. 5. The level of mode amplitudes is 15 dB higher in average at the absorption frequency, implying the major role of high order modes in the absorption process. The high level of mode amplitudes at the absorption frequency results in a component which has an -directed (Fig. 1) complex strong valued propagation wavenumber, producing damped standing wave pattern (damped oscillation in time domain) under the re-

Fig. 5. Comparison of the normalized (to the incident wave) amplitude 1) at the absorption frequency of higher order modes (ModeNumber (f = 5:53 GHz) and out of absorption band (f = 3:5 GHz).



sistive strip(s). In other words, the absorption is due to transfer of power from the incident TEM mode to a lossy cavity (arrays of cavities) supporting TM mode at the resonant frequency. This idea is demonstrated and verified in the following sections. A. Time Domain Analysis As the ratio decreases the non-transverse component of strengthens and results in a standing wave the electric field pattern under the resistive element at absorption frequency. This phenomena can be seen clearly in the time domain. A strip grating ultra thin absorber is considered with the frequency response of Fig. 7 (the solid curve). The absorber is illuminated by a wideband Gaussian signal, see the dashed curve of Fig. 6. In the time domain solver a test point is selected close to the edge of the resistive element inside the substrate. The time domain component of the scattered field at the test response of the point is shown in Fig. 6. It should be noted that the electric field of the input signal is oriented in the direction but for the ultra the scattered field at thin absorber of our example the test point is dominated by the -component. The damped oscillatory behavior of the output signal at test point) implies a lossy resonance that can be modeled by a complex pole in the -plane. This can be verified by observing the normalized spectrum of the output signal in Fig. 7 (the curve with circles). An interesting feature of Fig. 7 is the simultaneous occurrence of absorption (reflection coefficient dip) and peak of the test point signal spectrum. This emphasizes the fact that the absorption is due to a transition of the real valued -directed wavenumber of the incident wave to a complex -directed wavenumber that dissipates the power. The complex pole of the time domain oscillation and the complex wavenumber of the standing wave are estimated accurately by eigenmode analysis of the cavity models of the next section. III. TM MODE CAVITY MODEL It was shown that in an ultra thin absorber a resonance occurs in the substrate. This implies that a relation must exist be-

KAZEMZADEH AND KARLSSON: ON THE ABSORPTION MECHANISM OF ULTRA THIN ABSORBERS

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Fig. 8. The structure of the square patch ultra thin absorber.

Fig. 6. Time domain response of the non-transverse component of E-field (E ) at the test point. The dashed curve (input signal) indicates the incident field in the transverse plane.

Fig. 7. Frequency response of the absorber (solid curve) and the normalized spectrum of the E component at test point (curve with circles).

tween the resistive element dimensions and the resonance wavelength. An example is provided to verify this relationship. Consider an ultra thin absorber with a periodic array of resistive square patches. A finite extent of the absorber is shown in Fig. 8. For fixed values of width, substrate thickness and periodicity mm, mm, mm) the absorber frequency ( response is calculated for different substrate permittivities. The narrowband frequency response of the absorber can be modeled by a complex pole in the frequency plane. For two different substrate permittivities the complex poles are given in Table I . Next a single perfect conductor, square patch antenna is considered with the same width, substrate thickness and permittivity as above. There are many cavity models for estimating the resonance frequency of the patch antenna [1], [2], [7]. Among the models, the cavity model with the perfect magnetic conducting (PMC) walls is the simplest one, which results in real

TABLE I COMPARISON OF THE RESONANT FREQUENCY OF CAVITY MODEL (GHZ) FOR A SINGLE SQUARE PATCH AND ABSORPTION FREQUENCY OF THE CORRESPONDING ULTRA THIN ABSORBER (GHZ)

value wave-numbers [1]. An enhanced cavity model is proposed by [7], which is based on Carver and Coffey’s [2] design equation using modal-expansion technique. The method considers an admittance condition at the walls (taking the radiation resistance into account), resulting in a more accurate complex wave-number. This model is applied to the square patch of our mm, mm). The values obtained for the example ( resonance frequencies are tabulated in Table I. Comparison of the absorption frequency of the ultra thin absorber to the resonance frequency of the cavity model for the corresponding single element patch antenna, reveals many facts. First of all, it verifies the idea of occurrence of a TM mode resonance (excicomponent) at small ratios in ultra tation of a dominant thin absorbers. Secondly, it provides a reliable estimation of the absorption frequency (less than 3% error), which is significantly better than the TML model, see Fig. 3. This small error is expected since in the cavity model the resistivity of the patch elratios not close to 1, where ement is ignored. Finally, for the capacitance formed at the gap of neighboring elements is comsmaller than the capacitance formed by the dominant ponent of the scattered field and the ground plane, the absorption frequency of the periodic array can be estimated accurately from the resonance of a single element patch. This suggests that ultra thin absorbers should be able to operate in finite size array configurations, independent of neighboring elements. This is examined in the next section. IV. FINITE SIZE ULTRA THIN ABSORBER It was shown that for sufficiently large gaps between neigh, the cavity model of a single patch boring elements can estimate the absorption frequency of the periodic array ultra thin absorber accurately. It suggests that the mutual coupling between neighboring elements are imperceptible and the dominant field distribution is formed by the eigenmode of the cavities. Since the TM mode resonance has a localized field distribution in the vicinity of the resistive element, such an ultra thin

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Fig. 9. The reflection coefficient of the periodic square patch absorber plus the 6:59 GHz). monostatic RCS of finite size arrays. (f



Fig. 11. Comparison of RCS for single element patch absorber at and out of resonance frequency band. Direction of incidence is indicated by the arrow.

Fig. 10. Comparison of the field distribution at the resonant frequency inside the substrate (z = 0:5; y = 0) for the periodic array (top) and single element absorber (bottom). The edges of the resistive patch are located at x = 7:5.

Fig. 12. Comparison of RCS for 3 by 3 elements patch array absorber at and out of resonance frequency band. Direction of incidence is indicated by the arrow.

absorber should be able to absorb power even in finite size array configurations. This feature of ultra thin absorbers is important for practical applications. To illustrate the above statements a square patch periodic abmm, mm, mm, sorber is considered , see Fig. 8. The frequency response of the periodic absorber is plotted in Fig. 9. The resonance frequency of the abGHz. Next a finite size array formed by sorber is the unit cell of the periodic structure is considered. The unit cell is composed of a resistive square patch with width over a finite size square shape ground plane of width (periodicity of the ultra thin absorber). The monostatic radar cross sections of 3 3 elements and the single element absorbers are calculated over the frequency interval 4–8 GHz. The frequency response of the monostatic radar cross sections are also plotted in Fig. 9, to make comparison possible. As seen from Fig. 9, the reflection coefficient of the periodic array and the monostatic radar cross sections have their minima at the same absorption frequency.

This demonstrates that the field distribution in the periodic array is independent of the presence of the neighboring elements and is a localized cavity mode field distribution. This can be verified by comparing the electric field distribution inside the substrate for the single element and the periodic array absorber. The normalized E-field distributions (to the maximum are shown in Fig. 10 at the resonant frequency. As of the component has almost a cosine proseen from the figure, the file underneath the resistive patch which is the characteristic behavior of the first TM cavity mode with perfect magnetic wall component is only significant approximation [1]. Also the at the close vicinity of the resistive patch edges and vanishes rapidly in the direction of the adjacent element in the periodic array. Finally to prove that the reduction of the monostatic radar cross section of the finite size arrays are due to absorption of power and not directing the scattered field in directions other than the backscattering one, bistatic radar cross sections of the

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KAZEMZADEH AND KARLSSON: ON THE ABSORPTION MECHANISM OF ULTRA THIN ABSORBERS

finite size arrays are calculated at the absorption frequency and outside the absorption band. The bistatic radar cross section of the single element array and the RCS of the 3 3 elements array are shown in Figs. 11 and 12. V. CONCLUSION Ultra thin absorbers are studied in detail and some new features of their operation mechanism are introduced. It is shown that the conventional TML model approach is not applicable for this type of absorbers. Therefore, full wave analysis both in the time and frequency domains are applied to provide a correct explanation of the absorption physics. It is shown that large amplitudes of higher order Floquet modes and consequently excitation of a perceptible non-transverse electric field at the absorption frequency are the main reasons for the TML model inaccuracy. Time domain simulation of the absorber demonstrates the occurrence of a resonance phenomena in the vicinity of the resistive element. The cavity model of a single element patch antenna can estimate the absorption frequency accurately and verify the idea of a TM mode resonance. For sufficiently large gaps between array elements it is shown that the absorber can operate even in finite size extensions. ACKNOWLEDGMENT The authors would like to thank the Swedish Research Council for their support of this project.

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[5] N. Engheta, “Thin absorbing screens using metamaterial surfaces,” in Proc. Antennas Propag. Society Int. Symp., 2002, vol. 2, pp. 392–395, IEEE 2002. [6] R. L. Fante, M. T. McCormack, T. D. Syst, and M. A. Wilmington, “Reflection properties of the salisbury screen,” IEEE Trans. Antennas Propag., vol. 36, no. 10, pp. 1443–1454, 1988. [7] Y. B. Gan, C. P. Chua, and L. W. Li, “An enhanced cavity model for microstrip antennas,” Microw. Opt. Technol. Lett., vol. 40, pp. 520–523, 2004. [8] Q. Gao, Y. Yin, D. B. Yan, and N. C. Yuan, “Application of metamaterials to ultra-thin radar-absorbing material design,” Electron. Lett., vol. 41, pp. 936–, 2005. [9] A. K. Zadeh and A. Karlsson, “Capacitive circuit method for fast and efficient design of wideband radar absorbers,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2307–2314, Aug. 2009. [10] A. Kazemzadeh, A. Karlsson, and M. Gustafsson, “Theory, design, and characterization of ultra thin absorbers,” in Proc. 4th Swedish Conf. on Computational Electromagnetics - Methods and Applications (EMB07), Lund, Sweden, Oct. 18-19, 2007, pp. 19–26. [11] D. J. Kern and D. H. Werner, “A genetic algorithm approach to the design of ultra-thin electromagnetic bandgap absorbers,” Microw. Opt. Technol. Lett., vol. 38, no. 1, pp. 61–64, 2003. [12] H. Mosallaei and K. Sarabandi, “A one-layer ultra-thin meta-surface absorber,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2005, vol. 1, pp. 615–618, IEEE, 2005. [13] B. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. Alireza Kazemzadeh was born in 1977, in Tehran, Iran. He received the Bachelor’s degree in telecommunication engineering from Iran University of Science and Technology, Tehran, the Master’s degree in microwave engineering from Sharif University of Technology, Tehran, and the Ph.D. degree from Lund University, Lund, Sweden, in 2010. His research interests are wave propagation and scattering, periodic structures, frequency selective surfaces and stealth technology.

REFERENCES [1] C. A. Balanis, Antenna Theory, 2nd ed. New York: Wiley, 1997. [2] K. R. Carver, “Theoretical investigation of the microstrip antenna,” Las Cruces. Physical Science Lab., New Mexico State Univ., Tech. Rep., Jan. 1979. [3] L. J. Du Toit, “The design of Jauman absorbers,” IEEE Antennas Propag. Mag., vol. 36, no. 6, pp. 17–25, 1994. [4] L. J. Du Toit and J. H. Cloete, “Electric screen Jauman absorber design algorithms,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2238–2245, 1996, Part 1.

Anders Karlsson was born in 1955, in Göteborg, Sweden. He received the M.Sc. degree in engineering physics and the Ph.D. degree in theoretical physics from Chalmers University of Technology, Gothenburg, Sweden, in 1979 and 1984, respectively. Since 2000, he has been a Professor in the Department of Electroscience, Lund University, Lund, Sweden. His research activities include scattering and propagation of waves, inverse problems, and time-domain methods. Currently he is involved in projects concerning absorbers for microwaves, antennas, and design of passive components on silicon.

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Scattering and Radiation from/by 1-D Periodic Metallizations Residing in Layered Media Dries Vande Ginste, Member, IEEE, Hendrik Rogier, Senior Member, IEEE, and Daniël De Zutter, Fellow, IEEE

Abstract—An efficient technique is proposed to compute the scattering or radiation from/by 1-D periodic structures residing in a layered background medium. The technique is based on a mixed potential integral equation (MPIE) combined with the method of moments (MoM), solving for the unknown current density flowing within a unit cell of the periodic structure. The formalism requires the knowledge of the pertinent layered medium Green’s functions with 1-D periodicity. Here, these Green’s functions are derived in closed-form by invoking the perfectly matched layer (PML)-paradigm. The stationary phase method is applied to determine the far field of the infinite, periodic structure, leading to a series of Floquet modes. In addition, approximating this series leads to an efficient technique to estimate the scattering or radiation from/by large, but finite, periodic structures with a 1-D periodic character. The theory is illustrated and validated by means of various examples, stemming from scattering and radiation applications from/by antenna arrays residing on microstrip substrates. The efficiency of the method is also demonstrated. Index Terms—Antenna arrays, electromagnetic radiation, electromagnetic scattering, Green’s function, integral equation, perfectly matched layer (PML), periodic structures, method of moments (MoM), stationary phase method.

I. INTRODUCTION UMEROUS simulation techniques were proposed for the analysis and design of one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) periodic microwave, millimeter wave, and optical wave devices. The range of applications of these periodic structures comprises, among others, frequency selective surfaces, electromagnetic and photonic bandgap structures, metamaterials, antenna arrays, and leaky-wave antennas. Techniques for 1-D periodic devices have rightfully attracted a lot of attention, e.g., [1]–[6]. In [7] an overview of 2-D periodic structures is given. A 3-D periodic metamaterial is described in [8]. The Floquet-Bloch theorem allows to analyze a periodic structure with infinite extent by merely considering a representative unit cell. Integral equation based techniques—describing unknown fields or current distributions within this unit cell—then rely on the proper periodic Green’s function of the background medium under consideration. During the past decades, a lot of attention has been paid to the construction and the efficient computation of these periodic Green’s functions, both for 2-D and 3-D configurations [9]–[12]. Some

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Manuscript received November 09, 2009; revised February 14, 2010; accepted March 24, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The authors are with the Electromagnetics Group, Department of Information Technology, Ghent University, B-9000 Gent, Belgium (e-mail: vdginste@intec. UGent.be). Digital Object Identifier 10.1109/TAP.2010.2055811

more recent contributions comprise a mathematical study of quasi-periodic Green’s functions for the free-space Helmholtz and Laplace equation with the 3-D Green’s function with 1-D periodicity as a particular case [13], an efficient computation of 1-D periodic 3-D Green’s function in free space, making use of an Ewald splitting technique to improve the convergence [14], a comparison between several techniques to accelerate the convergence for 2-D and 3-D Green’s functions with 1-D and 2-D periodicity [15], an algorithm based on the spectral Kummer-Poisson method for the calculation of 2-D Green’s functions with 1-D and 2-D periodicity [16], an interpolation technique for the determination of layered media Green’s functions [17], and a calculation method for 2-D and 3-D scalar and dyadic free-space Green’s functions with 1-D periodicity [18]. As this paper focuses on a frequency-domain technique, the above cited work refers to frequency-domain periodic Green’s functions. Nevertheless, for completeness—but without going into much detail—it needs to be mentioned that important research on periodic Green’s functions and periodic structures in time domain has been conducted, e.g., [19], [20]. Obviously, real-life devices always have a finite extent. So, much research has been devoted to the prediction of the edge effects of finite and semi-infinite periodic structures. Focusing on antenna arrays, earlier work in this field is described in [21], [22]. A detailed study of the Green’s functions for periodic, semi-infinite structures was given in [23], [24]. Important new results, concerning the development of free-space Green’s functions for finite-by-infinite periodic configurations and their application to simulate edge and corner effects occurring in antenna arrays, have been presented in [25]–[29]. This paper focuses on 1-D periodic metallizations residing in layered media, and in particular on antenna arrays on top of a microstrip substrate. The technique requires the knowledge of the 1-D periodic Green’s functions of the layered background medium. A very intuitive and natural way to derive closed-form solutions of (periodic and non-periodic) layered medium Green’s functions, is by application of the Perfectly Matched Layer (PML)-paradigm [30], [31]. In combination with a MoM [32], this paradigm has proven to be very useful for non-periodic 2-D and 3-D structures [33]–[35]. It was also successfully applied to determine periodic layered medium Green’s functions with 1-D periodicity in closed-form for 2-D [36] and 3-D [37] configurations, but so far, to the authors’ knowledge, the analysis of real-life scattering and radiation applications using 1-D periodic layered medium PML-based Green’s functions was not yet attempted. Therefore, in this work, a Mixed Potential Integral Equation (MPIE) approach for 3-D layered media configurations with 1-D periodicity is

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VANDE GINSTE et al.: SCATTERING AND RADIATION FROM/BY 1-D PERIODIC METALLIZATIONS RESIDING IN LAYERED MEDIA

adopted, which is then solved by means of the MoM. The MPIE is constructed using the pertinent 1-D periodic layered medium Green’s functions for the magnetic vector potential and the electric scalar potential. The unknown current distribution flowing on the metallizations within one unit cell is determined and further used to compute the scattering or radiation characteristics of the infinite, periodic structure under consideration. This is done by describing the far field in terms of a series of radiating Floquet modes. The edge effect cannot be captured by this series. However, by approximating it, an efficient and accurate simulation technique for finite, large antenna arrays with a periodic character is developed. The outline of this paper is as follows. In Section II the theoretical framework of the technique is described. After presenting a representative microstrip geometry (Section II.A), the MPIE and the MoM are constructed (Section II.B), based on the pertinent closed-form PML-series for the 1-D periodic layered medium Green’s functions (Section II.C). A Floquet series for the far-field is derived in Section II.D, allowing to analyze the radiating modes of the infinite periodic structure. An approximation of this series is presented in Section II.E resulting in an efficient formalism to estimate the far-field patterns of large antenna arrays with a finite extent. This theory is validated and illustrated in Section III. The examples comprise both scattering (Section III.A) and radiation (Section III.B) applications, demonstrating the efficiency of the method. Conclusions are summarized in Section IV. In the sequel, all sources and fields are assumed time-harmonic with angular frequency and time dependencies are suppressed. Also, transverse to restrictions of vectors are denoted ; here , and are unit Cartesian vectors.

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E

Fig. 1. A 1-D periodic microstrip structure, illuminated by a plane wave or excited by currents at the input ports, inducing a scattered or radiated field respectively.

E

surface currents on . The total electric field comprises both the incident and the scat. tered/radiated fields, viz. B. Integral Equation and Method of Moments An MPIE is constructed to determine the above dewithin one unit scribed, unknown current distribution . This MPIE follows from the boundary condicell , where tion , viz. is that part of the metallization residing in the unit cell . This boundary condition expresses that the total electric field transverse to vanishes . Using the proper expressions (see at the metallization ) for Section II.C) for the periodic Green’s functions the magnetic vector potential and for the electric scalar potential, the boundary condition is cast as

II. FORMULATION OF THE TECHNIQUE A. Geometry Consider a 1-D periodic microstrip structure with periodicity along the -axis (Fig. 1). This microstrip structure consists of perfect electrically conducting (PEC) planar elements (traces and/or patches), residing in unit cells , on top of an infinite PEC-backed substrate of thickness , permittivity , and permeability ; here and denote the permittivity and permeability of the air-filled , and and are the relative permittivity half-space and relative permeability of the substrate. For lossy substrates, and are complex. Impressed sources, radiating in the presence of the PEC-backed substrate, cause an incident electric field . This incident field can be due to (i) a plane wave , impinging upon the substrate or (ii) it can be excited by enforcing currents at the input ports of each unit cell . Hence, in both cases (i) and (ii), each unit cell is excited in exactly the same way, apart from a phase difference or , respectively. In response, a scattered or radiated electric field , respectively, is generated by electric

(1) with . To solve integral equation (1) is apby means of the Method of Moments (MoM) [32], proximated by a (potentially nonuniform) rectilinear mesh with interior edges. Next, is expanded into a set of vector rooftop basis functions [38] , with support comprising two patches that are joined by the mesh’s th interior edge (2) Inserting (2) into (1) and applying a Galerkin testing procedure yields an linear system: (3) The

-vector

contains the unknown expansion coefficients , and the elements of the -vector and the matrix are given by (4)

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(5) Linear system (3) is solved using direct or iterative schemes. C. 1-D Periodic Layered Medium Green’s Functions The 1-D periodic layered medium Green’s functions and are the crux of the above described formalism and they need to be determined carefully. As in [37], the PML-paradigm is invoked to determine these Green’s functions. In this section the gist of the PML-paradigm is briefly described. A classic solution method relies on the following 2-D Fourier and its inverse Transform

(6)

Fig. 2. A 1-D periodic configuration of point sources on the substrate-air ind. The air-filled half-space is closed by a PEC-backed PML at terface z complex distance .

=

D

that, by complex coordinate stretching, the air-PML combination can be treated as one single air layer with complex thickness [30], [31], [44]–[47]. The PML-closed substrate mimics the behavior of the original, open substrate and the original modal spectrum is now replaced by a discrete set of TE- and TM-polarized modes of the PML-closed substrate. The propagation constants of these modes are denoted and . This approach leads to a series expansion for the spatial 3-D Green’s function for a 1-D periodic grid of point sources (see also [37])

(7) with

(10) (8)

into converting 1-D quasi-periodic spatial functions and vice versa, respectively. the spectral functions The inverse Fourier Transform (7) involves a discrete sum of Floquet modes along the direction of periodicity. It is correctly stated in [39] that in case of the 1-D periodic microstrip configuration of Fig. 1, no analytical expressions are available for the terms in the spectral-domain series

(11) Here, zeroth-order Hankel function of the second kind, and

is the

(12) (9)

This series (9) converts the well-known spectral Green’s funcand (see [40] and also further, tions and (15) and (16)) into spatial Green’s functions . The fact that no analytical expressions are available is due to the presence of the semi-infinite layer of air (Fig. 1). This layer corresponds to a continuous set of radiation modes in the modal spectrum of the microstrip substrate, necessitating the cumbersome, numerical evaluation of Sommerfeldintegrals [41]–[43]. In this paper, this problem is tackled by the application of the PML-paradigm. To this end, the air-filled halfspace is closed by a PEC-backed PML with material parameand (Fig. 2). Previous investigations demonstrated ters

(13) with and . In theory, an infinite number of PML-modes is needed in (10) and (11). and In practice, however, as all propagation constants , are located in the fourth quadrant of the complex plane, only a limited set of modes needs to be retained. In [37], it was shown that the convergence of the series (10) and (11) can be accelerated. Other rapidly convergent series were also presented. In conclusion, it can be stated that the PML-paradigm allows to compute the Green’s functions very efficiently, this in contrast to more classical Sommerfeld-approaches.

VANDE GINSTE et al.: SCATTERING AND RADIATION FROM/BY 1-D PERIODIC METALLIZATIONS RESIDING IN LAYERED MEDIA

D. Far-Field Calculation Often a stationary phase method [41] is used to compute the far field that is scattered or radiated from/by microstrip structures. This method is applied here as well. However, it needs to be adapted to take the periodicity and the infinite extent of the geometry into account and the end result needs to be carefully interpreted. Consider again the 2-D Fourier Transform (6) and its inverse (7). The scattered or radiated field in the spectral domain and is then given by for

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are omitted for simplicity. In (24)–(26) the arguments Invoking the inverse Fourier Transform (7), the scattered/radiated electric field in the spatial domain is given by (27) We switch to polar coordinates

in the

-plane: (28) (29)

Defining (30)

(14) with

, and . In contrast to the MPIE formulation (1), (14) also takes the -dependency into account because the goal here is to derive expressions for the far field. For the microstrip structure of Fig. 1, the periodic Green’s funcare given by tions in the spectral domain and for

and applying the stationary phase method to (27), evaluating the integrand in its stationary point , yields the electric field for large (31) with (32)

(15)

(16) with (17) (18) (19) (20) (21) (22) With the above definitions (15)–(22), the electric field in the spectral domain (14) can be written as (23) with (24)

The diacritical mark indicates that does not correspond to the ‘classical’ far-field of a finite structure and that (31) should be interpreted in a different manner. Indeed, the far-field of finite, 3-D structures is usually expressed as a function of as it is evaluated on a sphere in the the spherical angles Fraunhofer region of the structure under consideration. Here, in (31) the stationary phase method gives rise to an evaluation of the field on a cylinder in the far-field. This sphere and cylinder are indicated in Fig. 3, together with the corresponding coordiis nate systems. Notice that the coefficient of the th Floquet mode in the series . Depending on the particular values for the period and the phase difference , only a limited set of the lower-order Floquet modes radiate into the far field, i.e., , and thus . Also only these modes for which in contrast to classical far field patterns of finite structures, the far-field’s radial component w.r.t. to the cylinder, i.e., along the direction , is not necessarily zero. Rewriting the far field as (33) and , (25) with . For all other values of and (26) show that only , a radial component can be expected. Nevertheless, although this might seem counterintuitive at first sight, a lot of information can be derived from these Floquet coefficients. This will be demonstrated in Section III. E. Far-Field Calculation of a Finite Periodic Array

(25) (26)

The Floquet series (31) can also be approximated to analyze and design finite arrays with a periodic character. Application of the Poisson summation to (31) and, once more, invoking a

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=

Fig. 4. Unit cell with period b 10 mm or b = 100 mm comprising a single patch of dimensions 5 mm 5 mm.

2

Fig. 3. Cylinder (full line) and sphere (dashed line) in the Fraunhofer region and the pertinent coordinate systems.

stationary phase technique after switching to the usual spherical (see Fig. 3), yields directions of observation (34) with (35)

F

Fig. 5. Far-field  (;  ) for the configuration of Fig. 4 with b = 10 mm and illuminated perpendicularly.

(36) Approximation

(34) is valid for large . The factor corresponds to the array factor of a uniformly spaced linear array of an infinite number of currents with equal amplitudes and with a uniform progressive phase factor [48]. The factor represents the far field produced by the current flowing within one unit cell, where all mutual coupling is correctly taken into account. Hence, large, but finite, 1-D periodic structures, unit cells, can be efficiently analyzed by consisting of introducing a finite array factor (37) with (38) (39) Apart from edge effects, induced by periodic structures with a finite extent, the technique leads to very accurate results, that are obtained in an efficient way (see Section III). III. NUMERICAL VALIDATION AND EXAMPLES In this section the above described formalism is demonstrated and validated. It is shown that it can be successfully applied to efficiently analyze the scattering and radiation from/by 1-D periodic antenna arrays residing on a microstrip substrate. (Of course, it is readily seen that the above theory can easily be extended to general multilayered structures with planar and non-planar metallizations on different levels.) Comparisons

are made with the 3-D full-wave SVD-PML-MLFMA solver presented in [34], [35], [49], which by itself has already been extensively validated and tested on various microstrip configurations. All simulations were carried out on a Linux-based 64-bit AMD Opteron 270 computer with 8 GB of RAM running at 2 GHz. A BiCGstab iterative solver was used to solve linear system (3). A. Scattering From 1-D Periodic Antenna Arrays In this section the scattering from periodic antenna arrays is computed with the above presented technique. Consider a non-magnetic substrate of thickness mm and relaon which a linear array of square tive permittivity PEC metallic patches of dimension 5 mm 5 mm resides. The top view of one unit cell is shown in Fig. 4. A plane wave with an angular frequency GHz impinges perpendicularly upon the structure. First, we compute (32) for two values of the period, i.e., the scattered field mm and mm. The results are shown in Figs. 5 mm (Fig. 5), only the lowest order mode and 6. For radiates (for all other modes ). Given the perpen(see Section II.A), leading to dicular incidence, we have . Therefore, the radial component . In this particular case, the current flowing in the -direction is . Together very small compared to the -directed current with (24) and , this results in a very low . Hence, this component is not shown in Fig. 5. Increasing the period to mm, leads to the results shown in Fig. 6. Because of the , are radiating. In larger period, 7 Floquet modes, this particular case, , so

VANDE GINSTE et al.: SCATTERING AND RADIATION FROM/BY 1-D PERIODIC METALLIZATIONS RESIDING IN LAYERED MEDIA

 (;  ) for the configuration of Fig. 4 with b = 100 mm and Fig. 6. Far-field F illuminated perpendicularly (a = ; ; x).

Fig. 7. Far-field F (; 0 ) in the xz -plane for the configuration of Fig. 4 with b = 10 mm and N = 5.

only the four modes are shown. The higher order modes , correspond to grating lobes (see further). These modes comprise a significant part of the scattered electric field. Compared to Fig. 5, this results in less radiation along the specular direction. unit cells. Now consider arrays of finite extent containing (see (37)) in the - and the In Figs. 7–10 the far-field -plane for the two configurations ( mm and mm) are shown for . These results are in good agreement with the far field of a finite array of patches, computed using the SVD-PML-MLFMA. As expected, for the mm, only one maximum in the -plane low value is observed, corresponding to the specular reflection . For the large period mm, apart from the specular reflection, also six grating lobes are observed at angles , being , and . The far-field pattern of the finite array is the usually expressed using spherical coordinates, (Fig. 3). We i.e., underline that the radial component w.r.t. the sphere is always . The far-field pattern of equal to zero, i.e.,

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Fig. 8. Far-field F (; 90 ) in the yz -plane for the configuration of Fig. 4 with b = 10 mm and N = 5.

Fig. 9. Far-field F (; 0 ) in the xz -plane for the configuration of Fig. 4 with b = 100 mm and N = 5.

Fig. 10. Far-field F (; 90 ) in the yz -plane for the configuration of Fig. 4 with b = 100 mm and N = 5.

the infinite periodic array, however, can have a significant radial component w.r.t. to the cylinder, i.e., along the direction .

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TABLE I COMPARISON BETWEEN (I) THE NUMBER OF UNKNOWN EDGE CURRENTS N , (II) THE NUMBER OF ITERATIONS NEEDED TO REACH 6 DIGITS OF ACCURACY, AND (III) THE SOLUTION TIMES

=

Fig. 11. Unit cell with period b 40 mm comprising three patches of dimensions 10 mm 10 mm, equispaced in the y -direction with a spacing of 40 mm.

2

F

Fig. 12. Far-field (; 0 ) in the xz -plane for the configuration of Fig. 11 with b = 40 mm and N = 5.

Now we demonstrate the scattering of a plane wave impinging upon an antenna array under oblique incidence. Consider the same microstrip substrate as before, but with a much larger unit cell consisting of three square patches, each of dimension 10 (Fig. 11). The mm 10 mm, at the substrate-air interface three patches are equidistantly spaced along the -axis, corresponding to a period along the -direction of 40 mm. The same period is chosen along the -axis. The structure is illuminated by a plane wave with an angular frequency GHz under an oblique incidence of 30 , i.e., . , and hence for a 5 3 array of patches, the For -component of the scattered far-field (see (37)) in the -plane is presented in Fig. 12 and compared with the SVD-PML-MLFMA, again demonstrating good agreement. Apart from the specular reflection at , two grating lobes at and are expected. These are, of course, theoretical values only based on the array factor (39). Since the actual far-field is the product of the far-field of the unit cell (35) with the array factor, the is somewhat tilted, i.e., it appears at a grating lobe at smaller angle . This effect is accurately predicted by the technique described in this paper. From the above, it is shown that the new technique presented in this paper can be successfully applied to analyze the scat-

Fig. 13. Unit cell with period b = 23:58 mm, i.e., one “arm” of the antenna array first detailed in [50].

tering from infinite and large, but finite arrays with a periodic character in one dimension. Moreover, it is interesting to mention that, due to the smaller system matrix (see (5)), the solution time of the linear system (3) is much smaller compared to the reference technique, the SVD-PML-MLFMA, although the latter is especially designed to accelerate the matrix-vector multiplications required by the iterative BiCGstab solver. In Table I some numbers that illustrate this efficiency are summarized. The SVD-PML-MLFMA and the iterative solver are set to reach 6 digits of accuracy. B. Radiation by 1-D Periodic Antenna Arrays In this section the technique is used to compute radiation by antenna arrays. We consider an antenna array similar to the one first presented in [50]. The array resides on a non-magnetic microstrip substrate of thickness mm and with relative permittivity . The unit cell corresponds to one “arm” of four patches, detailed in Fig. 13. The periodicity is mm. First, the array is fed by a current at its input port with angular frequency GHz and with a fixed phase, i.e., (see Section II.A). The radiated field (32) is normalized to 0 dB, i.e., , and shown in Fig. 14. In this case, only one Floquet mode

VANDE GINSTE et al.: SCATTERING AND RADIATION FROM/BY 1-D PERIODIC METALLIZATIONS RESIDING IN LAYERED MEDIA

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Fig. 14. Normalized far-field F (;  ) for the configuration of Fig. 13 with b = 23:58 mm and with  = 0 .

Fig. 16. Normalized radiation pattern F (; 90 ) in the yz -plane for the configuration of Fig. 13 with b = 23:58 mm and with  = 0 .

Fig. 15. Normalized radiation patterns F (; 0 ) and F (; 0 ) in the xz -plane for the configuration of Fig. 13 with b = 23:58 mm and with  = 0 (a = ; ).

Fig. 17. Normalized radiation patterns F (; 0 ) and F (; 0 ) in the xz -plane for the configuration of Fig. 13 with b = 23:58 mm and with a uniform progressive phase shift  = 266:7 (a = ; ).

radiates, due to the particular excitation . is not shown because it is very small. The component The slight asymmetry w.r.t. the -plane can be easily explained by the fact that the metallization (Fig. 13), and the currents flowing on it, are also not symmetrical. Again, the technique is compared with the SVD-PML-MLFMA, using (37). Choosing , as the original array consisted of eight “arms” [34], [35], [50], leads to the results presented in Figs. 15 and 16. The radiation pattern is normalized w.r.t. to its largest value, i.e., . Again, excellent agreement is observed. In the last example the same metallization as detailed in Fig. 13 is used, but now it is converted into a scanning array [48] by changing the phase of the current fed at the input. The GHz, but angular frequency of the current is still is now, a uniform progressive phase factor adopted. Only taking the array factor into account, the main and a grating lobe should shift to should lobe at appear. A comparison with the SVD-PML-MLFMA is per. formed in Figs. 17 and 18 for only three “arms”, i.e.,

Fig. 18. Normalized radiation pattern F (; 90 ) in the yz -plane for the configuration of Fig. 13 with b = 23:58 mm and with a uniform progressive phase shift  = 266:7 .

The combination of radiation close to the endfire direction and the very low number of unit cells, is especially chosen to verify

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TABLE II COMPARISON BETWEEN (I) THE NUMBER OF UNKNOWN EDGE CURRENTS , (II) THE NUMBER OF ITERATIONS NEEDED TO REACH 6 DIGITS OF ACCURACY, AND (III) THE SOLUTION TIMES

N

whether or not approximation (37) becomes critical. It is seen from Figs. 17 and 18 that the edge effects are more noticeable for this rather extreme example. But still, acceptable agreement is observed, again validating our approach. For completeness, it should be mentioned that because of the ground plane, the lobe is tilted again towards a smaller angle, and the predicted becomes dominant. lobe at For both examples, due to the smaller system matrix , the solution time is much smaller when using the new technique presented in this paper. A comparison with the SVD-PML-MLFMA is presented in Table II. The SVD-PML-MLFMA and the iterative solver are set to reach 6 digits of accuracy. IV. CONCLUSION In this paper an efficient technique is presented to compute the scattering or radiation from/by 1-D periodic structures residing in a layered background medium. The technique is based on flowing a MPIE, describing the unknown current density within a unit cell of the periodic structure. The MoM is used to solve this unknown current density. The MPIE-MoM formalism is constructed using the pertinent 1-D periodic layered ) for the magnetic vector medium Green’s functions potential and for the electric scalar potential. These Green’s functions are derived in closed form by invoking the PML-paradigm. Application of a stationary phase method leads to the far field of the infinite, periodic structure. This far field is actually a series of Floquet modes, which requires careful interpretation. Also, an approximation of this series leads to an efficient technique to estimate the scattering or radiation from/by large, but finite, periodic structures with a 1-D periodic character. Focusing on antenna arrays residing on microstrip substrates, various illustrative examples validate the method. The efficiency of the method is demonstrated. Obviously, due to the particular construction of periodic Green’s functions, edge effects induced by 1-D periodic structures with a finite extent cannot be exactly determined by the proposed method. A worst-case scenario could be the modeling of the standing wave pattern on a periodically meandering, but finite, piece of transmission line. So, therein lies a topic for further research. For antenna applications, techniques as described in [23]–[26] might lead to an accurate prediction of the effects introduced by the large, but still finite, dimensions of antenna arrays without having to resort to more classical techniques for large, finite structures.

REFERENCES [1] P. Petre, M. Swaminathan, G. Veszely, and T. K. Sarkar, “Integral equation solution for analyzing scattering from one-dimensional periodic coated strips,” IEEE Trans. Antennas Propag., vol. 41, no. 8, pp. 1069–1080, Aug. 1993. [2] J. Moore, H. Ling, and C. S. Liang, “The scattering and absorption characteristics of material-coated periodic gratings under oblique incidence,” IEEE Trans. Antennas Propag., vol. 41, no. 9, pp. 1281–1288, Sep. 1993. [3] H.-C. Chu, S.-K. Jeng, and C. H. Chen, “Reflection and transmission characteristics of single-layer periodic composite structures for the TE case,” IEEE Trans. Antennas Propag., vol. 45, no. 7, pp. 1065–1070, July 1997. [4] T. X. Wu and D. L. Jaggard, “Scattering of chiral periodic structure,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1859–1870, Jul. 2004. [5] R. Rodríguez-Berral, F. Mesa, P. Baccarelli, and P. Burghignoli, “Fast numerical analysis of a 1D array of microstrip patches,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Honolulu, HI, Jun. 9–15, 2007, vol. 1–12, pp. 1677–1680. [6] S. Paulotto, P. Baccarelli, F. Frezza, and D. R. Jackson, “A novel technique for open-stopband suppression in 1-D periodic printed leakywave antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1894–1906, Jul. 2009. [7] P. Baccarelli, S. Paulotto, and C. Di Nallo, “Full-wave analysis of bound and leaky modes propagating along 2D periodic printed structures with arbitrary metallisation in the unit cell,” IET Microw. Antennas Propag., Special Issue on Metamaterials EBG, vol. 1, no. 1, pp. 217–225, 2007. [8] M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 4, pp. 1418–1430, Apr. 2005. [9] R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microwave Theory Tech., vol. 33, no. 8, pp. 734–736, Aug. 1985. [10] S. Singh and R. Singh, “Application of transforms to accelerate the summation of periodic free-space Green’s function,” IEEE Trans. Microwave Theory Tech., vol. 38, no. 11, pp. 1746–1748, Nov. 1990. [11] R. M. Schubair and Y. L. Chow, “Efficient computation of the periodic Green’s function in layered dielectric media,” IEEE Trans. Microwave Theory Tech., vol. 41, no. 3, pp. 498–502, Mar. 1993. [12] A. W. Mathis and A. F. Peterson, “A comparison of acceleration procedures for the two-dimensional Green’s function,” IEEE Trans. Antennas Propag., vol. 44, no. 4, pp. 567–571, Apr. 1996. [13] A. Moroz, “Quasi-periodic Green’s functions of the Helmholtz and Laplace equations,” J. Phys. A: Math. General, vol. 39, no. 36, pp. 11 247–11 282, Sep. 2006. [14] F. Capolino, D. R. Wilton, and W. A. Johnson, “Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Comput. Phys., vol. 223, no. 1, pp. 250–261, Apr. 2007. [15] G. Valerio, P. Baccarelli, P. Burghignoli, and A. Galli, “Comparative analysis of acceleration techniques for 2-D and 3-D Green’s functions in periodic structures along one and two directions,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1630–1643, Jun. 2007. [16] A. L. Fructos, R. R. Boix, F. Mesa, and F. Medina, “An efficient approach for the computation of 2-D Green’s functions with 1-D and 2-D periodicities in homogeneous media,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3733–3742, Dec. 2008. [17] G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, “Regularization of mixed-potential layered-media Green’s functions for efficient interpolation procedures in planar periodic structures,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 122–134, Jan. 2009. [18] D. Van Orden and V. Lomakin, “Rapidly convergent representations for 2D and 3D Green’s functions for a linear periodic array of dipole sources,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1973–1984, July 2009. [19] N.-W. Chen, M. Lu, F. Capolino, B. Shanker, and E. Michielssen, “Floquet-wave-based analysis of transient scattering from doubly periodic discretely planar, perfectly conducting structures,” Radio Sci., vol. 40, no. 4, Aug. 2005. [20] J. Gao and B. Shanker, “Time domain Weyl’s identity and the causality trick based formulation of the time domain periodic Green’s function,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1656–1666, Jun. 2007.

VANDE GINSTE et al.: SCATTERING AND RADIATION FROM/BY 1-D PERIODIC METALLIZATIONS RESIDING IN LAYERED MEDIA

[21] A. Ishimaru, R. J. Coe, G. E. Miller, and W. P. Geren, “Finite periodic structure approach to large scanning array problems,” IEEE Trans. Antennas Propag., vol. 33, no. 11, pp. 1213–1220, Nov. 1985. [22] A. K. Skrivervik and J. R. Mosig, “Analysis of finite phase arrays of microstrip patches,” IEEE Trans. Antennas Propag., vol. 41, no. 8, pp. 1105–1114, Aug. 1993. [23] F. Capolino, M. Albani, S. Maci, and L. B. Felsen, “Frequency-domain Green’s function for a planar periodic semi-infinite phased array—Part I: Truncated Floquet wave formulation,” IEEE Trans. Antennas Propag., vol. 48, no. 1, pp. 67–74, Jan. 2000. [24] F. Capolino, M. Albani, S. Maci, and L. B. Felsen, “Frequency-domain Green’s function for a planar periodic semi-infinite phased array—Part II: Diffracted wave phenomenology,” IEEE Trans. Antennas Propag., vol. 48, no. 1, pp. 75–85, Jan. 2000. [25] C. Craeye, A. B. Smolders, D. H. Schaubert, and A. G. Tijhuis, “An efficient computation scheme for the free space Green’s function of a two-dimensional semiinfinite phased array,” IEEE Trans. Antennas Propag., vol. 51, no. 4, pp. 766–771, Apr. 2003. [26] C. Craeye and F. Capolino, “Accelerated computation of the free space Green’s function of semi-infinite phased arrays of dipoles,” IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 1037–1040, Mar. 2006. [27] C. Craeye, “Fast computation and extrapolation of the effects of array truncation in broadband antenna arrays,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Columbus, OH, Jun. 22–27, 2003, vol. 4, pp. 19–22. [28] C. Craeye, A. G. Tijhuis, and D. H. Schaubert, “An efficient MoM formulation for finite-by-infinite arrays of two-dimensional antennas arranged in a three-dimensional structure,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 271–282, Apr. 2004. [29] C. Craeye and X. Dardenne, “Element pattern analysis of wide-band arrays with the help of a finite-by-infinite array approach,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 519–526, Feb. 2006. [30] F. Olyslager, “Discretization of continuous spectra based on perfectly matched layers,” SIAM J. Appl. Math., vol. 64, no. 4, pp. 1408–1433, May 2004. [31] F. Olyslager and H. Derudder, “Series representation of Green dyadics for layered media using PMLs,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2319–2326, Sep. 2003. [32] R. F. Harrington, Field Computation by Moment Methods. Piscataway, NJ: IEEE Press, 1993. [33] D. Vande Ginste, H. Rogier, D. De Zutter, and F. Olyslager, “A fast multipole method for layered media based on the application of perfectly matched layers—The 2-D case,” IEEE Trans. Antennas Propag., vol. 52, no. 10, pp. 2631–2640, Oct. 2004. [34] D. Vande Ginste, E. Michielssen, F. Olyslager, and D. De Zutter, “An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1538–1548, May 2006. [35] D. Vande Ginste, E. Michielssen, F. Olyslager, and D. De Zutter, “A high-performance upgrade of the perfectly matched layer multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1728–1739, Jun. 2009. [36] H. Rogier and D. De Zutter, “A fast converging series expansion for the 2-D periodic Green’s function based on perfectly matched layers,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 4, pp. 1199–1206, Apr. 2004. [37] H. Rogier, “New series expansions for the 3D Green’s function of multi-layered media with 1D periodicity based on perfectly matched layers,” IEEE Trans. Microwave Theory Tech., vol. 55, no. 8, pp. 1730–1738, Aug. 2007. [38] J. Sercu, N. Faché, F. Libbrecht, and P. Lagasse, “Mixed potential integral equation technique for hybrid microstrip-slotline multilayered circuits using a mixed rectangular-triangular mesh,” IEEE Trans. Microwave Theory Tech., vol. 43, no. 5, pp. 1162–1172, May 1995. [39] P. Baccarelli, C. D. Nallo, S. Paulotto, and D. R. Jackson, “A fullwave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 4, pp. 1350–1362, Apr. 2006. [40] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part I: Theory,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 335–344, Mar. 1990. [41] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Piscataway: IEEE Press, 1994. [42] N. Faché, F. Olyslager, and D. De Zutter, Electromagnetic and Circuit Modelling of Multiconductor Transmission Lines. New York: Oxford Univ. Press, 1993.

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[43] F. Olyslager, Electromagnetic Waveguides and Transmission Lines. New York: Oxford Univ. Press, 1999. [44] W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett., vol. 7, no. 13, pp. 599–604, Sep. 1994. [45] W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate system as a generalized absourbing boundary condition,” in Proc. 13th Ann. Review of Progress in Applied Electromagnetics, Monterrey, CA, 1997, vol. 2, pp. 909–914. [46] H. Derudder, F. Olyslager, and D. De Zutter, “An efficient series expansion for the 2D Green’s function of a microstrip substrate using perfectly matched layers,” IEEE Microwave Guided Wave Lett., vol. 9, no. 12, pp. 505–507, Dec. 1999. [47] H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas Propag., vol. 49, no. 2, pp. 185–195, Feb. 2001. [48] R. S. Elliot, Antenna Theory and Design—Revised Edition. Hoboken, NJ: Wiley, 2003, IEEE Press Series on Electromagnetic Wave Theory. [49] D. Vande Ginste, L. Knockaert, and D. De Zutter, “Error control in the perfectly matched layer based multilevel fast multipole algorithm,” J. Comp. Phys., vol. 228, no. 13, pp. 4811–4822, July 2009. [50] C.-F. Wang, F. Ling, and J.-M. Jin, “A fast full-wave analysis of scattering and radiation from large finite arrays of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 10, pp. 1467–1474, Oct. 1998.

Dries Vande Ginste (M’05) was born in 1977. He received the M.S. degree and the Ph.D. degree in electrical engineering from Ghent University, Gent, Belgium, in 2000 and 2005, respectively. From October 2000 until March 2006, he was with the Department of Information Technology (INTEC), Ghent University, as a Doctoral and Postdoctoral Researcher, where his research focused on fast techniques for the modeling of layered media. In June and July 2004, he was a Visiting Scientist at the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign (UIUC). From April 2006 to May 2007, he was active as a Senior Consultant in a private company, i.e., Applied Logistics N.V., where he was mainly involved in the modeling of material handling systems, feasibility studies, technical-economical evaluations for clients, and where he also had commercial responsibilities. In June 2007, he joined the Department of Information Technology (INTEC) again, first as a Technology Developer in the field of high-frequency technologies for ICT applications and since July 2009 as a Postdoctoral Researcher and Lecturer. His current research interests comprise computational electromagnetics, electromagnetic compatibility, and antenna design.

Hendrik Rogier (SM’08) was born in 1971. He received the Electrical Engineering and the Ph.D. degrees from Ghent University, Gent, Belgium, in 1994 and in 1999, respectively. He is currently a Postdoctoral Research Fellow of the Fund for Scientific Research Flanders (FWO-V), Department of Information Technology, Ghent University, where he is also Associate Professor with the Department of Information Technology. From October 2003 to April 2004, he was a Visiting Scientist at the Mobile Communications Group of Vienna University of Technology. He authored and coauthored about 55 papers in international journals and about 80 contributions in conference proceedings. His current research interests are the analysis of electromagnetic waveguides, electromagnetic simulation techniques applied to electromagnetic compatibility (EMC) and signal integrity (SI) problems, as well as to indoor propagation and antenna design, and in smart antenna systems for wireless networks. Dr. Rogier is serving as a member of the Editorial Boarding of IET Science, Measurement Technology and acts as the URSI Commission B representative for Belgium. Her was twice awarded the URSI Young Scientist Award, at the 2001 URSI Symposium on Electromagnetic Theory and at the 2002 URSI General Assembly.

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Daniël De Zutter (F’90) was born in 1953. He received the M.Sc. degree in electrical engineering and the Ph.D. degree from Ghent University, Gent, Belgium, in 1976 and 1981, respectively, where, in 1984, he completed a thesis leading to a degree equivalent to the French Aggrégation or the German Habilitation. From 1976 to 1984, he was a Research and Teaching Assistant at Ghent University. From 1984 to 1996. he was with the National Fund for Scientific Research of Belgium. He is now a Full Professor of electromagnetics. Between 2004 and 2008 he served as the Dean of the Faculty of Engineering of Ghent University and is now the head of the Department of Information Technology. Most of his earlier scientific work dealt with the electrodynamics of moving media. His research now

focusses on all aspects of circuit and electromagnetic modelling of high-speed and high-frequency interconnections and packaging, on electromagnetic compatibility (EMC) and numerical solutions of Maxwell’s equations. As author or coauthor he has contributed to more than 140 international journal papers and 150 papers in conference proceedings. In 1993, he published the book Electromagnetic and circuit modelling of multiconductor transmission lines in the Oxford Engineering Science Series. Dr. Zutter was a co-recipient of the 1990 Montefiore Prize of the University of Liège and the 1995 IEEE Microwave Prize Award from the IEEE Microwave Theory and Techniques Society for best publication in the field of microwave. In 1990, he was elected as a Member of the Electromagnetics Society. In 1999 he received the Transactions Prize Paper Award from the IEEE EMC Society. In 2000, he was elected to the grade of Fellow of the IEEE. He is an Associate Editor for the IEEE MICROWAVE THEORY AND TECHNIQUES TRANSACTIONS.

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Analysis of Antenna Coupling in Near-Field Communication Systems Yen-Sheng Chen, Shih-Yuan Chen, Member, IEEE, and Hsueh-Jyh Li, Senior Member, IEEE

Abstract—A simple mathematical formula is proposed for computing the coupling coefficient between two arbitrary antennas that are placed within the other’s near-field region. The information required by this expression consists of the associated normalized vector far-field patterns, their relative orientations, and the antenna spacing. To validate our proposed expression, coupling coefficients in several near-field scenarios are computed, including for the case of a practical near-field ultra-high frequency (UHF) radio-frequency identification (RFID) system. These results are then compared to practical measurements and to the outcome of full-wave simulations generated using Ansoft HFSS. All results are in good agreement. Additionally, in this paper, it is shown that several factors may influence the coupling coefficient, such as the impedance matching of the receiving antenna and the directivities of both antennas. With the aid of our formula, the near-field read range can be determined and near-field coupling phenomena can be investigated. The results thus obtained may be useful in near-field communication systems. Index Terms—Electromagnetic coupling, power transmission, RFID, UHF antennas.

I. INTRODUCTION N past years, there has been increasing research interest in near-field communication systems, and the emergent technology has been deployed in various applications. For instance, the near-field ultra-high frequency (UHF) radio-frequency identification (RFID) system in the 860–960 MHz band has been used in item-level tagging in pharmaceuticals and retailing [1]–[3]. Low frequency (LF) and high frequency (HF) RFID systems have been extensively used in access control and public transportation ticketing. Considerable attention is also given to the Near Field Communication (NFC) system [4]–[6] that enables contactless payment via any hand-held device (e.g., a mobile phone). Still many other applications exist, such as health monitoring [7], mCoupons [8], and magnetic resonance

I

Manuscript received April 10, 2009; revised February 01, 2010; accepted March 28, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported by the National Science Council, Taiwan, under Contracts NSC 96-2221-E-002-013MY2 and NSC 99-2221-E002-059, and in part by the National Taiwan University under Excellent Research Project NTU-ERP-99R80302. Y.-S. Chen and S.-Y. Chen are with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]). H.-J. Li is with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055782

imaging (MRI) [9]. In order to successfully design and optimize near-field communication systems, it is critical to investigate the antenna coupling that occurs when antennas are placed in close proximity. In the lower frequency range, such as the LF (125–134 KHz) and the HF (13.56 MHz) bands, depending on antenna type, it is either the electric or the magnetic field that is more pronounced in the antenna’s near zone. For example, the magnetic field becomes more significant in the near zone of an electric loop antenna, and, therefore, transmitting and receiving antennas used in the near-field magnetic (inductive) coupling system are mostly loop antennas. Some attempts have been made to compute the LF/HF inductive coupling power transfer [10]–[15]. However, in the UHF band and higher, such as the 860–960 MHz, 2.4 GHz, and 5.8 GHz bands, the field distribution in the same near zone becomes more complex and may also include an electrostatic or magnetostatic component. Although some empirical and experimental approaches have been employed to evaluate the performance of UHF RFID and NFC devices [16], [17], to the authors’ best knowledge, few theoretical studies have been conducted that emphasize the generality for different antenna types and relative orientation of the antennas in calculating near-field antenna coupling in the microwave region. In this paper, an analytical form is proposed to compute the near-field coupling coefficient as a function of the spacing between two arbitrary antennas. This form is based primarily on the coupling quotient expressed in terms of the antenna far fields [18]. However, the associated numerical complexity, due to the usage of the fast Fourier transform (FFT) and tedious truncation methods, has been greatly reduced. In the proposed method, the three-dimensional (3D) vector far-field patterns, the relative orientation of the transmitting and receiving antennas, and the antenna spacing are required to calculate the coupling coefficient. One may use closed-form expressions, if any, for the 3D far-field patterns or data acquired via simulation or measurement. The proposed formula is a near-field counterpart of the Friis transmission equation, and is applicable to any antenna type used in near-field communication systems. For verification purposes, we used this formula to calculate the coupling coefficients of several near-field scenarios. The results were compared to those of full-wave simulations generated using Ansoft HFSS, and they agreed well. In addition, a near-field UHF RFID system is implemented. Again, the results computed using the proposed method were in good agreement with those measured and those simulated by HFSS. Additionally, some factors were discovered to be critical in improving the coupling level. This paper is organized as follows: Section II presents the formula used to calculate the near-field coupling coefficient. In

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Fig. 1. Arbitrarily oriented receiving antenna in the near field of a transmitting antenna.

Section III, three typical communication scenarios are simulated using Ansoft HFSS, and the results are compared with those computed via the proposed method. In Section IV, a near-field UHF RFID system is considered as an example. Results obtained through the proposed formula, measurement, and HFSS simulation are shown and compared. Finally, some observations and design guidelines are summarized in Section V. II. FORMULATION Let us consider a receiving antenna placed in the near field of a transmitting antenna as depicted in Fig. 1. The incident and emergent waveguide mode coefficients for the transmitting and ( and ) respectively. (receiving) antenna are The coupling quotient between the transmitting and receiving [18]. This can be interpreted as antennas is defined as the signal coupled into the receiving antenna when a unit signal is fed into the transmitting antenna. This, in turn, is identical to the definition of the forward transmission coefficient of the , when the transmitting and scattering parameters, namely receiving antennas and the region in between are considered a two-port network. Since we are more interested in the coupled is used instead. In this paper, power level, is expressed in decibels and is referred to as the power coupling level or the coupling coefficient . The coupling quotient between the transmitting and receiving antennas can be expressed as [18]

is the where is the intrinsic impedance of free space, characteristic impedance of the feed line of the receiving anand are the reflection coefficients of the retenna, and ceiving antenna and its passive termination, respectively. Note from (1) that the coupling quotient is a function of the position vector . The double integral in (1) is computed over the and , and transverse components of propagation vector the inner product of the two vector far-field patterns in the integrand represents the interaction between the transmitting and signifies that receiving antennas. The integral interval only the propagating waves are considered, which corresponds to the real part of complex power. The same assumption has been made in the near-field measurement [18], which can be considered as the reciprocal problem to the one discussed here. In the near-field measurement, the far-field pattern of an antenna under test (AUT) is obtained by measuring the near-field coupling between the AUT and the probe antenna. In most situations, this probe antenna is placed in the radiating near field of the AUT, and the induced tangential electrical fields on it are measured [19]. The coupling measured in the radiating near field is then converted into the far-field pattern of the AUT. Conversely, in our proposed method, the far-field patterns are converted into the coupling in the radiating near field due to the fact that in near-field communication systems the receiving antenna is always placed in the radiating near field of the transmitting antenna, and hence, beyond its reactive near field. The proposed formula is thus applicable to the coupling coefficients between antennas placed in the radiating near field of each other. In fact, the evanescent part of the spectrum contributes mainly to the reactive near fields but has only negligible effects on the coupling in the radiating near field, and therefore, it can be neglected. to (1) yields Applying the Laplacian operator

(3) Rearranging (3), we obtain (4)

(1)

where and are the normalized vector far-field patterns for the receiving and transmitting antennas, respectively. is the propagation vector, and , where is is the free-space wavelength. the position vector of the receiving antenna with respect to the is a mismatch constant defined as transmitting antenna. (2)

which indicates that the coupling quotient satisfies the scalar Helmholtz equation [20]. As a result, the coupling quotient can be expressed as

(5) where , , and are the corresponding spherical coordinates of the position vector . and are the spherical Hankel functions of the first kind and the associated Legendre polynois the spherical wave coefficient. mials, respectively, and Here, the coupling quotient is expanded by a set of known basis

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functions. To evaluate the spherical wave coefficient , one could exploit the orthogonality relationships of these basis functions and have [20]

(6)

Using the series expansion method, the singularity that occurs when approaches zero can be avoided, and the integraand are changed to and , which tion variables results in a simpler double integral. To further simplify (5) and , the coordinate system in Fig. 1 is facilitate evaluation of rotated around the origin, the phase center of the transmitting antenna, such that the phase center of the receiving antenna lies on the -axis of the rotated coordinate system. This orientation and is depicted in Fig. 2. In this new coordinate system . The relative orientation between the transmitting and receiving antennas can then be accounted for by simply rotating the far-field patterns accordingly. In addition, it is known that , in the case of the associated Legendre polynomial, . Thus, its argument being unity, is nonzero only when (5) and (6) can be rewritten as (7) and (8), respectively

(7)

(8)

and are the largest dimensions of the transmitwhere ting and receiving antennas, respectively, and is the free-space wavelength. The constraints of were derived in the process of reducing the range of integration of (1) [18]. The coupling quotient in (7) is now a function of the antenna spacing . Given the 3D vector far-field patterns for both transmitting and receiving antennas and the relative orientation, the associated inner product in the integrand of (8) can readily be calculated. It , Yaghjian exploited an must be pointed out that, to evaluate FFT algorithm to convert the double integral into summations [18]. In this work, a simple numerical integration is adopted to directly from (8). Substituting thus obtained, calculate into (7) yields the desired coupling quotient. Although an infinite series is needed to compute the coupling quotient based on (7), it has been observed that the series would converge in less than ten terms. In the formulation, only the direct or zeroth-order interaction between the transmitting and receiving antennas is accounted for, and other higher-order interactions are neglected. It has been , all higher-order interactions beshown [21] that for come negligible. In the near-field UHF RFID and NFC systems,

Fig. 2. Rotated (primed) coordinate system with receiving antenna on the -axis.

+z

the receiving antenna is always placed in the radiating near field . of the transmitting antenna, satisfying the criterion, Therefore, we only consider the direct interaction in our proposed formula. On the other hand, the source distribution of an antenna is more or less changed in the presence of another antenna, primarily due to the higher order interactions between them. Since all the higher order interactions are neglected in the proposed formula, the source distribution or the far-field pattern of the isolated antenna is therefore used. As one may expect, the accuracy of the formula may be enhanced if the modification to the far-field pattern is taken into account. However, it is pertinent to mention that it is a very challenging task to accurately acquire the modified far-field patterns, especially for the antenna separations considered in this work.

III. VERIFICATION WITH SIMULATION In order to verify the applicability of the proposed expression, we consider three classic near-field scenarios in this section. Three different operating frequencies from the UHF band (433 MHz, 915 MHz, and 2.4 GHz) are used in each of these scenarios. The frequencies are chosen due to their widespread use in near-field or short-range communication systems [2], [22], [23]. The associated coupling coefficients between the transmitting and receiving antennas are computed and compared to those obtained from simulations using Ansoft HFSS. It must be is used noted that, in the HFSS simulation, the quantity for comparison, and is obtained by assuming port 2, namely the receiving antenna in our cases, is perfectly matched to its load impedance. For consistency, this condition can be included in . our formula simply by setting A. Side-by-Side, Parallel Half-Wave Dipoles Consider two identical, y-directed half-wavelength dipole antennas, one of which is placed at the origin and the other on the -axis with a separation . We consider the former the transmitting antenna, and the latter to be the receiving antenna. It is fairly easy to derive the normalized vector far-field pattern of an

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Fig. 3. Coupling coefficient versus normalized antenna separation for polarization-matched dipoles.

Fig. 4. Coupling coefficient versus normalized antenna separation for polarization-mismatched dipoles.

ideal y-directed half-wavelength dipole based on its well-known z-directed counterpart, and this can be denoted as

(9)

(10) Substituting (9) and (10) into (8) yields the desired spherical wave coefficients . The computed coefficient, , diminishes drastically for higher order terms , causing (7) to rapidly converge. The near-field coupling coefficient as a func, computed by our protion of normalized antenna spacing posed method and that obtained via HFSS are shown in Fig. 3. The results agree well with those obtained using our formula.

Fig. 5. Coupling coefficient versus normalized antenna separation for polarization-matched square loop and dipole.

B. Side-by-Side, Polarization-Mismatched Half-Wave Dipoles In the preceding subsection, the two dipoles were polarization-matched, which is the ideal case in a two-dipole system. However, in most cases, the dipoles are arbitrarily oriented. To further verify our formulation, we consider a scenario having polarization-mismatched dipoles. In the current case, the receiving dipole lying on the y-z plane is rotated by 20 around its phase center. This can be accounted for in our expression simply by transforming accordingly the coordinate system of the vector thus obtained, simfar-field pattern of the receiving dipole. leading to rapid converilarly diminishes drastically for gence of (7). The coupling coefficients obtained are plotted in Fig. 4. One can see that the coupling coefficients are smaller here than in the preceding case because of the polarization mismatch between the two dipoles. C. Polarization-Matched Square Loop and Half-Wave Dipole We next use a square loop antenna with perimeter equal to a wavelength to replace the transmitting dipole in Section III-A. The loop antenna is centered at the origin with the loop lying on the x-y plane, oriented such that the resultant polarization

is aligned with the y-directed receiving dipole. The coupling coefficient for this setup can thus be computed as a function of the antenna spacing. The results are compared with those from simulation in Fig. 5. In the three scenarios, as mentioned above, the agreement between the computed results and those simulated by HFSS indicates that the proposed method can be effectively utilized to determine the near-field coupling coefficient if the relative orientation, the antenna spacing, and the far-field patterns of the transmitting and receiving antennas are known. Also, it must be noted from (7) that the formula is theoretically applicable within the radiating near field. As a result, the coupling coefficients shown in Figs. 3 and 4 and Fig. 5, respectively, can be predicted and , and acaccurately for curacy degrades outside that particular range. However, one can , the formula is still also see that for being as small as able to predict the coupling coefficient with satisfactory accuregardless racy. Therefore, results are presented for of the constraints. It is also important to note that the differences between the simulated results for different resonant frequencies

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Fig. 6. Simplified architecture of near-field RFID systems.

are all kept within 1 dB in the three scenarios. Higher accuracies could be obtained by properly placing the radiation boundaries and further refining the mesh in the HFSS simulations. IV. APPLICATION IN NEAR-FIELD UHF RFID SYSTEM After validating our expression with these typical scenarios, we apply it to the case of a near-field UHF RFID system for verification against experimental results. Near-field UHF RFID systems are preferable to those operating in the LF and HF bands due to the higher data rate, smaller antenna size, and lower manufacturing cost. A near-field UHF RFID system is composed of a reader and a tag just as in ordinary RFID systems. A simplified system architecture is depicted in Fig. 6. The power generated , is transferred to the reader anby the reader circuitry, tenna, and then acquired by the tag antenna through near-field , can be excoupling. The power absorbed by the tag chip, pressed as [24]

Fig. 7. Geometry of two-element square loop array with a back reflector.

(11) and are the power transmission coefficients where of the reader and tag, respectively, between its front-end ciris the coupling cocuitry and the associated antenna, and efficient between the reader and tag antennas. With the aid of the proposed formula, the coupling coefficient may be calculated. To experimentally verify the results, a near-field UHF RFID system is implemented. A broadband, square loop array with a back reflector is designed, fabricated, and deployed as the reader antenna. Two different tag antenna designs [25], [26] are used individually in the system. The associated coupling coeffiat the terminals of cient can be obtained by measuring the the reader and tag antennas. Also, it must be pointed out that the 3D far-field patterns needed in the formula require a finer sampling step of 1 due to numerical integration, and, therefore, they are obtained by transforming the near-field measurement data. A. Reader Antenna The geometry of the proposed reader antenna for the near-field UHF RFID system is shown in Fig. 7. Two printed square loop antennas, with perimeters equal to a wavelength, are back-to-back and connected by a coplanar strip (CPS) of . The two arms of the CPS are connected respeclength tively at their midpoints to the inner and outer conductors of the feeding coaxial cable. The coaxial cable is fed from the direction normal to the antenna plane. Although a balun could be added to slightly improve radiation performance, the proposed design, i.e., direct feed via a coaxial cable, can still provide satisfactorily higher gain and a well-shaped radiation pattern. As one may expect, the antenna radiates bi-directionally; however, most RFID reader antennas require a unidirectional radiation

Fig. 8. Simulated and measured radiation patterns of the proposed reader antenna at 920 MHz. (a) x-z plane and (b) y-z plane.

pattern. To produce a unidirectional pattern and further increase the antenna gain, an electrically large, planar conducting sheet is utilized as a back reflector for the loop array as shown in Fig. 7. The spacing between them, , is set to approximately a quarter of a wavelength in free space. A prototype antenna designed at around 915 MHz was fabricated on an FR-4 substrate with dielectric constant and thickness . A copper sheet of dimensions is used as the back reflector and placed at a distance from the loop array. The design is well matched within a wide frequency range, and the measured 10-dB return loss bandwidth is 19.1% (848–1022 MHz). The peak gains measured at 920 MHz and 930 MHz are 10.2 and 10.1 dBi, respectively. Since the radiation pattern remains nearly the same throughout the return loss bandwidth, for the sake of simplicity, Fig. 8 depicts the x-z and y-z plane patterns measured at 920 MHz only. B. Tag Antennas Two different tag antenna designs, referring to [25] and [26] respectively, are implemented and deployed in our near-field experimental setup. The first one is the folded dipole with a closed loop [25], the main advantage of which is its tunable input impedance that achieves a conjugate match for various commercial tag chips. A prototype antenna designed at 915 MHz is shown in Fig. 9. We would like to point out that, to

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Fig. 9. Photograph of the folded dipole antenna. (Area: 64:4

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 10, OCTOBER 2010

2 27 6 mm :

).

Fig. 11. Measurement setup for the near-field RFID system.

Fig. 10. Photograph of the meandered circular loop antenna. (Radius: 27 mm).

facilitate measuring the coupling coefficient through the vector network analyzer (VNA), the antenna is designed for 50 instead of being conjugate matched to the highly capacitive tag chips and is fed by a section of CPS connected to a balun. Due to fabrication errors and uncertainty of the dielectric constant of FR4, the resonant frequency of the prototype antenna slightly shifts from the design frequency of 915 MHz to 920 MHz. The peak gain measured at 920 MHz is 2.2 dBi. The other design used is a meandered circular loop [26]. A prototype antenna designed at 915 MHz is shown in Fig. 10. It is also designed for 50 , and the same feeding structure is used for easy measurement. Similarly, the resonant frequency of the prototype antenna shifts to 930 MHz. The peak gain measured at this frequency is 2.7 dBi. C. Results The aforementioned reader and tag antennas are utilized to measure the coupling coefficient in the near-field UHF RFID system. The schematic of the experimental setup is shown in Fig. 11. The measurements are performed in an anechoic chamber, and the associated coupling coefficients are obat the antenna terminals using the tained by measuring HP8753D VNA. Initially, the coupling coefficients are measured for various antenna separations, , and two sets of data are acquired using the two tag antennas individually. During the measurement, the reader antenna is fixed, while the tag antenna to 400 mm with incremental steps is moved from of 5 mm. The antennas are also kept polarization-matched with the main beam maximum aimed at each other. A photograph of the experimental setup is shown in Fig. 12. The coupling coefficients for the case where the folded dipole serves as tag antenna are measured at 920 MHz, i.e. the resonant frequency of the fabricated folded dipole. Results obtained by practical

Fig. 12. Photograph of the measurement setup in anechoic chamber.

measurement, and calculation from our formula, and from full-wave simulation are plotted in Fig. 13. The agreement between them is excellent. The same measurement procedure is repeated, and the folded dipole is replaced by the meandered circular loop. The experiment is conducted at 930 MHz, i.e. the resonant frequency of the meandered circular loop. The results obtained are plotted in Fig. 14. Again, a satisfactory agreement can be observed among them, once more validating our proposed formula and demonstrating its accuracy. Following the same process, the coupling coefficient is also investigated at frequencies other than the resonant frequency of the tag antenna. The measurements are conducted individually at 910, 920, 930, and 940 MHz, spanning only a small fraction of the wide bandwidth of the reader antenna. The radiation performance and input matching of the reader antenna remain nearly the same throughout the frequency range. The coupling coefficients measured and computed via the formula are compared and plotted in Fig. 15. For off-resonance operation, the associated coupling coefficient decreases mainly due to the impedance mismatch of the tag antenna. The more the operating frequency deviates from resonance, the lower the coupling coefficient will be. Additional measurements are performed using the folded dipole as the tag antenna and the meandered circular loop (previously used as the tag antenna) as the reader antenna. Fig. 16 shows the measured and computed coupling coefficients along with those for the case where the proposed high-gain reader antenna is used. The latter exhibits a higher coupling coefficient

CHEN et al.: ANALYSIS OF ANTENNA COUPLING IN NEAR-FIELD COMMUNICATION SYSTEMS

Fig. 13. Coupling coefficient versus antenna separation for the near-field RFID setup at 920 MHz. (Tag antenna: folded dipole).

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Fig. 16. Coupling coefficient versus antenna separation for the near-field RFID setup at 920 MHz. (Receiving antenna: folded dipole).

is preferable in near-field UHF RFID systems. And, the same holds true for the directivity of the tag antenna because of the reciprocity theorem. Finally, coupling coefficients are also obtained for the case where the tag antenna is placed at various points on a transverse plane spaced apart from the reader antenna by . In this setup, the reader antenna is fixed, while the tag antenna “scans” on a transverse plane with a constant antenna orientation. Fig. 17 shows the tag antenna located at an off-axis point with trans) from the on-axis point A. The transverse verse offsets ( , offsets can be converted into the relative orientation for the antennas by the following:

Fig. 14. Coupling coefficient versus antenna separation for the near-field RFID setup at 930 MHz. (Tag antenna: meandered circular loop).

Fig. 15. Coupling coefficient versus antenna separation for the near-field RFID setup at 910, 920, 930, and 940 MHz. (Tag antenna: folded dipole).

because of the more directive far-field pattern of the reader antenna. Therefore, a reader antenna that has a higher directivity

(12)

Given the tag antenna position and the 3D patterns of the reader and tag antennas, the associated coupling coefficient can be computed by rotating the 3D patterns in accordance with the relative antenna orientation. Also, it must be pointed out that the antenna spacing in the formula should be instead of . In this case, the folded dipole is chosen as the tag antenna, while the square loop array with a back reflector is used as the reader antenna. The measurements are made in 25-mm increments along both x- and y-axes employing a near-field planar scanner system developed by Nearfield Systems Inc. The entire . The measured and calculated scanning area is , 200, coupling coefficients for the transverse planes at and 300 mm are plotted in Figs. 18–20, respectively. Again, the measured results agree very well with those computed using our proposed formula. There are some minor errors, which may be attributed to the round-off error introduced in manipulating (12). During the calculation, the transverse offsets are first converted into the relative orientations (in degrees) between the transmitting and receiving antennas. The relative orientations are then rounded off to integer values because the measured patterns of

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Fig. 20. Coupling coefficient versus transverse displacements for d . (a) Calculated and (b) measured 3D surface plots.

300 mm

Fig. 17. Receiving antenna placed at an arbitrary point on a transverse plane spaced apart from the transmitting antenna by d.

=

With the aid of the proposed formula, it is fairly straightforward to determine the read range for near-field RFID systems. We rearrange (11) as (13) (13)

Fig. 18. Coupling coefficient versus transverse displacements for d . (a) Calculated and (b) measured 3D surface plots.

100 mm

=

is the minimum threshold power necessary to power where up the tag chip provided by the chip vendors. For simplicity, it is assumed that the reader circuitry (tag chip) and the antenna . are perfectly matched such that Given the output power of the reader circuitry, , and , one can find the corresponding read range from the curve of coupling coefficient, , calculated using our proposed formula, and considering the coupling coefficient as a function of antenna separation for the particular reader and tag antennas used. As an example, we consider one of the above scenarios where the folded dipole is the tag antenna and the square loop array with back reflector is the reader antenna. For a given and , one can see from Fig. 13 that the corresponding read range for coupling coefficient equals 300 mm. This information implies that an output power greater than 10 dBm may enable the reader to detect the tags outside the read range in addition to those that lie within it. V. CONCLUSION

Fig. 19. Coupling coefficient versus transverse displacements for d . (a) Calculated and (b) measured 3D surface plots.

200 mm

=

the antennas are sampled with a step of 1 . This will inevitably introduce some errors. Note from the figures that the maximal coupling occurs at the on-axis point A, in contrast to the minimal coupling near the corners of the scanning area. However, near the corners, the coupling coefficient slightly increases as is increased because the angular offset from the on-axis point A. Therefore, as can be observed from these figures, the spatial distribution of the coupling coefficient is more uniform for a larger . D. Near-Field Read Range In near-field (UHF) RFID systems, it is desirable for tags to be detectable within a specified read range and not beyond that.

A simple formulation has been proposed in this paper for computing the coupling coefficient between two antennas that are arbitrarily oriented and placed in the near field of each other. The formula, in a sense, can be regarded as a near-field counterpart of the Friis transmission formula. It has been validated by considering three typical scenarios, and the results have been shown to agree well with those obtained from simulation. In addition, an experimental setup for a near-field UHF RFID system has been designed and implemented. The coupling coefficients obtained via our formula, experimental measurement, and HFSS simulation are all in good agreement. Additionally, some factors critical to the coupling coefficient have been identified, including impedance matching of the receiving antenna, directivities and relative orientation of the transmitting and receiving antennas. Another advantage of our formula is that the computation time is greatly reduced as compared to full-wave simulation. It is believed that the proposed formula may prove useful for design and optimization of near-field communication systems.

CHEN et al.: ANALYSIS OF ANTENNA COUPLING IN NEAR-FIELD COMMUNICATION SYSTEMS

ACKNOWLEDGMENT The authors would like to thank the Department of Electrical Engineering, National Taiwan University of Science and Technology (NTUST) for providing measurement facilities and Dr. I. Lin for his technical assistance. REFERENCES [1] P. Harrop, “Near field UHF vs. HF for item level tagging,” IDTechEx article [Online]. Available: http://www.eurotag.org/?Articles_and_Publications [2] D. Desmons, “UHF Gen2 for item-level tagging,” presented at the RFID World 2006, [Online]. Available: http://www.impinj.com/files/Impinj_ILT_RFID_WORLD.pdf [3] C. Ajluni, “Item-level RFID takes off,” RF Design Mag., Sep. 2006. [4] S. Ortiz, Jr., “Is near-field communication close to success?,” Computer, vol. 39, pp. 18–20, Mar. 2006. [5] J. Walko, “A ticket to ride,” IET Commun. Eng., vol. 3, pp. 11–14, Feb.–Mar. 2005. [6] C. Evans-Pughe, “Close encounters of the magnetic kind,” IEE Rev., vol. 51, pp. 38–42, May 2005. [7] S. Esko, K. Jouni, P. Juha, Y. Arto, and K. Ilkka, “Application of near field communication for health monitoring in daily life,” in Proc. IEEE 28th Annu. Int. Conf. Engineering in Medicine and Biology Society, Aug. 2006, pp. 3246–3249. [8] S. Dominikus and M. Aigner, “mCoupons: An application for near field communication (NFC),” in Proc. 21st Int. Conf. on Advanced Information Networking and Applications Workshops, May 2007, vol. 2, pp. 421–428. [9] G. A. Wright, “Magnetic resonance imaging,” IEEE Signal Processing Mag., vol. 14, no. 1, pp. 56–66, Jan. 1997. [10] K. Fotopoulou and B. W. Flynn, “Optimum antenna coil structure for inductive powering of passive RFID tags,” in Proc. IEEE Int. Conf. on RFID, Mar. 2007, pp. 71–77. [11] P. Cole, “Coupling relations in RFID systems” and “Coupling relations in RFID systems II: Practical performance measurements,” white papers, 2003 [Online]. Available: http://www.autoidlabs.org/publications/page.html [12] H. Schantz, “A near-field propagation law and a novel fundamental limit to antenna gain versus size,” in Proc. IEEE AP-S Int. Symp. and URSI Radio Science Meeting, Jul. 2005, vol. 3B, pp. 134–137. [13] D. C. Yates, A. S. Holmes, and A. J. Burdett, “Optimal transmission frequency for ultralow-power short-range radio links,” IEEE Trans. Circuits Syst., vol. 51, no. 7, pt. I, pp. 1405–1413, 2004. [14] S. Iliev, Z. A. Abou Chahine, J.-F. Luy, and R. Weigel, “A hybrid method for determining the coupling coefficient of near field identification systems,” in Proc. 38th Eur. Microwave Conf., Oct. 2008, pp. 1054–1057. [15] H. C. Jing and Y. E. Wang, “Capacity performance of an inductively coupled near field communication system,” presented at the IEEE AP-S Int. Symp. and URSI Radio Science Meeting, San Diego, CA, Jul. 2008. [16] S. R. Aroor and D. D. Deavours, “Evaluation of the state of passive UHF RFID: An experimental approach,” IEEE Syst. J., vol. 1, pp. 168–176, Dec. 2007. [17] J. Langer, C. Saminger, and S. Grunberger, “A comprehensive concept and system for measurement and testing near field communication devices,” in Proc. IEEE EUROCON, May 2009, pp. 2052–2057. [18] A. D. Yaghjian, “Efficient computation of antenna coupling and fields within the near-field region,” IEEE Trans. Antennas Propag., vol. 30, no. 1, pp. 113–128, Jan. 1982. [19] A. D. Yaghjian, “An overview of near-field antenna measurements,” IEEE Trans. Antennas Propag., vol. 34, no. 1, pp. 30–45, Jan. 1986. [20] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: Wiley, 2001, pp. 264–276. [21] B. Derat, A. Cozza, and J.-C. Bolomey, “Influence of source-phantom multiple interactions on the field transmitted in a flat phantom,” in Proc. 18th Int. Zurich Symp. on Electromagnetic Compatibility, Munich, Germany, Sep. 2007, pp. 139–142.

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[22] L. Nagy, “Indoor propagation modeling for short range devices,” in Proc. 2nd Eur. Conf. on Antennas and Propagation, Nov. 2007, pp. 1–6. [23] X. Chen, W.-G. Yeoh, Y.-B. Choi, H. Li, and R. Singh, “A 2.45-GHz near-Field RFID system with passive on-chip antenna tags,” IEEE Trans, Microw. Theory Tech., vol. 56, no. 6, pp. 1397–1404, Jun. 2008. [24] P. V. Nikitin, K. V. S. Rao, and S. Lazar, “An overview of near field UHF RFID,” in Proc. IEEE Int. Conf. on RFID, Mar. 2007, pp. 167–174. [25] S.-L. Chen and K.-H. Lin, “A folded dipole with a close loop antenna for RFID applications,” in Proc. IEEE AP-S Int. Symp. and URSI Radio Science Meeting, Honolulu, HI, Jun. 2007, pp. 2281–2284. [26] H.-K. Ryu and J.-M. Woo, “Size reduction in UHF band RFID tag antenna based on circular loop antenna,” in Proc. 18th Int. Conf. on Applied Electromagnetics and Communications, Oct. 2005, pp. 1–4.

Yen-Sheng Chen was born in Taichung, Taiwan, on January 4, 1985. He received the B.S. degree in electrical engineering in 2007 and the M.S. degree in communication engineering in 2009 from the National Taiwan University, Taipei, where he is working toward the Ph.D. degree. His research interests include the analysis and design of RFID systems and near-field communication systems and the design of RFID tag/reader antennas.

Shih-Yuan Chen (M’05) was born in Changhua, Taiwan, in May 1978. He received the B.S. degree in electrical engineering in 2000, and the M.S. and Ph.D. degrees in communication engineering in 2002 and 2005, respectively, all from the National Taiwan University, Taipei. From 2005 to 2006, he was a Postdoctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University, working on the 60-GHz switched-beam circularly-polarized antenna module. In July 2006, he joined the faculty of the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, and served as an Assistant Professor. From August 2008 to July 2009, he visited the Department of Electrical and Computer Engineering, Michigan State University, East Lansing. His current research interests include the design and analysis of slot antennas/arrays, dielectric lens antennas, microstrip antennas, RFID tag/reader antennas, near-field communication systems, and metamaterial-inspired antennas.

Hsueh-Jyh Li (SM’10) was born in Yun-Lin, Taiwan, China, on August 11, 1949. He received the B.S.E.E. degree from National Taiwan University, Taipei, Taiwan, in 1971, the M. S. E. E. degree in 1980, and the Ph.D. degree from the University of Pennsylvania, Philadelphia, in 1987. Since 1973, he has been with the Department of Electrical Engineering, National Taiwan University, where he is a Professor. He was the Director of the Communication Research Center at NTU, from 1995 to 2000, and Chairman of the Graduate Institute of Communication Engineering, from 1997 to 2000. His main research interests are in microstrip antennas, radar scattering, microwave imaging, and wireless communication. Dr. Li received the Distinguished Research Award from the National Science Council, Republic of China, in 1992.

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Uncoupled Matching for Active and Passive Impedances of Coupled Arrays in MIMO Systems Michael A. Jensen, Fellow, IEEE, and Buon Kiong Lau, Senior Member, IEEE

Abstract—Impedance matching for coupled antenna arrays has received considerable attention in the research community. Because optimal matching requires implementation of a coupled network, leading to high complexity and often narrow operation bandwidth, new research has focused on the development of uncoupled matching networks with good performance. This paper explores traditional uncoupled impedance matching techniques for coupled arrays, specifically matching to the array active and passive impedances, within the context of multiple-input multiple-output communication. The concept of matching to the array active impedance is extended to the case where the propagating field is specified stochastically, and the performance of this solution is compared to that of traditional solutions using simulations. While emphasis is placed on matching for maximum power transfer, the paper concludes with a discussion on matching for minimum amplifier noise figure. Index Terms—Antenna array mutual coupling, impedance matching, MIMO systems.

I. INTRODUCTION URRENT interest in using multi-antenna technology to enhance wireless communication performance coupled with the small antenna separation mandated by compact mobile devices have led to vigorous interest in impedance matching techniques that compensate for the degradation created by antenna mutual coupling. Optimal solutions to this problem require coupled matching networks, with examples being the well-known optimal multiport conjugate match (MCM) for maximum power transfer [1]–[3] and the corresponding result for minimum noise figure [4]–[6]. Unfortunately, such coupled networks are typically complicated and often result in narrowband matching performance [7]. The challenges associated with implementation of optimal matching motivate the identification of uncoupled matching networks that achieve near-optimal performance. In traditional array research, this is typically accomplished by matching to the array self, passive [8], [9], or active [10] impedance. However, when it comes to multiple-input multiple-output (MIMO)

C

Manuscript received October 30, 2009; revised March 03, 2010; accepted March 25, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by Telefonaktiebolaget LM Ericssons Stiftelse för Främjande av Elektroteknisk Forskning and in part by the U. S. Army Research Office under the Multi-University Research Initiative (MURI) under Grants W911NF-04-1-0224 and W911NF-07-1-0318. M. A. Jensen is with the Electrical and Computer Engineering Department, Brigham Young University, Provo, UT 84602 USA (e-mail: [email protected]. edu). B. K. Lau is with the Department of Electrical and Information Technology, Lund University, SE-221 00 Lund,Sweden (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055803

systems, where the objectives of the array processing generally differ from those for traditional or beamforming arrays, the exploration of these uncoupled matching techniques has been limited, with most work considering either numerically-optimized solutions for a given open-circuit covariance [11], [12] or input impedance matching [7], which respectively represent forms of matching to the active and passive impedances. In this MIMO context, therefore, we lack a common theoretical framework for designing such networks, a detailed understanding of their different behaviors, and a careful comparison of their relative performance. This paper focuses on uncoupled matching for MIMO systems by formulating these solutions under a common theoretical framework and discussing each approach in the context of what is known about the propagation channel. Furthermore, because the channel is typically time-variant, we demonstrate how to extend the concept of matching to the active array impedance to the case of stochastically-specified fields. The ergodic channel capacity achieved with these matching techniques is compared to that resulting from a numerically-optimized impedance match. Simulations with closed-form and numerically-generated antenna characteristics illustrate that active impedance matching provides good beamforming gain and optimal MIMO capacity for small signal-to-noise ratio (SNR) or high antenna coupling, while passive matching achieves superior performance for high SNR and low coupling. While the paper focuses on matching for maximum power transfer, it concludes with an approach for applying the methods to achieve minimum amplifier noise figure.

II. MATCHING FOR POWER TRANSFER A common design goal is to maximize the power transferred either from the transmit power amplifiers to the antennas or from the receiving antennas to the terminating loads. The goal of this section is therefore to formulate the uncoupled terminations that achieve this maximum power transfer. Throughout this analysis, boldface lowercase and uppercase symbols denote vectors and matrices respectively, while script versions of the symbols inis the th element dicate elements of the vector or matrix ( of the vector ). An overbar indicates a vector electromagnetic quantity. Let the th antenna in an -element array be characterized by an open-circuit radiation pattern (pattern with all other ele, where ments terminated in an open circuit) denoted as with and representing the elevation and azimuth angles in a spherical coordinate frame, as shown in Fig. 1. If the

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, with being the real part of , under that , the assumption of reciprocal antennas that satisfy is the transpose. Solving this relationship for and where substituting the result into (2) leads to

Fig. 1. Geometry showing a two-element z -oriented dipole array and the incident electric field in the system coordinate frame.

(4) where

is the

th element of the vector inside the brackets.

B. Stochastic Field Matching

Fig. 2. Impedance parameter equivalent network model of a coupled antenna array at the (a) receiver and (b) transmitter.

electric field incident on the array is voltage on this antenna is given as

, the open-circuit

Achieving optimal power transfer for a specific incident field is only practical if the termination can adapt to the changing incident field. When this is infeasible, it is useful to consider the termination that functions over a range of incident field profiles. Consider a discrete set of incident fields, with the th field generating the open-circuit voltage and current . Finding the load impedance that on average matches the active impedances associated with all excitations means finding the solution minimizing the objective function

(5) (1)

If the array is further characterized by a full impedance matrix and is terminated in a load having an impedance matrix , then the equivalent circuit for the receiving system is that shown in Fig. 2(a) [13]. In this network, and respectively represent the voltage across and the current through the termination. It has been well-established that maximum power is transferred , where indito the load for the MCM [3], or cates a conjugate transpose. This requires a coupled termination (or matching network), which typically leads to implementation complexity and reduced bandwidth [7]. A. Deterministic Field Matching: Active Impedance Maintaining maximum power transfer while avoiding the complexities associated with the MCM is possible for a specific incident field (or equivalently value of ) using the notion of matching to the array active impedance, the goal of which is to maintain the voltages and currents associated with the optimally-terminated network [6], [10]. This is accomplished through the relations

(2) (3) is the active impedance, and are diagonal where is the conjugate. Using (2) in (3) leads to matrices, and , where . Note

where is the active impedance for the th excitation. Setequal to zero and ting the derivative of (5) with respect to solving (with the help of (2) and (3)) leads to

(6) (7)

where and denotes the th element of the matrix inside the brackets. appears within , we solve (6) by first initializing Since the load impedance (for example, ) and then using (6), and constructing and using (7). We compute use this new value to update and . This procedure repeats iteratively until it achieves convergence. , the sum in (7) becomes a true exIn the limit as pectation over the stochastic set of excitations. We assume that the fields are zero-mean Gaussian random processes obeying the angular correlation model

(8) where is the dyadic power andenotes gular spectrum (PAS) of the incident field and

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the expectation. Substitution of (1) into the expression for shown in (7) yields the covariance of given as [14]

(9)

However, when considering stochastic fields, we likely wish to take the average of this quantity, which is difficult given its rational form. However, we may approximate the average -factor by taking the ratio of averages. Specifically, using that and that, as discussed in connection with (7) and (9), , this approximate average -factor can be expressed as

C. Unknown Field Matching Systems often operate in a variety of scenarios, making it impossible to impedance match even to the stochastic nature of the fields. If the system cannot adapt its termination to changing statistics, then the termination should be designed to accommodate all incident fields. One simple approach is to set in (9) (field arriving from all angles) and use this PAS to design , which is the the matching network. In this scenario, mutual resistance of the coupled array [6], [15]. Since any dif. ference in scaling is removed in (6), we can use An alternate approach is to match to the array passive impedance [8], [9], which is the input impedance seen looking into each antenna port. While a closed-form derivation of the terminations that allow all ports to be simultaneously matched to their input impedances is simple for the case of two identical antennas [7], [16], the solution becomes tedious for larger or inhomogeneous arrays. Therefore, consider the antenna array used in transmit mode as depicted in Fig. 2(b) for which and . We excite the array with a voltage on the th port and zero voltage on the other ports and subsequently compute the input impedance seen looking into the coupled array from the th port. The load impedance for this port is then chosen as the conjugate of this input impedance, which can be expressed as

(10) Since appears in the expression for , this equation can be solved iteratively using the procedure outlined in Section II-B. Fundamentally, the passive impedance represents a special case of the active impedance. Specifically, matching to the active impedance maximizes the received power for a specific beamformer [6]. The passive impedance is the active impedance when the beamformer has only one non-zero weight (i.e. processes the signal from only one antenna port).

(11)

where is the trace. We also need to limit the allowable superdirectivity in the solution, and we therefore adopt the pragmatic approach of incorporating antenna loss and spatially-white noise in the model [19]. Specifically, if the th radiating element has a radiation , then its associated self-impedance can efficiency be modified to have resistance . Similarly, the open-circuit covariance matrix should contain the spatially-white noise generated by the antenna loss. For simplicity, we assume that the ratio of the open-circuit noise to open-circuit signal squared voltages is proportional to the ratio of the loss to the radiation resistances, leading to the regularization . E. S-Parameter Analysis The developments detailed in this section can be formulated using the full S-parameter matrix of the antenna and the uncoupled reflection coefficient matrix representing the load. For matching to the active impedance given a deterministic field or to the passive impedance as discussed in Sections II-A and II-C, the solutions based on Z-parameters and S-parameters are identical. However, when trying to simultaneously match to a range of active impedances by minimizing the cost function in (5), the relative weight of each term in the cost function for Z-parameters differs from that for S-parameters because of their nonlinear mathematical relationship, and therefore slight differences can occur in the two sets of solutions. In the cases considered in this paper, these slight differences change neither the behavioral trends nor the fundamental conclusions drawn from the results. Furthermore, while the derivation of the corresponding techniques using S-parameters is straightforward, their inclusion requires definition of new notation and terminology. Motivated by these observations and for the sake of conciseness, we therefore forego S-parameter analysis in this paper.

D. Superdirective Solutions The concept of selecting a termination based on the incident field characteristics emphasizes that the uncoupled termination itself serves as a beamformer. As a result, for closely-spaced antennas the terminations may produce superdirective behavior [17], and therefore we should take measures to remove these solutions when designing practical systems. To assess the level of superdirectivity caused by a termina, tion, we compute the array -factor defined as where [18]. This equation is suitable when considering the active impedance match for a specific field.

III. COMPUTATIONAL RESULTS A. MIMO System Capacity While impedance matching for maximum power transfer is applicable to many scenarios, our focus is on comparing the difrepreferent terminations in terms of MIMO capacity. Let sent the transimpedance transfer matrix, or

(12)

JENSEN AND LAU: UNCOUPLED MATCHING FOR ACTIVE AND PASSIVE IMPEDANCES OF COUPLED ARRAYS

where is the vector of transmit currents and is the receiver noise referred to the open-circuit antenna terminals. The instantaneous signal power received by the loads is

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. Since the structure of leads to , the covariance of is given as

becomes

(16) (13) where is the diagonal matrix of load resistances. We assume that the transmit array has large element spacing (no coupling, no correlation) so that the transmit currents have the covariance , where is the identity matrix. We . note that Given an effective channel matrix relating the transmit currents to the signals at the loads, the capacity is given by

The properly-normalized off-diagonal elements of this matrix represent the commonly-used correlation coefficients between antennas. However, for this study, we instead use the eigenvalues of this covariance, as this directly translates the correlation into the average power for each of the communication modes. This can be explicitly seen in the upper bound on the ergodic capacity that is computed as [16]

(14) where is the noise variance at each load and is the determinant. Since the last term in the determinant represents received SNR, comparison with (13) indicates that

(15) This formulation allows us to express the average power re. For all ceived in the loads as computational examples, we present the ergodic capacity computed as an average over 250 random channel realizations for single transmit and normalized by the average capacity receive antennas in the same environment. B. Channel Matrix and Spatial Covariance To construct the channel matrix for the simulations, we first matrix whose entries are independent realize an zero-mean unit-variance complex Gaussian random variables. Using our assumption of uncorrelated transmit signals outlined in Section III-A and assuming that the signals at the receive array have the covariance given by (9), we can use the well-known Kronecker model for the covariance to write . This model is known to have deficiencies for small element separation or large arrays [20], but since our goal is to compare the relative performance of different terminations, its use here is reasonable. must preserve the impact of The normalization used for element spacing and load impedance on the capacity. To acin (15) assuming an inficomplish this, we first construct nite antenna separation ( is diagonal) and with . (or ) so that the Frobenius norm of We then normalize is , meaning that is the single-input single-output (SISO) SNR as detailed in [16]. We finally confor different terminations using this normalized struct with an antenna efficiency of . version of In evaluating the performance of MIMO systems, it is also useful to understand the correlation structure of the signals across the loads. Given our formulation for the capacity such that (13) and channel matrix, this signal is

(17) where

is the

th eigenvalue of

.

C. Simulation Scenarios In the computations, the PAS is described by a truncated and with an Gaussian function in elevation centered at angle spread of 10 . The distribution in azimuth is described either as a constant (uniform distribution) or as a single cluster represented by a truncated Laplacian function with an angle . These PAS funcspread of 40 and centered at the angle tions are computed using the methods in [21]. We use arrays of half-wavelength dipole antennas oriented in the -direction as shown in Fig. 1, so that these PAS functions represent the average power in the polarization. In all of the cases considered, we first compute the terminations for each of the closed-form techniques (match to the active impedance for the PAS, match to the passive impedance, etc.). We then choose the termination achieving the largest avas a seed for a numerical optimization whose erage capacity goal is to find the uncoupled termination that maximizes this average capacity. Since, however, the capacity as a function of the terminations has local maxima, we generate 250 different starting points by randomly varying the real and imaginary parts of the seed termination uniformly over a range of 25%. For each of these starting points, we complete a Nelder-Mead simplex optimization that determines the termination achieving the local maximum average capacity. Because each of these local maxima potentially represents a unique value of , the termiis selected as the optination achieving the largest value of mization outcome. D. Two-Element Dipole Array As a starting point in our analysis, we assume a linear array of two half-wave dipole antennas at the receiver. The open-circuit radiation patterns are assumed to be identical to the isolated dipole element patterns computed using the simple formula in [22]. The impedance matrix is also computed using the closed-form expressions in [22]. While this antenna characterization approach is approximate, it provides maximum flexibility

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Fig. 3. Normalized average capacity as a function of element spacing for a PAS described by a single Laplacian cluster arriving on a two-element array at broadside for different matching conditions and two different SNR levels.

Fig. 4. Normalized average capacity as a function of element spacing for a PAS described by a single Laplacian cluster arriving on a two-element array at endfire for different matching conditions and two different SNR levels.

in sweeping antenna parameters and therefore allows us to explore basic behaviors before adding the complexity associated with numerical antenna characterization. Fig. 3 plots the capacity as a function of the element spacing (broadside) using the different for the Laplacian PAS at terminations and for two different values of the SISO SNR. The performance for a perfect MCM is shown for comparison. Fig. 4 plots the same results when the Laplacian PAS is centered at (endfire). Despite the simplicity of these simulations, they teach some clear principles. For example, when the SNR is low, the termination obtained by actively matching to the incident PAS is optimal. In this scenario, only one of the two possible communication modes can be efficiently used for MIMO communication, and therefore the termination that matches to the incident PAS functions as a beamformer that enhances the quality of the dominant mode. The benefit of this beamforming is less pronounced for broadside excitation, since in this case all solutions consist of identical terminations on the two antennas due to the problem symmetry, and therefore the active match offers little benefit over other terminations.

K

Fig. 5. Normalized eigenvalues of the covariance as a function of element spacing for a PAS described by a single Laplacian cluster arriving on a twoelement array at endfire and for SNR = 20 dB.

When the SNR is high, matching to the active impedance maximizes the capacity when the element spacing is small. Here, the high signal correlation creates a scenario where only one mode is useful, and therefore the termination-induced beamforming enhances the communication. As the element separation increases, however, it becomes beneficial to equalize the two modes rather than enhance one mode at the expense of the other. The termination resulting from matching to the passive impedance does not attempt to beamform for the PAS and therefore better accomplishes this mode equalization. The relationship between beamforming gain and capacity is . reinforced by examining the eigenvalues of the covariance Fig. 5 plots these two eigenvalues, normalized so that the maximum value is unity, for the case of the PAS arriving at (endfire) and for an SNR of 20 dB. These results clearly show that the active match to the PAS maximizes the dominant eigenvalue at the expense of the other, reinforcing its nature as a beamformer. The eigenvalues for the numerically optimized termination mirror this behavior for small element separation but then abruptly jump to become more equalized as the spacing increases. Despite this abrupt change in eigenvalues, the capacity of the optimized solution is smooth, showing that this change represents a transition from the optimality of beamforming enhancement of a single communication mode to the exploitation of multiple modes. The power delivered to the loads for this scenario is shown in Fig. 6, with a behavior that reinforces these concepts. The superiority of the passive impedance match for high SNR is consistent with recent work demonstrating that this match maximizes the upper bound on the capacity when the SNR is in each term of the calarge enough to satisfy pacity expression of (17) [16]. When interpreting the results in Figs. 3 and 4, it is important to recognize that even when the SNR is 20 dB, the second eigenvalue for small element spacing is so weak that this high-SNR approximation does not apply, which explains why the passive impedance match does not maximize capacity in this regime. Fig. 6 also shows the level of supergain, as measured by the effective -factor, for the different terminations. These results

JENSEN AND LAU: UNCOUPLED MATCHING FOR ACTIVE AND PASSIVE IMPEDANCES OF COUPLED ARRAYS

Q

Fig. 6. Effective -factor and average power delivered to the loads as a function of element spacing for a PAS described by a single Laplacian cluster arriving on a two-element array at endfire and for SNR = 20 dB.

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Fig. 8. Normalized average capacity as a function of element spacing for a PAS described by a single Laplacian cluster arriving at  = 0 on linear and triangular three-element arrays characterized using MoM for SNR = 0 dB.



E. Three-Element Dipole Arrays Fig. 8 plots the capacity for a SISO SNR of 0 dB for linear and triangular arrays of three dipoles, again characterized using the MoM, as shown in the figure inset. The PAS is a single (endfire to the linear array), and Laplacian cluster at only key curves are plotted to simplify the discussion. In this case, the larger array aperture perpendicular to the cluster for the triangular array enables a higher overall capacity, while the beamforming enabled by the active PAS match provides a larger relative benefit for the linear array than the triangular array for moderate element separation. F. Applicability to Other Antennas Fig. 7. Normalized average capacity as a function of element spacing for a uniform PAS for different matching conditions and two different SNR levels with the elements characterized using MoM.

reveal that the terminations produced by MCM, numerical optimization, and active matching to the PAS yield relatively large effective -factors for small element separation, consistent with their nature as beamformers. We have also characterized the array and the corresponding reference of isolated antennas numerically using the method of moments (MoM) implementation of [23] for half-wave dipoles . We find that the results and therefore our of diameter analysis closely match what we have observed using the closedform antenna characteristics. As one example, Fig. 7 shows the capacity for two SNR levels as a function of element spacing when the PAS is uniform in azimuth. In this case, not surprisingly, the behavior for the termination for active matching to the PAS matches that obtained for the termination for active matching to a spherically uniform PAS, since the two PAS structures are the same in azimuth. This indicates that for a dipole array in this orientation, the resulting termination is relatively insensitive to the elevation structure of the PAS.

Naturally, the framework presented here can be used to formulate uncoupled matching networks for any coupled antenna topology. However, we can also use the results for dipoles to estimate what might occur for other antennas. Specifically, the eigenvalues of the covariance matrix include the impact of signal correlation and the power transfer achieved by the match. Therefore, if one can find a dipole spacing for which the eigenvalues in Fig. 5 are similar to those of the target antenna, then using the capacity from Fig. 4 corresponding to the selected spacing will give an estimate of the capacity for the MIMO system using the target array. IV. MATCHING FOR NOISE FIGURE The discussion on matching to maximize power transfer has provided valuable insights into the behavior of different matching topologies. However, this discussion would be incomplete without considering impedance matching for practical receivers where the front-end amplifiers represent a dominant noise source. In this situation, some amplifier noise is coupled between ports due to the antenna coupling, and the matching between the antennas and the amplifiers directly controls the front-end noise figure and system capacity.

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A. Matching Implementation The challenge of identifying an uncoupled termination that achieves minimum receiver noise figure is that it is based on a theory of optimally mismatching the antennas to the front-end amplifiers. In this case, rather than use the simple receiver model of Fig. 2(a), we introduce a matching network between the antennas and amplifiers using the model and theory derived in [4]. For the sake of conciseness, this analysis will not be repeated here, and we only indicate that the goal of the matching network is to transform the antenna impedance such that it appears to the amplifiers as the optimal noise figure termination with di. Naturally, this precise condition agonal impedance matrix is only satisfied either for a specific incident field or for a coupled matching network. Given the different matching techniques outlined in Section II, the challenge is to determine the mechanism for specifying the characteristics of the matching network to achieve the goal of minimum noise figure. In the context of the theory presented in [4], we have found that the following sequence of steps produces reasonable results. 1) Use the theory outlined in Section II to design the load that achieves maximum power transfer. 2) Given this load, assume that the active impedance seen . looking into the antenna terminals is 3) Using the theory in [4], design the uncoupled matching network that transforms this uncoupled antenna active . Note that, since the theory in [4] uses impedance to the S-parameter representation, must be converted to a diagonal reflection coefficient for this computation. This approach works for the active impedance matching techniques as well as the passive impedance match. However, rather than use this approach for the active impedance match assuming a uniform PAS, we instead use the theory in [6] that presents a similar solution achieving the goal of minimum noise figure for the uniform PAS. B. Example Computation Computations performed using this theory generally demonstrate that the basic observations and conclusions made in connection with the results presented in Section III apply to the case of optimal noise matching. As an example, consider again the two-element dipole array with the Laplacian cluster arriving at array broadside and the elements characterized using MoM. The transistors forming the amplifiers have noise parameters , , , (optimal reflection coefficient), and an input impedance of 50 (see [4] for a detailed discussion on how these are used in the design and simulations). Fig. 9 plots the capacity resulting from this analysis for two different values of SISO SNR, where the “Minimum Noise” match is that obtained from [6]. As can be seen, under this procedure, the conclusions for high SNR are similar to those obtained during the analysis of matching for maximum power transfer. Specifically, when the SNR is high, matching to the active impedance (including the solution from [6]) leads to optimal performance only for high coupling, while matching to the passive impedance is optimal elsewhere.

Fig. 9. Normalized average capacity as a function of element spacing for a PAS described by a single Laplacian cluster arriving on a two-element array at broadside for different matching conditions and two different SNR levels with the elements characterized using MoM.

However, when the SNR is low, the numerically optimum solution outperforms all other uncoupled solutions for high antenna coupling. Closer investigation of this case reveals that, despite the problem symmetry that suggests identical matching on the antenna ports, the numerical solution provides asymmetric matching to achieve these results. In fact, when the optimization is constrained to produce symmetric matching, the numerically-optimized solution follows the analytical curves as observed in all other capacity plots in this paper. This means that at low SNR, degrading the quality of one output port to enhance the quality on the other provides benefit. We also point out that the port selected to achieve improved performance is arbitrary. Given the symmetric nature of this problem, it is currently unclear as to how to develop a closed-form matching strategy to achieve this behavior. However, it is important to recognize that this case of high coupling and low SNR is impractical for most realistic communication scenarios. V. CONCLUSIONS This paper uses a common framework to develop and analyze uncoupled impedance matching for coupled array antennas. Specifically, it discusses matching to the array active impedance for deterministic and stochastically-specified electromagnetic fields, and shows that such active matching in effect creates a beamformer that maximizes received power. It also discusses a previously-proposed technique for matching to the antenna passive impedance, also referred to as input impedance matching, known to be optimal in certain circumstances. Simulation results of MIMO capacity using different propagation environments demonstrate that for low SNR or high coupling, active matching outperforms passive matching due to the associated beamforming gains. However, for moderate coupling or high SNR, passive impedance matching enables better use of the multiple communication modes. The discussion concludes by demonstrating the application of the matching techniques for minimizing the system noise figure.

JENSEN AND LAU: UNCOUPLED MATCHING FOR ACTIVE AND PASSIVE IMPEDANCES OF COUPLED ARRAYS

REFERENCES [1] H. A. Haus and R. B. Adler, Circuit Theory of Linear Noisy Networks. New York: Wiley, 1959. [2] J. B. Andersen and H. H. Rasmussen, “Decoupling and descattering networks for antennas,” IEEE Trans. Antennas Propag., vol. AP-24, no. 6, pp. 841–846, Nov. 1976. [3] J. W. Wallace and M. A. Jensen, “Mutual coupling in MIMO wireless systems: A rigorous network theory analysis,” IEEE Trans. Wireless Commun., vol. 3, pp. 1317–1325, Jul. 2004. [4] M. L. Morris and M. A. Jensen, “Network model for MIMO systems with coupled antennas and noisy amplifiers,” IEEE Trans. Antennas Propag., vol. 53, pp. 545–552, Jan. 2005. [5] K. F. Warnick and M. A. Jensen, “Effects of mutual coupling on interference mitigation with a focal plane array,” IEEE Trans. Antennas Propag., vol. 53, pp. 2490–2498, Aug. 2005. [6] K. Warnick, E. Woestenburg, L. Belostotski, and P. Russer, “Minimizing the noise penalty due to mutual coupling for a receiving array,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1634–1644, Jun. 2009. [7] B. K. Lau, J. B. Andersen, G. Kristensson, and A. Molisch, “Impact of matching network on bandwidth of compact antenna arrays,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3225–3238, Nov. 2006. [8] P. W. Hannan, “The element-gain paradox for a phased-array antenna,” IEEE Trans. Antennas Propag., vol. 12, no. 4, pp. 423–433, Jul. 1964. [9] D. Schaubert, A. Boryssenko, A. van Ardenne, J. Bij de Vaate, and C. Craeye, “The square kilometer array (SKA) antenna,” in Proc. IEEE Intl. Symp. on Phased Array Systems and Technology, Boston, MA, Oct. 14–17, 2003, pp. 351–358. [10] C. Craeye, B. Parvais, and X. Dardenne, “MoM simulation of signal-tonoise patterns in infinite and finite receiving antenna arrays,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3245–3256, Dec. 2004. [11] J. B. Andersen and B. K. Lau, “On closely coupled dipoles in a random field,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 73–75, 2006. [12] B. K. Lau, J. B. Andersen, G. Kristensson, and A. F. Molisch, “Antenna matching for capacity maximization in compact MIMO systems,” in Proc. 3rd Int. Symp. Wireless Commun. Syst., Valencia, Spain, Sep. 2006, pp. 253–257. [13] W. Geyi, “Derivation of equivalent circuits for receiving antenna,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1620–1624, Jun. 2004. [14] B. T. Quist and M. A. Jensen, “Optimal antenna radiation characteristics for diversity and MIMO systems,” IEEE Trans. Antennas Propag., vol. 57, no. 11, pp. 3474–3481, Nov. 2009. [15] R. Vaughan and J. B. Andersen, “Antenna diversity in mobile communications,” IEEE Trans. Veh. Technol., vol. 36, no. 4, pp. 149–172, Nov. 1987. [16] Y. Fei, Y. Fan, B. K. Lau, and J. Thompson, “Optimal single-port matching impedance for capacity maximization in compact MIMO arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3566–3575, Nov. 2008. [17] M. Uzsoky and L. Solymar, “Theory of super-directive linear antennas,” Acta Physica Hungarica, vol. 6, no. 2, pp. 185–205, 1956. [18] Y. T. Lo, S. W. Lee, and Q. H. Lee, “Optimization of directivity and signal-to-noise ratio of an arbitrary antenna array,” Proc. IEEE, vol. 54, pp. 1033–1045, Aug. 1966. [19] N. W. Bikhazi and M. A. Jensen, “The relationship between antenna loss and superdirectivity in MIMO systems,” IEEE Trans. Wireless Commun., vol. 6, no. 5, pp. 1796–1802, May 2007. [20] H. Özcelik, M. Herdin, W. Weichselberger, J. Wallace, and E. Bonek, “Deficiencies of ‘Kronecker’ MIMO radio channel model,” Electron. Lett., vol. 39, pp. 1209–1210, Aug. 7, 2003.

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[21] L. Schumacher, K. I. Pedersen, and P. E. Mogensen, “From antenna spacings to theoretical capacities—Guidelines for simulating MIMO systems,” in Proc. IEEE 13th Int. Symp. on Personal, Indoor and Mobile Radio Comm., Lisboa, Portugal, Sep. 15–18, 2002, vol. 2, pp. 587–592. [22] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Wiley, 1997. [23] S. M. Makarov, Antenna and EM Modeling With MATLAB. New York: Wiley, 2002.

Michael A. Jensen (S’93–M’95–SM’01–F’08) received the B.S. (summa cum laude) and M.S. degrees in electrical engineering from Brigham Young University (BYU), Provo, UT, in 1990 and 1991, respectively, and the Ph.D. in electrical engineering from the University of California, Los Angeles, in 1994. Since 1994, he has been in the Electrical and Computer Engineering Department at BYU where he is currently a Professor and Department Chair. His research interests include antennas and propagation for communications, microwave circuit design, and multi-antenna signal processing. Dr. Jensen is currently Editor-in-Chief of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He has been an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, Chair of the Joint Meetings Committee for the IEEE Antennas and Propagation Society, a member of the society AdCom, and Co-Chair and Technical Program Chair for five symposia sponsored by the society. In 2002, he received the Harold A. Wheeler Applications Prize Paper Award in the in recognition of his research on multi-antenna communication.

Buon Kiong Lau (S’00–M’03–SM’07) received the B.E. degree (with honors) from the University of Western Australia, Crawley, and the Ph.D. degree from Curtin University of Technology, Perth, Australia, in 1998 and 2003, respectively, both in electrical engineering. During 2000–2001, he took a year off from his Ph.D. studies to work as a Research Engineer with Ericsson Research, Kista, Sweden. From 2003 to 2004, he was a Guest Research Fellow at the Department of Signal Processing, Blekinge Institute of Technology, Sweden. In 2004, he was appointed a Research Fellow in the Department of Electrical and Information Technology, Lund University, Sweden, where he is now an Assistant Professor. During 2003, 2005 and 2007, he was also a Visiting Researcher at the Department of Applied Mathematics, Hong Kong Polytechnic University, China, the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, and Takada Laboratory, Tokyo Institute of Technology, Japan, respectively. His research interests include array signal processing, wireless communication systems, and antennas and propagation. Dr. Lau is an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and is an active participant of EU COST Action 2100, where he is the Co-Chair of Subworking Group 2.2 on “Compact Antenna Systems for Terminal.”

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Elevation Plane Beam Scanning of a Novel Parasitic Array Radiator Antenna for 1900 MHz Mobile Handheld Terminals Md. Rashidul Islam, Student Member, IEEE, and Mohammod Ali, Senior Member, IEEE

Abstract—A novel internal antenna system is introduced for the GSM 1900 MHz frequency band which utilizes the electrically steerable parasitic array radiator (ESPAR) concept to steer the elevation plane beam in specific angular directions. The proposed antenna consists of a planar inverted-F antenna (PIFA) as the driven element and two inverted L antennas (ILAs) as parasitic elements. The ILAs are terminated with capacitive impedances at their bases. Full-wave analysis shows that the radiation beam can be scanned in space when the antenna is in free space as well as next to human head and hand phantoms. Measured results using varactor diodes controlling the base reactances show good return loss bandwidth. Index Terms—Beam steering, ESPAR antenna, PIFA, SAR.

I. INTRODUCTION

T

O efficiently use channels and to improve signal to interference ratio (SIR) currently spatial division multiple access (SDMA) technique is used in base stations to steer the beam of an antenna system in an intended direction [1]. The same has not been the case for handheld terminals (e.g., mobile phones, wireless personal digital assistants (PDAs) etc.) primarily due to the limited space available in such devices and the associated cost and complexity. Instead, to compensate for the weak signal in a multipath fading environment researchers have proposed and studied diversity antennas for handheld terminals [2]–[7]. One of the schemes is termed the pattern diversity scheme in which multiple antennas provide orthogonal or uncorrelated radiation patterns to discriminate against incoming signals over a certain angular space [8]. This is primarily achieved by spatially separating multiple antennas and then sequentially turning them on and off. Significantly better radiation performance can be achieved by implementing the electrically steerable parasitic array radiator (ESPAR) concept in a portable handheld terminal. Typically, an ESPAR antenna consists of a single driven element surrounded by a number of parasitic elements whose states are either open or short. Some earlier ESPAR antennas reported in [9]–[11] consisted of monopoles in which the parasitic monopoles were eiManuscript received September 23, 2009; revised April 03, 2010; accepted April 13, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The authors are with the Department of Electrical Engineering, University of South Carolina, Columbia, SC 29208 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055798

ther open or shorted to the ground plane using diode or semiconductor switches. Harrington [12] introduced the concept of a reactively controlled directive array with one driven dipole and six parasitic dipoles which surrounded the driven dipole in a circular fashion. In [12] it was demonstrated that the antenna beam could be scanned over an azimuth angle of 60 by adding variable reactances at the bases of the parasitic dipoles. Other examples of ESPAR antenna for base stations and wireless network terminals include [13]–[17]. Kawakami et al. [17] designed a seven-element ESPAR array using vertical monopoles on a ground skirt. The parasitic monopoles were connected to the ground skirt via varactor diodes which in turn provided reactive impedances. To our knowledge, the only work which incorporated a somewhat similar concept to the ESPAR concept is [18]. In [18] a four-element whip array (a driven vertical monopole was surrounded by a fixed short circuited parasitic element and two switchable parasitic elements), was placed on top of a metal box. By switching the states of the switchable parasitic elements the antenna could direct the radiation beam in two different directions. Given that almost all portable handheld wireless devices these days contain internal antennas it is important that the ESPAR concept is explored and exploited from that perspective. To that end, recently we presented the preliminary design of an ESPAR antenna consisting of three planar inverted-F antennas (PIFAs) for a mobile phone terminal [19]. Our present work builds on our previous conference presentation and substantially develops it. In this paper we present a reactively controlled internal ESPAR antenna system consisting of a driven planar inverted-F antenna and two adjacent inverted-L antennas for operation at the 1900 MHz GSM frequency band. This paper is organized as follows. First, the antenna geometry, the appropriate range of capacitances and the distance between the driven and the parasitic antenna elements are defined. Second, radiation pattern results using HFSS [20] are presented followed by simulated and measured results of return loss. Third, SAR and antenna gain computed using XFDTD [21] for a true anatomical head model are presented. Finally, the effect of the capacitances is studied for a dual band antenna with beam scanning capability at 1900 MHz. II. ANTENNA CONFIGURATION A. Geometrical Details for HFSS Simulation The proposed ESPAR antenna system consists of a PIFA and two inverted L antennas (ILA) as shown in Fig. 1. The PIFA is

0018-926X/$26.00 © 2010 IEEE

ISLAM AND ALI: ELEVATION PLANE BEAM SCANNING OF A NOVEL PARASITIC ARRAY RADIATOR ANTENNA

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Fig. 3. Variation of 0 with capacitance C : (a) magnitude of 0 and (b) phase of 0. Parameters: l = 29 mm, l = 2 mm, h = 6 mm, w = 3 mm, f = 1:85 GHz, 1.92 GHz and 1.99 GHz. Fig. 1. The three-element electrically steerable parasitic array radiator (ESPAR) antenna.

B. Baseline Effective Capacitance Range Using transmission line theory Schlub et al. [13] determined the range of capacitances that affected the radiation characteristics of a seven element monopole ESPAR antenna. To determine and for the proposed the effective range of capacitances ESPAR antenna we also used the transmission line theory. Consider the ILA shown in Fig. 2 and assume it to be a microsrtip can be caltransmission line. The characteristic impedance, culated as [22]

Fig. 2. An ILA with its capacitor arrangements in the proposed ESPAR antenna.

the driven element while the ILAs are parasitic elements. Since an ILA is a low-profile variant of a monopole antenna suitable for development on a mobile handheld terminal we selected it to take place of the parasitic elements. As a starting point, first, a single PIFA without the ILAs was designed on a 100 mm by 40 mm ground plane using HFSS for resonance at around 1900 MHz. The PIFA consisted of a top metal strip with 32 mm length and 3 mm width, a vertical feed with 6 mm height and 1 mm width, and a vertical shorting pin with 6 mm height and 1 mm width. The distance between the feed and the shorting pin was 1 mm. Afterwards, each of the two ILAs shown in Fig. 1 was added at a distance from the driven PIFA. Each ILA consisted of a top metal strip with 32 mm length and 3 mm width, a connecting pin with 6 mm height and 1 mm width, and a lumped capacitor at and the base of the connecting pin. Each capacitor ( for for ) was connected between the corresponding connecting pin and the ground plane. A narrow notch (4 mm in the X direction and 2 mm in the Y direction) was cut on the ground plane to accommodate the capacitors. Each connecting pin was 2 mm offset from the edge of the ILA (the offset distance, is shown in Fig. 2). In real application we envision the use of varactor diodes to adjust the capacitances. For practical testing the ground plane will be supported by a dielectric substrate underneath, which will house all necessary components. The capacitors have an important role to play. By adjusting their values the input impedances of the parasitic ILAs can be controlled which will then enable one to control the current distributions on them. By controlling the current distributions one can control the radiation patterns.

(1) is the effective dielectric constant which is 1 for air, where and are the height and width of the ILA respectively. Then and (Fig. 2) are calculated as

(2) where is the phase constant and and are the respective is the equivline lengths (29 mm and 2 mm here). Now if and alent impedance of the parallel combination of there will be an impedance mismatch between and the capacitive reactance provided by the capacitor attached between the connecting pin and the ground plane (see Fig. 2). The reflection coefficient, that quantifies this mismatch is calculated as follows:

(3) where is the reactance provided by either capacitor or . Using (1)–(3) was calculated for by varying capac. Fig. 3 shows the magnitude and phase of versus itance . As seen there is no change in when the capacitance exis ceeds 100 pF (Fig. 3(a)). Conversely, the only change in the step transition from 180 to 0 as the capacitance reaches 2 is varied. Since pF (Fig. 3(b)). The same thing happens when is high and nearly flat when pF and pF we consider the range of 0.1 pF–100 pF as the effective capacitances that would affect the radiation characteristics of the proposed antenna.

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C. Effective Distance Fig. 1 defines the distance, between the driven PIFA and each of the ILAs. To determine a preliminary study on beam scanning was performed. First, using HFSS the radiation patterns of the single PIFA (which had only the driven PIFA without the parasitic ILAs) was computed at 1.92 GHz which showed plane the beam maximum was in the that in the direction. For the ESPAR antenna system the folpF and pF. lowing parameters were fixed: Then was varied between 5 mm to 65 mm at 5 mm steps and the radiation patterns were computed at 1.92 GHz. It was observed that for all nine values which were in the range of 10 mm to 50 mm all of the patterns in the plane pointed in the direction. Thus clearly the beam for the single PIFA to for the ESPAR shifted from mm all patantenna for all of the above values. For terns were very similar to the pattern of a single PIFA. Higher peak gain was achieved when was in the range of 15 mm to 20 mm. Since smaller would result in stronger mutual coupling and will reduce the antenna impedance bandwidth, mm was selected. Note that, among the three principal planes , and ) the proposed ESPAR an( plane only tenna is capable of scanning the beam in the because the single PIFA without any ILAs have a directive pattern only in the plane while the patterns in other patterns are mostly non-directional. planes as well as all

The locations of the elements in the coordinate plane are shown in Fig. 4. The unit vectors directed from the feed to the two parasitic elements are given by

(6) The angles

can be expressed as

(7) The array factor (AF) is given by [1],

(8) where and feed. Thus,

are the distances of points C and

from the

(9) In the

plane the AF is given by,

D. Theoretical Basis of Beam Scanning To elucidate the beam scanning concept using a driven and two capacitively terminated radiating elements the following simple approach can be used. In this approach the array factor consisting of the three elements is calculated considering the excitation currents defined below. The voltages and the currents in the three elements are related by the impedance (Z) matrix defined in (4)

(4)

where the subscripts 0, 1, 2 correspond to the driven PIFA, and respectively. The terms and represent the and respeccapacitive terminations at the bases of and the are the self and mutual tively. As apparent the impedances in the Z matrix. The PIFA excitation voltage is . The currents on the parasitic ILAs normalized to the current on the driven PIFA can be expressed as [23]

(10) Once the AF is known the overall radiation pattern can be obtained by multiplying it with the element factor. As indicated above, for the ESPAR antenna shown in Fig. 1 the elements are not oriented along the same axis; the PIFA is oriented along the y-axis while the ILAs are oriented along the x-axis. Thus their element patterns will differ. Nevertheless, the objective of this exercise is to get a basic intuitive understanding on whether beam scanning is theoretically possible. To that end, the matrix in (4) was calculated using HFSS (the capacitors at the base of the ILAs in Fig. 1 were replaced with voltage sources). The distance was kept at 20 mm (see Fig. 1) and simulations were performed at 1.92 GHz. Fig. 5 shows the AF for various capacitive terminations. As seen, the AF is nearly constant when pF. This is expected because the reactances at the bases of the ILAs will allow little or no currents in them. The other capacitive terminations result in different induced currents in the ILAs and hence different AFs. III. RESULTS: THE ESPAR ANTENNA IN FREE SPACE A. Radiation Patterns

(5)

The antenna geometry shown in Fig. 1 and described in Section II.A was considered. For comparison a single PIFA without any ILA or capacitors was also studied. Several cases

ISLAM AND ALI: ELEVATION PLANE BEAM SCANNING OF A NOVEL PARASITIC ARRAY RADIATOR ANTENNA

Fig. 4. Locations of isotropic elements in the coordinate plane for array factor calculation. The ground plane is on the xy plane with the driven element at the origin.

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E =0 C = 10 C = 0 1 C = 10 C = 30 C = 10 C = 5 C =C =1 C =1 C = 20 C = 20 C = 10 C =7 C =5

Fig. 6. Computed normalized patterns of the ESPAR antenna at 1.92 GHz in the  plane obtained with (a) pF, : pF; pF, pF, pF, pF (b) pF; pF, pF; pF, pF and pF, pF.

TABLE I . THE RANGE OF IS 0.1–100 PF. FOR BEAM SCANNING BY VARYING VALUES FOR WHICH PATTERNS ARE COMPUTED: EACH CASE DISCRETE : PF, 0.5 PF, 1 PF, 2 PF, 3 PF, 5 PF, 7 PF, 10 PF, 15 PF, 20 PF, 30 PF, 50 PF, 70 PF, 100 PF. FREQUENCY : GHZ

C

C =01

C

C

= 1 92

Fig. 5. Variation of array factor of the proposed ESPAR antenna as the capac;f : GHz. itive loads in base of the ILAs are varied, '

=0

= 1 92

were studied in which was varied between 0.1 pF to 100 was kept fixed and vice versa. Simulations were pF while performed using HFSS. Patterns were computed at 1.92 GHz and in all principal planes. The beam scanning property was plane. Simulation data listed in Tables I observed in the pattern scans the beam maximum and II demonstrate that the and in the at various angles between plane. Cases 5 and 6 from Tables I and II represent the widest beam scanning angle among all the cases. Computed radiation patterns at 1.92 GHz and in the plane are shown in Fig. 6(a) for Case 5 of Table I and in Fig. 6(b) for different cases extracted from Tables I and II. The patterns in Fig. 6(a) and (b) are normalized to the peak gain for the cases pF, pF and pF respectively. with and As seen the beam maximum scans between for different capacitive combinations. The patterns with a fixed of 10 pF and a variable of 0.1 pF, 30 pF and with a and 5 pF (Fig. 6(a)) point towards peak gain of 2.6 dBi, 2.3 dBi and 2.7 dBi respectively.

TABLE II . THE RANGE OF IS 0.1–100 pF. FOR BEAM SCANNING BY VARYING : pF, 0.5 pF, 1 pF, 2 pF, 3 pF, 5 pF, 7 pF, 10 pF, 15 pF, 20 EACH CASE pF, 30 pF, 50 pF, 70 pF, 100 pF. FREQUENCY : GHz

C =01

C

C

= 1 92

On the other hand, Fig. 6(b) shows the scanning of radiation and are varied. As seen, the pattern maximum when both pF has a beam maximum at . This for pattern resembles the pattern of the single PIFA without any and ILAs present. The beam maximum is in the directions for cases with pF, pF;

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Fig. 7. Computed normalized E patterns of the ESPAR antenna at (a) 1.85 GHz and (b) 1.99 GHz in the  = 0 plane obtained with C = 1 pF, C = 1 pF; C = 1 pF, C = 20 pF; C = 20 pF, C = 10 pF and C = 7 pF, C = 5 pF. Fig. 8. Fabricated prototype of the proposed ESPAR antenna. Two varactor diodes (SMV 1248) provided the variable capacitances.

pF, pF; and pF, pF respectively. The peak gain for these four cases are 3.6 dBi, 1.9 dBi, 2.7 dBi and 2.6 dBi respectively. It is noteworthy that in some cases the phenomenon of beam formation was not observed as distinctively as it was observed with the capacitances mentioned in Fig. 6(a)–(b). In fact, the radiation patterns in these cases (e.g., Case 4 of Table I and Case 3 of Table II) showed multiple lobes with approximately equal magnitudes. When the capacitive terminations were varied, one lobe became larger with the presence of the rest of the other lobes. This is why the beam scanning angles for these cases were discrete. Radiation patterns were also computed at 1.85 GHz and 1.99 GHz as shown in Fig. 7. Comparing Fig. 7(a) with Fig. 6(b) it can be seen that the beam maxima in the upper hemisphere with pF, pF and pF, pF have shifted about 10 clockwise. Also, in Fig. 7(a) higher peak gain occurs with a different capacitance combination than pF, in Fig. 6(b). A butterfly shaped pattern resulted for pF. The peak gain for the four cases at 1.85 GHz are dBi, 2.3 dBi and 0.2 dBi respectively. Compared 1.7 dBi, to Fig. 6(b) the patterns in Fig. 7(b) demonstrate that the beam maxima have shifted about 10 counter-clockwise. The peak gain for the four cases at 1.99 GHz are 2.7 dBi, 3.9 dBi, 3.3 dBi and 2.2 dBi respectively. B. Input Characteristics The proposed ESPAR antenna was fabricated on a 1.5 mm thick Duroid 6002 printed circuit board (PCB). The PCB had copper on each side. All four edges of the PCB were closed and soldered using copper tapes. The driven PIFA along with its feed and shorting pin was fabricated on a 6 mm thick piece of foam. Similarly the ILAs were also fabricated on a 6 mm thick piece of foam. The feed of the PIFA was connected to the inner conductor of a semi rigid coaxial cable while the shorting pin was connected to the PCB ground. The outer conductor of the coax was soldered to the PCB ground. Two varactor diodes (SMV 1248 [24]) were used to control the base capacitances of the two ILAs. By changing the bias voltages of the varactors or values between 1.3 pF and 22 pF can be obtained. The layout of the varactor diodes and their associated circuits can be seen in Fig. 8. As seen two DC blocking capacitors (in parallel) were used which were connected in series with the bias

Fig. 9. S plots of the ESPAR antenna using varactor diodes. Computed data correspond to the geometrical model of Fig. 1.

via an RF choke. Furthermore, for each ILA there was an additional RF choke and a resistor (670 ) connected in series with the negative terminal of the DC power supply (not shown in Fig. 8). (dB) data for Fig. 9 shows the computed and measured two different capacitance combinations. The computed closely follows the measured . For the case with pF, pF the antenna exhibits dual resonance characteristic. The dual resonance phenomenon is clear in the computed data. The resonance in the higher frequency band is due to the driven PIFA whereas the resonance in the lower frequency band is due to the parasitic ILAs. The ESPAR antenna with pF has a 3:1 VSWR bandwidth of 150 MHz. The bandwidth pF, pF is 240 MHz. The single PIFA with without any ILAs has a bandwidth of 130 MHz. The ESPAR antenna for the other capacitive terminations also provides bandwidths wider than the single PIFA which was confirmed by simulations. The wider return loss bandwidth attained with the ESPAR antenna is simply because of multiple resonant elements operating at slightly different frequencies. Similar phenomena have been reported in [25], [26]. The bandwidth pF was not as wide as the obtained with bandwidth obtained with pF and pF as because of smaller currents induced for the former case than the latter.

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TABLE III PEAK 1g AND 10g AVERAGED SAR IN THE HEAD MODEL DUE TO THE ESPAR ANTENNA AND THE SINGLE PIFA. INPUT POWER: 250 mW, FREQUENCY: 1.92 GHz, DISTANCE: 10 mm

Fig. 10. Computed normalized E patterns of the ESPAR antenna at 1.92 GHz in the 8 = 0 plane for Case II.

IV. RESULTS: THE ESPAR ANTENNA IN PRESENCE OF HEAD AND HAND MODELS In addition to the free-space performance analysis the ESPAR antenna was also studied for SAR (specific absorption rate) and gain. For this part of the study a finite difference time domain solver XFDTD [21] was used. An anatomically correct heterogeneous head model named the Duke head model [27] was used. Duke is a 34 year old male model from the Virtual Family models [27]. The Duke head model was developed from the whole body model using Varipose [21]. The head model consists of 47 different tissue types, the dielectric constant and conductivity of which were obtained from [28], [29]. The 18 mm thick ear was compressed to 4 mm to mimic a realistic cell phone conversation. Two cases were investigated: • Case I: The antenna was placed vertically against the Duke head model (see Table III). • Case II: A heterogeneous hand model consisting of 12 tissues was added (see Table III). The hand was cropped from the Visible Human Body [30] using Varipose [21]. The tissue properties of the hand model were also taken from [28], [29]. The distance between the antenna ground plane and the head model was 10 mm for both cases. A. SAR For XFDTD simulations the cell size used in all directions was 1 mm. The driven PIFA was operating at 1.92 GHz. Eight layers of perfectly matched layers (PML) absorbing boundary were used. All SAR values were computed considering 250 mW of continuous wave power. SAR data for the two cases defined above are listed in Table III. SAR data for the single PIFA (no ILAs) are also listed. As seen the ESPAR antenna with pF results in the highest peak 1g avg. SAR. In contrast, pF and pF. The the lowest SAR occurs when SAR induced with pF, pF is always lower than that induced by the single PIFA. The inclusion of the hand model in Case II results in decreased SAR. A portion of the power radiated by the antenna is absorbed in the hand which causes the SAR to decrease. A

maximum of 33% SAR reduction from Case I to Case II was observed. These results corroborate similar observations reported pF the in [31], [32]. For Case II except with SAR induced by the ESPAR antenna is always lower than that induced by the single PIFA. B. Radiation Patterns patterns for Case II are shown in Fig. 10 which Computed demonstrate that the beam scanning capability of the ESPAR antenna is retained in the presence of the head and hand. Patterns pF. are normalized to the peak gain attained with Unlike the free-space patterns shown in Fig. 6(b) these patterns show significant head blockage in the lower hemisphere ( to ). The patterns rotated 5 –25 clockwise due to the presence of the head and hand. The peak gain ranges from 1.2 to 3.7 dBi. The peak gain ranges from 2.6 to 4.3 dBi without the hand. The total 3D power gain patterns for Case II are shown in Fig. 11. The 3D patterns also demonstrate the beam scanning and are changed. capability of the ESPAR antenna as Another parameter of interest here is the mean effective gain (MEG). The definition of MEG can be found in [33]. Computed dBi and MEG for Case II shows that it lies between dBi as opposed to dBi for the single PIFA. It is clear that the ESPAR antenna provides better MEG compared to the single PIFA. C. Input Characteristics in the Presence of Head and Hand In order to understand the effects of the head and the hand data were computed on the antenna input characteristics for Case II (Fig. 12). The ESPAR antenna model shown in Fig. 1 was used. The 3:1 VSWR bandwidths are in the range of 135 MHz–250 MHz for all four capacitance combinations. pF, pF is narrower than The bandwidth with the free space bandwidth. The bandwidth with pF is about the same as the free space bandwidth. These results are indicate that the effects of the head and hand on the not uniform for different capacitance combinations. This is because of the location of the hand with respect to the position of the ILAs on the ground plane. In the model the hand was

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Fig. 11. Computed total power gain pattern of the ESPAR antenna for Case II: (a) C = 1 pF, C = 1 pF (b) C = 1 pF, C = 20 pF, (c) C = 20 pF, C = 10 pF (d) C = 7 pF, C = 5 pF. The white arrow indicates the direction of the beam maximum. The gain is normalized to 3.7 dBi which is the maximum gain of the four combinations.

Fig. 12. Computed input characteristics of the ESPAR antenna for Case II.

covering (Fig. 12). Thus when pF the reactance is larger than that with pF. This at the base of makes the pF, pF and pF pF combinations far less sensitive to the surrounding hand than the other capacitance combinations. In contrast, the smaller pF or pF causes higher reactance due to currents to be induced in . Hence the hand will have for these cases. Similarly and greater influence on the will affect these antennas if a left handed person holds the device. For practical applications industry engineers will have to select optimum location, spacing and capacitance values to ensure satisfactory input, radiation, and SAR performance.

Fig. 13. (a) A dual band antenna with ESPAR concept applied in the high frequency band, (b) detail dimensions of the driven PIFA.

V. A DUAL BAND ANTENNA WITH HIGH BAND BEAM SCANNING PROPERTY Finally, the ESPAR concept was studied in the context of a dual-band antenna design. Thus a dual-band antenna consisting of 900 and 1900 MHz elements was studied. A generic dualband antenna (investigated in this work) is shown in Fig. 13. For the driven PIFA the metal plate parallel to the outer edge of the ground plane is responsible for operation in the 1900 MHz band while the metal plate containing the meandered element is responsible for operation in the 900 MHz band. The antenna height was changed to 10 mm from 6 mm to satisfy the 880 MHz–960 MHz bandwidth requirement. data of the dual band ESPAR antenna Computed input and do not afare shown in Fig. 14. The capacitances fect the low band performance as expected. The antenna has a 3:1 VSWR bandwidth of 10.4% in the 900 MHz band. The 3:1 VSWR bandwidth in the 1900 MHz band varies between 6.2% and 12%. in the plane at Computed radiation patterns 0.92 GHz and 1.92 GHz are shown in Figs. 15(a) and (b) respectively. Fig. 15(a) also shows the radiation pattern of a dual band antenna which consists only of the driven PIFA shown in and have almost no effect on the Fig. 13(b). It is clear that patterns at 0.92 GHz. The patterns shown in Fig. 15(b) reconfirm the beam scanning capability at 1.92 GHz (the beam maximum to ). The peak gain varies from 1.7 dBi shifts from to 3.2 dBi. Comparing these patterns with those in Fig. 6(b) it is evident that the beam scanning directions remained the same. VI. CONCLUSION The efficacy of the ESPAR concept in mobile terminal antenna design is demonstrated by analyzing an internal antenna

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ACKNOWLEDGMENT The authors would like to thank Dr. C. Penney from Remcom Inc. for his help with the hand model used in this work. REFERENCES

Fig. 14. Computed input characteristics of the dual band antenna of Fig. 13(a) for different capacitive combinations.

Fig. 15. Computed E patterns of the dual band antenna of Fig. 13(a) in the  = 0 plane at (a) 0.92 GHz and (b) 1.92 GHz.

system consisting of a driven PIFA and two parasitic ILAs. A simple transmission line model to determine the effective range of capacitances is presented. Simulation results clearly demonstrate that the radiation pattern of the antenna can be scanned plane by varying the base capacitances of the in the data using varactor diodes two ILAs. Measured return loss demonstrate that the antenna bandwidth is a strong function of the base capacitances. With specific capacitance values bandwidths in excess of 85% is achieved. Full-wave FDTD analysis with the Duke head phantom shows that the ESPAR antenna induces lower SAR than a single PIFA when certain capacitive terminations are chosen. The mean effective gain (MEG) is also found to be higher with the ESPAR antenna than the single PIFA. The ESPAR idea is also validated for a dual band design. The ESPAR antenna system consisting of driven and reactively terminated internal antennas and with its unique radiation and input characteristics is very promising for future portable/wearable wireless devices.

[1] C. A. Balanis, Antenna Theory Analysis and Design. New York: Wiley, 2005, pp. 945–958. [2] J. S. Colbum, Y. R. Samii, M. A. Jensen, and G. J. Pottie, “Evaluation of personal communications dual-antenna handset diversity performance,” IEEE Trans. Ve. Technol., vol. 47, no. 3, pp. 737–746, Aug. 1998. [3] M. G. Douglas, M. Okoniewski, and M. A. Stuchly, “Planar diversity antenna for handheld PCS devices,” IEEE Trans. Veh. Technol., vol. 47, no. 3, pp. 747–754, Aug. 1998. [4] B. M. Green and M. A. Jensen, “Diversity performance of dual-antenna handsets near operator tissue,” IEEE Trans. Antennas Propag., vol. 48, no. 7, pp. 1017–1024, Jul. 2000. [5] S. C. K. Ko and R. D. Murch, “Compact integrated diversity antenna for wireless communications,” IEEE Trans. Antennas Propag., vol. 49, no. 6, pp. 954–960, Jun. 2001. [6] Y. Ding, Z. Du, K. Gong, and Z. Feng, “A novel dual-band printed diversity antenna for mobile terminals,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 2088–2096, Jul. 2007. [7] A. T. Sayem, S. Khan, and M. Ali, “A miniature spiral diversity antenna system with high overall gain coverage and low SAR,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 49–52, 2001. [8] C. B. Dietrich, Jr., K. Dietze, J. R. Nealy, and W. L. Stutzman, “Spatial, polarization, and pattern diversity for wireless handheld terminals,” IEEE Trans. Antennas Propag., vol. 49, no. 9, pp. 1271–1281, Sep. 2001. [9] L. Himmel, S. H. Dodington, and E. G. Parker, “Electronically controlled antenna system,” U.S. Patent 3725938, Feb. 2, 1971. [10] S. H. Black and R. B. Formeister, “Direction finder system,” U.S. Patent 3725938, Apr. 3, 1973. [11] M. Gueguen, “Electronically step-by-step rotated directive radiation beam antenna,” U.S. Patent No. 3846799, Nov. 5, 1974. [12] R. F. Harrington, “Reactively controlled directive arrays,” IEEE Trans. Antennas Propag., vol. 26, no. 3, pp. 390–395, 1978. [13] R. Schlub, J. Lu, and T. Ohira, “Seven-element ground skirt monopole ESPAR antenna design from a genetic algorithm and the finite element method,” IEEE Trans. Antennas Propag., vol. 51, no. 11, pp. 3033–3039, Nov. 2003. [14] J. Lu, D. Ireland, and R. Schlub, “Dielectric embedded ESPAR (DEESPAR) antenna array for wireless communications,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2437–2443, Aug. 2005. [15] R. Schlub and D. V. Thiel, “Switched parasitic antenna on a finite ground plane with conductive sleeve,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1343–1347, May 2004. [16] D. V. Thiel and S. Smith, Switched Parasitic Antennas for Cellular Communications. Boston, MA: Artech House, 2001, pp. 127–134. [17] H. Kawakami and T. Ohira, “Electrically steerable passive array radiator (ESPAR) antennas,” IEEE Antennas Propag. Mag., vol. 47, no. 2, pp. 43–49, Apr. 2005. [18] R. Vaughan, “Switched parasitic elements for antenna diversity,” IEEE Trans. Antennas Propag., vol. 47, no. 2, pp. 399–405, Feb. 1999. [19] R. Islam and M. Ali, “A multi-element enhanced bandwidth PIFA for beam steering in a mobile phone at 1900 MHz,” in IEEE Antennas Propag. Society Int. Symp., Charleston, SC, Jun. 2009. [20] Ansoft Corporation [Online]. Available: www.ansoft.com [21] Remcom Inc. [Online]. Available: www.remcom.com [22] D. M. Pozar, Microwave Engineering. New York: Wiley, 2005, p. 145. [23] L. Petit, L. Dussopt, and J. Laheurte, “MEMS-switched parasitic-antenna array for radiation pattern diversity,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2624–2631, Sep. 2006. [24] Skyworks Solutions, Inc. [Online]. Available: www.skyworksinc.com [25] M. Ali, G. J. Hayes, H.-S. Hwang, and R. A. Sadler, “Design of a multi-band internal antenna for third generation mobile phone handsets,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1452–1461, Jul. 2003. [26] M. Ali, M. Okoniewski, M. A. Stuchly, and S. S. Stuchly, “Dual-frequency strip-sleeve monopole for laptop computers,” IEEE Trans. Antennas Propag., vol. 47, no. 2, pp. 317–323, Feb. 1999. [27] IT’IS Foundation [Online]. Available: http://www.itis.ethz.ch

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[28] Italian National Research Council [Online]. Available: http://niremf. ifac.cnr.it [29] Federal Communications Commission [Online]. Available: http://www.fcc.gov [30] U.S. National Library of Medicine [Online]. Available: http://www. nlm.nih.gov [31] O. Kivekas, J. Ollikainien, T. Lehtiniemi, and P. Vainikainen, “Bandwidth, SAR, and efficiency of internal mobile phone antennas,” IEEE Trans. EM Compat., vol. 46, no. 1, pp. 71–86, Feb. 2004. [32] J. T. Rowley and R. B. Waterhouse, “Performance of shorted microstrip patch antennas for mobile communications handsets at 1800 MHz,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 815–822, May 1999. [33] M. Karaboikis, C. Soras, G. Tsachtsiris, and V. Makios, “Compact dual-printed inverted-F antenna diversity systems for portable wireless devices,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 9–14, 2004. Md. Rashidul Islam (S’10) received the B.Sc. degree in electrical engineering from the Bangladesh University of Engineering and Technology, Dhaka, in 2004 and the M.E. degree in electrical engineering from the University of South Carolina, Columbia, in 2010. From February 2004 to July 2007, he worked as a microwave transmission Network Designer at Telekom Malaysia International, Bangladesh. Since 2007, he has been a Graduate Research Assistant at the Microwave Engineering Laboratory in the University of South Carolina. His research interests include antenna design for handheld terminals, ESPAR antenna, and electromagnetic interaction of portable wireless devices with human body. He is the author/coauthor of several conference papers.

Mohammod Ali (M’93–SM’03) received the B.Sc. degree in electrical and electronic engineering from the Bangladesh University of Engineering and Technology, Dhaka, in 1987, and the M.A.Sc. and Ph.D. degrees, both in electrical engineering, from the University of Victoria, Victoria, BC, Canada, in 1994 and 1997, respectively. He was with the Bangladesh Institute of Technology, Chittagong, from 1988 to 1992. From January 1998 to August 2001, he was with Ericsson Inc., Research Triangle Park, NC. Since August 2001, he has been with the Department of Electrical Engineering, University of South Carolina, Columbia, where currently he is an Associate Professor. He had also held appointments as a Visiting Research Scientist with the Motorola Corporate EME Research Laboratory, Plantation, FL, during June to August 2004. He established the Microwave Engineering Laboratory at the University of South Carolina in 2001. He is the author/coauthor of over 115 publications and 5 granted US patents. His research interests include miniaturized packaged (embedded) antennas, metamaterials and their antenna applications, distributed wireless sensors, and portable/wearable antennas and their interactions with humans (SAR). Dr. Ali is the recipient of the 2003 National Science Foundation Faculty Career Award. He is also the recipient of the College of Engineering and Information Technology Young Investigator Award and the Research Progress Award from the University of South Carolina in 2006 and in 2009 respectively. He was the Technical program Co-Chair of the IEEE Antennas and Propagation Society’s International Symposium in Charleston, SC in 2009. He has also served as a member of the Technical Program Committee for the IEEE Antennas and Propagation Society’s International Symposium for a number of years. He has served as a reviewer and panelist for grant proposals for a number of federal and local funding agencies. He is an Associate Editor for the journal IEEE Antennas and Wireless Propagation Letters.

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Depth and Rate of Fading on Fixed Wireless Channels Between 200 MHz and 2 GHz in Suburban Macrocell Environments Kyle N. Sivertsen, Student Member, IEEE, Anthony Liou, and David G. Michelson, Senior Member, IEEE

Abstract—Various bands between 200 MHz and 2 GHz have recently been reallocated to multipoint fixed wireless services. The links in such systems are usually obstructed by buildings and foliage and are susceptible to fading caused by windblown trees and foliage. To date, there have been relatively few efforts to characterize either the depth of fading in bands below 1.9 GHz or the rate of fading in any of these bands. We transmitted CW signals in the 220, 850 and 1900 MHz bands from a transmitter located 80 m above ground level in a typical suburban macrocell environment and collected time-series of received signal strength at distances between 1 and 4 km from the site. We reduced the data to show how the depth and rate of fading depend on the frequency band, time-averaged wind speed and distance in such an environment. Our most significant finding is that the rate of signal fading is very similar in all three bands. In particular, it is not proportional to carrier frequency, as a simplistic model involving moving scatterers might suggest. These results will provide useful guidance to those who seek to simulate, or develop detailed physical models of, fade dynamics in such environments. Index Terms—Channel model, fading channels, macrocell environment, radiowave propagation, radiowave propagation-meteorological factors.

I. INTRODUCTION

I

N recent years, as: (i) common carriers seek methods for providing either fixed or nomadic network access services to residential households without the expense of deploying wireline connectivity over the last mile [1], [2] and (ii) public utilities seek methods that will allow them to: (a) detect and report outages, (b) monitor usage, and (c) implement strategies that encourage customers to limit consumption and adopt sustainable practices [3]–[5], the possibility of deploying fixed wireless multipoint communication systems in suburban macrocell environments has attracted considerable interest. In order to provide developers with the insights required to design effective sysManuscript received April 16, 2009; revised February 27, 2010; accepted March 31, 2010. Date of publication July 08, 2010; date of current version October 06, 2010. This work was supported in part by grants from Bell Canada, British Columbia Hydro and Power Authority, Tantalus Systems and Western Economic Diversification Canada. K. N. Sivertsen and D. G. Michelson are with the Radio Science Laboratory, Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail: [email protected]; [email protected]). A. Liou was with Radio Science Laboratory, Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada. He is now with the Universal Scientific Industrial Company, Taiwan (e-mail: [email protected]) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055783

tems, several groups in Canada, the United States, the United Kingdom, Chile, Australia and elsewhere have conducted measurement campaigns which have aimed to characterize the depth of signal fading observed in such environments, e.g., [6]–[13]. In macrocell environments, the base station antenna is mounted well above the local rooftop or treetop level and the remote terminal antenna is mounted below the local rooftop or treetop level. As a result, the wireless links are usually obstructed by intervening obstacles and a large fraction of the signal that reaches the receiver does so as a result of scattering and diffraction by objects in the environment. Because both the transmitting and receiving antennas in such applications are fixed, signal fading is caused solely by the motion of objects in the environment that scatter and diffract the signal. In suburban macrocell environments, a large fraction of those objects are trees and foliage with leaves and branches that sway when blown by the wind. The vast majority of previous studies of fixed wireless channels in suburban macrocell environments focused on individual frequency bands at 1.9 GHz and above, including the PCS band at 1.9 GHz, the ISM band at 2.45 GHz, the Fixed Wireless Access (FWA) band at 3.5 GHz and the U-NII and ISM bands at 5.2 and 5.8 GHz. However, spectrum regulators have recently begun to reallocate frequency bands below 2 GHz in order to help meet the requirements for broadband wireless access for urban and rural areas and/or narrowband telemetry for public utilities. In Canada, spectrum in frequency bands near 700 MHz has been proposed for fixed wireless broadband use in rural areas [18] and may find application in distribution automation by the electrical power industry. Frequency bands such as 220–222 MHz, 1429.5–1432 MHz and 1800–1830 MHz have recently been designated for utility telemetry and distribution automation [19]. Regulators are increasingly designating multiple primary allocations within individual frequency bands, as well as proposing more flexible licensing schemes, in an attempt to accommodate different users and services in the same spectrum. Both the amount of radio spectrum, and the choice of frequency bands available for fixed wireless use, will almost certainly increase in coming years. The manner in which path loss, or its reciprocal, path gain, is affected by the carrier frequency, the heights of and separation between the base station and mobile terminal in suburban macrocell environments over the range from 200 MHz to 2 GHz has been well-studied over the years and has been captured by several standard models [20]–[22]. However, existing channel models do not provide a description of either the

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depth or rate of signal fading that fixed wireless channels will experience over this frequency range in suburban macrocell environments. Although previous efforts to characterize the fade dynamics of propagation through vegetation provided useful insights, the amount of data collected was limited [23]–[25].This lack of information places those charged with planning, simulating or deploying fixed wireless systems in suburban macrocell environments at a severe disadvantage when asked to predict the performance of data link protocols (including handshaking schemes) and opportunistic schedulers that attempt to synchronize transmission with favourable channel conditions. Here, we take the first steps to determine how both the depth and rate of fading on fixed wireless channels in a typical suburban macrocell environment vary with carrier frequency, wind speed and distance across the frequency range from 200 MHz to 2 GHz. We established a transmitting site atop an 18-story office tower located in the middle of a large suburban area. We simultaneously broadcast single carrier signals in the 220, 850 and 1900 MHz bands and collected time-series of the received signal strength observed in each band at fixed locations at ranges between 1 and 4 km. The frequencies that we employed bracket the majority of the bands that have been allocated to fixed wireless access and SCADA (supervisory control and data acquisition) applications. Although our results strictly apply to narrowband channels, they are also relevant to single carriers in multicarrier modulation schemes. The remainder of this paper is organized as follows: In Section II, we discuss common representations of fading on fixed wireless channels. In Section III, we summarize the essential aspects of our second-order model of fading on narrowband channels. In Section IV, we describe our measurement setup and test site. In Section V, we present our results and suggest how these results can be used in system-level simulations. In Section VI, we summarize our findings and contributions and discuss the implications of our results.

, and is much less efthe Ricean -factor is high, e.g., is often fective in suburban macrocell environments where . In [13], it was found that the measured level crossing rate (LCR) and/or average fade duration (AFD) distributions seen on fixed wireless links can be fitted to expressions that are normally justified only for mobile wireless links. This allows one to express the time variation on the link in terms of just three parameters: the mean signal strength, the Ricean -factor, and an effective maximum Doppler frequency which is referred to in [13] and which we will simply refer to as . The as details are described in Section III. III. A SECOND-ORDER FADING CHANNEL MODEL , If the complex envelope of the time-varying path gain, experienced by either a mobile or fixed links is given by the sum of a fixed component and a zero-mean complex Gaussian and , the first-order statistics of will process follow a Ricean distribution where

(1) is the average envelope power and is the zero order modified Bessel function of the first kind. In such cases, the Ricean -factor is given by (2) where is the power in the time-varying component. Various methods for estimating have been proposed. Here, we use the moment-based method described in [26] where (3)

II. SIGNAL FADING ON FIXED WIRELESS CHANNELS Because fading on fixed wireless channels in macrocell environments normally follows a Ricean distribution, the depth of fading is typically expressed in terms of the Ricean -factor. The rate at which signal fading occurs may be characterized either by the level crossing rate (LCR) and average fade duration (AFD) at selected thresholds above and below the mean signal strength [13] (which depend on both the first- and second-order statistics of the fading signal) or by a Doppler power spectrum [14], [15] (which depends only upon the second-order statistics). Although the latter representation is particularly useful because it is a key input for algorithms used to simulate (or emulate) fading channels, e.g., [16], [17], estimation of the Doppler power spectrum from measured data generally requires coherent time series data (amplitude and phase). Fading on fixed wireless links occurs so slowly, however, that lack of phase coherence between the local oscillators in the widely separated transmitter and receiver can severely distort the measurement. Although a method for estimating the Doppler spectrum from amplitude-only measurement data was proposed in [15], it is mainly intended for use on short-range line-of-sight paths where

and is the rms fluctuation of the envelope about , i.e., the . standard deviation of In cases where the base station is fixed, the terminal is in motion, and scattering is two-dimensional and isotropic, the Doppler spectrum of the time-varying component is given to by Clarke’s U-shaped spectrum and ranges from . The frequency offset of the carrier that corresponds to the fixed component is determined by the direction of the propagation path relative to the velocity of the terminal. In such cases, the LCR and AFD are given by

(4) and

(5)

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where is the threshold voltage, is the threshold is the Marcum-Q function normalized to the rms envelope, [13]. and, in this case, corresponds to In cases where both the base station and the terminal are fixed, time variation is entirely due to the motion of scatterers in the environment and the corresponding Doppler spectrum generally exhibits a sharp peak at the carrier frequency and rapidly decays as the frequency offset increases, e.g., [14]. In [13], it was shown that for cases where the time derivative of the envelope is independent of , the expressions for LCR and AFD given in (4) and (5) do not depend on the shape of the Doppler spectrum. estimated using (3) to In particular, applying the value of the expressions for LCR and AFD given by (4) and (5), and will often provide a good choosing an appropriate value for approximation to the LCR and AFD characteristics observed on fixed wireless links. Further, it was reported that a good estimate of can often be obtained by considering only the zero crossing , i.e., rate (ZCR), which is defined as the value of LCR for

(6) , this expression is virtually insensitive to the actual For value of , yielding the convenient approximation (7) is now less clear given that it no longer The significance of applies to the maximum frequency component of Clarke’s U-shaped spectrum. In Section IV-B, we recount a possible interpretation of the physical significance of . In the sections that follow, we describe our efforts to characterize the depth and rate of fading experienced over fixed wireless links across a broad frequency range from 200 MHz to 2 GHz in a typical suburban macrocell environment.

Fig. 1. (a) The tri-band transmitter that was deployed at the base station and (b) the tri-band receiver that was carried aboard the propagation measurement van.

TABLE I LINK BUDGET PARAMETERS FOR THE TRI-BAND CHANNEL SOUNDER

IV. THE MEASUREMENT SETUP A. Tri-band Channel Sounder Our tri-band channel sounder consists of three continuous wave (CW) transmitters and three corresponding receivers that operate in the 220, 850 and 1900 MHz frequency bands. A block diagram of the CW transmitter is shown in Fig. 1(a). The signal source portion of the transmitter contains a pair of Marconi 2022 RF signal generators, each of which is capable of supplying a CW signal up to 6 dBm over the range 10 kHz to 1 GHz, and a Marconi 2031 RF signal generator capable of supplying a CW signal up to 13 dBm over the range 10 kHz to 2.7 GHz. The signal generators are locked to a 10 MHz reference signal supplied by a Stanford Research Systems PRS10 Rubidium frequency standard. It, in turn, is disciplined by the 1 pulse per second (PPS) signal supplied by a Trimble Resolution-T GPS receiver that has been designed for such applications. The amplifier portion contains three power amplifiers: (i) a TPL Communications LMS series RF power amplifier capable of delivering between 20 and 100 W at 220 MHz, (ii) a Unity Wireless Dragon RF power amplifier capable of delivering up to 30 W between 869 and 894 MHz and (iii) a Unity Wireless Grizzly RF power amplifier capable of delivering up to 35 W

between 1930 and 1990 MHz. During data collection, all three amplifiers were configured to deliver 20 W signals to their respective feedlines. A wireless remote control device that operates near 150 MHz allowed the data collection team to remotely enable or disable the power amplifiers at the start or end of a measurement session. The 220, 850 and 1900 MHz transmitting antennas are omnidirectional and have gains of 8.1, 6.1 and 5.0 dBi, respectively. The remaining parameters used in the system link budget for each band are given in Table I. A block diagram of our multiband receiver is shown in Fig. 1(b). The receiving antennas are omnidirectional and all have the same nominal gain of 1 dBi. When used in NLOS configurations, fixed wireless antennas are typically mounted at heights between 0.5 m (e.g., for nomadic applications) and 4 m (e.g., for permanent installations). As a compromise, we

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mounted the antennas on the roof of our propagation measurement van at a height of 2.3 m. The multiband receiver consists of: (i) a pair of Anritsu MS2651B spectrum analyzers that operate over the range from 9 kHz to 3 GHz with a selectable IF bandwidth, (ii) an Anritsu MS2721A spectrum analyzer that operates over the range from 100 kHz to 7.1 GHz with a selectable IF bandwidth, (iii) a Stanford Research Systems PRS10 Rubidium frequency standard that generates a 10 MHz reference signal to which the spectrum analyzers can be locked and (iv) a Trimble Resolution-T GPS receiver that supplies the 1 PPS signal used to discipline the frequency standard. External low-noise pre-amplifiers with 30 dB and 26 dB gain were used to increase the sensitivity of the spectrum analyzers that measure the received strength of the 850 and 1900 MHz signals, respectively. We used a laptop computer equipped with a GPIB adapter to: (i) configure the spectrum analyzers and (ii) collect data from them. We geocoded the data with a nominal circular error probability (CEP) of less than 5 metres using location information supplied by a u-blox Antaris 4 SuperSense GPS receiver. B. Verification Protocol Before we collected any field data, we verified the function and operation of our tri-band CW channel sounder using a Spirent SR5500 channel emulator. We set the relevant narrowband channel parameters, including path gain and Ricean -factor, to various values over a broad range and, in each case, confirmed that we were able to correctly estimate each of the parameters. We verified the transmitted power levels using a Bird Model 5000EX digital wattmeter. C. Weather Instruments We measured the wind speed, wind direction, rain rate and outdoor temperature using a Davis Vantage Pro 2 wireless weather station that we mounted on a mast located about 30 metres away from the transmitting antennas. Internally, the weather station samples the relevant weather parameters every few seconds. Once per minute, it logs the average values of these parameters over the previous minute to an internal database. We used a custom software tool to match the received signal strength time series collected at a given location to the relevant weather data. Because previous work has shown that variations in average wind speed at tree top level or above are well correlated over mesoscale distances of several kilometers [27], we concluded that collecting wind data at a single location near the base station would be adequate for our purposes. D. Test Area Our transmitting antennas were installed atop the 18-story office tower at BC Hydro’s Edmonds facility in Burnaby, BC at a height of 80 m above ground level. The test area consisted of suburban neighborhoods with generally flat terrain, light to moderate foliage and one- and two-story houses. We collected measurement data at 92 fixed measurement locations that were situated within an annular sector between 1 and 4 km from the transmitter site. Almost all the motion in the environment arose from windblown foliage; few, if any, cars, people or other moving scatterers were in the vicinity of the receiver when we

collected measurement data. Most of the foliage in the area is deciduous and between 4 and 7 m in height but at least one-third is coniferous and up to 15 m in height. E. Scope and Limitations Due to the nature of our measurement setup, our results apply strictly to suburban macrocell environments with high transmitting sites and moderate foliage. Development of a broadly applicable model will require additional data collected at other sites with transmitters at other heights. The duration of the measurement campaign was too short to permit observation of the effects of seasonal variations in the foliage. All of our data was collected with leaves on the trees. In many fixed wireless deployments, the terminal antennas are directional. Because our primary objective is to compare the behavior of the channel at different frequencies, we elected to simplify the data collection protocol by collecting the measurement data using omnidirectional antennas. If the remote terminal antenna’s beamwidth decreased or its height increases, previous work suggests that the path gain and/or the Ricean -factor will also tend to increase [7]. F. Data Collection Protocol Our data collection protocol comprised the following steps. First, we conducted a rapid survey of the proposed measurement locations in order to ensure that the strength of the received signal would be adequate at all locations. Next, over a span of several days, the operator drove the propagation measurement van to each of the fixed measurement locations that we had selected in advance. At each location, the operator collected simultaneous time series of the received strength of the 220, 850 and 1900 MHz CW signals. The measured data were collected in the form of fifteen successive 24-second sweeps. For the two higher bands, the pair of Anritsu MS2651B spectrum analyzers were used to record fifteen sweeps of 501 samples each, yielding 7515 received signal strength samples at each location and a sampling rate of 20.9 samples/sec. For the 220 MHz band, the Anritsu MS2721A spectrum analyzer was used; it yielded 551 samples per sweep or 8265 samples at each location and a sampling rate of 23.0 samples/sec. The sampling rates were chosen to be far greater than the anecdotal estimates of the maximum observed Doppler frequency reported previously, e.g., [28], [29]. As reported in the next section, our estimates of the effective maximum Doppler frequency, which is always less than the maximum observed Doppler frequency, were all significantly lower than 10 Hz. V. RESULTS A. Estimation of the Effective Maximum Doppler Frequency We processed the time series data that we collected at 92 locations as follows: First, we estimated using (3) and the zero-crossing rate ZCR, which is defined as the value of LCR for . This corresponds to the case where the threshold is equal to the mean value of the fading envelope. If , we estimated using (7). Oththe effective maximum Doppler frequency erwise, we estimated using (6). We assessed the accuracy of the results by substituting our estimates of and into (4) and

SIVERTSEN et al.: DEPTH AND RATE OF FADING ON FIXED WIRELESS CHANNELS BETWEEN 200 MHz AND 2 GHz

Fig. 2. A good fit between theoretical and measured fading distributions for: (a) Measured time series, (b) average fade duration (AFD), (c) level crossing rate (LCR), where  is the threshold normalized to the rms envelope.

(5) to yield the theoretical LCR and AFD distributions, respectively, and then superimposing them on the corresponding LCR and AFD distributions obtained by directly processing the time series. This allowed us to determine how transient signal fading, transient signal enhancement and non-stationary channel behavior affect the performance of the estimator, an issue not considered in [13]. An example where the theoretical and experimental AFD and LCR distributions are a close match is given in Fig. 2. Reduction of time series data collected in the 850 MHz band at a disand tance of 1555 m from the base station yielded . Inspection of the time series suggests that both the depth and rate of fading is consistent across the 6-minute duration of the observation. We conclude that the model given by (4) applies. A counterexample where the theoretical and experimental AFD and LCR curves do not match particularly well is given in Fig. 3. Reduction of time series collected in the 220 and MHz band at a distance of 3170 m yielded . However, inspection of the time series reveals that the depth and rate of fading are not consistent across the duration of the observation. Instead, the signal is virtually flat for the first 100 seconds (with the exception of a brief fade and seconds) then begins to experience rapid enhancement at and consistent scintillation during the remainder of the observation. We interpret this as a transition between two channel states. We produced plots of individual time series and the corresponding AFD and LCR distributions similar to those presented in Figs. 2 and 3 for all of our measurement locations. Deviations from stationary Ricean fading were identified by discrepancies between the empirical and theoretical AFD and LCR curves exist due to changes in channel state during the observation period. These deviations are quite obvious and easily discernible by visual inspection of both the AFD and LCR curves and the original RSSI time series data. The results are summarized in Table II. In the vast majority of cases (69% at 220 MHz, 75% at 850 MHz, and 85% at 1900 MHz), the depth and rate of fading

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Fig. 3. A poor fit between theoretical and measured fading distributions for: (a) Measured time series, (b) average fade duration (AFD), (c) level crossing rate (LCR), where  is the threshold normalized to the rms envelope.

TABLE II DATA QUALITY SUMMARY IN PERCENTAGES

in the time series were consistent across the duration of the observation and the theoretical and experimental curves matched well. Transient signal enhancement, possibly due to reflections from passing vehicles, was the most common impairment. Slow fading superimposed upon an otherwise consistent fading signal was the next most common impairment. Neither of these was observed to be dependent on distance. Slow fading tended to occur more often when the channel experienced high values of . This suggests that the slow fading was the direct result of fading of the fixed component of the signal. In both cases, the experimental AFD curves were far more affected by fading and enhancement of the signal and deviated far more from their theoretical counterparts than did the experimental LCR curves. Between 4 and 9% of the time series in each band displayed either single or multiple transitions between channel states. In such cases, even the experimental LCR curves tended to deviate significantly from their theoretical counterparts. Because the parameters estimated from such time series would not be meaningful, we did not process them further. B. Significance of the Equivalent Maximum Doppler Frequency If the remote terminal is in motion and scattering is two-dimensional and isotropic, the Doppler spectrum of the fading signal follows Clarke’s model and in (4) and (5) is given by (8)

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where . If the scattering is non-isotropic and/or the terminal is not in motion, the shape of the Doppler spectrum will be quite different. During the calibration and validation protocol described in Section IV-B, we determined the value of that applies to various Doppler spectrum shapes. We found that as the fraction of energy in the high frequency portion of the spectrum decreases, so does . In particular, the 6-dB classic, flat and rounded spectra described in [30] yielded , 0.74 and 0.58, respectively. Further work will be required to determine the corresponding relationship for spectra more typical of those observed in fixed wireless environments, e.g., [14], [15]. C. Joint-Distribution of Equivalent Maximum Doppler Frequencies Over the 92 measurement locations and in all three frequency bands, the effective maximum Doppler frequency distributions are well approximated by lognormal distributions (i.e., normal in dBHz). Therefore, these effective maximum Doppler frequency values at 220, 850, and 1900 MHz bands can be cast as a three-element vector of jointly random Gaussian processes which are completely specified by the means, standard variations, and mutual correlation coefficients. The mean values of the effective maximum Doppler frequency at 220, 850, and 1900 MHz bands are 1.62, 2.46, and 0.34 dBHz (or 1.45, 1.76, and 1.08 Hz, respectively.) The standard deviations of the effective maximum Doppler frequency in these bands are 2.03, 2.99 and 2.87 dBHz, respectively. The correlation matrix between the Doppler frequencies observed in these bands is given by (9) where the rows and columns correspond to the bands in the sequence given above. It is apparent that the marginal distributions of the effective maximum Doppler frequencies are very similar among the three frequency bands. In particular, the rate of signal fading is not proportional to carrier frequency, as a simplistic model involving moving scatterers might suggest, e.g., [14]. This constraint will provide useful guidance to those who seek to develop detailed physical models of fade dynamics on fixed wireless channels in suburban macrocell environments. D. Ricean -factor and Equivalent Maximum Doppler Frequency vs. Average Wind Speed From previous work, it is well known that the Ricean -factor drops as the average wind speed increases. However, the corresponding relationship between the effective maximum Doppler frequency and the average wind speed, and the effect and of carrier frequency on the relationship between and the average wind speed has not been previously revealed. and vs. the average wind speed in the Our results for 220, 850 and 1900 MHz bands are presented in Figs. 4 and 5 respectively. We estimated the regression line that best fits our measured data, the correlation coefficient between each parameter and the average wind speed, and the location variability of the parameter, i.e., the variation of the parameter about the regression line

K

Fig. 4. Ricean -factors observed at (a) 220 MHz, (b) 850 MHz, (c) 1900 MHz vs. average wind speed.

at a given average wind speed. A regression line is the simplest model and, in the absence of a clear indication to the contrary, is a reasonable first choice when evaluating the relationship between two parameters. For completeness, we also evaluated the goodness of fit of a quadratic polynomial in each case but did not observe any improvement. The regression line for and , and the corresponding correlation coefficients and location variabilities in each frequency band are given by

(10)

(11)

(12)

SIVERTSEN et al.: DEPTH AND RATE OF FADING ON FIXED WIRELESS CHANNELS BETWEEN 200 MHz AND 2 GHz

Fig. 5. Effective maximum Doppler frequency observed at (a) 220 MHz, (b) 850 MHz, (c) 1900 MHz vs. average wind speed.

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Fig. 6. Ricean -factors observed at (a) 220 MHz, (b) 850 MHz, (c) 1900 MHz vs. distance.

E. Ricean -factor and Equivalent Maximum Doppler Frequency vs. Distance

and

(13)

(14)

(15) , is expressed respectively, where the average wind speed, and are weakly but negatively in km/h. In general, both correlated with the average wind speed in all three bands. Here, we say that the correlation is weak if the mean value of is occurs less than 0.3. We say that no correlation exists if in the interval within one standard deviation from the mean of and the average wind speed are rho. In the 220 MHz band, effectively uncorrelated.

From previous work, it is well known that the Ricean -factor tends to present a slight negative correlation with distance. However, the corresponding relationship between the effective maximum Doppler frequency and distance, and the and effect of carrier frequency on the relationship between and distance has not been previously revealed. Our results for and vs. distance in the 220, 850 and 1900 MHz bands are presented in Figs. 6 and 7 respectively. We estimated the regression line that best fits our measured data, the correlation coefficient between each parameter and the distance, and the location variability of the parameter, i.e., the variation of the parameter about the regression line at a and and the corgiven distance. The regression line for responding correlation coefficients and location variabilities in each frequency band are given by

(16)

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Fig. 7. Effective maximum Doppler frequencies observed at (a) 220 MHz, (b) 850 MHz, (c) 1900 MHz vs. distance.

K

Fig. 8. Ricean -factor vs. effective maximum Doppler frequency observed at (a) 220 MHz, (b) 850 MHz, (c) 1900 MHz.

F. Joint Dependency of the Ricean Maximum Doppler Frequency (17)

(18) and

-factor and Equivalent

We found that the Ricean -factor (in dB) and the effective maximum Doppler frequency (in dBHz) both present normal distributions. This suggests that the two may be cast as jointly Gaussian random variables with specified mean, standard deviation and mutual correlation coefficient. Scatter plots of and in the 220, 850 and 1900 MHz bands are presented in Fig. 8 together with the corresponding regression lines and correlation coefficients given by (22) (23) (24)

(19)

(20)

that best fit the data in a least-squares sense. The mean and stan(in dB) in the 220, 850 and 1900 MHz dard deviations of bands are given by (25) (26)

(21) respectively, where distance, , is expressed in km. In general, are correlated with distance. neither nor

The corresponding mean and standard deviations of in (19)–(21).

are given

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G. Effect of Transmitter Height Although our measurement setup did not permit direct evaluation of the effect of transmitter height, physical reasoning suggests that as the transmitter height decreases, we can expect to see lower values of due to a weaker direct signal and greater interaction with vegetation (i.e., more scattering). However, we to change because, although the magnitude do not expect of the fixed component is expected to increase, the physical process that leads to time variation does not change. Verification of these predictions is a task for future work. H. Physical Interpretation The results presented here tend to support a physical model proposed in [23] in which the vegetation mass may be considered as a diffraction aperture with a random aperture pattern. Although the objective of that work was to determine how changes in the random aperture due to wind blowing through leaves and branches affects the spatial distribution of fading at some distance beyond the vegetation mass, it can also be used to predict the effect of carrier frequency on both the depth and rate of fading. In particular, the model proposed in [23] correctly predicts that fading will be more severe at higher frequencies, but the rate of fading is strictly a function of the rate at which the random apertures open and close and not be dependent on the carrier frequency. Moreover, the results presented here suggest that the assumptions upon which the model is based apply well below 1 GHz. Detailed comparison of the model to measurement is a topic for future work.

erage wind speed and is lognormally distributed about its mean value, which typically falls between 1 and 2 Hz. Although the lognormal distribution suggests that the randomness is the result of a multiplicative process, determining the precise details is a task for future work. The results presented here will provide useful guidance to those who seek to: (i) simulate channels encountered in suburban macrocell environments with high transmitting sites and moderate foliage or (ii) develop detailed physical models of propagation in such environments. Although our test site is typical of suburban neighborhoods with light to moderate foliage, other sites that are not as homogenous or that have more or less vegetation may yield slightly different results. We believe it is unlikely that observations of the depth and rate of channel fading at other sites will show large deviations from the trends presented here, however, development of measurement-based models applicable to a broad range of environments will require additional data collected: (i) at other sites, (ii) with transmitters at different heights and (iii) at additional frequencies within the range of interest. ACKNOWLEDGMENT The authors thank BC Hydro Telecom Services for providing them with access to the radio room and rooftop facilities atop Edmonds tower that were used at their transmitting site. They also thank UBC Student Housing and Conferences for providing them with access to the Walter Gage Residence, East Tower, during their equipment development and validation runs.

VI. CONCLUSION Our results corroborate Feick et al.’s observation [13] that even though the fixed Doppler spectrum assumes a much different shape than it does in mobility scenarios, substituting an appropriate value for what would normally be the maximum Doppler frequency (and which we refer to here as the effective maximum Doppler frequency) into the theoretical expressions for the LCR and AFD distributions often yields a good match to the fixed wireless observations. Further, we have shown how transient peaks, fades, or nonstationary behavior in the fading signal affect the fit of the measured LCR and AFD curves to their theoretical counterparts and have provided convincing evidence that fitting the theoretical LCR curve to the measured curve provides the most robust and reliable results. Finally, we recount preliminary results that suggest that the ratio of the effective maximum Doppler frequency to the maximum Doppler frequency: (i) is determined by the shape of the Doppler spectrum and (ii) decreases as the fraction of energy in the high frequency components of the Doppler spectrum decreases. Our most significant finding is that the effective maximum Doppler frequency observed at a given location is not proportional to the carrier frequency as: (i) a model based upon the radial motion of moving scatterers would predict and (ii) what others have observed in conventional indoor and mobility environments. This suggests that the random aperture model proposed in [23] is correct and is valid at frequencies below 1 GHz. Further, we found that the effective maximum Doppler frequency is effectively independent of either distance or av-

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REFERENCES [1] W. Webb, “Broadband fixed wireless access as a key component of the future integrated communications environment,” IEEE Commun. Mag., vol. 39, no. 9, pp. 115–121, Sep. 2001. [2] K. Lu, Y. Qian, and H.-H. Chen, “Wireless broadband access: WIMAX and beyond,” IEEE Commun. Mag., vol. 45, no. 5, pp. 124–130, May 2007. [3] S. S. Venkata, A. Pahwa, R. E. Brown, and R. D. Christie, “What future distribution engineers need to learn,” IEEE Trans. Power Syst., vol. 19, pp. 17–23, Feb. 2004. [4] G. Simard and D. Chartrand, “Hydro-Quebec’s economic and technical approach to justify its distribution automation program,” in Proc. IEEE PES’07, Jun. 24–28, 2007, pp. 1–5. [5] S. Jim, W. Carr, and S. E. Collier, “Real time distribution analysis for electric utilities,” in Proc. IEEE REPC’08, Apr. 27–29, 2008, pp. B5–B5. [6] D. G. Michelson, V. Erceg, and L. J. Greenstein, “Modeling diversity reception over narrowband fixed wireless channels,” in Proc. IEEE MTT-TWA’99, Feb. 21–24, 1999, pp. 95–100. [7] L. J. Greenstein, S. S. Ghassemzadeh, V. Erceg, and D. G. Michelson, “Ricean -factors in narrowband fixed wireless channels: Theory, experiments and statistical models,” in Proc. WPMC’99, Sep. 21–23, 1999. [8] S. Perras and L. Bouchard, “Fading characteristics of RF signals due to foliage in frequency bands from 2 to 60 GHz,” in Proc. WPMC’02, Oct. 27–30, 2002, pp. 267–271. [9] M. J. Gans, N. Amitay, Y. S. Yeh, T. C. Damen, R. A. Valenzuela, C. Cheon, and J. Lee, “Propagation measurements for fixed wireless loops (FWL) in a suburban region with foliage and terrain blockages,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 302–310, Apr. 2002. [10] E. R. Pelet, J. E. Salt, and G. Wells, “Effect of wind on foliage obstructed line-of-sight channel at 2.5 GHz,” IEEE Trans. Broadcasting, vol. 50, no. 3, pp. 224–232, 2004. [11] D. Crosby, V. S. Abhayawardhana, I. J. Wassell, M. G. Brown, and M. P. Sellars, “Time variability of the foliated fixed wireless access channel at 3.5 GHz,” in Proc. IEEE VTC 2005 Spring, Sep. 25–28, 2005, pp. 106–110.

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[12] H. Suzuki, C. D. Wilson, and K. Ziri-Castro, “Time variation characteristics of wireless broadband channel in urban area,” in EuCAP’06, Nice, France, Nov. 2–6, 2006. [13] R. Feick, R. A. Valenzuela, and L. Ahumada, “Experiment results on the level crossing rate and average fade duration for urban fixed wireless channels,” IEEE Trans. Commun., vol. 9, pp. 175–179, Jan. 2007. [14] S. Thoen, L. V. D. Perre, and M. Engels, “Modeling the channel timevariance for fixed wireless communications,” IEEE Commun. Lett., vol. 6, no. 8, pp. 331–333, Aug. 2002. [15] A. Domazetovic, L. J. Greenstein, N. B. Mandayam, and I. Seskar, “Estimating the Doppler spectrum of a short-ranged fixed wireless channel,” IEEE Commun. Lett., vol. 7, no. 5, pp. 227–229, May 2003. [16] K. E. Baddour and N. C. Beaulieu, “Accurate simulation of multiple cross-correlated Rician fading channels,” IEEE Trans. Commun., vol. 52, pp. 1980–1987, Nov. 2004. [17] B. Natarajan, C. R. Nassar, and V. Chandrasekhar, “Generation of correlated Rayleigh fading envelopes for spread spectrum applications,” IEEE Communic. Lett., vol. 4, no. 1, pp. 9–11, Jan. 2000. [18] “Policy for the Use of 700 MHz Systems for Public Safety Applications and Other Limited Use of Broadcasting Spectrum,” Industry Canada Radio Systems Policy, 2006, RP-006—Issue 1. [19] “Proposals and Changes to the Spectrum in Certain Bands Below 1.7 GHz,” Industry Canada Gazette Notice DGTP-004–05, 2005. [20] M. Hata, “Empirical formula for propagation loss in land mobile radio services,” IEEE Trans. Veh. Technol., vol. 29, pp. 317–325, Aug. 1980. [21] E. Damosso, Digital Mobile Radio Toward Future Generation Systems—Final Report 1996. [22] V. Erceg, L. J. Greenstein, S. Y. Tjandra, S. R. Parkoff, A. Gupta, B. Kulic, A. A. Julius, and R. Bianchi, “An empirically based path loss model for wireless channels in suburban environments,” IEEE J. Sel. Areas Commun., vol. 17, no. 7, pp. 1205–1211, Jul. 1999. [23] D. A. J. Pearce, A. G. Burr, and T. C. Tozer, “Modelling and predicting the fading performance of fixed radio links through vegetation,” in Proc. IEE NCAP’99, Apr. 31, 1999, pp. 263–266. [24] P. Lédl, P. Pechaˇc, and M. Mazánek, “Time-series prediction of attenuation caused by trees for fixed wireless access systems operating in millimeter waveband,” in Proc. IEE ICAP’03, Mar.–Apr. 31–3, 2003, pp. 646–649. [25] M. Cheffena and T. Ekman, “Modeling the dynamic effects of vegetation on radiowave propagation,” in Proc. IEEE ICC’08, May 19–23, 2008, pp. 4466–4471. [26] L. J. Greenstein, D. G. Michelson, and V. Erceg, “Moment-method estimation of the Ricean -factor,” IEEE Commun. Lett, vol. 3, pp. 175–176, Jun. 1999. [27] S. R. Hanna and J. C. Chang, “Representativeness of wind measurements on a mesoscale grid with station separations of 312 m to 10 km,” Boundary-Layer Meteorology, vol. 60, pp. 309–324, 1992. [28] D. S. Baum, D. A. Gore, R. U. Nabar, S. Panchanathan, K. V. S. Hari, V. Erceg, and A. J. Paulraj, “Measurements and characterization of broadband MIMO fixed wireless channels at 2.5 GHz,” in Proc. IEEE ICPWD’00, Dec. 17–20, 2000, pp. 203–206. [29] V. Erceg et al., “Channel models for fixed wireless applications,” IEEE 802.16 Broadband Wireless Access Working Group, Jun. 2003. [30] SR5500 Wireless Channel Emulator Operations Manual. Eatontown, NJ, Spirent Communications, 2006, pp. 3–16.

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Kyle N. Sivertsen (S’08) received the B.A.Sc. degree in electrical engineering from the University of British Columbia (UBC), Vancouver, BC, Canada, in 2007, where he is currently working toward the M.A.Sc. degree. His main research interests include propagation and channel modeling for fixed wireless communications.

Anthony Liou received the B.A.Sc. and M.A.Sc. degrees in electrical engineering from the University of British Columbia (UBC), Vancouver, BC, Canada, in 2006 and 2009, respectively. His thesis project focused on propagation and channel modeling for fixed wireless communications. He recently joined Universal Scientific Industrial Co., Taiwan, where he is working as an engineer-intraining within the RF branch.

David G. Michelson (S’80–M’89–SM’99) received the B.A.Sc., M.A.Sc., and Ph.D. degrees from the University of British Columbia (UBC), Vancouver, BC, Canada, all in electrical engineering. From 1996 to 2001, he served as a member of a joint AT&T Wireless Services (Redmond, WA) and AT&T Labs—Research (Red Bank, NJ) team concerned with development of propagation and channel models for next generation and fixed wireless systems. The results of this work formed the basis for the propagation and channel models later adopted by the IEEE 802.16 Working Group on Broadband Fixed Wireless Access Standards. From 2001–2002, he helped to oversee deployment of one of the world’s largest campus wireless LANs at the University of British Columbia while also serving as an Adjunct Professor in the Department of Electrical and Computer Engineering. Since 2003, he has led the Radio Science Lab at UBC where his current research interests include propagation and channel modeling for fixed wireless, UWB and satellite communications. Prof. Michelson serves as Chair of the IEEE VT-S Technical Committee on Propagation and Channel Modeling, as an Associate Editor for Mobile Channels for the IEEE Vehicular Technology Magazine and as an Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. From 1999 to 2007, he chaired the IEEE Vancouver Section’s Joint Communications Chapter. Under his leadership, the Chapter received Outstanding Achievement Awards from the IEEE Communications Society in 2002 and 2005, and the Chapter of the Year Award from IEEE Vehicular Technology in 2006. He received the E.F. Glass Award from IEEE Canada in 2009 and currently serves as Chair of Vancouver Section. Under his leadership, the Section received the Outstanding Large Section Award from the IEEE in 2010.

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Internal Broadband Antennas for Digital Television Receiver in Mobile Terminals Jari Holopainen, Outi Kivekäs, Clemens Icheln, and Pertti Vainikainen

Abstract—The Implementation of internal broadband antennas for digital television receiver (DTV) in mobile terminals operating at the lower UHF band is studied comprehensively. Certain challenges such as inherently narrow impedance bandwidth of electrically small antennas and limited volume available for the antennas inside mobile terminals are identified and handled. The proposed design principle for DTV antennas is to decrease the total efficiency of the antenna to a level which is just good enough to ensure a sufficient signal-to-noise ratio and that way make the size of the antenna sufficiently small. The limits for the size of broadband capacitive coupling element-based DTV antenna structures inside handsets of different sizes are studied. In the end, a manufactured prototype and simulated design are presented and compared with the limits studied earlier in the paper. The results show that the studied antenna concept is a promising candidate for broadband DTV antennas in mobile terminals. The work also increases general understanding on the implementation of antennas based on the radiation of the finite ground plane. Index Terms—broadband antennas, broadband communication, broadcasting, digital TV, DVB-H, impedance matching, microstrip antennas, mobile antennas, receiving antennas.

I. INTRODUCTION ECENTLY, there has been a significant increase in the number of different functions and radio systems built into handheld devices. In addition to the traditional cellular systems, many other radios have been introduced in mobile terminals, such as FM radio, digital television (DTV), 3 G, GPS, Bluetooth, and WLAN. The size of the terminal has remained roughly the same despite the increased number of radio systems and at the same time internal antennas are preferred. Thus, the volume available for antennas is very limited inside a mobile terminal and the size of the antennas is a very critical issue. The wavelength of DTV at the UHF frequencies (0.47–0.75 GHz in mobile terminals which include also E-GSM) is 400–640 mm, and as the typical handset size is 100–130 mm, an internal DTV antenna inside such a terminal is electrically rather small. Due to the physical limitations [1]–[4], the implementation of small broadband DTV antennas is very challenging with the typical matching criterion (6 dB return loss) of current cellular antennas.

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Manuscript received May 07, 2009; revised February 22, 2010; accepted April 08, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by the Academy of Finland and Tekes through the Center-of-Excellence Programme. The work of J. Holopainen was supported in part by the Graduate School of Electrical and Communications Engineering, Nokia Foundation, HPY:n tutkimussäätiö, and Emil Aaltosen säätiö. The authors are with the the SMARAD Center of Excellence, Department of Radio Science and Engineering, Aalto University School of Science and Technology, FI-00076 Aalto, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055786

The available internal broadband DTV antenna solutions can be divided into two main categories according to the instantaneous frequency bandwidth. The first group is antennas with fixed broadband matching covering the whole band. The main challenge is to cover the whole band with sufficient performance. Broadband solutions are introduced e.g., in [5]–[7]. The other main group is formed by electrically tunable antennas which create an instantaneous resonance at a suitable frequency so that at least a single 8-MHz channel is covered. Electrically tunable antennas are introduced e.g., in [8], [9] and [10]. One challenge is the non-linearity caused by the semiconductor tuning component, which becomes a problem during the simultaneous use of E-GSM transmission when a part of the transmitted E-GSM signal is coupled to the DTV antenna. In addition, the tuning circuit needs a control voltage, might consume valuable power, and increases the complexity of the whole DTV receiving system. According to [11] another challenge arises when the terminal is doing a handover from one transmitter to another. While receiving signal from the first transmitter, the receiver needs to simultaneously make a new channel-scan, which might be complicated to perform with a single tunable antenna. Due to the inherently narrow impedance bandwidth of small antennas, tunable antennas seem like a reasonable choice for DTV antennas, but on the other hand the high linearity and fixed matching of passive implementation with a simple antenna structure motivates the use of broadband DTV antennas. It is also expected that the effect of the user on the matching is smaller on broadband antennas. Therefore, this paper concentrates only on implementation and characteristics of such antennas. The ground planes of the printed circuit board (PCB), EMC shieldings and other conductive structures (such as the display) of a handheld device create a solid RF counterpoise, here called a chassis. This electrically conductive chassis has a significant effect on the antenna operation because it operates as the main radiator at lower UHF frequencies, i.e., below 1 GHz [12]. Therefore, the size of the chassis has a significant effect on the achievable bandwidth and also on the size of the antenna element [12], [13], especially at the DTV frequencies [14]. In this work the chassis is expected to be a solid piece of metal and thus any direct feed-based antenna structures as presented e.g., in [6], [7] are not considered. To the authors’ knowledge there exist no scientific publications with a systematic study of implementation of internal broadband DTV antennas. The purpose of this paper is to present a thorough study of the implementation of such DTV antennas in mobile terminals. First, the used antenna concept, capacitive coupling element structures, are briefly introduced.

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Fig. 1. Capacitive coupling element (CCE) structure.

After that the essential design and performance parameters are derived and discussed. In this paper DVB-H (digital video broadcasting—handheld) is used as a DTV standard since it is a typical broadband DTV standard for mobile terminals and most probably used at least in Europe. Next, the design of a simplified full-metal DTV antenna is presented. Minimum height and volume required of a DTV antenna to reach a certain performance inside mobile terminals of different sizes are presented. Some methods to reduce the thickness of the antenna are also handled. In the end the results are compared and verified with a prototype and simulated designs. II. CAPACITIVE COUPLING ELEMENT ANTENNA STRUCTURES AND MATCHING CIRCUITS Traditional mobile terminal antennas, such as PIFA, create the antenna resonance and couple currents to the surface of the chassis [12], [13]. In principle, traditional terminal antennas could be used as DTV antennas but the bandwidth would be rather narrow or the antenna volume would be too large to be placed inside a handset. Since the antenna element itself is only a minor radiator below 1 GHz, the volume occupied by the element can be decreased significantly by introducing capacitive coupling elements (CCE) whose principal function is only to couple currents to the chassis [12], [13]. The resonance is then created with a separate matching circuitry outside the coupling element e.g., using well-known basic matching methods presented in text books, see e.g., [15], [16]. One possible capacitive coupling element antenna structure is shown in Fig. 1. In order to achieve the largest possible impedance bandwidth, the location and shape of the capacitive coupling element need to be chosen in such a way that the coupling between the coupling element and the chassis dominant wavemode is maximized [13]. For the strongest possible coupling, the coupling element needs to be placed beyond the edge of the chassis with the element bent over the shorter edge of the chassis, see Fig. 1. The vertical part of the element could be further lengthened on the upper side of the chassis but that place is typically reserved for other components (such as connectors, buttons, microphone/earpiece, camera, etc.) in real terminals. For the structure shown in Fig. 1 it is shown in [17] that with constant distance the maximum impedance bandwidth equals . Furat the DTV frequencies is achieved when ther increase in the bandwidth can only be achieved at the expense of larger volume occupied by the coupling element, i.e., and/or the height increasing the length of the element of the CCE from the upper surface of the chassis. Since DTV is a receive-only-system the specific absorbtion rate (SAR) values can be neglected.

The matching circuit can be implemented using lumped or distributed technologies [5],[13], [17]. Below 1 GHz the advantage of high-Q lumped elements (such as inductors and capacitors) over distributed elements is the significantly smaller required matching circuit area [17]. According to [13], the matching circuit might be integrated into the RF front-end module in commercial mobile terminals. Another option is introduced in [18] where the coupling element and the matching circuit are integrated into a single component using the LTCC technology. The bandwidth can be increased with a multi-resonant matching circuit, which can be implemented with coupled high-Q resonators, see e.g., [19]. In practice the number of matching resonators in a matching circuit is limited by design complexity, losses and the fact that the bandwidth increase from each additional resonator gets smaller [20]–[22]. However, the matching circuit needs to be optimized for each capacitive coupling element and chassis combination [13]. III. DESIGN PARAMETERS OF BROADBAND INTERNAL DTV ANTENNAS A. Performance of DTV Receiving Antennas A basic requirement for radio systems is to reach a sufficient signal-to-noise ratio over the frequency band of operation. In order to minimize the size of the antenna element, the design principle of DTV antennas is to provide a performance which is just enough for guaranteeing the operation with a certain reliability level. As was discussed, it is difficult to cover the whole DTV frequency band with a good matching level (e.g., 6 dB return loss). As is commonly known, the impedance bandwidth of an antenna can be increased by sacrificing a part of the total efficiency [4]. Even though the method is not commonly recommended, in small internal DTV antennas the total efficiency needs to be sacrificed in order to make the size of the antenna feasible for today’s mobile terminals [23], [24]. This is possible since DTV is a receive-only-system and lower total efficiency compared to transceiver antennas can be accepted [25]. In order to estimate the lowest required total efficiency of a DTV antenna, an equation is derived in [26]: (1) is the sensitivity of the receiver, is the far where is the typical field wave impedance in the free space 377 , minimum electric field strength guaranteed by the broadcasting network, is the speed of light, is the directivity of the antenna and is the frequency used. According to [23], the sensitivity of a typical DTV (DVB-H) receiver can be expected to be . In [27] it has been reported that at least the minimum electric field strength in the DTV indoor reception . The directivity is typically better than is about 2 dBi for small antennas [28]. These roughly approximated values and (1) yield that the minimum total efficiency required in a DTV antenna is in the order of over the band.

HOLOPAINEN et al.: INTERNAL BROADBAND ANTENNAS FOR DIGITAL TELEVISION RECEIVER IN MOBILE TERMINALS

The expected DTV antenna performance in the DVB-H standard followed in this paper has been given in terms of realized gain which consists of the directivity and the total efficiency [29]. In the DVB-H system specifications the realized gain of an antenna inside a mobile terminal is expected to be in the at 0.47 GHz and it increases linearly in dB’s order of at 0.75 GHz. When the directivity (2 dBi) to about is excluded, the total efficiency is expected to be in the order according to the above-mentioned realized of gain. Concluding the calculations above, the expected performance is at least 3.5 dB higher than the estimated minimum required total efficiency in the previous paragraph. In the following the presented realized gain limit is considered as the performance specification for internal DTV antennas. The specification is given for an antenna inside a real terminal [23], [24]. In this work the antenna structures include only metal chassis (and in the prototype low-loss printed circuit board), antenna element and matching circuit. Thus, an extra margin to the specification is needed in the designs in this work because in the final commercial products the printed circuit board (typically FR4), plastic covers, display, battery, earpiece, microphone and other lossy parts of the terminal cause additional losses.

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Fig. 2. Conceptual graph on how a narrow-band antenna can be matched so that a larger bandwidth is covered at the expense of the total efficiency.

B. Broadening Impedance Bandwidth of DTV Antennas An inherently narrow-band antenna can be matched in such a way that the impedance bandwidth is significantly increased at the expense of the total efficiency [21], [29], see Fig. 2. One can understand the idea considering that the available area below the narrow-band matching curve is redistributed over a larger impedance bandwidth. However, one should note that the areas below the narrow-band and broadband matching curves in Fig. 2 are not exactly equal because the radiation quality factor of the antenna changes as a function of the frequency. The remaining questions are how much and which way the total efficiency can be sacrificed in a DTV antenna. There are basically three options. Firstly, one can use resistive loading of the antenna (lowering radiation efficiency), secondly, one can use resistive matching (attenuators) or, thirdly, one can accept higher mismatch between the antenna and the receiver (lowering matching efficiency). Comparing these three cases, the third option can be shown to provide the widest bandwidth for a given decrease of the total efficiency [17]. The radiation efficiency is close to 100% for a low-loss fullmetal (only chassis and antenna element) antenna structures. In that case the only parameter affecting the realized gain is the return loss of the antenna. Now, the realized gain can be plotted at the edge of the impedance band as a function of the return loss matching criterion, see Fig. 3. As can be seen, in the low-loss case the matching of 1.5 dB return loss yields a realized gain of approximately . Since the realized gain specification is between and over the DTV band, the minimum margin to the specification is at least 3 dB, which can be reserved for the roughly approximated implementation losses introduced by the other components in practical antennas. Thus, 1.5 dB return loss is used in this work as the design parameter for the ultimate lowest acceptable matching criterion for low-loss internal DTV antenna designs. As is well known, the above-mentioned implementation losses consequently improve the matching level

Fig. 3. Realized gain estimation of a DTV antenna as a function of the return loss matching criterion.

and thus the matching of antenna structures of complex commercial mobile terminal and metallic low-loss antenna structures studied in this paper are generally not directly comparable. One should also note that the used 3-dB margin to the realized gain specification in the low-loss case is just one possible choice, and based on the typical total efficiencies reported for cellular antennas of commercial terminals [26], [30], [32]. IV. DESIGN OF BROADBAND DTV ANTENNAS WITH CAPACITIVE COUPLING ELEMENTS A. Simulated Example Design The purpose of this section is to demonstrate a simplified low-loss capacitive coupling element-based DTV antenna design for a typical-size monoblock terminal. An example antenna is designed with dimensions , and , see Fig. 1. The height of the antenna element from the surface of the chassis is a variable. The input impedance of the antenna structure (without the matching circuit) with several values of height (0.25 mm steps) against 50 reference impedance was simulated from the CCE feed point across 0.3–1.5 GHz with a method of moments-based electromagnetic simulator IE3D [32]. The impedance curves for , 10 and 15 mm) are shown on the capacicertain cases ( tive half of the Smith chart in Fig. 4. As can be seen, the input are fairly low and the reactances resistances are highly capacitive at the DTV

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Fig. 6. (a) Triple-resonant and (b) dual-resonant matching circuits and their component values for the example design.

Fig. 4. Impedance curves at 0.3–1.5 GHz when h is 5, 10, and 15 mm.

Fig. 5. General triple-resonant matching circuit with series and parallel resonators in turn.

frequencies. One can also note the lowest order wavemode/res, 10 onance of the chassis at 1.1, 1.05 and 1.0 GHz for and 15 mm, respectively [12]. Unfortunately the resonances are out-band for the DTV system frequencies and they can not be exploited in the matching of the antenna. The next task is to match the antenna across the DTV band with a multi-resonant matching circuit. In this work dual-resonant and triple-resonant matching circuits are used. With more additional resonators the bandwidth increase gained from each resonator saturates and also in practice the matching circuit becomes rather complex. In [33] it is reported that a capacitive coupling element antenna structure is inherently a series-type resonator and thus, the first resonator in the combination of the antenna and the matching circuit is series-type. Since CCE structures are highly capacitive at the DTV frequencies (see Fig. 4 ), the resonant frequency of the first resonator can be tuned to the DTV frequencies with a rather large-value series inductor. In order to achieve the lowest possible complexity of triple-resonant matching circuit (five matching circuit components), the second resonator is a shunt parallel resonator and the third is again a series resonator, see Fig. 5 [19]. Dual-resonant matching circuits can be implemented either by leaving out the third resonator inFig. 5 (three matching circuit components) or by using coupled series-type resonators, in which the impedance inverters can be partly integrated into resonators, as presented in [35] (four matching circuit components). of the first inductor in the tripleThe inductance value resonant matching circuit (see Fig. 5 ) can be estimated rather of the easily. The input reactance at the center frequency DTV band is (see Fig. 4 ) and thus the inductor can be estimated from equation , where value and thus . Even though the theory for general multi-resonant matching circuits is known [20]–[22], there exist no closed-form design formulas and thus the rest of

Fig. 7. Simulated (a) return loss in the Cartesian coordinate system and (b) the complex impedance on the Smith chart for the example design.

Fig. 8. Simulated realized gains of the example design.

the matching circuit needs to be designed with an automatic optimization tool of a circuit simulator. In this work APLAC [22] was used. The matching circuits were tuned to be optimal from the bandwidth point of view [22]. At this stage ideal (i.e., lossless) lumped elements were used in the matching circuits. The height of the coupling element was chosen to be the smallest possible which enables at least 1.5 dB return loss with a low-loss full-metal antenna structures across the DTV band. (The effect of the distance is to be handled later.) However, the smallest possible heights are surprisingly high, and with triple-resonant matching and dualresonant matching, respectively. Thus the volumes of the couand . The corresponding pling elements are ideal matching circuits are illustrated in Fig. 6. The return losses and realized gains against 50 reference impedance for both studied cases are shown in Figs. 7 and 8. On the Smith chart in Fig. 7(b) the reflection coefficients have in both cases the inner loops circulating the center of the Smith chart symmetrically indicating optimal frequency response from the bandwidth point of view [22]. The realized

HOLOPAINEN et al.: INTERNAL BROADBAND ANTENNAS FOR DIGITAL TELEVISION RECEIVER IN MOBILE TERMINALS

TABLE I TYPICAL TOLERANCES OF THE MURATA CHIP INDUCTORS AND CAPACITORS

Fig. 9. Effect of the variation of the matching circuit component values on the realized gain.

gains are on average which agrees very well with the estimation presented in Fig. 3. The minimum margin to the realized gain specification in both cases is 3.1 dB, which can also be expected for this low-loss design according to Fig. 3. At lower DTV frequencies the margin is much larger than 3.1 dB, even 6.5 dB, which is actually very useful because at the lower DTV frequencies the radiation resistance of the CCE antenna structure is rather low (see Fig. 4 ) and thus the losses in real antenna structures will decrease the radiation efficiency more than at the upper DTV frequencies where the radiation resistance is larger. The simulated antenna structure and matching circuit provide a good starting point for the implementation process of a more realistic DTV antenna. Next, the change of the realized gain due to the typical statistical variation of the matching circuit component values is studied. The analysis is performed for both studied cases. The typical tolerances of inductance and capacitance values have been obtained for Murata high frequency chip components in [35] and they are shown in Table I. ”Small” component value means smaller than 10 nH for inductors and smaller than 5 pF for capacitors. Because of lots of differences in the tolerances of the available Murata inductor series, the ”best” inductors mean the tightest tolerance available and the ”worst” inductors vice versa. For capacitors the tolerances are typically the same independent of series. The analysis is performed in such a way that the effect of the statistical variation of the matching circuit component values on the return loss is calculated with the Monte Carlo analysis tool in APLAC. After that the change of the realized gain is calculated from the change of the return loss. The results are shown in Fig. 9. The results can be interpreted the following

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TABLE II TERMINAL DESIGNATIONS AND DIMENSIONS.

way: If we tolerate e.g., maximum change of the realized gain, 90% of the products (acceptance level 90%) in the worst change in the triple-resonant case will have less than realized gain. At the same time if we use dual-resonant matching with the 90% acceptance level, we have the maximum change of . On the other hand, we have the realized gain of only also 4 mm thicker antenna element than in the triple-resonant case. As can be seen, first of all, the dual-resonant and triple-resonant cases have surprisingly large difference in the acceptance levels with a given change of the realized gain although in the triple-resonant matching circuit there is only one shunt capacthan in the dual-resonant matching circuit. Secitor more ondly, there is a big difference in the acceptance levels between the best and worst tolerance inductors. In the end, the decision of the used components and topology of the matching circuit are to be dictated by the size and performance of the antenna, and also the cost of the components. B. Height and Volume of the Capacitive Coupling Element Versus Handset Dimensions This section provides information about the height and volume occupied by the coupling element in different size terminals. As discussed, the size of the terminal, which determines the size of the chassis, affects significantly the size of the capacitive coupling element. When reducing the terminal dimensions we need a larger antenna element to obtain the same performance and vice versa [12]–[14]. Hence, there is a motivation to study what is the smallest possible antenna element with given outer dimensions of a terminal. In this paper four different terminal dimensions ( and in Fig. 1 ) are used, three of them are for a monoblock terminal and one is for a tablet-size terminal, see Table II. Other terminals types are not considered here because the bar-type (monoblock or tablet) terminal is considered to be the most challenging case and the mechanics is also pretty well known. For example in an open clamshell the total length of the chassis is rather long (even 160 mm) and thus, the implementation of a DTV antenna within a small volume is less problematic [14]. The minimum height of the capacitive coupling element enabling at least 1.5 dB return loss is studied for each of the four terminal sizes the same way it was done in the previous subsection. The minimum outer dimensions ( , and ) of the terminal are supposed to enclose the chassis, antenna element and other relevant components of the antenna structure, see Fig. 10. The matching circuit topologies are similar to those presented in Fig. 6, only the component values are slightly different in each case. In principle, the -parameters of real matching circuit components could be used in the simulations but non-idealities

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Fig. 10. Minimum outer dimensions of the different-size terminals. Gray color illustrates the capacitive coupling element and the chassis. The sketch is not in the real scale.

Fig. 12. Effect of different terminal sizes on the minimum height of the coupling element in dual-resonant and triple-resonant matching cases. . d l

=

= 10 mm

TABLE IV VOLUME OCCUPIED BY THE ANTENNA ELEMENTS IN THE DUAL-RESONANT AND TRIPLE-RESONANT MATCHING CASES AND THE OF THE CHASSIS d l AVAILABLE GROUNDED AREA

[cm ]

[cm ]

=

= 10 mm

Fig. 11. Effect of different terminal sizes on the minimum height of the coupling element in dual-resonant and triple-resonant matching cases. d l P .

=

= 5 mm

TABLE III OCCUPIED BY THE ANTENNA ELEMENTS IN THE VOLUME DUAL-RESONANT AND TRIPLE-RESONANT MATCHING CASES AND THE OF THE CHASSIS d l AVAILABLE GROUNDED AREAS

[CM ]

[CM ]

=

= 5 MM

such as losses, discrete component values and parasitics would make the comparison of the results very difficult and actually inconsistent. As a result of the simulations the minimum height of the antenna element is plotted as a function of the length of the terminal in Fig. 11 and the volumes occupied by the element and the available grounded areas of the chassis in different cases are reported in Table III. Based on the results in Fig. 11 and Table III, as expected, the size of the chassis has generally very significant effect on the size of the antenna element. While very big antenna elements are needed to reach the 3-dB margin to the realized gain specification with fairly short monoblock terminals, the tablet-size terminal can allow a rather small antenna element. This is because the radiation quality factor of the chassis increases when its size (both length and width) is decreased and thus the available bandwidth of the combination of the (unchanged) antenna element and the chassis is reduced. In order to maintain the available bandwidth, a smaller chassis can be compensated with a

larger antenna element which has both smaller radiation quality factor and increased coupling to the chassis lowest order wavemode [12]. Thus, the more the chassis size is decreased, the more the antenna element size needs to be increased. Triple-resonant matching instead of dual-resonant matching is shown to decrease the minimum height of the coupling element several millimeters, especially in smaller terminals. This can also be expected since in the triple-resonant case a smaller antenna element is enough to guarantee sufficient bandwidth compared to the dual-resonant case. Nevertheless, the heights of e.g., 15 mm of the antenna elements derived in the previous paragraph are too big for today’s commercial monoblock terminals. As discussed in Section II, the bandwidth of the coupling element antenna can be increased by increasing the distance of the coupling element. Vice versa, if the bandwidth of the antenna is kept equal, the increased makes it possible to decrease the height of the antenna element. In the following the minimum height of the coupling element has been studied with the antenna structure having (see Fig. 1 ). One should now notice that since the total length of the whole structure is held constant, the chassis becomes 5 mm shorter than in the case. The procedure has been similar to the previous study. The results are illustrated in Fig. 12 and Table IV. As can be seen, the case. same trend can be noticed as for the When compared to the previous case, the height of the coupling element can now be made much smaller at the expense of larger

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total volume occupied by the element and the slightly smaller available grounded area of the chassis. C. Capacitive Coupling Element Height Versus Performance of the Antenna The antenna elements even with a height of only 9 mm (see Figs. 11 and 12 ) are still fairly big for today’s terminals. As presented, while the performance is maintained, the height of the antenna element can be made smaller by making the distance of the coupling element larger. However, in current mobile terminals the distance and the volume reserved for antennas is very limited and the volume of the antenna cannot be increased very much. Thus, obviously the only way to make the height of the antenna element smaller with the used antenna concept is to further sacrifice a part of the performance of the antenna. Earlier it was presented that worsening the return loss is the most efficient way to increase the bandwidth (or consequently decrease the size of the antenna element) with a given decrease of the total efficiency. However, the choice made in this publication was that 1.5 dB return loss is the ultimate lowest acceptable matching level and thus the return loss will not be worsened anymore. Therefore, if we want to make the height of the antenna element smaller and compensate the inherently narrowed bandwidth, we are required to have additional resistive losses in the antenna structure, i.e., lower radiation efficiency. That can be done either by using lossier materials/components or by placing a small resistor between the antenna structure and the matching circuitry. In this work we use the latter method because it is a more systematic way to control the additional losses. One should note that the resistance of the resistor used here is a design variable and it can be partly or totally omitted in real antennas because the relatively large series inductor ( , see Fig. 6 ) in the matching circuit contains a series resistance in its equivalent circuit [35]. Two cases are to be handled in this study: typical-size monoblock in which the distance is 5 or 10 mm with the triple-resonant matching. The starting values for the coupling for the case and element heights are for the case, see Figs. 11 and 12. The height of the element is decreased using 2 mm steps. The smallest possible resistor, which yields 1.5 dB return loss matching over the DTV band is used. The realized gain is also simulated for each case and the margin to the specification is calculated. As a result of the study the minimum margin to the realized gain specification is plotted as a function of the height of the coupling element, see Fig. 13. As can be seen, the height of the antenna element can be decreased several millimeters by sacrificing the minimum margin case 3 dB by less than 1 dB. For example in the we can drop the height from 9.25 mm to 6 mm by sacrificing only 0.3 dB or to only 4 mm by sacrificing 0.6 dB. After that the minimum margin drops quite quickly, though. V. ANTENNA PROTOTYPES AND SIMULATED DESIGNS This section presents a certain capacitive coupling elementbased antenna prototype and simulated design. The results will be compared with those of Section IV.

Fig. 13. Minimum margin to the realized gain specification versus the height of the element (triple-resonant matching) for typical-size monoblock. x.x is the resistance of the resistor added between the antenna element and the matching circuit.

Fig. 14. Photograph of the tablet-size DTV prototype. The shape and dimensions (in mm) of the antenna are similar to shown in Fig. 10.

A. Manufactured Prototype for a Tablet-Size Terminal This section introduces a manufactured and measured DTV antenna prototype intended for a tablet-size mobile terminal. The dimensions of the coupling element are , and , see Figs. 10 and 14, and thus . The the total volume occupied by the element is only prototype is implemented on low-loss Rogers RT Duroid 5870 printed circuit board. The matching is implemented with a dualresonant lumped-element matching circuit, see Fig. 15. The inductors in the matching circuit are from Murata’s LQW18A series, the capacitor is from Murata’s GRM18 series [35] and the components are modeled in simulations with the parameter available in [36]. The simulated and measured return loss are shown in Fig. 15. They are at least 2 dB across the DTV band. The difference between the simulated and measured results is expected to be caused by the slight differences in the simulation model and the manufactured prototype and the statistical variation of the matching circuit component values from the nominal values. As discussed in Sections III-A and III-B, in real terminals the implementation losses would further improve the return loss at the expense of the realized gain. The realized gain of the

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Fig. 17. Structure of the simulated design for typical-size terminal. Dimensions are in mm.

Fig. 15. Simulated and measured return loss of the tablet-size prototype (a) in the Cartesian coordinate system and (b) on the Smith chart.

Fig. 18. Matching circuits for simulated (a) DTV and (b) E-GSM antennas.

coupling element is selected to be 1 mm larger, i.e., 4 mm, due to practical reasons. Firstly, due to the effect of the user, the usable bandwidth is typically downtuned [14], [38]. Thus, it is useful if there is some extra bandwidth available above 0.75 GHz. Secondly, in real antennas the matching circuit cannot be designed as easily as with ideal components. Thus, some design margin is required to compensate for the effect of non-idealities such as discrete component values and the statistical variation of the component values from the nominal ones, see Fig. 9. The prototype presented can be seen to be a very promising candidate for the DTV antenna of a tablet-size terminal. Fig. 16. Simulated and measured realized gain of the tablet-size DTV prototype.

B. Design for a Typical-Size Terminal Including E-GSM

prototype was measured with the Satimo Stargate antenna measurement system [37] for which the measurement inaccuracy is given by the manufacturer. Due to the ripple the measured results are mathematically interpolated with a 5th order polynomial. The simulated and measured realized gain is shown in Fig. 16. Across the DTV band the minimum margin to the realized gain specification is 3.5 dB. Because the chassis is the main radiator, the far field directional pattern is similar to that of a dipole antenna and the directivity in the direction of the main lobe is 2 dBi across the DTV band. Thus, the total efficiency is 2 dB lower than the realized gain in Fig. 16, and in the and at minimum the (simulated) total efficiency is 0.47 and 0.75 GHz, respectively. However, the total efficiency is at least 7 dB larger than the minimum required total efficiency estimated in Section III-A. The far field is practically linearly polarized, i.e., the long axis of the chassis oriented parallel to the -axis generates the far field electric field parallel to the elevation (theta) unit vector of the standard spherical coordinate system. The minimum polarization discrimination in the main lobe is 22 dB, and not very large improvement can be expected at the frequencies lower than 1 GHz with handheld-size terminals [23]. Thus, in the worst case rather low polarization efficiency can be expected. According to Fig. 11 the minimum height of the coupling element is 3 mm in the lossless dual-resonant case, which covers 0.47–0.75 GHz. In the manufactured prototype the height of the

The purpose of this subsection is to demonstrate a simulated DTV antenna design for a typical-size terminal with realistic, non-ideal matching circuit implementation. The design takes place with the DTV antenna element dimensions of and see Fig. 17. The volume . According to of the DTV coupling element used is Fig. 13 the margin to the realized gain specification is 2.8 dB additional resistor and ideal matching circuit when a components are used. The structure is considered to be a good compromise between the size and the performance. In addition, the effect of the E-GSM antenna mounted on the same chassis is taken into account in this design. The capacitive coupling element-based E-GSM antenna is designed according to [13]. The volume occupied by the E-GSM coupling element is , see Fig. 17. The realistic matching circuit components are chosen from the Murata’s selection [35]. The inductors are from LQW18A and LQW15A series and the capacitors are from GJM15 and GJM03 series. The components are modeled with parameters available in [36]. The E-GSM matching circuit is designed according to [33]. Both matching circuit topologies and the nominal values of the components are shown in Fig. 18. As discussed above, the relatively large-value series inductor in the DTV matching circuit has relatively large series resistance can be left out. [35] and thus, the additional resistor The simulated parameters for DTV and E-GSM antennas are shown in Fig. 19. The realized gain of the DVB-H antenna and the total efficiency of the E-GSM antenna are shown in

HOLOPAINEN et al.: INTERNAL BROADBAND ANTENNAS FOR DIGITAL TELEVISION RECEIVER IN MOBILE TERMINALS

Fig. 19. Simulated S parameters of the typical-size terminal design (a) in the Cartesian coordinate system and (b) on the Smith chart.

Fig. 20. Simulated realized gain of the DVB-H antenna and the simulated total efficiency of the E-GSM antenna of the typical-size terminal design.

Fig. 20. All the simulations are performed against reference impedance. As can be seen, the return loss is at least 2.8 dB across the DTV band. The improvement of the matching level from 1.5 to 2.8 dB is due to the losses of the matching circuit components. On the Smith chart the reflection coefficient looks fairly optimal from the bandwidth point of view [22]. For the E-GSM antenna the return loss is at least 9 dB across the E-GSM band (0.88–0.96 GHz). Below 0.5 GHz the realized gain of the DTV antenna drops several decibels compared to the ideal matching circuit implementation, e.g., in Fig. 8. Instead, at the upper DTV frequencies the realized gain remains roughly on the same level as in the ideal designs. This is due to two main reasons: First of all, at the lower DTV frequencies the radiation resistance is rather small (see Fig. 4 ) and thus the resistive losses in the matching circuit decrease the radiation efficiency more than at the upper DTV frequencies, where the radiation resistance is larger. Thus, it is very important that the losses of the matching circuit components are modeled realistically with the parameters of real components. Secondly, the matching circuit is optimized in such a way that a part of the available margin at the lower DTV frequencies is transferred to the upper frequencies, where the margin to the realized gain is typically lower. Thus, the margin to the realized gain specification is optimized across the DTV band and it is at least 2.5 dB. Again, the total efficiency is 2 dB lower than the realized gain. The total efficiency of the E-GSM antenna is higher than (50%) across the E-GSM band. Relatively high losses

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are caused also here by the matching circuit components. The total efficiency can be improved using components with lower losses in the matching circuit. Because the chassis is the main radiator below 1 GHz, both antenna elements mounted on the chassis are electromagnetically strongly coupled to the lowest order wavemode of the chassis and thus the elements are also strongly coupled to each other through the chassis [12]. Since the antennas are matched at different frequency bands, the matching circuits operate also as band pass filters [14]. Because the transition from the pass band to the reject band cannot be very sharp, the minimum isolations between the antennas are only 29 and 22 dB at the DTV and E-GSM bands, respectively, see Fig. 19. Thus, if the E-GSM transmits at its maximum power level, i.e., 33 dBm, 11 dBm is coupled to the DTV receiver and the operation of DTV would be blocked [23], [24]. Therefore, the matching circuits do not provide enough isolation and in order to make the simultaneous operation possible more isolation is needed. According to [24] 61 dB isolation between DTV and E-GSM at the E-GSM band is required, the next subsection is devoted for that issue. The isolations between DTV and the other transmitting systems (such as GSM1800, UMTS, Bluetooth and WLAN) are not problematic since the corresponding frequency bands are far enough from DTV band [14]. C. Improvement of the Isolation Between DTV and E-GSM Basically there are two alternative ways how to improve the isolation between DTV and E-GSM. Firstly, one can use circuit technology such as filters, or secondly, one can increase the electromagnetic isolation, concept introduced in [39], by modifying the 3D antenna structure. The first way exploiting an available GSM reject filter in the input of the DTV receiver is an easy solution from the antenna point of view. An alternative approach has been introduced in [40] where a combined matching and filtering circuit comprises a strong GSM trap which attenuates the GSM signals by more than 40 dB. The circuit-based solutions seem feasible but, however, they always introduce a certain insertion loss also for the DTV signals and thus they decrease the efficiency of the DTV antenna. Using the second way the antenna structure might be modified in such a way that the electromagnetic isolation, which does not depend on the matching, is maximized. The current placing of the DTV and E-GSM antenna elements in the opposite ends of the chassis in Fig. 17 is not obviously optimal from the electromagnetic isolation point of view since both elements couple strongly to the same wavemode created by the longitudinal currents of the chassis. However, the current placing in the typical-size terminal is chosen in order to get enough impedance bandwidth for both systems. By reshaping the elements they might be placed in the corners of the same end of the chassis and hence the electromagnetic isolation might increase because the principal current paths for both elements are diagonally from corner to opposite corner and thus have some diversity. Other solutions might include, e.g., quarter-wavelength wavetraps [41] but it is possible that they are physically too large at the lower UHF frequency band to be placed inside typical-size

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terminals, or neutralization line which might produce opposite coupling as proposed in [42]. The best solution might be to somehow maximize the electromagnetic isolation according to the boundary conditions of a given mobile environment, and then provide the rest of the required isolation with the help of the circuit technology. However, finding an optimal way to fulfill the isolation requirement requires further work. VI. DISCUSSION The capacitive coupling element-based broadband DTV antenna structures offer certain significant advantages. Firstly, the design of the studied antenna concept is relatively straightforward because the antenna matching is created with the matching circuit and thus the type, location and shape of the antenna element can be optimized according to the given boundary conditions of the mobile terminal environment. Secondly, the broadband antenna structures provide rather simple implementation compared e.g., to electrically tunable antennas based on a semiconductor component. Thirdly, the passive implementation of the broadband antenna structures does not cause any distortion problems when using the cellular system transmitters simultaneously with DTV. The implementation of broadband DTV antennas is a compromise between the size and the performance of the antenna. In this work a 3-dB margin to the realized gain specification was reserved for the implementation losses introduced in commercial products. When applying the 3-dB margin to the specification it was shown with electromagnetic simulations that relatively high capacitive coupling elements are required especially in today’s typical-size terminals. In order to make the antenna elements thinner without increasing the total volume of the elements, the performance of the antenna has to be slightly sacrificed. The results of the paper also show a few apparent challenges of the studied antenna concept. Firstly, the efficiency of broadband DTV antennas is not as good as usually in mobile terminal transceiver antennas of cellular systems. However, the lower efficiency level can be accepted because DTV is a receive-only-system. Secondly, the isolation between the DTV and E-GSM systems of the simulated design for a typical-size terminal is not high enough to fulfill the specification given in [24] and thus additional filtering between DTV and E-GSM radios is required. Thirdly, the chosen matching circuit topology and the used components affect relatively much the performance of the antenna through the losses especially at the low DTV frequencies and the variation of the component values. In this paper a traditional way to evaluate mobile terminal antennas has been used: the performance is calculated against 50 ohms reference impedance in terms of the total efficiency or realized gain. This approach is especially useful in academic publications since the comparison between different antennas is thereby rather easy to perform. However, a different approach is available for receive-only antennas. As well known, for receiving systems the sufficient signal-to-noise ratio (SNR) is the fundamental requirement. The SNR of a highly capacitive unmatched antenna and a given high-impedance low noise amplifier can be optimized to implement a broadband SNR response, as demonstrated in [43], [44]. The design principles of low noise

amplifiers (LNA), introduced e.g., in [16], can be used by considering the input of the antenna as the source of the LNA. The method might also be useful for evaluating a CCE-based broadband DTV antenna and low noise amplifier receiver system. In addition, the method might also provide a new approach how to optimize the DTV antenna structure. VII. CONCLUSIONS The implementation and design of broadband internal digital television antennas for handheld multisystem mobile terminals is studied systematically. Since the volume that can be reserved for antennas inside a mobile terminal is very restricted, the design principle of (receive-only) digital television antennas is to sacrifice the total efficiency of the antenna to a level which is just enough to ensure a sufficient signal-to-noise ratio and that way make the size of the antenna element sufficiently small. Thus, the lower limit for the size of broadband capacitive coupling element-based antenna structures inside terminals of different size was studied with electromagnetic simulations. The results indicate that DTV antenna elements for today’s small and typical size monoblock terminals are relatively high and large when reserving a 3-dB margin for the implementation losses of commercial terminals. It was shown that the height of a DTV antenna element can be made clearly smaller by sacrificing only slightly the 3-dB margin. On the other hand, for terminals whose chassis is longer than in typical size terminals, such as in tablet-size terminals, the height of a DTV antenna element can be significantly smaller. The derived results are demonstrated both with a prototype and a simulated design. The results show that the studied antenna concept is a promising candidate for broadband digital television antennas in mobile terminals. The results also increase understanding on the implementation of antennas based on the radiation of the finite ground plane, so, the presented methods could also be applied at other frequencies for other systems, e.g., for the spectrum sensing antenna of the cognitive radio [45]. ACKNOWLEDGMENT The authors want to thank Mr. R. Valkonen, Dr. J. Villanen, and Mr. M. Kyrö for valuable comments. REFERENCES [1] L. J. Chu, “Physical limitations on omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, 1948. [2] R. C. Hansen, “Fundamental limitations in antennas,” Proc. IEEE, vol. 69, pp. 170–182, Feb. 1981. [3] J. S. McLean, “A re-examination of the fundamental limits on the radiation of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, pp. 672–676, May 1996. [4] H. F. Pues and A. R. van de Capelle, “An impedance-matching techniques for increasing the bandwidth of microstrip antennas,” IEEE Trans. Antennas Propag., vol. AP-37, pp. 1345–1354, Nov. 1989. [5] J. Holopainen, J. Villanen, M. Kyrö, C. Icheln, and P. Vainikainen, “Antenna for handheld DVB terminal,” in Proc. IEEE iWAT 2006 Small Antennas and Novel Metamaterials, White Plains, NY, Mar. 6–9, 2006, pp. 077–077, CD-ROM (ISBN 0–7803-9444–5). [6] K.-L. Wong, Y.-W. Chi, B. Chen, and S. Yang, “Internal DTV antenna for folder-type mobile phone,” Microw. Opt. Technol. Lett., vol. 48, no. 6, pp. 1015–1019, Jun. 2006. [7] J. Holopainen, J. Villanen, C. Icheln, and P. Vainikainen, “Mobile terminal antennas implemented using direct coupling,” presented at the EuCAP 2006 European Conf. on Antennas and Propagation, Nice, France, Nov. 6–10, 2006, CD-ROM SP-626 (92–9092-937–5), paper: OA17 349858jh.pdf.

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[8] Z. D. Milosavljevic, “A varactor-tuned DVB-H antenna,” IEEE iWAT 2007 Small Antennas and Novel Metamaterials, pp. 124–127, Mar. 2007. [9] M. Komulainen, M. Berg, H. Jantunen, and E. Salonen, “Compact varactor-tuned meander line monopole antenna for DVB-H signal reception,” Electron. Lett., vol. 43, no. 24, pp. 1324–1326, Nov. 2007. [10] L. Huang and P. Russer, “Electrically tunable antenna design procedure for mobile applications,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 2789–2797, Dec. 2008. [11] B. S. Collins, “Small antennas for the reception of future mobile television services,” presented at the IEEE iWAT 2009 Small Antennas and Novel Metamaterials, Santa Monica, CA, Mar. 2009, paper: 1569175875.pdf. [12] P. Vainikainen, J. Ollikainen, O. Kivekäs, and I. Kelander, “Resonator-based analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, pp. 1433–1444, Oct. 2002. [13] J. Villanen, J. Ollikainen, O. Kivekäs, and P. Vainikainen, “Coupling element based mobile terminal antenna structures,” IEEE Trans. Antennas Propag., vol. 54, pp. 2142–2153. [14] J. Holopainen, “Handheld DVB and Multisystem Radio Antennas,” Licentiate, Helsinki Univ. Technology, Dept. Radio Sci. Eng., Espoo, Finland, 2008. [15] A. V. Räisänen and A. Lehto, Radio Engineering for Wireless Communication and Sensor Applications. Boston, MA: Artech House, 2003, pp. 366–366. [16] D. M. Pozar, Microwave Engineering. New York: Wiley, 1998, pp. 716–716. [17] J. Holopainen, “Antenna for Handheld DVB Terminal,” Master’s thesis, Helsinki Univ. Technology, Radio Lab., Espoo, Finland, 2005. [18] D. Manteuffel, M. Arnold, Y. Makris, and Z.-N. Chen, “Concepts for future multistandard and ultra wideband mobile terminal antennas using multilayer LTCC technology,” presented at the IEEE iWAT 2009 Small Antennas and Novel Metamaterials, Santa Monica, CA, Mar. 2009, paper: 1569175881.pdf. [19] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance Matching Networks and Coupling Structures. New York: McGraw-Hill, 1964. [20] H. W. Bode, Network Analysis and Feedback Amplifier Design. NY: Van Nostrad, 1945. [21] R. M. Fano, “Theoretical limitation on the broad-band matching of arbitrary impedances,” J. Franklin Institute, vol. 249, pp. 57–83, Jan. 1950. [22] J. Ollikainen, “Design and implementation techniques of wideband mobile communication antennas,” Doctoral dissertation, Helsinki Univ. Technology, Radio Lab., Espoo, Finland, 2004. [23] EICTA: Mobile and Portable DVB-T/H Radio Access—Interface Specification, Version 2.0 2007. [24] DVB-H Implementation Guidelines, DVB Document A092 Rev. 2 DVB Project—Standards & Technology (cited 23.6.2008), 2007 [Online]. Available: http://www.dvb.org [25] R. E. Collin, Antennas and Radiowave Propagation. New York: McGraw-Hill, 1985, pp. 293–336. [26] D. Manteuffel and M. Arnold, “Considerations for reconfigurable multi-standard antennas for mobile terminals,” in Proc. IEEE iWAT 2008 Small Antennas and Novel Metamaterials, Chiba, Japan, 2008, p. 127. [27] P. Hannula, “Planning Parameters for DVB-T Portable Indoor Reception,” Master’s thesis, Helsinki Univ. Technology, Radio Lab., Espoo, 2004. [28] J. D. Kraus and R. J. Marhefka, Antennas for All Applications, Third Edition. New York: McGraw-Hill, 2002, pp. 938–938. [29] IEEE Standard Definitions of Terms for Antennas, , 1993, IEEE STD145. [30] K. R. Boyle, “Mobile phone antenna performance in the presence of people and phantoms,” in Proc. Inst. Elect. Eng. Technical Seminar Antenna Meas. and SAR (AMS 2002), U.K., 2002, p. 4. [31] K. R. Boyle, “The performance of GSM 900 antennas in the presence of people and phantoms,” in Proc. 12th Int. Conf. Antennas Propagat. (ICAP 2003), Exeter, U.K., 2003, pp. 35–38. [32] IE3D, a Method of Moments Based Commercial Electromagnetic Simulator, Ver. 12.2 Zeland Software, Inc., Fremont, CA [Online]. Available: http://www.zeland.com (cited 24.6.2008) [33] J. Villanen and P. Vainikainen, “Optimum dual-resonant impedance matching of coupling element based mobile terminal antenna structures,” Microw. Opt. Technol. Lett., vol. 49, no. 10, pp. 2472–2477, Oct. 2007.

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[34] A Commercial Circuit Simulator, Ver. 8.21, AWR-APLAC Oy APLAC, Espoo, Finland [Online]. Available: http://www.aplac.com/ (cited 29.1.2009) [35] Murata Products [Online]. Available: http://www.murata.com/products/index.html (cited 2.4.2009) [36] Murata S Parameter Download Service [Online]. Available: http://www.murata.co.jp/sparameter/ (cited 2.4.2009) [37] Satimo Stargate, a Commercial Antenna Measurement System [Online]. Available: http://www.satimo.fr/eng/ (cited 5.5.2009) [38] J. Holopainen, J. Poutanen, C. Icheln, and P. Vainikainen, “User effect of antennas for handheld DVB terminal,” presented at the ICEAA 2007 International Conference on Electromagnetics in Advanced Applications Turin, Italy, Sep. 17–21, 2007, CD-ROM (1–4244-0767–2), paper: 313.pdf. [39] J. Rahola and J. Ollikainen, “Removing the effect of antenna matching in isolation analyses using the concept of electromagnetic isolation,” in Proc. IEEE iWAT 2008 Small Antennas and Novel Metamaterials, Chiba, Japan, 2008, pp. 335–335. [40] V. Rambeau, H. Brekelmans, M. Notten, K. R. Boyle, and J. van Sinderer, “Antenna and input stages of a 470–710 MHz silicon TV tuner for portable applications,” in Proc. Eur. Solid-State Circuit Conf. (ESSCIRC), Sep. 2005, pp. 239–242. [41] P. Lindberg and E. Öjefors, “A bandwidth enhancement technique for mobile handset antennas using wavetraps,” IEEE Trans. Antennas Propag., vol. 54, Aug. 2006. [42] A. Diallo, C. Luxey, P. L. Thuc, R. Staraj, and G. Kossiavas, “Study and reduction of the mutual coupling between two mobile phone PIFAs operating in the DCS1800 and UMTS bands,” IEEE Trans. Antennas Propag., vol. 54, pt. 1, pp. 3063–3074, Nov. 2006. [43] P. Lindberg, S. Irmscher, and A. Kaikkonen, “Electricall small receiveonly resonant antennas with wideband performance for FM radio reception in mobile phones,” presented at the IEEE iWAT 2010 Small Antennas and Novel Metamaterials, Lisbon, Portugal, Mar. 1–3, 2010, paper: PS1.13. [44] K. F. Warnick and M. A. Jensen, “Signal and noise analysis of small antennas terminated with high-impedance amplifiers,” presented at the 2nd Eur. Conf. on Antennas and Propagation, EuCAP 2007, Edinburgh, UK, Nov. 11–16, 2007. [45] P. S. Hall, P. Gardner, J. Kelly, E. Ebrahimi, M. R. Hamid, and F. Ghanem, “Antenna challenges in cognitive radio,” presented at the ISAP 2008 Int. Symp. on Antennas Propag., Taipei, Taiwan, Oct. 27–30, 2008, paper: 1645173.pdf.

Jari Holopainen was born on June 8, 1980 in Siilinjärvi, Finland. He received the M.Sc. degree (with distinction) and Lic.Sc. degree in electrical engineering from the Helsinki University of Technology, Espoo, Finland, in 2005 and 2008, respectively. He is currently working toward the D.Sc. degree at Aalto University, Aalto, Finland. Currently he is a Research Engineer in the Department of Radio Science and Engineering, School of Science and Technology, Aalto University School of Science and Technology. His current scientific interests include research of small mobile terminal antennas and teaching of radio science and engineering.

Outi Kivekäs received the M.Sc., Lic.Sc., and D.Sc. (with distinction) degrees from the Helsinki University of Technology (TKK), Espoo, Finland, in 1999, 2001, and 2005, respectively. She works as a Researcher at the SMARAD Center of Excellence, Department of Radio Science and Engineering, School of Science and Technology, Aalto University (formerly TKK), Aalto, Finland. Her main research interests include mobile terminal antennas and their user interaction.

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Clemens Icheln was born in Hamburg, Germany, in 1968. He received the M.Sc. degree in electrical engineering from Hamburg-Harburg University of Technology, Germany, in 1996, the Licentiate degree in radio engineering and the Doctor of Science in technology degree from Helsinki University of Technology (TKK), Finland, in 1999 and 2001, respectively. He is currently working as Lecturer in the Department of Radio Science and Engineering, School of Science and Technology, Aalto University (former TKK), Aalto, Finland. His research interests include design of mobile terminal antennas and their evaluation methods.

Pertti Vainikainen received the degree of Master of Science in technology, Licentiate of Science in technology and Doctor of Science in technology from Helsinki University of Technology (TKK), Espoo, Finland, in 1982, 1989 and 1991, respectively. From 1992 to 1993, he was Acting Professor of Radio Engineering, since 1993 Associate Professor of Radio Engineering, and since 1998, Professor in Radio Engineering, all in the Radio Laboratory (since 2008 Department of Radio Science and Engineering) of TKK (since 2010 Aalto University). From 1993 to 1997, he was the Director of the Institute of Radio Communications (IRC), TKK, and in 2000 a Visiting Professor at Aalborg University, Denmark, and in 2006 at the University of Nice in France. His main fields of interest are antennas and propagation in radio communications and industrial measurement applications of radio waves. He is the author or coauthor of six books or book chapters and about 340 refereed international journal or conference publications and the holder of 11 patents.

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3375

Communications Design of Yagi-Uda Antenna Using Biogeography Based Optimization Urvinder Singh, Harish Kumar, and Tara Singh Kamal

Abstract—Biogeography based optimization (BBO) is a new inclusive vigor based on the science of biogeography. Biogeography is the schoolwork of geographical allotment of biological organisms. BBO employs migration operator to share information between the problem solutions. The problem solutions are identified as habitat and sharing of features is called migration. In this communication, BBO algorithm is matured to optimize the element length and spacing for Yagi-Uda antenna. The gain of Yagi-Uda is a multimodal function and is hard to optimize because of its reliance on changes in lengths and spacings. To confirm the capabilities of BBO, Yagi-Uda antenna is optimized for three different design objectives which include gain, input impedance and side lobe level (SLL). During optimization, NEC2—a method of moment’s code evaluates the performance of each design generated by BBO algorithm. The results obtained by BBO are compared with the genetic algorithm (GA), evolutionary programming (EP), comprehensive learning particle swarm optimization (CLPSO), simulated annealing (SA) and computational intelligence (CI). Index Terms—Antenna, biogeography based optimization, genetic algorithms, optimization, Yagi-Uda.

I. INTRODUCTION A Yagi-Uda antenna is familiar as the commonest kind of terrestrial TV antenna to be found on the rooftops of homes. It is usually in employment at frequencies between 30 MHz to 3 GHz, or a wavelength range of 10 meters to 10 centimeters. The Yagi-Uda array is named after its inventors (in practice it is usually called a “Yagi”). It is also used for radar and low cost communications at microwave frequencies and millimeter wavelength in the form of quasi-Yagi antennas [1]. The insertion of copious parasitic elements at high gain makes very intricate to optimize Yagi-Uda antenna. Since its inception in 1920s, it has been optimized numerous times using diverse optimizing techniques. It is hard to obtain a perfect model using closed form expressions because of the parasitic elements of this antenna. In 1959, when e-brain did not exist, Ehrenspeck and Poehler optimized Yagi-Uda antenna and obtained higher values of gain experimentally by varying the lengths and spacings of the elements [2]. The availability of powerful software and hardware at affordable prices make it possible to optimize antennas numerically. Cheng et al. [3]–[5] voted a substantial quality of effort using gradient techniques to optimize the gain of Yagi-Uda antennas. They established that the Yagi-Uda antenna gain function is highly non-linear. Conclusively they published: “There are many minor maxima for directivity in the multidimensional (parameter) space. Hence, the initial choice is expected to affect the end result. Manuscript received February 02, 2009; revised March 10, 2010; accepted March 28, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. U. Singh is with the Department of ECE, GNDP College, Ludhaina, Punjab, India (e-mail: [email protected]). H. Kumar is with the Department of ECE, SLIET, Longowal, Sangrur, Punjab, India (e-mail: [email protected]). T. S. Kamal is with the Department of ECE, DIET, Kharar, Punjab, India (e-mail:[email protected]). Digital Object Identifier 10.1109/TAP.2010.2055778

It is recommended to start with an initial array that is known to have a good directivity either from empirical or experimental data” [4]. Altshuler and Linden [6] applied genetic algorithms (GA) for the design of wire antennas and the optimized designs were also confirmed experimentally to meet up the design specifications. Jones and Joines [7] lucratively established binary coded genetic algorithms for the design of Yagi-Uda antennas to maximize the gain and achieved the desired antenna characteristics. They formulated this design as an unconstrained single objective maximization problem by combining different objectives using scaling factors. Kumar and Ahmad [8] worked with the evolutionary programming (EP) to Yagi antenna and found the better results than Jones and Joines [7]. Venkatarayalu and Ray offered single and multi-objective formulations for the design of Yagi-Uda antennas using a computational intelligence (CI) method. Now, a designer can handle constraints and objectives separately via Pareto ranking by this means the need for scaling and aggregation had been eliminated [9], [10]. They obtained improved results than GA optimized antennas, but the number of objective function evaluations performed was approximately 20 times higher. The foremost advantage of their design was that the weighting coefficients are not involved and there was no necessitate tuning required by running the problem many times. Baskar et al. [11] optimized the Yagi antenna with comprehensive learning particle swarm optimization (CLPSO) and achieved good quality results. Recently, simulated annealing (SA) was used for the Yagi-Uda antenna optimization and better results had been obtained at the cost of time taken for the optimization [12]. This paper presents a way of using BBO algorithm to optimize the element spacings and lengths of Yagi-Uda antennas. NEC2—a method of moments code—was used to simulate and evaluate the antenna designs during optimization process generated by the BBO. To the best of the literature available, the performance of BBO in design optimization of Yagi-Uda antenna has not been investigated yet. After a brief introduction in the Section I, the paper is structured as follows: In the Section II, Yagi-Uda antenna design representation has been offered which is followed by introduction to the BBO in the Section III and BBO based optimization algorithm has been discussed in Section IV. In Section V, design examples of three antennas simulated with MATLAB are presented and the comparisons of the results obtained have been carried out with other algorithms. Finally Conclusions have been drawn in Section VI. II. DESIGN REPRESENTATION OF A YAGI-UDA ANTENNA Yagi-Uda antenna consists of one driven element plus one or more directors on one side and a reflector on the other as shown in Fig. 1 which is a six-element antenna. An incoming field sets up resonant currents on all the dipole elements. This causes the passive elements to re-radiate signals. These re-radiated fields are then picked up by the driven element. Hence the total current induced in the driven dipole is a combination of the direct field striking it and the re-radiated contributions from the directors and reflector [13]. The goal of a designer is to obtain a Yagi-Uda antenna which satisfies particular performance criteria like gain, input impedance, side lobe level, beam width etc. This can be achieved by varying lengths and spacings of Yagi antenna. An N-element Yagi-Uda antenna consists of 2N-1 variables apart from radius of elements. Therefore, the design variables of N-element Yagi-Uda antenna are x = [L0 ; L10000000000 LN01 ; S0 ; S1......... SN02 ] where 2Li is

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Fig. 1. Six-element Yagi-Uda antenna.

the length of ith element and th (i + 1) element.

Si

is the spacing between the

th

i

and Fig. 2. Habitat migration rate vs. habitat suitability index.

III. BIOGEOGRAPHY THEORY Biogeography is the study of distribution of biodiversity over space and time. The science of biogeography was sown by naturalists like Alfred Wallace and Charles Darwin. Till 1967, biogeography was mainly a descriptive study. But the work carried out by R. MacAurthur and E. Wilson changed this perception. They were able to construct a mathematical model for biogeography and made it feasible to predict the number of species in a habitat. In the science of biogeography, a habitat is an ecological area that is inhabited by a particular plant or animal species and which is geographically isolated from other habitats. Each habitat is classified by habitat suitability index (HSI). Areas or habitats which are well suited as living places for biological species have high HSI while habitats that are not good have low HSI. The value of HSI depends upon many features of habitat like rainfall, temperature, diversity of vegetation, land area, safety and security. If each of the features is assigned a value, HSI is a function of these values. Each of these features that characterize habitability is known as suitability index variables (SIV). SIVs are the independent variables while HSI are the dependent variables. Habitats with high HSI have large population, high emigration rate , simply by virtue of large number of species that migrate to other habitats. The immigration rate  is low for these habitats as these are already saturated with species. On the other hand, habitats with low HSI have high immigration rate , low emigration rate  because of sparse population. The value of HSI of low HSI habitat may increase with the influx of species from other habitats as suitability of a habitat is function of its biological diversity. But if HSI does not increase and remains low, species in that habitat go extinct and this leads to additional immigration. For sake of simplicity, it is safe to assume a linear relationship between a habitat HSI and its immigration and emigration rate and also that the rates are same for all the habitats. The immigration and emigration rate depends upon the number of species in the habitats. These relationships are shown in Fig. 2. The values of emigration and immigration rates are given as:  =I  =

1

0n k

E n

(1) (2)

where I is the maximum possible immigration rate; E is the maximum possible emigration rate; k is the number of species of the k-th individual and n is the number of species. IV. BIOGEOGRAPHY-BASED OPTIMIZATION BBO is a novel population-based global optimization algorithm stimulated by the science of biogeography. The candidate solutions for a problem are considered as habitats. Each solution is associated with

the fitness which is analogous to HSI of a habitat. A good solution is analogous to a habitat with high HSI and a poor solution represents a habitat with a low HSI. Good solutions share their features with poor solutions by means of migration. Good solutions have more resistance to change than poor solutions. On the other hand, poor solutions are more dynamic and accept a lot of new features from good solutions. Consider a global optimization problem and population of possible solutions. In an N-dimensional optimization problem, a habitat is a 1X Nvar array. Each solution is represented by N-dimensional integer vector as [SI V1.........; SI VNvar ]. In BBO, a SIV is a solution feature corresponding to “gene” while habitat is similar to “chromosome” in other population-based optimization algorithm like GA. The variable values or SIVs in the habitat are represented as floating numbers. The set of all such vectors is the search space from which the optimum solutions are to be found. The value of HSI of a habitat is value of objective function associated with that solution. The value of HSI is found by evaluating the cost of function at the variables [SI V1;......... SI VNvar ]. Therefore HSI = f(Habitat) = f(SI V1 ... SI VNvar ):

These solutions are made to share features among themselves by applying migration operator. For each SIV, in each solution, it is decided probabilistically whether or not to immigrate. If immigration is selected for a given solution feature, the emigrating habitat is selected for a given solution probabilistically using roulette wheel normalized by . The mutation operator is probabilistically applied to the habitat which tends to increase the biological diversity of the population. The mutation rate m is inversely proportional to the solution probability which is given by: m = mmax

1

P 0 Pmax

(3)

where mmax is a user-defined parameter. As in other population-based optimization algorithms, elitism is introduced so that the best solutions are retained in the population from one generation to the next. The algorithms for mutation and migration process of BBO are shown in Fig. 3 and Fig. 4 respectively. Mutation and migration operators in BBO are similar to GA and PSO and therefore, it is also applicable to same type of problems as GA and PSO are used for. BBO is different in some respects with the other global optimization techniques, e.g., as compared with GA it does not involve reproduction and it keeps the solution set while moving from one iteration to the next [14].

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TABLE I RESULTS OF GAIN OPTIMIZED SIX-ELEMENT YAGI-UDA ANTENNA DESIGNS

Fig. 3. Migration process of BBO.

Fig. 5. Convergence characteristics of BBO.

Fig. 4. Mutation process of BBO.

V. DESIGN EXAMPLES In this section, the BBO is applied to three different design problems of Yagi-Uda antennas using MATLAB programming environment and NEC2 software [15]. The results are compared to previously published results from GA [7], EP [8] CI [9], [10] and CLPSO [11]. NEC-2, a method of moments code has been used in the analysis of the antenna. MATLAB-based interface is used for editing structures and viewing simulation results. The number of segments per element was taken as seven and was same as taken by GA [7], which gave satisfactory results. A. Example 1 In the first example six-element Yagi-Uda antenna optimized for gain only is presented. The geometry of a 6-element Yagi-Uda array has been shown in Fig. 1. The distances between consecutive elements and the length of each element are the parameters to be optimized. Each vector solution in the population has 11 SIVs which is made up of 6 lengths and five spacings in the vector

x = [L0 ; L1 . . . L5 ; S0 ; S1 . . . S4 ]:

(4)

The spacing between elements was allowed to vary between 0:10 and 0:45, and the length of each element was allowed to vary between 0:15 and 0:35. The cross-section radius was the same for all elements and was set equal to 0:003369 at 300 MHz. The segment size for all elements was fixed at 0:1, where  is the wavelength corresponding to 300 MHz. The source element for excitation was specified to be the middle segment of the driven element. The x location of the driven element was always set to 0. The antenna was analyzed in free space. For BBO, the following parameters were taken: • No. of habitats or population: NP = 70; • Generations = 50; • Number of SIVs per habitat = 11; • Habitat modi cation probability = 1; • Mutation probability: mmax = :05; • Elitism parameter = 2; • Maximum Migration Rates E = 1 and I = 1: The BBO algorithm was applied on the Yagi-Uda antenna problem which consisted of migration operator followed by mutation. The duplicates solutions were removed at each generation and restored with random mutations. At the last, elitism operation was applied for preserving two fittest habitats from each generation. The maximum gain achieved after ten runs was 13.84 dBi. The BBO algorithm took around 8 minutes to complete the optimization while the GA algorithm took 9 minutes and SA converged in 1 hour and 5 minutes. All these algorithms were run on Celeron 1.4 GHz machine with 512 MB of RAM. Clearly for this optimization problem, BBO was faster than the SA and a little better than GA in converging. The corresponding lengths and spacings are shown in Table I along with the results from the published work. It can be seen that the maximum gain obtained by BBO is more than obtained by GA [7] and EP [8]. The convergence characteristics of the optimization are shown in Fig. 5.

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TABLE II RESULTS OF GAIN AND IMPEDANCE OPTIMIZED SIX-ELEMENT YAGI-UDA ANTENNA DESIGNS

TABLE III RESULTS OF GAIN, IMPEDANCE AND SIDELOBE LEVEL OPTIMIZED FOUR-ELEMENT YAGI-UDA ANTENNA DESIGNS

B. Example 2 Yagi-Uda antenna generally suffers from the problem of having low real input impedance as its driven element is a dipole. This can also observed from the Table I, the real part of impedance of antenna obtained is low and imaginary part is high. This makes these antennas difficult to match. To overcome this problem, an antenna is designed to have real part of impedance of 50 and having imaginary part equal to zero. The following objective function rewards an antenna design x for having high gain G and at the same time penalizes the design if the real part of impedance Z is not equal to 50 or imaginary part is not equal to zero

O(x) = aG(x) 0 b j50 0 Re (Z (x))j 0 c jIm (Z (x) j :

(5)

The constants a, b, c are the weighting coefficients and their values 40, 2, 1 yielded satisfactory results. The constraints on the length and spacing of each element were same as in previous example. Along with this antenna, previous published results have been listed. Clearly, it can be seen from the Table II that the results of BBO technique are better than GA [7] and CLPSO [11] both in terms of gain and impedance.

Fig. 6. H-pattern of four-element Yagi-Uda antenna.

In this example, a four-element Yagi-Uda antenna is optimized for gain and impedance and low side lobe level (SLL). Here side lobes are lobes other than main lobe including back lobe. For this, objective function is changed and another term is added to it which penalizes the antenna for having large side lobes. For this optimization the radius of each element was taken as :00225, spacing was varied between 0:10 and 0:35 and the length of each element was allowed to vary between 0:15 and 0:35. The objective function for this design problem is given as follows:

The values of a, b, c were taken as that taken by GA [7], i.e., 40, 1, 1, 3 respectively. The value of d was equal to 3 which was different from GA [7] in which the same was equal to 2. After several runs, many good results were obtained but the solution with the lowest side lobe level is listed in the Table III. The BBO obtained lower SLL and good impedance match as compared to other listed results though the gain has suffered a bit. This resulted from the larger value of weighting factor d as compared to given by GA [7]. The results obtained are listed in Table III and are compared with previous obtained results by GA [7], CI [9], [10] and CLPSO [11]. The H-plane radiation patterns of the BBO, GA and CI optimized four element Yagi-Uda antenna are shown in Fig. 6.

O(x) = aG(x) 0 b j50 0 Re (Z (x))j (6) 0c jIm (Z (x))j 0 MaxSLL(x):

BBO is a new algorithm in electromagnetics optimization and is applied for the optimization of Yagi-Uda antenna. It has been endeavor

C. Example 3

VI. CONCLUSIONS

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of this paper to introduce this novel optimization technique to electromagnetic community. BBO has been able to achieve superior results and has been in conversant with the available literature. The findings address that they are better or in equality with the GA and CLPSO. As compared to SA, BBO has been proved to be a faster and leads for better results. Further, it is easy to implement and has been consistent in giving solutions i.e. deviation in results obtained after multiple runs has been found to be lesser. Conclusively, BBO has been a good choice for optimizing Yagi antenna and can be tried on other antennas like antenna arrays as well.

REFERENCES [1] Y. X. Qian, W. R. Deal, N. Kaneda, and T. Itoh, “Microstrip-fed quasiYagi antenna with broadband characteristics,” Electron. Lett., vol. 34, no. 23, pp. 2194–2196, 1998. [2] H. W. Ehrenspeck and H. Poehler, “A new method for obtaining maximum gain from Yagi antennas,” IRE Trans. Antennas Propag., vol. AP-7, pp. 379–386, Oct. 1959. [3] D. K. Cheng and C. A. Chen, “Optimum element spacings for Yagi-Uda arrays,” IEEE Trans. Antennas Propag., vol. AP-21, pp. 615–623, Sep. 1973. [4] C. A. Chen and D. K. Cheng, “Optimum element lengths for Yagi-Uda arrays,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 8–15, Jan. 1975. [5] D. K. Cheng, “Gain optimization for Yagi-Uda arrays,” IEEE Antennas Propag. Mag., vol. 33, pp. 42–45, Jun. 1991. [6] E. E. Altshuler and D. S. Linden, “Wire-antenna designs using genetic algorithms,” IEEE Antennas Propag. Mag., vol. 39, no. 2, pp. 33–43, 1997. [7] E. A. Jones and W. T. Joines, “Design of Yagi-Uda antennas using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 45, pp. 1386–1392, Sep. 1997. [8] K. Chellapilla and A. Hoorfar, “Evolutionary programming: An efficient alternative to genetic algorithms for electromagnetic optimization problems,” presented at the IEEE AP-S Intl. Symp. and USNC/URSI Meeting, Atlanta, GA, 1998. [9] N. V. Venkatarayalu and T. Ray, “Single and multi-objective design of Yagi-Uda antennas using computational intelligence,” in Proc. Cong. Evolutionary Computation, Canberra, Australia, 2003, pp. 1237–1242. [10] N. V. Venkatarayalu and T. Ray, “Optimum design of Yagi-Uda antennas using computational intelligence,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1811–1818, 2004. [11] S. Baskar, A. Alphones, P. M. Suganthan, and J. J. Liang, “Design of Yagi-Uda antennas using comprehensive learning particle swarm optimisation,” IEE Proc.: Microw., Antennas Propag., vol. 152, no. 5, pp. 340–346, 2005. [12] U. Singh, M. Rattan, and N. Singh, “Optimization of Yagi antenna for gain and impedance using simulated annealing,” in Proc. IEEE, ICECOM, Croatia, Sep. 2007, vol. 24–26, pp. 1–4. [13] C. A. Balanis, Antenna Theory and Design. New York: Harper and Row, 1982. [14] D. Simon, “Biogeography-based optimization,” IEEE Trans. Evolutionary Computation, vol. 12, no. 6, pp. 702–713, Dec. 200. [15] G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code (NEC)—Method of Moments Lawrence Livermore Lab., Livermore, CA, Rep. UCID18834, Jan. 1981.

3379

A Wideband Circularly Polarized H-Shaped Patch Antenna Kwok L. Chung

Abstract—A wideband H-shaped patch antenna for circular polarization is described. By incorporating a long probe inside a substantial high cavity ( 0 26 ), a circularly polarized (CP) antenna in broadside direction is achieved. The horizontal arm of the probe is implemented by means of a printed monopole that is diagonally coupled to a small H-shaped copper plate. The CP antenna features a wide operational bandwidth of 19.4%. The maximum gain and the efficiency of this small antenna are recorded as 5 dBic and 80%, respectively, across the bandwidth. The dimension 0 3 whereas the ground plane size is of the H-shaped patch is 0 3 18 18 . Index Terms—Axial-ratio bandwidth, H-shaped patch antenna, returnloss bandwidth, singly-fed circularly polarized antenna.

I. INTRODUCTION With the rapid development of wireless communication systems, circularly polarized (CP) wideband antennas using single-patch have received considerable attention within the antenna community. Suspended plate/patch antenna (SPA) is one of the simple yet cost-effective approaches. A copper or aluminum plate directly suspended over a ground plane at a large fractional height of the free-space wavelength, which not only just minimizes the fabrication cost but also eradicates the dielectric loss [1], [2]. In general, SPA uses air as the dielectric substrate so that the cavity height can be easily adjusted for obtaining an optimal impedance matching. The H-shaped patch antenna is advantageously small compared to the conventional rectangular patch for a given resonant frequency [3], [4]. Recently, a conventional probe-fed CP antenna using a square patch with a perturbation in form of dual slits was proposed. It was named as H-shaped microstrip antenna [5]. It is well-known that a linearly polarized rectangular patch antenna laid in xy -plane, can be excited at the fundamental resonant mode TM01 if a feed is located along the y -axis, or operated in the TM10 mode if the feed is moved to a point along the x-axis. However, when a square patch is fed at its diagonal line, either 45 or 135 with respect to the x-axis, the two near degenerative orthogonal modes TM01 and TM10 are excited simultaneously. Therefore, a circularly polarized patch antenna using a single-feed point can be produced, whilst its sense of CP radiation (left-hand or right-hand) in the broadside direction is determined by the feed location as well as the sign of perturbation. The near-square patch is understood as a square patch with a strip (positive perturbation) added on one side whereas the H-shaped patch can be regarded as two rectangles subtracted from a square patch (negative perturbation). However, the operational bandwidth of the singly-fed CP patch antennas is usually constrained by its narrow axial-ratio bandwidth. The 2.45-GHz CP square patch with dual slits reported in [5] has a small operating frequency range of 1.3%, which is the overlapped Manuscript received August 18, 2009; revised April 06, 2010; accepted April 07, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by the University Grants Committee of the Hong Kong Special Administrative Region, Research Grants Council General Research Fund PolyU 5162/09E. The author is with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055794

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bandwidth between the return-loss bandwidth (RLBW) and the boresight axial-ratio bandwidth (ARBW) of CP antenna. This narrow bandwidth is attributed to a high unloaded -factor of the parallel-plate cavity when a high dielectric constant material is used. This can be verified by using the closed-form expressions for the ARBW and RLBW, respectively, as given by [6]

Q

p 01 ARBW = AR ARQ

(1)

0 1) : RLBW = 2(V SWR Q

(2)

and

Here, we define the maximum achievable bandwidths ratio with specific criteria for a CP patch antenna as

RLBW : BRjRL=AR = ARBW

(3)

AR p BR

When the 3-dB axial-ratio ( = 2) and the 10-dB return-loss (  2) are used as the bandwidths criteria in CP antennas, (3) gives a bandwidths ratio as j10=3  4. That is, the maximum achievable 10-dB RLBW is about 4 times higher than that of the 3-dB ARBW for a given structure of CP patch cavity. Likewise, the bandwidths ratio would reduce to 2.87 for a maximum achievable 14-dB RLBW ( = 1 5), viz., j14=3 = 2 87. The above BR values can be justified by the measured bandwidths of the CP patch antennas reported in [5] and [7], respectively. However, these are the maximum achievable values when considering ARBW and RLBW separately. The relation between their mean (referenced) frequencies is not known. Nonetheless, it is known that the total unloaded -factor is directly related to the dielectric constant of the medium that made up the cavity, but is inversely proportional to the electrical height of the cavity. For a singly-fed CP patch antenna, the perturbation amount inversely varies with the total unloaded -factor [8]. Therefore, a suspended patch structure with an electrically large cavity height gives a small -factor, which in turn leads to a wide bandwidth. The art of the CP antenna design is to tailor the mean frequency of its ARBW close to that of the RLBW so that a wide overlapped/operational bandwidth of the CP antenna can be attained. Based on the aforementioned theories, this communication delivers a presentation on the design of a wideband singly-fed CP patch antenna using a printed monopole fed by a long probe. The printed monopole is regarded as the horizontal arm which gives the same effect of the inverted -probe feeding technique [9]. Because of the large amount of negative perturbation—the rectangle cuttings, the squared patch becomes an H-shaped patch. Numerical simulations are confirmed by experimental results that the proposed singly-fed H-shaped patch antenna features high gain and high efficiency within a wide operational bandwidth.

V SWR

V SWR

:

:

BR

Q

Q

Q

L

Fig. 1. Geometry of the proposed CP H-shaped patch antenna: S = 37, H = 32, a = 13, b = 14, d = 4, F = 24, F = 5:0, F = 2:0. Unit: mm.

S : 

: 

H-shaped patch has a side of = 37 mm (0 30 at 2.45 GHz), and = 32 mm (0 26 ) from the ground plane. In order a height of to excite a circularly polarized wave with a left-hand sense in the broadside direction, two large rectangles perturbation ( 2 ) were cut away from the top and bottom sides of the square patch whilst the monopole is placed at 45 w.r.t. the -axis. It is worthy of note that the printed monopole is arranged towards the patch corner instead of patch centre. This is in contrast to the conventional -probe fed patch cases, at which the horizontal arm is placed inwards, e.g. the antennas presented in [9], [10]. The introduction of capacitive effect on the horizontal monopole allows for compensation to the feed inductance due to the long-probe. Hence, this feed technique gives a better impedance matching. The use of printed monopole instead of a 90 -bended copper wire provides extra degrees of freedom on the impedance matching, so that the mean frequencies can be adjusted closely. Moreover, the use of the feed layer (RO4003C) and nylon bolts allows a tight control on the alignment between the feed and the small patch. The use of nylon bolts over the foam also gives an advantage on feasible fine-adjustment of the air-gaps during the measurement phase. The ground plane of this antenna is implemented by using a 1-mm thick aluminum plate having a size of 220 mm 2 220 mm.

H

a b

x

L

II. ANTENNA CONFIGURATION The proposed CP antenna has a simple structure as depicted in Fig. 1: a single H-shaped copper plate of 0.5 mm thick is suspended, by using three nylon bolts, over a very thin dielectric layer: 0.2-mm = 0 0021). This feed layer is used thick RO4003C ( r = 3 38, for the printing of a horizontal stripline (monopole) which diagonally excites the patch. The height of the feed layer, V = 24 mm, is the length of the vertical probe, whereas the horizontal monopole has a width and a length of w = 2 mm and H = 5 mm, respectively. According to the aforementioned singly-fed CP patch theory, the

"

: tan

:

F

F

F

III. RESULTS AND DISCUSSION To investigate the performance of the proposed antenna configuration in terms of good boresight axial-ratio bandwidth and return-loss bandwidth, the EM simulator Ensemble was used for full-wave simulations in the design and optimization phases. A systematic tuning method [11] has been used to obtain the wideband CP characteristics. The aim is to achieve an overlapped bandwidth by getting the individual mean frequencies close to each other, viz., at 2.45 GHz. A photograph of the assembled small H-shaped CP antenna is shown in Fig. 2. It can

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Fig. 2. Photograph of the wideband CP H-shaped patch antenna.

Fig. 4. Measured maximum gain and antenna efficiency versus frequency.

Fig. 3. Measured return-loss and on-axis axial-ratio compared to simulated results.

of RLBW. One can see that the measured return-loss has a readable 20-dB RLBW despite the small bandwidths ratio (BRj10=3 ) of 1.16, which is less than the theoretical maximum limit of 4. Moreover, the bandwidths ratio and the ARBW obtained in this design are comparable to the results recently reported by another research team [10], in which a square patch with truncated corners fed by a bent-wire L-probe. Fig. 4 shows the measured maximum gain and efficiency of this small antenna. The use of thick air-substrate and low-loss mechanisms in this design leads to a maximum gain of 5.7 dBic and a high efficiency of 95% at 2.45 GHz. These have been recorded from a near-field SATIMO chamber. Over the CP bandwidth, an efficiency of not less than 80% and a maximum gain of not less than 5 dBic can be warranted. Fig. 5 presents a comparison on the surface currents flow on three types of singly-fed CP patch antennas at 2.45 GHz. Fig. 5(a) shows a near-square patch with a positive perturbation included on the bottom edge of a square patch. The one shown in Fig. 5(b) is a square patch with a negative perturbation—the narrow-band H-shaped patch antenna presented in [5]; and the one shown in Fig. 5(c) is the proposed wideband H-shaped patch antenna. All these CP patches are 45 diagonally (w.r.t. x-axis) excited with a zero-degree phase. It is worthwhile to note that the current flows of the two near-degenerative modes on the patches of Fig. 5(a) and Fig. 5(b) is opposite due to using the opposite signs of perturbation. Hence these CP patch antennas produce the opposed senses of CP wave (RHCP vs. LHCP) in the broadside direction. As the amount of perturbation is much larger for the wideband H-shaped patch shown in Fig. 5(c), the current flows are constrained into the horizontal (from left to right) and vertical directions. The corresponding measured far-field patterns of the wideband H-shaped patch at 2.45 GHz in two principal (xz and yz ) planes are shown in Fig. 6. A left-hand sense of CP as the co-polarization with an offset directive is observed. The on-axis cross-polarization discrimination (XPD) of 30 dB (corresponds to 0.5 dB axial-ratio) is verified, nonetheless, the off-axis cross-polarization levels have gone up rapidly due to the effect of radiation from the vertical long-probe.

be seen that the copper plate and the dielectric feed layer were mounted together by means of nylon bolts. The simulated and measured returnlosses and boresight axial-ratios for the proposed antenna are shown in Fig. 3. As can be seen, the measured results are well corresponded with the theoretical simulations. The small H-shaped antenna has a wide 3-dB axial-ratio bandwidth (ARBW) of 2.3–2.77 GHz (18.5%) in simulation and of 2.28–2.77 GHz (19.4%) from the measurement. From the same figure, a 10-dB return-loss bandwidth of 2.22–2.64 GHz (17.2%) has been obtained in simulation and that of 2.21–2.77 GHz (22.5%) has been measured on a vector network analyzer, Agilent E8363C. These wideband characteristics are attributed to the inclusion of a thick air substrate ( 0:26) in the design, which has significantly reduced the Q-factor of the cavity. It is noted that both the measured values on the axial-ratio and return-loss bandwidths are slightly larger than the simulated ones. These are possibly attributed to the nylon bolts and the small holes on the patch that have not been included in the simulation, whereas the fabrication tolerances may also contribute to the discrepancy. Indeed, it is possible to have a wider RLBW by further optimizing the antenna parameters. However, by knowing the operating bandwidth of a CP antenna is limited by the ARBW, we have put the first objective of this design on the maximization of ARBW with an effort on the mean frequencies close to 2.45 GHz. The second objective was set as the impedance matching around 2.45 GHz, rather than a maximization

IV. CONCLUSION An H-shaped geometry of suspended patch antenna for circular polarization is presented in this communication. The proposed CP antenna is singly-fed by a long probe in conjunction with a printed monopole. The measured results illustrate a good agreement with the numerical simulations. A wide axial-ratio bandwidth of 19.4% is achieved, which is entirely fallen into the measured 22.5% return-loss

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Fig. 6. Measured far-field radiation patterns at 2.45 GHz: (a) xz -plane, (b) yz -plane.

patch antenna attained a maximum gain of 5–6 dBic and over 80% efficiency across the operational bandwidth. ACKNOWLEDGMENT The author would like to thank the Microwave and Wireless Communication Laboratory at the Chinese University of Hong Kong, for the assistance in the antenna measurement. Sincere thanks also go to the anonymous reviewers who gave the useful, constructive comments on this work.

REFERENCES

Fig. 5. A comparison of surface current plots on three types of CP radiators: (a) near-square patch antenna with RHCP, (b) narrow-band H-shaped patch with LHCP [5], (c) wideband H-shaped patch antenna with LHCP operation.

bandwidth. In spite of the small bandwidths ratio, it has demonstrated the essence on the design of a wideband CP antenna: the mean frequencies of axial-ratio and return-loss bandwidths are tailored at the desired frequency of 2.45 GHz. Moreover, the present H-shaped

[1] Z. N. Chen and M. Y. W. Chia, “Broadband suspended plate antenna with probe-fed strip,” IEE Proc. Microw., Antennas Propag., vol. 148, pp. 37–40, 2001. [2] P. H. Rao, M. R. Ranjith, and N. Lenin, “Offset fed broadband suspended plate antenna,” IEEE Trans. Antennas Propag., vol. 53, pp. 3839–3842, Nov. 2005. [3] V. Palanisamy and R. Garg, “Rectangular ring and H-shaped microstrip antennas—Alternatives to rectangular patch antenna,” Electron. Lett., vol. 21, pp. 874–876, Sep. 1985. [4] D. Singh, C. Kalialakis, P. Gardner, and P. S. Hall, “Small H-shaped patch antennas for MMIC applications,” IEEE Trans. Antennas Propag., vol. 48, pp. 1134–1141, Jul. 2000. [5] W. C. Liu and P. C. Kao, “Design of a probe-fed H-shaped microstrip antenna for circular polarization,” J. Electromagn. Waves Appl., vol. 21, no. 7, pp. 857–864, 2007. [6] W. L. Langston and D. R. Jackson, “Impedance, axial-ratio, and received-power bandwidths of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 25, pp. 2769–2773, Oct. 2004.

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[7] F. S. Chang, K. L. Wong, and T.-W. Chiou, “Low-cost broadband circularly polarized patch antenna,” IEEE Trans. Antennas Propag., vol. 51, pp. 3006–3009, Oct. 2003. [8] Y. T. Lo and W. S. Richards, “Perturbation approach to design of circularly polarized microstrip antenna,” Electron. Lett., vol. 17, pp. 383–385, May 1981. [9] C. L. Mak, K. M. Luk, K. F. Lee, and Y. L. Chow, “Experimental study of a microstrip patch antenna with an L-shaped probe,” IEEE Trans. Antennas Propag., vol. 48, pp. 777–783, May 2000. [10] S. L. S. Yang, K. F. Lee, A. A. Kishk, and K. M. Luk, “Design of wideband single feed truncated corner microstrip patch antennas for circularly polarized applications,” IEEE Antennas Propag. Symp., pp. 1–4, Jul. 2008. [11] K. L. Chung and A. S. Mohan, “A systematic method to obtain broadband characteristics for singly-fed electromagnetically coupled patch antennas for circular polarization,” IEEE Trans. Antennas Propag., vol. 51, pp. 3239–3248, Dec. 2003.

Polarization Reconfigurable U-Slot Patch Antenna Pei-Yuan Qin, Andrew R. Weily, Y. Jay Guo, and Chang-Hong Liang

Abstract—A compact U-slot microstrip patch antenna with reconfigurable polarization is proposed for wireless local area network (WLAN) applications. PIN diodes are appropriately positioned to change the length of the U-slot arms, which alters the antenna’s polarization state. Two antenna prototypes with identical dimensions are designed, fabricated and measured. The first antenna prototype enables switching between linear and circular polarization by using a PIN diode and a capacitor located on the U-slot. The second antenna prototype uses two PIN diodes to switch between the two circular polarization senses. A good impedance match 10 dB) for both linear and circular polarization is achieved (S from 5.725 to 5.85 GHz, a band typically used for WLAN applications, and the 3 dB axial ratio bandwidth is greater than 2.8%. Details of the simulated and measured reflection coefficient, axial ratio, gain and radiation patterns are presented. Index Terms—Circular and linear polarization, microstrip antennas, reconfigurable antennas, slot antennas.

I. INTRODUCTION Circularly polarized (CP) antennas have been widely used in wireless and satellite communication systems to reduce multipath effects and the need for accurate polarization alignment between transmitting and receiving antennas [1]. In contrast to the traditional techniques including extra stubs and chamfering corners of the square patch, which have the drawback of limited 3 dB axial ratio bandwidth, printed slot antennas are gaining popularity for CP radiation design since they have the potential to improve the CP bandwidth without increasing the antenna size. Recently, a single-layer probe-fed U-slot patch antenna with Manuscript received November 02, 2009; revised February 04, 2010; accepted March 27, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported by the DIISR Australia-China special fund CH080270 and the China Scholarship Council (CSC). P.-Y. Qin is with the National Key Laboratory of Antennas and Microwave Technology, Xidian University, Xi’an 710071, China, and also with the CSIRO ICT Centre, Epping, NSW 1710, Australia (e-mail:[email protected]). A. R. Weily and Y. J. Guo are with the CSIRO ICT Centre, Epping, NSW 1710, Australia. C.-H. Liang is with the National Key Laboratory of Antennas and Microwave Technology, Xidian University, Xi’an 710071, China. Digital Object Identifier 10.1109/TAP.2010.2055808

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circular polarization was presented, which significantly increases both the impedance and the 3 dB axial ratio bandwidth [2]. Polarization reconfigurable antennas, also known as polarization agile antennas, have attracted considerable attention due to their ability to improve the performance of the communication systems through polarization diversity and frequency reuse. Polarization reconfigurable antennas also have applications in multiple-input multiple-output (MIMO) systems. Most of the previous work for polarization reconfigurability concentrates on switching between right hand circular polarization (RHCP) and left hand circular polarization (LHCP) [3]–[8]. Relatively few antennas have been presented that can switch between linear and circular polarization because it is difficult to simultaneously realize a good impedance match for circular and linear polarization [4]. The reason is that CP radiation is generated by two degenerate orthogonal linear modes, the input impedance of which is different from that of one resonant mode used to generate linearly polarized (LP) radiation. However, it will make the antenna more versatile if switching between linear and circular polarization can be achieved [9]. Several interesting designs have been proposed to solve this problem [9]–[11]. In [9], four pin-diodes were used on a corner-truncated square patch to produce LP and CP radiation with a small impedance bandwidth (2.5%). In [10], a perturbed square-ring slot antenna using four pin diodes was designed that allows operation in both CP and LP modes. Unfortunately, the biasing and control circuits were not physically implemented. In [11], a ring-slot-coupled microstrip circular patch antenna, fabricated on two single FR4 substrates separated by a piece of foam, was proposed that can switch between LP and CP modes. However, the overlapped operating frequency bandwidth is 2.2% and it is difficult to integrate such an antenna in a compact wireless device due to its large volume. A good summary of research on polarization reconfigurable antennas can be found in [12]. In this communication, a microstrip U-slot patch antenna is proposed to allow switching either between linear and circular polarization or between two circular polarization senses. The antenna is compact and can cover the WLAN frequency band with good impedance bandwidths for both CP (11.8%) and LP (6.1%) modes and 3 dB axial ratio bandwidth (>2:8%) for CP mode. The design is based on the circularly polarized U-slot antenna [2]. Compared to [2], [11], however, a single layer microwave substrate is used for ease of fabrication. Two beam-lead PIN diodes embedded into the slot at specific positions enable the length of the U-slot arms to be varied. Two orthogonal modes with quadrature phase can be excited by the asymmetrical U-slot to generate circular polarization. By turning the diodes on or off, the U-slot becomes either symmetrical or asymmetrical, which allows the patch antenna to switch between linear and circular polarization states. The validity of this concept is demonstrated by two identical antennas that achieve good agreement between the simulated and measured results. The complete dc bias network and equivalent circuits of the PIN diodes have been included in the antenna design. A bias-tee is used to superimpose the bias voltage on the RF signal; hence, the dc-bias circuit for the PIN diodes is relatively simple. II. RECONFIGURABLE U-SLOT ANTENNA DESIGN The configuration of the proposed reconfigurable U-slot patch antenna is shown in Fig. 1. The parameters and dimensions of the antenna are given in Table I. The length of the patch, denoted as L4, is  0:35 g (14.2 mm), where g is the guided wavelength. This value has been obtained from the parametric analysis in Section III. g is given by 0 =p"e , where 0 is the wavelength in free-space, "e  ("r + 1)=2 is the effective dielectric constant of the microstrip patch, and "r = 2:2 is the dielectric constant of the substrate. A U-slot

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Fig. 1. Schematics of the reconfigurable U-slot antenna (a) top view and (b) side view. TABLE I DIMENSIONS OF THE PROPOSED RECONFIGURABLE U-SLOT ANTENNA

TABLE II POLARIZATION STATES OF THE PROPOSED RECONFIGURABLE U-SLOT ANTENNA Fig. 2. Simulated performance of the proposed antenna as a function of L4 (a) input reflection coefficient for State 1 and (b) input reflection coefficient for State 2.

the y-axis in Fig. 1. The possible polarization states (RHCP, LHCP and LP) and the corresponding diode states are summarized in Table II. is inserted into a rectangular patch which is printed on a microwave substrate. As the length of the PIN diode is smaller than the width of the slot, conducting pads are placed in the gaps of the U-slot for PIN diodes attachment. The feeding probe connected to the U-slot patch through the ground plane and substrate is offset from the top edge of the patch by L5. It should be noted that in the antenna prototype, another thinner slot is cut on the top of the U-slot, which separates the patch into two parts to ensure the dc isolation. Three 30 pF capacitors are placed across this thinner slot to maintain RF continuity. The outer part of the patch is dc grounded by a shorting pin through an inductor used as an RF choke. Both the dc bias voltage and the RF signal are simultaneously fed through the coaxial probe by using a bias-tee. Beam lead PIN diodes (MA4AGBLP912) are used as switching elements in the U-slot. According to the PIN diode datasheet [13], the diode represents a resistance of 4 for the ON state and a parallel circuit with a capacitance of 0.025 pF and a resistance of  4 k for the OFF state. The length of the U-slot arm can be changed by changing between the different states of the diodes. When the left diode is on and the right diode is off, the RF current can flow across the left arm of the U-slot. In this case, the left arm of the U-slot is shorter than the right arm. In addition, the asymmetrical U-slot can excite two orthogonal modes in the patch. Adjusting the location of the PIN diodes can make the two modes have the same magnitude and a phase difference of 90 at a given frequency, thus enabling the antenna to generate CP radiation with a low axial ratio. The antenna radiates LHCP when the left arm of the U-slot is longer than the right arm. RHCP can be achieved if the right arm is longer than the left arm. The U-slot becomes symmetrical when both of the diodes are on or off, which enables the antenna to radiate linear polarization [2]. In this case, the electric field polarization is parallel to

III. PARAMETRIC STUDY OF THE PATCH LENGTH Two important parameters which affect the input reflection coefficient and axial ratio of the proposed reconfigurable U-slot antenna are the patch length L4 and the ratio of the length of the asymmetrical U-slot arms. Since a parametric study of the latter has already been reported in [2], we only examine the effects of the former in this communication. The other parameters remain constant and have the values given in Table I. To complete the parametric analysis for five different patch lengths, full-wave simulations using the commercial software CST Microwave Studio [14] have been performed on the antenna shown in Fig. 1. As the antenna structure is basically symmetrical along the yz-plane, except for the bias line, the RHCP and LHCP should have similar axial ratio bandwidth. Therefore, for the analysis of the axial ratio, only results for LHCP are presented. In addition, the effects on the input reflection coefficient of both LP states are examined. Fig. 2(a) and (b) show the effects of L4 on the resonant frequencies of State 1 and State 2, respectively. From this figure, it is observed that the resonant frequencies of both LP modes decrease as the length of the patch increases. For LP State 1, the antenna is not suitable for most applications as the input reflection is too large. For LP State 2, the impedance match is acceptable for WLAN applications as long as L4 is larger than 14.2 mm. Fig. 3(a) and (b) show the effects of L4 on the axial ratio and input reflection coefficient of the CP mode, respectively. As seen from Fig. 3(a), the 3 dB axial ratio bandwidth increases as L4 decreases. However, L4 can not be smaller than 14.2 mm if the LP mode is desired. This can be viewed as a compromise in order for the antenna to provide linear polarization; hence the optimum value of L4 is chosen to be 14.2 mm. The 3 dB axial ratio bandwidth is 2.8% with respect to the center frequency of 5.75 GHz when L4 is 14.2 mm. From Fig. 3(b),

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Fig. 3. Simulated performance of the proposed antenna as a function of L4 (a) axial ratio and (b) input reflection coefficient for the CP mode.

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Fig. 4. Configurations of the proposed polarization reconfigurable U-slot antennas (a) Antenna I and (b) Antenna II.

it is seen that the CP operating frequency bands are all located within the range from 5.6 to 6.3 GHz as L4 is varied, which is wide enough to cover the 5.725–5.85 GHz frequency band. IV. SIMULATED AND MEASURED ANTENNA PERFORMANCE According to the above analysis and Table II, a single U-slot antenna that switches between linear polarization, RHCP or LHCP can be designed as long as the diodes are independently biased. Unfortunately, the bias network required will be relatively complicated. Since the aim of the communication is to validate the concept, two polarization reconfigurable U-slot antennas with the same dimensions shown in Table I are designed and measured. The antennas can switch either between linear and circular polarization (denoted as Antenna I) or between two circular polarization senses (denoted as Antenna II). Fig. 4(a) and (b) present the configurations of Antennas I and II, respectively. The antennas are printed on a 3.175-mm-thick RT/duroid 5880 substrate (dielectric constant "r = 2:2; tan  = 0:0009). For Antenna I, a 30 pF capacitor is soldered onto the right pad to create a short circuit across the right arm of the U-slot. Meanwhile, a PIN diode D1 is mounted across the other arm using electrically conductive silver epoxy. When a forward bias is applied to turn on D1 , the diode acts as a resistor of 4 and the RF current can flow across the slot. In this case, the two arms of the U-slot are of the same length, which leads to linear polarization mode. LHCP mode can be obtained when D1 is turned off by changing the polarity of the dc voltage. Antenna I can also provide the switching between RHCP and linear polarization if the positions of the capacitor and the pin diode are interchanged. Simulated and measured input reflection coefficients versus frequency for the CP and LP modes are shown in Fig. 5(a). Simulated and measured axial ratios at boresight for the CP mode are given in Fig. 5(b). From the experimental results, it is observed that the impedance bandwidths

Fig. 5. Simulated and measured (a) input reflection coefficient for the LP and CP modes and (b) axial ratio for the CP mode of Antenna I.

for the LP and CP modes are 6.1% and 13.5%, respectively, with almost the same center frequency of 5.9 GHz, which can cover the entire

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Fig. 6. Simulated and measured (a) input reflection coefficient for the LHCP and RHCP modes and (b) axial ratio for the LHCP and RHCP modes of Antenna II.

5.725–5.85 GHz WLAN band. The measured 3 dB axial ratio bandwidth at boresight extends from 5.7–5.86 GHz. For Antenna II, two pin diodes (D2 , D3 ) are embedded across the two arms of the U-slot. The orientation of the two pin diodes is shown in Fig. 4(b). As the bias voltage is supplied from the coaxial probe, reversed dc voltage polarities are applied to D2 and D3 . If a positive voltage is supplied, D2 is turned off and D3 is turned on and the antenna radiates LHCP. When a negative voltage is supplied, D3 is turned off and D2 is turned on and the antenna radiates RHCP. Fig. 6(a) presents the simulated and measured reflection coefficients versus frequency for the LHCP and RHCP modes. It can be observed that the frequency bandwidths of both CP senses are similar to the result of the CP mode of Antenna I in Fig. 5. However, it is clear that the impedance match of LHCP is better than that of RHCP, which can be mostly attributed to the existence of microstrip bias line. Simulations show that if we put the bias line on the other side of the patch, the impedance match of RHCP will be better than that of LHCP. Radiation patterns were measured for the LP and CP modes in a spherical near-field (SNF) system. For the CP modes, the axial ratio was measured as well. Fig. 6(b) shows the simulated and measured axial ratio for both CP senses. The 3 dB axial ratio bandwidths of LHCP and RHCP are 3.1% and 2.8% respectively with the same center frequency of 5.77 GHz. In Figs. 7–10, the simulated and measured normalized radiation patterns of both co- and cross-polarization at 5.7875 GHz are compared. The radiation plots are as follows: Figs. 7 and 8 show radiation patterns for LP and CP modes of Antenna I, respectively; Figs. 9 and 10 show radiation patterns for LHCP and RHCP modes of Antenna II, respectively. It can be observed from Fig. 7 that the cross polarization level for the linear polarization remains below 012 dB. Additionally, realized gain is measured by using the gain comparison technique. The

Fig. 7. Simulated and measured (a) y-z plane and (b) x-z plane normalized radiation patterns at 5.7875 GHz for the LP mode of Antenna I.

Fig. 8. Simulated and measured (a) y-z plane and (b) x-z plane normalized radiation patterns at 5.7875 GHz for the CP mode of Antenna I.

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Fig. 11. Simulated and measured gains for (a) Antenna I and (b) Antenna II.

Fig. 9. Simulated and measured (a) y-z plane and (b) x-z plane normalized radiation patterns at 5.7875 GHz for the LHCP mode of Antenna II.

losses of the cable and bias tee have been calibrated out of the gain measurement. The measured gains for the LP and CP modes are plotted in Fig. 11. From the figure, it is observed that the measured gains vary from 6.3–7.5 dBic and 6.25–7.3 dBi for CP and LP modes across the WLAN band respectively, which agree reasonably well with the simulated results. It can be noted that for Antenna II the gain of RHCP is 0.6 dB less than that of LHCP. The reason for the difference is that the values of the resistance of the two PIN diodes for the ON state are not exactly the same; hence the losses caused by the PIN diodes are different. The loss of the PIN diode also affects the efficiency of the antenna. The measured efficiency is obtained from the difference between the measured gain and directivity. The measured efficiencies of Antenna I for the LP and CP modes at 5.7875 GHz are 73% and 82%, respectively. The difference is due to the fact that for the LP mode the PIN diode acts as a 4 resistor but for the CP mode the PIN diode is turned off. Therefore, the loss for the CP mode is less than that of the LP mode. In addition, the measured efficiencies of Antenna II for the LCHP and RHCP modes at 5.7875 GHz are 86% and 78%, respectively. The efficiencies of Antenna II are different from those of Antenna I since the loss elements for Antenna I and Antenna II are different, and there are also slight variations in the resistance of individual diodes. V. CONCLUSION

Fig. 10. Simulated and measured (a) y-z plane and (b) x-z plane normalized radiation patterns at 5.7875 GHz for the RHCP mode of Antenna II.

A U-slot microstrip patch antenna with polarization reconfigurability has been demonstrated. PIN diodes are used to enable the antenna to radiate linear polarization, RHCP and LHCP. To verify the design concept, two antenna prototypes with reconfigurable polarization have been fabricated and measured. The first prototype switches between linear and circular polarization, while the second prototype switches between the two senses of circular polarization. The measured impedance bandwidth extends from 5.6 to 6.3 GHz and 5.72 to 6.08 GHz for CP and LP modes respectively, which can cover the

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entire WLAN band. An axial ratio bandwidth of 2.8% with the same center frequency of 5.77 GHz for CP modes was achieved. A simple dc bias network controlling the polarization states is applied to the antenna prototypes. The reconfigurable U-slot antenna has the further advantages of being compact and easy to manufacture, which makes it highly suited to the advanced wireless communication system. ACKNOWLEDGMENT The authors thank M. Shen for assisting with the attachment of the PIN diodes.

REFERENCES [1] I. J. Bahl and P. Bhartia, Microstrip Antennas. New York: Artech House, 1980. [2] K. F. Tong and T. P. Wong, “Circularly polarized U-slot antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2382–2385, Aug. 2007. [3] F. Yang and Y. Rahmat-Samii, “A reconfigurable patch antenna using switchable slots for circular polarization diversity,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 3, pp. 96–98, Mar. 2002. [4] M. K. Fries, M. Grani, and R. Vahldieck, “A reconfigurable slot antenna with switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 11, pp. 490–492, Nov. 2003. [5] A. Khaleghi and M. Kamyab, “Reconfigurable single port antenna with circular polarization diversity,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 555–559, Feb. 2009. [6] M. Boti, L. Dussopt, and J. M. Laheurte, “Circularly polarized antenna with switchable polarization sense,” Electron. Lett., vol. 36, no. 18, pp. 1518–1519, Aug. 2000. [7] S. H. Hsu and K. Chang, “A novel reconfigurable microstrip antenna with switchable circular polarization,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 160–162, 2007. [8] K. F. Tong and J. J. Huang, “New proximity coupled feeding method for reconfigurable circularly polarized microstrip ring antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 1860–1866, Jul. 2008. [9] Y. J. Sung, T. U. Jang, and Y. S. Kim, “A reconfigurable microstrip antenna for switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 11, pp. 534–536, Nov. 2004. [10] W. M. Dorsey and A. I. Zaghloul, “Perturbed square-ring slot antenna with reconfigurable polarization,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 603–606, 2009. [11] R. H. Chen and J. S. Row, “Single-fed microstrip patch antenna with switchable polarization,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 922–926, Apr. 2008. [12] S. Gao, A. Sambell, and S. S. Song, “Polarization-agile antennas,” IEEE Antennas Propag. Mag., vol. 48, no. 3, pp. 28–37, Jun. 2006. [13] MA4AGBLP912, M/A-COM. [14] CST Microwave Studio. Darmstadt, Germany, 2009.

High-Efficiency On-Chip Dielectric Resonator Antenna for mm-Wave Transceivers Mohammad-Reza Nezhad-Ahmadi, Mohammad Fakharzadeh, Behzad Biglarbegian, and Safieddin Safavi-Naeini

Abstract—A high radiation efficiency on-chip antenna is presented in a low-resistivity silicon technology. The proposed antenna configuration consists of a high-permittivity rectangular dielectric resonator excited by an H-slot antenna implemented in a silicon integrated circuit process. Using the Wheeler method an efficiency of 48% has been measured for the integrated antenna at 35 GHz. The maximum size of this low profile antenna mm) is close to (considering the dielectric resonator), and ( its radiation gain is around 1 dBi at 35 GHz. Moreover, the bandwidth of this antenna is 4.15 GHz (12%). Simulations and measurements show that by removing the passivation layer on top of the H-slot aperture the radiation efficiency increases by 10%.

=05

5

Index Terms—Antenna efficiency, millimeter wave antennas, on-chip antennas, silicon, SiGe, slot antenna.

I. INTRODUCTION On-chip antennas are essential for implementing fully integrated radio systems. An on-chip antenna significantly simplifies the matching network and improves the system performance through reducing the front-end loss and noise figure. Many of the on-chip antennas take advantage of low-cost semiconductor processing technologies, such as silicon-germanium (SiGe) and complementary metal oxide semiconductor (CMOS) due to their maturity and high integration capabilities [1], [2]. To radiate the maximum amount of the input power or extend the battery life the efficiency of the antenna must be as high as possible. On the other hand, the on-chip antenna dimension is the dominant factor in determining the chip area; hence, its size must be as small as possible to lower the fabrication cost. Antenna miniaturization can be performed by using compact configurations such as H-slot or employing high permittivity material. Miniaturizing the antenna while maintaining a high radiation efficiency is a challenge [3]. This communication reports the design procedure and the measured results of a high-efficiency electrically small integrated antenna. In the proposed configuration an H-slot aperture is used to excite a high-permittivity dielectric resonator (DR) placed on top of the radio chip. The effect of removing the passivation layer (PL) on top of the slot-antenna on the efficiency of the antenna is experimentally studied. The simulated efficiency of the proposed on-chip antenna is 59%. Measured results based on the Wheeler method verify that the radiation efficiency of this system is more than 48%. This is, to the authors’ knowledge, the highest measured efficiency for an on-chip antenna in low resistivity silicon . Manuscript received November 03, 2009; revised February 15, 2010; accepted April 12, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by the National Sciences and Engineering Research Council of Canada (NSERC), Research in Motion (RIM), and in part by CMC Microsystems. The authors are with the Electrical and computer Engineering Department, University of Waterloo, ON N2L 3G1, Canada, (e-mail: mrnezhad@maxwell. uwaterloo.ca). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055802

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Fig. 1. On-chip cavity backed slot antenna with dielectric resonator on top. (a) Cross section. (b) 3D structure (without the passivation layer).

II. ON-CHIP ANTENNA CONFIGURATION Any improvement in the antenna efficiency results in a larger transmission range for a fixed input power, or less power consumption for a fixed transmission range. This section proposes a configuration to increase the efficiency of the on-chip antenna and reduce the crosstalk to other circuits on the same substrate. A. Proposed On-Chip Antenna Structure Fig. 1 illustrates the proposed configuration for the integrated on-chip antenna, which consists of: silicon substrate, cavity and shield layer, H-slot aperture, the passivation layer, and DR. The thickness of silicon substrate is about 300 m, and its resistivity is 13.5 -cm. The slot aperture is implemented on the top metal layer (MT) of the silicon technology. The lowest metal layer (M1) is connected to the top ground plane through via holes. By shielding the antenna from the lossy silicon substrate, the antenna substrate is limited to silicon dioxide (SiO2 ) and the intermediate dielectric layers between metal layers. Also, a cavity is formed under the slot antenna. In Section III, it is shown that the PL can be removed for more efficient power coupling to the dielectric resonator. Finally, a dielectric layer with a large permittivity is placed on top of the chip to increase the radiation efficiency and improve the antenna matching. B. H-Slot Antenna and Dielectric Resonator H-slot is an aperture-type electrically small antenna. A slot aperture can be end-loaded to reduce its overall length at a given resonance frequency [4]. For example, in this work the maximum size of the slot antenna is 1.15 mm and the measured resonance frequency is 35 GHz. Thus, the overall length of the antenna is almost 0 =8, or 4 times smaller than a half-wavelength dipole operating at the same frequency. The radiation efficiency of such an electrically small antenna is small. To improve the radiation efficiency of the integrated antenna, a layer of high dielectric constant material is added on top of the slot aperture to create a rectangular DR antenna [5]. The slot behaves like a magnetic current, which excites the first order mode of the dielectric

Fig. 2. (a) Simulated reflection coefficient of the on-chip antenna with the DR. (b) 2D normalized radiation patterns. (c) Co-pol and Cross-pol patterns.

resonator in the proposed structure. Thus, the integrated antenna can radiate a larger portion of the input power. C. Simulation Results The H-slot was designed and optimized to have a resonance frequency close to that of the rectangular DR. The length, width, height, and dielectric constant of the DR are respectively 1.6 mm, 1.1 mm, 0.5 mm, and 38. Fig. 2(a) shows the simulated reflection coefficient of the on-chip antenna with DR for the TE11 mode. The resonant frequency is 34.5 GHz and the 10 dB bandwidth of the on-chip antenna is more than 3 GHz. The 2D radiation patterns in ' = 0 and ' = 90 planes of the antenna are shown in Fig. 2(b). The antenna pattern covers the upper half plane and the maximum gain of the antenna structure is 1.06 dBi. Furthermore, the half-power beamwidth of the on-chip antenna exceeds 130 in ' = 90 plane. Fig. 2(c) shows that the cross-polarization pattern is at least 20 dB below the co-polarization pattern at the resonant frequency. III. MEASUREMENT RESULTS Fig. 3(a) shows the die micrograph of the fabricated H-slot antenna and its dimensions. The H-slot antenna was implemented using the IBM SiGe5AM process. Two vertical slots in Fig. 3(a), which constitute the radiating sections of the antenna, have 500 m length and 100 m width. The horizontal sections on the top and bottom, which load the radiating slots, are 1 mm long and their width is 50 m. While the allocated silicon area for the implementation of antenna is 2 mm 2 1.5 mm, the largest antenna dimension is only 1.15 mm. The feed network, shown in Fig. 3(a), is a CPW line with 850 m length, which is probed with Micro-tech CPW probes for test purpose. In Fig. 3(b) a rectangular DR is placed over the slot to improve the radiation efficiency. The DR used in this work had a dielectric constant of 38 6 1 and a size of a = 1:6 mm, b = 1:15 mm, and h = 0:5

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Fig. 3. (a) The dimensions of the H-slot aperture. (b) Die micrograph of the H-shape slot with DR and CPW probe.

Fig. 5. Radiation pattern measurement set-up for the on-chip antenna. The inset shows the CPW probe coupled to the antenna under test surrounded by wave absorber.

Fig. 4. Measured reflection coefficient of the on-chip antenna with and without the passivation layer (with and without the DR).

mm. Nine samples of on-chip antenna were fabricated and measured: four samples with the PL and five without this layer. The DR was not glued to the H-slot antenna. In the following, the measured results of the different parameters of the fabricated antennas are presented. A. Resonance Frequency The first test comprised of measuring the reflection coefficient of the antenna with/without the DR and the PL. Fig. 4 shows the PL effect on the resonance frequency of the on-chip antenna. Removing the PL decreases the resonance frequency by less than 0.5 GHz. The resonance frequency of the sample without the PL shown in this figure is 34.8 GHz with the DR and 36.5 GHz without the DR.

Fig. 6. Normalized measured and simulated radiation patterns of the antenna at 35 GHz. The 3 dB circle is used to find the beamwidth of antenna.

0

of 130 . Part of this difference is due to the larger ground size of the fabricated samples compared to the simulated model. The rest can be explained by the limited number of the measured points (15 points), and the measurement error.

B. Radiation Pattern Measurement The radiation pattern measurement setup is shown in Fig. 5. To measure the radiation pattern, a standard gain horn antenna (SGH) was rotated above the on-chip antenna and the received power level by the horn antenna was recorded for each angle. The recorded data was calibrated using the SGH pattern, to find the radiation pattern of the desired structure. The rods and everything around the measurement setup were covered with wave absorbers to avoid any reflected signal. With this method, only the upper hemisphere radiation pattern of the antenna can be measured. Using the described test set-up the radiation pattern at ' = 90 plane (y-z plane) was measured. The simulation and measurement result are shown in Fig. 6. The measured half-power beamwidth of the on-chip antenna is close to 110 compared to the simulated value

C. Gain Measurement The set-up shown in Fig. 5 was used to measure the radiation gain of the antenna. Port 1 of the network analyzer was connected to the SGH, and Port 2 to the CPW probe which fed the on-chip antenna. All scattering parameters were measured with and without the DR. The return loss of the SGH was more than 10 dB over 30–40 GHz range. The measured S12 increased significantly by overlaying the DR on the H-slot antenna. Fig. 7 shows the improvement in the received power within the 10 dB bandwidth of the on-chip antenna (33–37 GHz). The improvement varied from 6 dB to 17 dB (after removing the effect of the measurement background noise). The average improvement over the whole 10 dB bandwidth was 12.7 dB

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Fig. 7. Measured improvement in the received power by the horn antenna after laying the DR on the H-slot, over the 10 dB bandwidth of the antenna.

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Fig. 8. The Wheeler test results. W1: with the PL and with cap, P1: with the PL-no cap, N1: no passivation-no cap, W2: no passivation with cap.

with the PL and 13.7 dB without the PL. The S-parameter measurements were used to estimate the gain of the on-chip antenna through the following relation [6]:

j

S21 j2

= (1 0 jS11 j2 )(1 0 jS22 j2 )

GHorn 1 GOn0Chip

 4R

2

(1)

where R is the distance between the horn and on-chip antennas (R = 19:5 cm in this test). In (1) it has been assumed that all sources of insertion loss from port 1 to port 2 of VNA have been compensated, otherwise the uncompensated insertion loss in the test set-up must be added to the right-side of (1). The unbiased product of horn and on-chip antenna gains derived from (1) is 15 dBi at 35 GHz. The maximum gain of horn antenna at 35 GHz is 15.5 dBi. There is 1 dB insertion loss for the coaxial to waveguide adapter, which connects the horn antenna to the RF cable of the network analyzer. Considering the adapter loss and the polarization mismatch the measured gain of the on-chip antenna at 35 GHz is above 0.5 dBi, which is in good agreement with the simulated gain of 1.06 dBi in Section II.C. D. Efficiency Measurement In this part, the Wheeler method is used to measure the efficiency of the on-chip antenna [7]. The Wheeler cap used in this work comprised of a cylindrical cavity, with 10 mm diameter and 8 mm depth, grooved in an Aluminum cube. A small opening was made for the CPW probe. To apply the Wheeler method, the reflection coefficient (S11 ) of the on-chip antenna system (with the DR) was measured twice: with and without the Wheeler cap. Then 10 dB fractional bandwidth of the antenna (BW) was calculated in each case, and then the Q factor was found from the following relation [8]:

Q=

v01 vBW

p

(2)

where v is the voltage standing wave ratio. Let QW and Q0 denote the Q factors of the DR antenna with and without the Wheeler cap.

Fig. 9. Simulated radiation efficiency of the entire structure (DR antenna plus chip) which shows a maximum at the resonance frequency.

Then the efficiency at the resonance frequency is calculated from [5]

 =10

Q0 : QW

(3)

Fig. 8 shows the results of the efficiency measurement test for the antenna with and without the PL. For the former case the values of Q0 and QW are calculated as 5.4 and 8.7. Thus, the efficiency for the DR antenna with the PL is more than 38%. The values of Q0 and QW for the antenna without the PL are calculated as 5.9 and 11.5. Thus, the radiation efficiency is more than 48%. These results show that removing the PL increases the on-chip antenna efficiency by 10%. To find the efficiency of the proposed on-chip antenna at other frequencies, an EM simulator (HFSS) is used to integrate the received power on a closed surface. Measured results were used to modify the antenna model. Fig. 9 shows the simulated efficiency with and without the PL for a frequency range of 33 to 37 GHz. The maximum efficiency with the PL is 49% at 35 GHz and 59% without the PL at 34.9 GHz. In full agreement with measured results the efficiency increases by 10% when the PL is removed. The 11% difference between the maximum

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simulated and measured efficiency is caused by the small opening in the Wheeler cap, Wheeler cap loss, possible shift of the DR after placing the cap atop the antenna, and the measurement/calibration errors.

Investigation Into the Effects of the Reflection Phase Characteristics of Highly-Reflective Superstrates on Resonant Cavity Antennas Alireza Foroozesh and Lotfollah Shafai

IV. CONCLUSION This communication introduced a high-efficiency on-chip antenna in SiGe MiMIC technology. The antenna consisted of an on-chip H-slot antenna and a rectangular dielectric resonator. A shielding mechanism was implemented to isolate the radiating section from the lossy silicon substrate. Using the Wheeler method a radiation efficiency of 48% was measured. This electrically small antenna has a relatively large bandwidth of 12% operating from 33 to 37 GHz. It was shown that adding the high-permittivity rectangular DR, improves the efficiency and matching of the antenna structure by 17 dB. Moreover, it was shown that removing the passivation layer on top of the slot improves the coupling between DR and H-slot antenna, which increases the radiation efficiency by 10%. This result was confirmed by both measurements and simulations.

REFERENCES [1] A. Shamim, L. Roy, N. Fong, and N. G. Tarr, “24 GHz on-chip antennas and balun on bulk Si for air transmission,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 303–311, Feb. 2008. [2] P. V. Bijumon, Y. Antar, A. P. Freundorfer, and M. Sayer, “Dielectric resonator antenna on silicon substrate for system on-chip applications,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3404–3410, Nov. 2008. [3] N. Behdad, “Single- and dual-polarized miniaturized slot antennas and their applications in on-chip integrated radios,” presented at the IEEE Int. Workshop on Antenna Technology, iWAT 2009, Mar. 2–4, 2009. [4] B. G. Porter and S. S. Gearhart, “Impedance and polarization characteristics of H and IHI slot antennas,” IEEE Trans. Antennas Propag., vol. 48, no. 8, pp. 1272–1274, Aug. 2000. [5] K. M. Luk and K. W. Leung, Dielectric Resonator Antennas. New York: Research Studies Press, Jun. 2002. [6] K. Payandehjoo and R. Abhari, “Characterization of on-chip antennas for millimeter-wave applications,” presented at the Int. Symp. Antennas Propag., Jun. 1–5, 2009. [7] H. A. Wheeler, “The radian sphere around a small antenna,” in Proc. IRE, Aug. 1959, vol. 47, no. 8, pp. 1325–1331. [8] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and Q of antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, Apr. 2005.

Abstract—First we describe two frequency selective surfaces (FSSs), one capacitive and the other inductive, that are designed to exhibit identical high-reflection magnitude at an arbitrary frequency. These two FSSs are then employed as the superstrate of two RCAs having identical microstrip patch source. In order to determine the resonant conditions and obtain approximate values for the antenna directivity, RCAs are initially analyzed using the well known simple ray-tracing method. Next, a full-wave analyzer (ANSOFT Designer v4.0), based on the method of moments (MoM), is utilized to thoroughly analyze the RCAs. Experimental results are provided to support the full-wave simulations, as well. In contrast to the prediction of the ray-tracing modeling, which is merely based on the reflection magnitude of the FSSs, it is pointed out that their phase properties have noticeable effects on the RCA gain. Second, two other RCAs are designed based on high permittivity and high permeability superstrates with identical contrast. There, too, it is shown that the reflection phases of the RCA superstrates determine the air-gap heights which in turn affect the RCA gains. Index Terms—Antenna gain, antenna input impedance, antenna radiation patterns, frequency selective surface (FSS).

I. INTRODUCTION A highly-reflective surface can be used as the antenna superstrate to substantially increase its directivity [1]. The phenomenon resulting in this significant gain enhancement is based on multiple reflections between the highly-reflective superstrate and the antenna ground plane similar to the Fabry-Perot resonator. Using a ray-tracing method, a simple formula has been derived in [1], which shows the relationship between the reflection magnitude of the superstrate and the relative increase in the antenna directivity by adding that superstrate. This simple relationship has been proven to be relatively accurate when highly-reflective capacitive FSSs, such as periodic patches or strips, have been employed as the RCA superstrates [2], [3]. However, the accuracy of this relationship has neither been studied nor verified in the literature, when inductive FSSs are used as the RCA superstrate. As it is known, the resonance length of the RCA is determined by the reflection phase of the FSS and the ground plane. It has been shown in [4] that for the RCAs whose superstrates are identical high-permittivity dielectrics but their ground planes are PEC and PMC surfaces, the RCA having PEC ground plane produces higher directivity and resonance length. Similar phenomena have been observed in [5], when metamaterial ground planes are utilized in designing RCAs. In [5], it has been shown that the antenna with shorter resonance length exhibits lower directivity. Therefore, the purpose of this communication is to investigate the effects of the reflection phase of the FSS superstrates, as opposed to the ground planes carried out in [4] and [5], on the RCAs directivity. This effect is also studied for the RCAs having high permittivity or permeability superstrate layers. Manuscript received November 06, 2009; revised March 12, 2010; accepted April 09, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by the Natural Science and Engineering Research Council of Canada (NSERC). A. Foroozesh and L. Shafai are with the department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T 5V6, Canada (E-mail: [email protected], [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055810

0018-926X/$26.00 © 2010 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 10, OCTOBER 2010

Fig. 1. RCA having (a) FSS superstrate and (b) high permittivity/permeability superstrate.

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Fig. 3. Unit cell of (a) capacitive (patch-type) FSS whose dimensions are p = 10 mm and s = 9:8 mm and (b) inductive (aperture-type) FSS whose dimensions are p = 10mm, e = 6:5mm. (c) Reflection coefficients of the capacitive and inductive FSSs shown in (a) and (b), respectively.

Fig. 2. Microstrip patch antenna acting as the excitation source of the RCAs.

The structure of the communication is as follows. In Section II, RCAs with FSS superstrates are analyzed and modeled using simple ray-tracing method. The characteristics of the RCAs, such as input impedance and gain versus frequency as well as radiation patterns at the resonant frequency are studied using a MoM-CAD software package (ANSOFT Designer v.4). Measurement results are shown in this section as well. RCAs having high permittivity or permeability are studied in Section III. Finally, conclusions are drawn in Section IV. II. RCAS WITH HIGHLY-REFLECTIVE FSS SUPERSTRATE Side view of the RCAs with highly-reflective FSS, whose excitation source is a microstrip patch antenna, is depicted in Fig. 1(a). The di= mensions of the microstrip patch source, shown in Fig. 2, are 8 5 mm and = 10 mm. The FSSs characterization as well as the analysis and experimental results of the designed RCA are presented in the following.

Fig. 4. Truncated FSSs (15-by-15) whose unit cells are shown in Fig. 3(a) and (b). Capacitive screen on the left and inductive screen on the right.

A. FSS Design and Characterization

directivity to that of the source rived in [5] and is given by

:

W

L

In this work, a capacitive patch-type (FSS1) and an inductive aperture-type (FSS2) are considered to be individually employed as the RCA superstrate. Shapes and dimensions of these FSSs are shown in Fig. 3(a) and (b). These metallizations are etched on the Arlon Diclad dielectric slab having thickness of 1.59 mm, relative permittivity of 2.5 and tangent loss of 0.0022. Reflection properties of these FSSs are shown in Fig. 3(c) in a relatively large frequency spectrum. As shown in Fig. 3(c), FSS1 and FSS2 exhibit identical reflection magnitude (0.94) at the frequency of 8.26 GHz. However, their reflection phases are almost equal in magnitude and opposite in sign. Reflection coefficient phase of the FSS1 (patch-type FSS) is about 0161 and that of the FSS2 (aperture-type FSS) is +161 . The reflection coefficient curves plotted in Fig. 3(c) indicate that, in the entire frequency range, FSS1 is capacitive whereas FSS2 is inductive [6] and [7]. The fabricated FSSs consisting of 15 2 15 unit cells are displayed in Fig. 4. B. Ray-tracing Modeling of the RCAs Ray-tracing model, first introduced by G. Von Trentini [1], gives an initial estimation for the RCA air-gap resonant length ( r ), and relative

l

(Dr ). Resonance length has been de-

 90 + 80 ; N = 0; 1; 2; . . . (1) lr = N 2 +4  where N is an integer number, lr is the distance between the ground

plane (including the substrate) and the FSS, 90 and 80 are the reflection phases of the FSS and the grounded dielectric slab, respectively, when they are subjected to the normal plane wave incidence. When Arlon Diclad dielectric with 1.59 mm thickness is used as the substrate, 80 is found to be 146.94 . Using this value and the aforementioned reflection phases of the FSS1 and FSS2, corresponding resonant lengths of 17.11 and 15.54 mm are obtained, respectively. Relative directivity of the RCA, r ( ), whose air-gap length is r and operates at frequency , has been expressed in [1] as

f

l

D f

(f; = 0)j Dr (f ) = 11 +0 j0j0FF SS SS (f; = 0)j

(2)

where 0F SS is the reflection coefficient of the FSS superstrate. As shown in Fig. 3, for normal incident angle ( = 0 ), the reflection



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Fig. 5. Simulation results of (a) jS j of the RCA1 for various air-gap lengths, (b) jS j of the RCA2 for various air-gap lengths, (c) jS j of the RCA1 for various probe positions, (d) jS j of the RCA2 for various probe positions.

magnitude of both FSS1 and FSS2 is 0.94. If a typical source, such as microstrip antenna, having directivity of 6.2 dBi is employed as the RCA feed, the directivity of the RCA1 and RCA2 will be 21.30 dBi. One should note that RCA1 and RCA2 correspond to the RCAs employing FSS1 and FSS2 as the superstrate, respectively. C. MoM Analysis of the RCAs A MoM-CAD software package (Ansoft Designer v4.0) is used to analyze the RCAs. Dielectric layers are considered to be infinite in the transverse direction in the MoM analysis utilized in this communication. The ground planes, too, are assumed to be infinite in the simulations throughout this work. The numbers of the FSS unit cells are truncated so that the area of the FSS superstrate becomes 15 2 15 cm2 . Input impedance characteristics and radiation properties of the RCAs are studied. Simulation results of the jS11 j of the RCA1 and RCA2 are plotted in Fig. 5(a) and (b), respectively, for various air-gap lengths. As can be seen, the resonant frequency of the antennas varies as the air-gap length changes. Fig. 5(c) and (d) show the sensitivity of the jS11 j of the RCA1 and RCA2, respectively, to the probe position when the air-gap length is fixed at their corresponding resonant lengths of 17.3 and 15.5 mm. Of course, this observation indicates that by adjusting the probe position, the input impedance of the RCAs antenna can be matched. Another interesting point is the existence of double and triple resonances in the shown frequency range that potentially makes these antennas wide-band in the input impedance sense. The input impedances of both antennas are well-matched at the above-mentioned resonant lengths and thus, their corresponding directivities are equal to the corresponding gains. Simulated gain of RCA1 and RCA2 versus frequency are plotted in Fig. 6(a) and (b), respectively. At the resonant length of 17.3 mm, RCA1 exhibits gain as high as 21.7 dBi at the frequency of 8.35 GHz. Yet, at the resonant length of 15.5 mm, RCA2 exhibits gain as high as 18.41 dBi at the frequency of 8.30 GHz. Although, the resonant lengths are close to those predicted by the ray-tracing method, obtained gains are considerably different. RCA1 demonstrate noticeable higher gain than RCA2, in contrast to ray-tracing prediction. This difference can be

Fig. 6. Simulation results of (a) gain of the RCA1 for different air-gap lengths, (b) gain of the RCA2 for different air-gap lengths, (c) typical radiation patterns of the RCA1 working at resonance, (d) typical radiation patterns of the RCA2 working at resonance.

explained by the leaky wave phenomenon occurring in RCAs due to the multiple reflections inside the cavity. Since the resonant air-gap length of the RCA1 is higher, the waves propagating from the center of the cavity to its corners and edges leak out more slowly and produce a uniform field distribution over a larger area above the RCA1 superstrate. The leak out happens more rapidly in the RCA2 because of its shorter resonant length and consequently results in a smaller area of aperture field distribution. This is consistent with the observation made for the RCAs with metamaterial ground planes reported in [5]. However, this phenomenon has not quantitatively been investigated for RCAs with different superstrates. This investigation reveals the imperative effects of the reflection phase of the FSS superstrates on the directivity of the RCAs. Typical radiation patterns of RCA1 and RCA2 are plotted in Fig. 6(c) and (d), respectively. Good overlap between the radiation patterns in both E- and H-planes is observed that makes the RCAs attractive for applications such as reflector feed. In addition to that their cross-polar levels are relatively low as well. D. Measurement Results of the RCAs Fig. 7(a) and (b) show the measurement results of the jS11 j of the RCA1 and RCA2, respectively, for various air-gap lengths. Both RCA1 and RCA2 are well matched at their corresponding resonant lengths, 17.5 and 15.5 mm, respectively. Interesting behaviors in the input impedance characteristics of these antennas are the appearances of double and triple resonances. This can better be explained using the plotted Smith Charts in Fig. 7(c) and (d), corresponding to the input impedance of the RCA1 and RCA2, respectively. The existence of the double loops in the Smith Chart curves implies the potential wide-band behavior of the RCAs having patch- or aperture-type FSS. The measured gains of the RCA1 and RCA2, for different air-gap lengths, are shown in Fig. 8(a) and (b), respectively. RCA1 exhibits gain as high as 20.07 dBi, for the resonant length of 17.5 mm, at the frequency of 8.45 GHz. RCA2 exhibits gain as high as 17.22 dBi, for the resonant length of 15.5 mm, at the frequency of 8.55 GHz. Resonant frequencies slightly shift upward in both cases. This can be due to

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phase. In fact, reflection phase of the FSS determines the RCA resonant length which in turn affects the RCA gain. III. RCAS WITH HIGH PERMITTIVITY OR PERMEABILITY SUPERSTRATE LAYER

Fig. 7. Measurement results of (a) jS j of the RCA1 for various air-gap lengths, (b) (a) jS j of the RCA2 for various air-gap lengths, (c) Smith chart representation of the S of the RCA1 for various air-gap lengths, (d) Smith chart representation of the S of the RCA2 for various air-gap lengths.

Fig. 8. Measurement results of (a) gain of the RCA1 for different air-gap lengths, (b) gain of the RCA2 for different air-gap lengths, (c) typical radiation patterns of the RCA1 working at resonance, (d) typical radiation patterns of the RCA2 working at resonance.

the imperfections in the fabrication because of the sagging of the superstrate layer at the center. One should note that such a central sagging causes the air-gap height in the middle of the RCA become shorter. It also causes phase error, which consequently distorts the radiation patterns slightly, as shown in Fig. 8(c) and (d). Another reason for the gain reduction in the measurement results, compared to the simulation, is the fact that in the simulation radiation occurs only in the upper half space. Also, superstrate dielectric support is assumed to be infinite in the transverse direction in the simulations. Measurement results also confirm that the RCA directivity depends not only on the FSS reflection coefficient magnitude but also on its

Employing either high permittivity or high permeability as the superstrate layer is one of the other well known methods of antenna gain enhancement [8]–[11]. Such an antenna is depicted in Fig. 1(b). The underlying phenomenology of the antenna functioning is similar to the RCAs and well explained in the literature [8], [9]. The resonance conditions of this type of antennas can be determined using transverse equivalent network modeling [8]. It has been shown in [8] that in order for such a multilayer substrate-superstrate antenna to produce maximum gain, superstrate thickness must be quarter-wavelength. The substrate thickness, however, must be half-wavelength if the superstrate has high relative permittivity and quarter-wavelength if the superstrate has high relative permeability. Here, wavelength is considered the effective wavelength of the corresponding medium. In some practical designs, microstrip patch antenna is chosen as the excitation source [9]. In these cases, the concept of the substrate used in [8] needs to be modified and extend to the combination of air-gap and grounded dielectric substrate. As can be seen in Fig. 1(b), the microstrip patch is placed at the interface of the substrate and air-gap layer. Therefore, in these cases, the thickness of the air-gap layer plus the thickness of the supporting substrate replaces the length of the one-layer substrate case, and should thus be around half-wavelength. It has been shown that using full-wave analyzer (MoM) the optimum height and resonance can easily be found with parametric studies [9]. In the previous section, it was observed that gain of the RCA, whose superstrate is FSS, reduced as the result of the air-gap height shortening. Therefore, it is worthwhile to investigate whether this phenomenon occurs in the RCAs with high permittivity or permeability superstrates layer, too. In the following, design and analysis of two RCAs, one having high permittivity and the other having high permeability superstrate are studied. All the analyses are carried out using Ansoft Designer software package (MoM-CAD). Thickness of t = 3:175 mm, shown in Fig. 1(b), is considered for the superstrate layer. RCA3 and RCA4 correspond to the RCAs having "r of 9.8 and r of 1.0 (high permittivity), and "r of 1.0 and r of 9.8 (high permeability), respectively. The aforementioned parameters correspond to the operating frequency of 7.55 GHz, for both RCA3 and RCA4. Both of these superstrates have reflection magnitude of 0.51. However, the reflection phase of the former is 0180 while that of the latter is 0 . The parameters of the dielectric substrate, shown in Fig. 1(b), are d = 0:79 mm, "d = 2:1 and d = 1:0. Dimensions of the microstrip patch source, shown in Fig. 2., are 11.6 2 17.0 mm2 . At the resonance, air-gap lengths of RCA3 and RCA4 are 8.8 and 18.9 mm, respectively. The gain and jS11 j of the antennas versus frequency are plotted in Fig. 9(a) and (b), for the RCA3 and RCA4, respectively. One should note that these are optimized results and they are obtained after several parametric studies and fine tuning. As shown in Fig. 9(a) and (b), both antennas are well matched at the operating frequency of 7.55 GHz, indicating that the gain and directivity are almost equal. The maximum gains of the RCAs having high permittivity and permeability are 15.78 and 14.29 dBi, respectively. Here, too, it is observed that the RCA having the shorter air-gap length exhibit lower gain. At the frequency of 7.55 GHz, the radiation patterns of RCA3 and RCA4 are plotted in Fig. 9(c) and (d), respectively. It is worthwhile to point out that cross-polar level is low even in the inter-cardinal plane. Another interesting point is the existence of the shoulder-like side lobes

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gain. The fact that the RCA height is directly related to the superstrate phase properties demonstrates the importance of the phase properties of the superstrates in addition to its reflection magnitude.

REFERENCES

Fig. 9. Simulation results of (a) gain and jS j of the RCA3, (b) gain and jS j of the RCA4, (c) typical radiation patterns of the RCA3 working at resonance, (d) typical radiation patterns of the RCA4 working at resonance.

arising around  = 70 , for the RCA3 whose superstrate is high permittivity dielectric, in the co-polar and cross-polar radiation patterns, in E- and H-planes, respectively. These two radiation patterns are calculated based on the  component of the electric far-field, or in other words, they are obtained for the TM-polarized cases [4]. Similar sidelobes can be observed in the radiation patterns presented in [9]. These side lobes do not exist for the case of RCA4 whose superstrate is high permeability material. Discussion on the appearance of these side-lobes is beyond the scope of this communication. IV. CONCLUSION Two different classes of resonant cavity antennas (RCAs), based on having either highly-reflective FSS or high permittivity/permeability superstrates, are studied. In the first class of the RCAs, involving highly-reflective FSS superstrates, two FSSs were introduced first, one capacitive and the other inductive. Both FSSs exhibited identical reflection magnitude but different reflection phase at the design frequency. Then, these two FSSs were used as the superstrate of two RCAs having identical microstrip patch antenna source. Using MoM analysis, it was illustrated that the RCA with capacitive FSS superstrate produces a higher gain than the one having inductive FSS. These simulation results were supported and validated by measurement results, as well. It was concluded that the simple ray-tracing method is incapable of predicting the correct directivity for some RCAs, since the directivity of the antennas in this method is calculated only based on the reflectivity of the FSS superstrate. In the second class of the RCAs, two different superstrate materials were considered. Relative permittivity and permittivity of the first one were 9.8 and 1.0, respectively, while those of the second one were 1.0 and 9.8, respectively. As expected, air-gap height of the RCA whose superstrate was made of high permeability material, is shorter height. However, its gain is also less than that of its counterpart. Although, the contrasts of both high permittivity and permeability materials are identical, the one whose air-gap height is longer produces higher gain. The above-mentioned investigations from two different types of RCAs, illustrate that height of the RCA has an important impact on its

[1] G. V. Trentini, “Partially reflecting sheet arrays,” IRE Trans. Antennas Propag., vol. AP-4, pp. 666–671, Oct. 1956. [2] A. P. Feresidis and J. C. Vardaxoglou, “High gain planar antenna using optimised partially reflective surfaces,” IEE Proc. Microw. Antennas. Propag., vol. 148, no. 6, pp. 345–350, 2001. [3] A. Foroozesh and L. Shafai, “Investigation into effects of the highlyreflective patch-type FSS superstrate on the high-gain cavity resonance antenna,” IEEE Trans. Antennas Propag., vol. AP-58, pp. 258–270, 2010. [4] A. Foroozesh and L. Shafai, “Effects of artificial magnetic conductors in the design of low-profile high-gain planar antennas with high-permittivity dielectric superstrate,” IEEE Antennas Wireless Propag. Lett,, vol. 8, pp. 10–13, 2009. [5] S. Wang, A. P. Feresidis, G. Goussetis, and J. C. Vardaxoglou, “Highgain subwavelength resonant cavity antennas based on metamaterial ground planes,” IEE Proc. Microw. Antennas. Propag., vol. 153, no. 1, pp. 1–6, 2006. [6] T. K. Wu, Frequency Selective Surface and Grid Arrays. New York: Wiley, 1995. [7] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [8] D. R. Jackson and N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas,” IEEE Trans. Antennas Propag., vol. AP-33, pp. 976–987, 1985. [9] X.-H. Shen, G. A. E. Vandenbosch, and A. R. Van de Capelle, “Study of gain enhancement method for microstrip antennas using moment method,” IEEE Trans. Antennas Propag., vol. AP-43, pp. 227–231, 1995. [10] H. Y. Yang and N. G. Alexopoulos, “Gain enhancement method for printed circuit antennas through multiple superstrates,” IEEE Trans. Antennas Propag., vol. AP-35, pp. 860–863, 1987. [11] A. Foroozesh, M. N. M. Kehn, and L. Shafai, “Application of artificial ground planes in dual-band orthogonally-polarized low-profile highgain planar antenna design,” Progr. Electromagn. Res., vol. PIER-84, pp. 407–436, 2008.

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Optimization of Uniaxial Multilayer Cylinders Used for Invisible Cloak Realization Branimir Ivsic, Tin Komljenovic, and Zvonimir Sipus

Abstract—Uniaxial cylindrical cloaks have recently been proposed to prevent scattering of electromagnetic waves, i.e., to render objects invisible. While it is possible to achieve that the rays follow the proper path inside the cloak, the proposed cloaks with reduced variation of constitutive parameters suffer from nonzero reflectance. The purpose of this paper is to improve invisibility performance of the original cloak structure in terms of total scattering width reduction and the operating frequency bandwidth. Therefore, the program for analyzing uniaxial cylindrical structures is merged with the global optimization program based on the particle swarm optimization algorithm. The results show that it is possible either to obtain the “perfect” cloak at the expense of bandwidth or to improve the operating frequency bandwidth at the expense of limited reduction of total scattering width. Index Terms—Electromagnetic propagation in anisotropic media, electromagnetic propagation in dispersive media, electromagnetic scattering by anisotropic media, electromagnetic scattering by dispersive media.

I. INTRODUCTION The realization of structures that do not scatter electromagnetic field, i.e., structures that appear invisible to EM waves, is not a new concept. The possibility of a plane wave passing through some structure without distortions (i.e., with zero scattered field) has been investigated theoretically since the 1960s (see, e.g., [1]–[4]). Recently, the possibility of cloaking objects using a metamaterial cover has extensively been studied ([5]–[7]). In this approach, metamaterial is used to render a volume effectively invisible to incident radiation, i.e., to squeeze space from the volume into a shell surrounding it. Coordinate transformations that are used for cloak design do not influence the form of Maxwell’s equations, but they affect permittivity and permeability tensors (" and  , respectively), making the needed materials spatially varying and anisotropic. When viewed externally, the concealed volume and the cloak both appear to have propagation properties of free space, i.e., they appear invisible to electromagnetic waves. The required anisotropy is supposed to be obtained by using metamaterials. As a sort of a canonical case, the cloaking of a PEC cylinder has been considered. This structure can be analyzed in the two-dimensional cylindrical coordinate system, which simplifies the calculations and provides more insight in cloak design. A sketch of the considered structure is given in Fig. 1. For the cloak design, a coordinate transformation which compresses free space from the cylindrical region 0 < r < b into the concentric cylindrical shell a < r 0 < b is applied, where a and b represent the cloak inner and outer radii, respectively. After using the coordinate transformation [5], it can be shown that all the components of permittivity and permeability tensors are functions of the radius, which requires a complicated metamaterial design. However, in [6], a simplified metamaterial cloak design is proposed, suitable for fabrication, which is designed to work properly at microwave frequencies when illuminated with a normal incident plane wave with the electric field polarized along z -axis (TMz cloak). In this Manuscript received July 01, 2009; revised April 01, 2010; accepted April 09, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The authors are with the Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb, Croatia (e-mail: [email protected]; tin. [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055789

Fig. 1. A sketch of the cross-section of the considered structure.

paper, this cloak is referred to as the Schurig cloak. This case benefits from a substantial simplification due to more flexibility in choosing functional forms for the electromagnetic material parameters (only the "z ; r and ' components are relevant). Thus it is possible to fix the values of "z and ' and to let only r to vary along the radial direction "z =

b

b

0a

2

;

r =

r

0a r

2

;

' = 1:

(1)

Note that the required radial anisotropy can be realized by using layers of metamaterials, in particular the split ring resonators, for the TMz cloak. In [7], the dual case (TEz cloak) was proposed, which can be realized by using metal wires embedded in a dielectric material. Therefore, such simplified cloaks are feasible and are of interest for further analysis and improvement. The achieved invisibility is quantified in mathematical terms in [8]. The analysis approach is based on the modal approach, i.e., the radial field distribution is described with a modified Bessel differential equation that takes into account the radial anisotropy of the proposed structures. The invisibility is characterized in terms of total scattering width (T ) of the cloaked cylinder, and the angular variation of bistatic scattering width (2D ) [9]. Furthermore, the total scattering width for a PEC cylinder without cloak is calculated as the referent case, which enables comparing different cloak realizations. The obtained invisibility for each cloak realization can also be simply described in terms of the total scattering width reduction (the ratio of total scattering widths of the referent case and the case with the cloak present), which in this paper is simply referred to as “invisibility gain.” In [8], the radial anisotropy of r or "r was approximated by piecewise constant functions representing the stepwise cloak realization (layers of metamaterial), and for all the analyzed structures it was shown that there was no significant improvement of the obtained invisibility gain for structures with more than 5 layers of metamaterial ([8]). Furthermore, it was shown that the obtained invisibility gain of the Schurig cloak was only around 3 [8]. The main reason for such a small gain was the reflection of the incident wave from the cloak surface ([6], [7]), due to the impedance mismatch. The bandwidth of simplified cloak realizations was also considered. In reality, the magnetic permeability in metamaterials is always frequency dependent and is described by the Lorentz model [10] e = 1

2 2 0 f 2 f0mpf 200f0j f :

(2)

0

Here, f is the frequency of the signal, fmp denotes the frequency at which e = 0 for the lossless case (null-point of the function), f0 is

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the frequency at which e diverges (the pole of the function), while the factor represents the losses. In all the calculations, fmp is approximated by fmp = 1:02 1 f0 , following the measured values of practical SRR-based metamaterials published in [10] and [11]. Thus the frequency variations of r also have an effect on the level of the achieved invisibility. It was shown that the invisibility bandwidth (relative frequency range where the invisibility gain is larger than 1) is only around 0.2% for the Schurig cloak [8]. Previous work in invisibility therefore gives rise to two questions: is it possible to realize a cloak with reduced variation of constitutive parameters that is entirely invisible (at least at the central frequency), and is it possible to enlarge the bandwidth of the cloak compared to the Schurig cloak. In order to answer these questions we have merged the program for analyzing uniaxial cylindrical structures (developed in [8]) with a global optimization program. The analysis focuses on the TMz cloak, where only the radial component of the permeability tensor (r ) varies along the radial direction. It should be noted that an attempt to optimize parameters of a 3-layer cloak was already made in [12], but in contrast to our approach, the authors analyzed a fully anisotropic case (with no indication how to practically realize it in microwave frequency range) and used a local search technique susceptible to initial guess.

TABLE I PARTICLE SWARM PARAMETERS

TABLE II RELEVANT CONSTITUTIVE PARAMETERS OF THE SCHURIG CLOAK AND THE OPTIMIZED CLOAKS

II. OPTIMIZATION RESULTS As a global optimization algorithm we have chosen the Particle Swarm Optimization (PSO) algorithm. This is an evolutionary algorithm similar to the genetic algorithm and to the simulated annealing, but it operates on a model of social interaction between independent agents and utilizes swarm intelligence to achieve the goal of the optimization problem. It has been chosen in this study since the associated algorithm is rather easy and straightforward to implement, and it has either the same or better performance compared to other global optimization algorithms [13]–[19]. We have used two kinds of PSO algorithms. The first is the basic PSO, outlined in detail in [13], with parameters that are very close to those suggested in the same reference. The boundaries are left open, and no fitness is calculated outside. The inertial weight w decreases linearly from 0.9 to 0.4 during one optimization run, and the values of c1 and c2 are fixed at 1.5. In contrast to the recommendation given in [13], the velocities are clamped and local best topology is used [15]. The second algorithm is a variant of PSO called Comprehensive Learning Particle Swarm Optimization (CLPSO) [17]. This technique uses a novel learning strategy that enables the diversity of the swarm to be preserved to discourage premature convergence. As a cost function, in both cases we use the obtained total scattering width (T ) in the frequency band of interest. The first aim is to optimize the cloak at a single frequency only (i.e., at 8.5 GHz—the central frequency of the Schurig cloak [6]) to obtain as large invisibility gain as possible. After optimizing for a single frequency, we have tried to expand the bandwidth by widening the frequency band of interest as much as possible while maintaining the same invisibility gain as in the Schurig cloak. In contrast to the recommendations for classical PSO given in [13], the results shown in [19] suggest that increasing the particle number provides better results since the search space is relatively large. Therefore, in the following optimizations where the classical PSO is used, the population size is increased to 300 and the number of iterations is limited to 5 000. The local neighborhood is set at 10 and is defined during the first evaluation based on normalized distance between particles. In case of CLSPO, the number of particles is kept at a reasonably small number (40), but the number of evaluations is set at 200 000. The refreshing gap m is set at 7 following the advice given in [17]. The mentioned steps are practical to implement, as the developed analyzing kernel is very fast, typically needing less than 0.1 seconds to

estimate total scattering width (T ). It should be noted that both algorithms have been able to obtain almost the same fitness values for all the cases (the differences where in fourth significant digit), so it is reasonable to assume that local extremes have been avoided. The radial anisotropy of r is approximated by discrete layers of metamaterial (since the results of the optimization process do not approximate some monotonous function of radial coordinate, we need more layers than suggested in [8] to obtain good results). The inner and outer radii of all considered cloaks are a = 2:71 cm and b = 5:89 cm, respectively. The central frequency is 8.5 GHz and all the results are compared to the ones obtained by the analysis of the Schurig cloak ([6], [8]). A. Invisibility Gain—Optimized Cloak Using the developed PSO algorithm we have optimized the relevant constitutive parameters (r and "z ) of each layer, in order to obtain a cloak that does not scatter the electromagnetic field at the central frequency, i.e., the “perfect” cloak. It should be noted that such a “perfect” cloak is designed with the assumption of lossless metamaterial ( = 0). The calculated relevant constitutive parameters of the Schurig cloak and the optimized cloak are given in Table II. The considered number of layers is 10, i.e., both the Schurig cloak and the optimized cloak consist of the same number of layers. The layer No. 1 represents the innermost layer while the layer No. 10 is the outermost one (Fig. 1).

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Fig. 2. Normalized total scattering width vs. frequency for the Schurig cloak and the invisibility gain—optimized cloak. The constitutive parameters of both cloaks are given in Table II.

The level of complexity of the optimized cloak construction is the same as in the construction of the Schurig cloak (if needed, the interval of obtainable permeability values can be easily adjusted in the optimization process if some of the values of r cannot be obtained in the practical cloak realization). However, the physical insight into wave propagation through the optimized cloak is lost. The calculated total scattering width of the optimized cloak, together with the comparison to the original Schurig cloak, is given in Fig. 2. The results are given in a narrow frequency range above the central frequency (8.5 GHz), since the cloak behaves quite symmetrically for frequencies lower than the central one. The invisibility gain is simply obtained by dividing the total scattering widths of the referent case (PEC only) and the case with the cloak present. It is evident that by using the optimization process it is possible to design a virtually invisible cloak with the obtained invisibility gain of about 1 400 (although the physical insight of the cloak is lost). Note that the bandwidth of the optimized cloak is reduced even more, compared to the Schurig cloak. While in the Schurig cloak the estimated invisibility bandwidth is approximately 0.2%, for the optimized cloak the bandwidth is only about 0.08%. The angular variation of the bistatic scattering width (2D ) has been calculated at the central frequency just to check if there is any direction with distinctively stronger scattered field (Fig. 3). As can be seen, the scattered field of the optimized cloak is around 20 dB lower than that of the Schurig cloak. B. Number of Layers The Schurig cloak consists of ten metamaterial layers, which is the starting point of our study. Here we shall study how the number of layers in the cloak, i.e., the subtlety of the approximation of radial permeability, relates to the level of invisibility obtained by optimization. All the results have been obtained by PSO optimization, keeping the total dimensions of the cloak intact (i.e., the inner and outer radii). From Fig. 4 it can be seen that the normalized total scattering width decreases as the number of layers increases. It is interesting to note that better invisibility performance than with the Schurig cloak can be obtained with only three metamaterial layers. Also, it should be stressed that the level of achieved invisibility is significantly increased once the tenth layer is added. With ten layers the cloak becomes practically invisible, although only at a single frequency.

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Fig. 3. Normalized bistatic scattering width of the considered cloak realizations at the central frequency (8.5 GHz). The constitutive parameters of both cloaks are given in Table II.

Fig. 4. Normalized total scattering width vs. number of layers. The parameters of layers were optimized with PSO at a single frequency of 8.5 GHz.

C. Losses and Tolerance In addition, we have estimated the influence of losses in the material on the maximum achievable invisibility gain. We have started from the lossless cloak and gradually increased losses (described with factor in (2)), running the optimization procedure for all the considered values of

independently. As expected, the total scattering width increases (and therefore the obtained invisibility gain decreases) as the losses increase (see Fig. 5). It has also been found that the losses of = 1003 f0 are still acceptable, for the invisibility gain of the optimized lossy cloak has fallen to approximately the same value as the invisibility gain of the lossless Schurig cloak (around 3). In the same figure we have plotted the influence of material losses on the bandwidth of the cloak (i.e., the frequency range where invisibility gain is larger than 1). It can be seen that the bandwidth increases with the introduction of the losses, while at the same time the maximum invisibility gain is reduced. Furthermore, sensitivity of the optimal solution to construction tolerances has been tested. We have run statistical analysis to estimate how much the variation (as a result of construction tolerances) of the

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Fig. 5. Dependence of obtained total scattering width and bandwidth on losses for the invisibility gain—optimized cloak.

Fig. 7. Normalized total scattering width vs. frequency for the Schurig cloak and the bandwidth optimized cloak. Lossless cloaks are considered ( = 0).

Fig. 6. Dependence of obtained total scattering width on construction tolerances for the invisibility gain—optimized cloak. The mean value of total scattering width is shown.

values of the radial component of permeability tensor r and of permittivity "z influences the obtained total scattering width. Fig. 6 shows the mean value of total scattering width if r and "z are varied randomly by 0% to 5% of their initial (ideal) value with uniform distribution (worst case). Two cases are shown—with and without losses. It can be seen that the cloak is not too sensitive to parameter tolerances and lossy cloaks are even less sensitive. We can conclude that cloaks that are quite invisible can be realized even with reasonably large parameter variances. The cloak with small losses is also easier to realize in practice since the variation of parameters is much smaller, i.e., all the values of radial permeability lie in range from 0 to 0.5 (Table II, column 3), so the design could proceed as in [6]. D. Bandwidth—Optimized Cloak We have also investigated the possibility of enlarging the bandwidth of the cloak while keeping the same level of the invisibility gain (i.e., the minimum of the total scattering width) as the one of the Schurig cloak. Two cases have been considered—the optimization of the lossless cloak ( = 0) and the optimization of the lossy cloak with small losses of = 1003 f0 , which have been shown to be still acceptable. The calculated total scattering widths are shown in Figs. 7 and 8, respectively. The comparison with the Schurig cloak (lossless and lossy,

Fig. 8. Normalized total scattering width vs. frequency for the Schurig cloak and the bandwidth optimized cloak. Lossy cloaks are considered ( = 10 f ).

respectively) is also given. The optimized bandwidth is around 0.4% for the lossless case and about 0.58% for the lossy case. Meanwhile, the Schurig cloak exhibits bandwidths of 0.2% (lossless case) and 0.24% (lossy case). In other words, the improvement in the bandwidth compared to the Schurig cloak is between 2 and 2.5 in both considered cases. III. CONCLUSION The purpose of this paper was to provide an answer to two questions: is it possible to realize a cylindrical cloak that is entirely invisible (at least at the central frequency) only with the radial component of permeability tensor being coordinate dependent, and is it possible to enlarge the bandwidth of such a cloak? The investigation was carried out numerically, by merging the program for analyzing uniaxial cylindrical structures with the PSO global optimization routine. The obtained results show that it is indeed possible to realize the “perfect” cylindrical cloak, but at the expense of bandwidth that is reduced to just about 0.08%. The discussion on the influence of the number of cloak layers shows that with only three layers it is possible to obtain a better cloak performance than the Schurig cloak (containing ten layers), and that

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with ten or more layers one can design a cloak that is practically invisible. Furthermore, it is shown that the bandwidth of the Schurig cloak can indeed be increased by up to a factor of 2.5, by optimizing the cloak layer parameters, while maintaining the same minimal value of the total scattered width. Finally, it is demonstrated that losses in the cloak increase the total scattering width, in other words, reduce the invisibility gain. However, for values of around = 1003 f0 the invisibility gain of the optimized cloak is still roughly equal to the invisibility of the Schurig cloak with no losses.

REFERENCES [1] L. S. Dollin, “On the possibility of comparison of three-dimensional electromagnetic systems with nonuniform anisotropic filling,” Izv. VUZov Radiofizika, vol. 4, no. 5, pp. 964–967, 1961. [2] M. Kerker, “Invisible bodies,” J. Opt. Soc. Am., vol. 65, pp. 376–379, Apr. 1975. [3] N. G. Alexopoulos and N. K. Uzunoglu, “Electromagnetic scattering from active objects: Invisible scatterers,” Appl. Opt., vol. 17, pp. 235–239, Jan. 1978. [4] P. S. Kildal, A. A. Kishk, and A. Tengs, “Reduction of forward scattering from cylindrical objects using hard surfaces,” IEEE Trans. Antennas Propag., vol. AP-44, pp. 1509–1520, Nov. 1996. [5] J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science, vol. 312, pp. 1780–1782, Jun. 2006. [6] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science, vol. 314, pp. 977–980, Nov. 2006. [7] W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with non-magnetic metamaterials,” Nature Photon., vol. 1, pp. 224–227, Apr. 2007. [8] B. Ivsic, Z. Sipus, and S. Hrabar, “Analysis of uniaxial multilayer cylinders used for invisible cloak realization,” IEEE Trans. Antennas Propag., vol. AP-57, pp. 1521–1527, May 2009. [9] E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section. New York: SciTech Publishing, 2004. [10] S. Hrabar, , N. Engheta and R. Ziolkowsky, Eds., “Waveguide experiments to characterize the properties of SNG and DNG metamaterials,” in Metamaterials: Physics and Engineering Explorations. Hoboken/ Piscataway, NJ: Wiley and IEEE, 2006, ch. 3. [11] S. Hrabar, L. Benic, and J. Bartolic, “Simple experimental determination of complex permittivity or complex permeability of SNG metamaterials,” in Proc. 36th Eur. Microwave Conf., Manchester, U.K., Sep. 2006, pp. 1395–1398. [12] B.-I. Popa and S. A. Cummer, “Cloaking with optimized homogeneous anisotropic layers,” Phys. Rev. A, vol. 79, no. 023806, 2009. [13] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, no. 2, pp. 397–407, Feb. 2004. [14] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–778, Mar. 2004. [15] J. Kennedy and R. Mendes, “Neighborhood topologies in fully informed and best-of-neighborhood particle swarms,” IEEE Trans. Syst. Man, Cybern. Part C, vol. 35, no. 4, pp. 515–519, Jul. 2006. [16] P. Demarcke, H. Rogier, R. Goossens, and P. De Jaeger, “Beamforming in the presence of mutual coupling based on constrained particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1655–1666, Jun. 2009. [17] J. J. Liang, K. Qin, P. N. Suganthan, and S. Baskar, “Comprehensive learning particle swarm optimizer for global optimization of multimodal functions,” IEEE Trans. Evol. Comput., vol. 10, no. 3, pp. 281–295, Jun. 2006. [18] H. Wu, J. Geng, R. Jin, J. Qiu, W. Liu, J. Chen, and S. Liu, “An improved comprehensive learning particle swarm optimization and its application to the semiautomatic design of antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3018–3028, Oct. 2009. [19] T. Komljenovic, R. Sauleau, Z. Sipus, and L. L. Coq, “Layered circular-cylindrical dielectric lens antennas—Synthesis and height reduction technique,” IEEE Trans. Antennas Propag., accepted for publication.

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A Hybrid Optimization Algorithm and Its Application for Conformal Array Pattern Synthesis Wen Tao Li, Xiao Wei Shi, Yong Qiang Hei, Shu Fang Liu, and Jiang Zhu

Abstract—Investigations on conformal phased array pattern synthesis using a novel hybrid evolutionary algorithm are presented. First, in order to overcome the drawbacks of the standard genetic algorithm (GA) and the particle swarm optimization (PSO), an improved genetic algorithm (IGA) and an improved particle swarm optimization (IPSO) algorithm are proposed by introducing novel mechanisms. Then, inspired by the idea of grafting in botany, a hybrid algorithm called HIGAPSO is proposed, which combines IGA and IPSO to take advantages of both methods. After that, a spherical array antenna using wide-band stacked patch antenna elements is selected as a synthesis example to illustrate the power of HIGAPSO in solving realistic optimization problems. Finally, HIGAPSO is used to optimize the amplitude of the element current excitation of the spherical conformal array. Experimental results show that the hybrid algorithm is superior to GAs and PSOs when applied to both the classical test function and the practical problem of conformal antenna array synthesis. Index Terms—Array synthesis, conformal antenna array, genetic algorithm, particle swarm optimization.

I. INTRODUCTION Conformal antenna arrays have attracted more and more attention in many applications where planar arrays or reflector antennas have definite drawbacks [1]–[3]. This is because conformal antenna arrays have advantages of visual unobtrusiveness, non-interference with the aerodynamic performance and antenna performance. Nowadays, many technologies have been proposed for their analysis and synthesis [4]–[7]. However, low side-lobe array pattern synthesis techniques developed for linear and planar arrays do not work well with conformal arrays, since, if the array is conformal to a curved surface, the radiating elements are directed in different directions, posing unique challenges in the synthesis of antenna arrays. Therefore, it is desirable to develop an algorithm which is robust and has excellent global optimization performance and fast convergence speed in the antenna synthesis field. Although great progress has been made in the development of the evolutionary computation recently [8]–[11] in such a way that they are more robust and efficient to solve real-world problems when compared with traditional computation systems, they show limitations in solving conformal array synthesis problems. Since both PSO and GA algorithms work with a population of solutions, recently some attempts have been made to combine them, but with a weak integration of these two methods [12]–[14]. In this communication, we addressed this problem from a different perspective. Manuscript received May 12, 2009; revised February 04, 2010; accepted February 05, 2010. Date of publication May 18, 2010; date of current version October 06, 2010. This work was supported by the National Science Foundation of China under Grant 60571057. W. T. Li, X. W. Shi, and S. F. Liu are with the National Key Laboratory of Science and Technology on Antennas and Microwaves, Department of Electronic Engineering, Xidian University, Xi’an 710071, China (e-mail: [email protected]). Y. Q. Hei is with the State Key Laboratory of ISN, Xidian University, Xi’an 710071, China (e-mail: [email protected]). J. Zhu is with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2050425

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Firstly, two new evolutionary algorithms, IGA and IPSO derive from GA and PSO by introducing some new mechanisms. Then, they are tightly combined into a new hybrid algorithm called HIGAPSO, in which the grafting principle is employed to take full advantages of these methods. Our current proposal is an improved version of the algorithm reported in [18], in which the novel mechanisms and the grafting idea are introduced to improve the the convergence speed and the optimum search ability. After that, a typical benchmark function is presented to validate the proposed algorithm. Furthermore, a stacked Minkowski fractal microstrip antenna is used to design the spherical conformal array. In this example, the proposed algorithm has demonstrated its capability to reduce the side-lobe levels by optimizing the element amplitude weights. The remainder of the communication is organized as follows: Section II presents the detailed architecture of the proposed hybrid algorithm. The low side-lobe pattern synthesis of a spherical array based on the proposed algorithm is given in Section III, while Section IV concludes this communication.

extrapolation, which results in offspring searching the rest of the domain. Then the potential offspring are capable of spreading over the entire domain. A similar crossover operator is presented in [15], however, certain search space is missed when compared with our scheme (see Appendix). 2) Mutation: To avoid overly fast converging to a local optimum domain, the offspring generated by crossover operation will undergo the mutation operation [16]. Define the original offspring and the mux1 1 1 1 xk 1 1 1 xn , respectively. The tated offspring as ~ x and ~ x0 randomly selected variable xk for mutation is defined as

=(

)

= xLk + xUk xLk w max =2; xmax xU xmin k = min xk +  xk k k max L min =2; xmin xk = max xk  xk xk k [0; 1] and  is determined by ( ) = 1 ( =T )[1 (=T )] xk

0

0

0

where w

0

Consider a global optimization problem:

min f (x) = f (x1 ; x2 ;

S:t: xmin i



xi



xn ) (i = 1; 2;

111

xmax i

111

min

)

n

(1) max

where n is the number of the optimized variables, xi and xi are the upper and lower bounds of xi respectively. In the following part, IGA and IPSO will be introduced, followed by the details of the proposed hybrid algorithm. After that a typical benchmark function is used to demonstrate the proposed hybrid algorithm. A. Improved Genetic Algorithm 1) Crossover: In natural biological evolution, several offspring may be generated by the two parents after crossover and inevitably a competition relationship exists among those offspring produced by the same parents. Motivated by that, new crossover operator introducing competition manipulation among the offspring of the same parent is adopted in IGA. Now define the two parent chromosomes as ~as xs1 xs2 1 1 1 xsn and ~at xt1 xt2 1 1 1 xtn , respectively, then the corresponding four offspring chromosomes are obtained according to:

=[

]

=[

]

1 = b11 b21 bn = w(~as + ~at )=2 + (1 w)max(~as ; ~at ) (2) 2 ~b2 = b12 b22 bn = (1 w)min(~as ; ~at ) + w(~as + ~at )=2 (3) 3 ~b3 = b13 b23 bn = ~amax (1 w) + max(~as ; ~at )w (4) 4 ~b4 = b14 b24 bn = ~amin (1 w) + min(~as ; ~at )w (5) max max ~amax = [xmax x x ] (6) 1 2 n min min min x2 xn (7) ~amin = x1 [0; 1], max(~as ; ~at ) denotes the vector with each element

~b1

111

0

111

0

111

0

111

0

111

111

where w 2 obtained by taking the maximum among the corresponding element of ~as and ~at . Inspired by survival of the fittest principle in Darwinian evolution theory, we choose the two offspring with better fitness values among f~b1 ; 1 1 1 ; ~b4 g as the output of the crossover operation. For our proposed crossover operator, (2) and (3) can be seen as the results of interpolation, which leads to offspring search in the domain between ~as and ~at . While (4) and (5) can be regarded as the results of

(9) (10)

2

0

0

II. HYBRID OF IGA AND IPSO

(8)

(11)

where T is the maximum number of iterations,  is the current iteration number, and b is the shape parameter. From (11), it is discovered the advantage of such mutation operator is that at the initial stage of evolution,    , the mutation domain is large. However, in the later evolution when  approaches to T ,    , the mutation domain becomes small and the individuals search in a local domain.

() 1

()

0

B. Improved Particle Swarm Optimization 1) Exceeding Boundary Control: Particles are commonly found lying outside the boundaries of the solution space during the position updating process. For such cases, the general methods are either to take the boundary as the coordinate of the new particle, or to keep the coordinate of the particle unchanged but to assign a poor fitness value to the particle. However, either approach may reduce the diversity of the particle as well as the global search ability of the algorithm. To keep the diversity of the particles, a novel approach is proposed

()

~vi  new

()

=

0

()

d 1~ vi  D

(12)

where ~vi  is the particle velocity, d is the distance between the particle and its violation boundary, and D is its variation range. The new velocity is determined by the violation distance, the variation range and the previous velocity, in such a way it can improve the diversity of the particles in the searching process and the global search ability of the algorithm. 2) Global Best Perturbation: PSO has been shown to converge rapidly at the initial stages of a global search, but slows down when the search is close to the global optimum. To this point, the global best perturbation operator is adopted in IPSO. In social society, the leader in a swarm can be always found to explore more regions in order to lead the swarm to achieve the target with a faster speed. Inspired by this behavior, the perturbation operation is applied to the global best particle on the basis of the extrapolation theory

( ) = p~g ( ) + ed

p ~g 

1

()

p ~g 

(13)

~g is the best global solution, ed is the extrapolation coefficient where p defined as ed

= 1 (1 + rd )e[1 0

(=T )]

0

(14)

rd is the uniformly distributed random number in [0,1]. By means of (13) and (14), the stagnant global best particle can be activated again so the global best can be found with a much higher probability.

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Fig. 1. Flow of key operations in HIGAPSO.

C. Hybrid Optimization Algorithm (HIGAPSO) Fig. 2. The average best fitness results over 20 independent runs.

Inspired by the principle that grafting in botany can integrate the superiority of the two original branches, the proposed hybrid algorithm combines IGA with IPSO to take the advantages of both methods. In each generation, based on the fitness values, the population is firstly divided into three parts: the best individuals, the better individuals and the worst individuals. The best individuals are directly reproduced to the next generation; while the following better individuals are enhanced by IPSO to generate the corresponding individuals of next generation. Those IPSO-enhanced individuals regarded as the grafting population together with the rest individuals, are evolved with IGA to generate the remaining individuals of next generation. For clarity, the flow of key operations is illustrated in Fig. 1 with sequential steps of the algorithm given below. Step 1) Randomly initialize a population of P individuals within the constraint range. Step 2) Calculate the fitness of each individual from the fitness function, and then sort the individuals in ascending order according to their fitness values. Step 3) Choose the top H individuals as the elites and directly reproduce them to the next generation. Step 4) Apply IPSO strategy to the S better individuals and those IPSO-enhanced individuals are regarded as the grafting population. Step 5) The rest P 0 H 0 S individuals together with the IPSO-enhanced individuals, are evolved with IGA. Then the best P 0 H 0 S individuals are selected as the remaining individuals of next generation. Step 6) Combine the three parts together as the population of the new generation and calculate their fitness values. Choose the best one among all the individuals obtained so far as the global best. Step 7) Repeat Steps 3 to 6 until a stopping criterion (i.e., a sufficiently good solution being discovered or a maximum number of generations being completed) is satisfied. The best scoring individual in the population is taken as the final solution. In the proposed algorithm, the better individuals are evolved with IPSO instead of IGA. This is because compared with GA, PSO has the advantages of memory efficiency and cooperation between particles. Therefore, IPSO is more reasonable to take the task in guiding evolution. Besides, IPSO has good flexibility in controlling the balance between local and global exploration of the problem space, which can readily overcome the premature convergence of elite strategy in IGA. Then, the combination of these two optimization mechanisms, not only improves the diversity of the offspring, but also maintains the balance of global search and local search. Therefore, the search ability of the algorithm can be enhanced.

D. Preliminary Numerical Experiments A typical test function is presented here to verify the efficiency of the hybrid algorithm. Algorithms of standard GA (SGA), IGA, PSO and IPSO are simulated for comparison. The test functions is

f (x) =

30 i=1

x2i 0 10 cos(2xi ) + 10 ; xi 2 [010; 10]:

(15)

The function defined in (15) is known as Rastigrin function with a global minimum of zero at the origin. This is a tough multimodal optimization problem, because the global minimum is surrounded by a large number of local minima, making the search of global minimum without being stuck at one of these local minima extremely difficult. The influence of parameters on the proposed algorithm has been investigated and we have found that the algorithm is sensitive to the population size and the percentage of S/P. Simulation results show that the optimum value of the population size is from round(2n=3) to round(7n=5) and the rational value of S/P changes from 0.18 to 0.35. For the test function, we choose population size P = 30, the maximum number of iterations T = 600, the number of elites H = 2, crossover probability pc = 0:8, the mutation probability pm = 0:02 and the acceleration constant c1 = c2 = 2:0. In IPSO, the number of population in each iteration is S = 0:2P . The average best fitness results obtained from the five algorithms under test through 20 independent simulations are illustrated in Fig. 2. From the simulation results, it can be concluded that: 1) Either SGA or PSO can hardly achieve the ideal results especially for the high dimensional problems, which is due to their inherent defects of evolutionary mechanisms. It is exactly these defects that make the algorithm prematurely converged or easily trapped in a local optimum. 2) Compared with SGA and PSO, the IGA and IPSO can obtain better results. This fact accounts well for those improvements on SGA as well as PSO are indeed efficient to overcome the drawbacks of GA and PSO to some extent. 3) The HIGAPSO has the strongest local search ability and the fastest convergence speed among the five previously mentioned algorithms. This implies that IGA and IPSO can take the advantages of grafting idea in our hybrid algorithm; hence, a superior performance can be achieved by HIGAPSO. 4) As for the complexity, HIGAPSO is approximately the same as IGA or GA. Although additional complexity afford of IPSO is introduced, the evolutionary iterations can be reduced. Therefore, the complexity of HIGAPSO is comparable to IGA and GA.

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current is Imn exp j mn , then the radiation pattern of the spherical array can be expressed as

(

)=

F F ; '

N M

Imn fmn (; ') n=1 m=1 2e(jkR(sin  sin  cos('0'

)+cos  cos  )+j

)

(16)

where fmn (; ') is the individual element pattern and k is the freespace wave number. The excitation current phase mn can be calculated by:

mn = 0kR [sin 0 sin n cos('0 0 'mn ) + cos 0 cos n ]

Fig. 3. Configuration of a coaxial-fed, stacked fractal patch antenna. (a) Top view of bottom layer. (b) Top view of top layer. (c) Side view. (All units are in mm). (d) Simulated and measured VSWR for the antenna.

III. PATTERN SYNTHESIS OF A CONFORMAL ANTENNA ARRAY A. Array Element Design Microstrip patch antenna is extensively utilized as the array element for their low profile, light weight, and low cost. Besides, they can be easily made conformal. In this communication, a compact stacked antenna is designed to have a center frequency of 3.2 GHz and a bandwidth exceeding 12% for VSWR  2. The Ansoft HFSS 11.0 is employed to perform the design. The geometry of the fractal stacked array element antenna, along with its optimized dimensions, is shown in Fig. 3, which is fabricated on two layers with relative permittivity of 2.65. The top layer consists of a one order quasi-Minkowski fractal patch and has a dimension of 28:2 2 28:2 mm2 . The bottom layer consists of an H-shaped patch and has a dimension of 24:6 2 24:6 mm2 . The distance between the two layers is 7 mm. The antenna is fed by a standard SMA coaxial connector from the bottom. Compared with the regular rectangular patch antenna with dimensions of 27:1 2 27:1 mm2 on the bottom layer and 31:3 2 31:3 mm2 on the top layer, the sizes of our patches are reduced by 17.6% and 18.83% respectively. Thus, the mutual coupling can be reduced. The impedance bandwidth (VSWR  2) is measured by Agilent N5230A network analyzer. From Fig. 3(d), it can be seen that the designed antenna covers from 3.0 to 3.51 GHz (15.67%) and the simulated result agrees with the measured one very well. Therefore, the designed antenna can fully satisfy with the design requirements. B. Conformal Antenna Array Configuration A half spherical array consisting of N = 8 concentric rings in the z-direction is investigated. The array is made up of a total number of 201 stacked fractal patch antennas on the sphere with radius R = 500 mm. Antennas are uniformly distributed along -direction (from    = 0 to  = 60 ) and '-direction at approximately 0:650 . Assume the spherical coordinate of the mth (m = 1; 2 1 1 1 ; Mn ) element on the nth concentric ring is (R; n ; 'mn ) and the corresponding excitation

(17)

where (0 ; '0 ) is the desired steering angle. The detailed array configuration can be found in [18]. Nevertheless, in this communication the element spacing has been changed to 0.65 wavelength to reduce the coupling effect, and experimental results reveal that the array gain has been greatly improved. C. Pattern Synthesis For conformal arrays, it is common to select the phase to focus the beam in the desired direction. Thus we only take the amplitude weights as optimization parameters. With the phase weights calculated in advance according to (17), the specified scan angle can be guaranteed. However, it should be noted that for some other synthesis problems when the maximum directivity is not sought after, the phases of the elements should also be optimized. Now considering that the optimization for excitation amplitude of each element on the spherical phased array may be a prohibitive task in practice, a previously proposed modified Bernstein polynomial for arc arrays [17] is used to synthesize conformal arrays. The modified Bernstein polynomial is defined as

+ A (1100AB) 2 N A (1 0 )N (10A) 0   10B 2+ A (10A) N A 2 (1 0 )N (10A)  1

B1

( )=

F U

B

U

U

;

U

U

;

U

A

A

(18)

U

where B1 , B2 , M1 , M2 and A are the parameters in the polynomial. By definition it can be deduced that the modified Bernstein polynomial does not have any oscillations, a common drawback of which always exists in the output of many optimizations and curve fitting routines. With the help of modified Bernstein polynomial, only five variables in our example need to be optimized for each concentric ring, and one variable for the top layer which only consists of only one element. That is to say, for a spherical conformal array consisting of N concentric circular arrays, the total number of variables to be optimized is reduced to 52N +1, which significantly reduces the overhead of the optimization. For the spherical conformal array mentioned above, the cost function to be minimized is defined as the arithmetic mean of the squares of the excess far field magnitude above the specified level. By applying rotations of the local co-ordinate system to the simulated polarized embedded individual radiation pattern located in the ' = 0 direction on each concentric ring, radiation patterns of the other elements on the same ring can be determined. Hence, the total field can be obtained by a coherent summation of radiation patterns of each patch in the array. By virtue of using a modified Bernstein polynomial, only 41 variables are optimized for the pattern synthesis of the entire spherical array. In

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TABLE I PERFORMANCE COMPARISONS OF DIFFERENT ALGORITHMS

The shape of the main-beam is also not optimized, only the beam-width in the calculated plane is rejected to the same value. Fig. 5 shows the comparison between the radiation patterns obtained by the proposed method and the full wave simulation results from Ansoft HFSS v11.0. The good agreement between those two results validates our method. Besides, the average maximum side-lobe level (MSLL) and cost function evaluations (Nf ) of the improved algorithms over 10 independent runs are given in Table I. Obviously the proposed algorithm achieves a lower average maximum side-lobe level at the scan angles with a faster convergence speed. It is worth pointing out that more factors such as side-lobes and cross-polarization component should also be calculated and optimized throughout the visible region, when our method is applied to the practical design process. IV. CONCLUSION

Fig. 4. Radiation pattern for the conformal array after rotation. (a) Scan direction of (0 ; 0 ). (b) Scan direction of (25 ; 0 ). (c) Scan direction of (25 ; 45 ).

A novel hybrid evolutionary optimization algorithm is proposed and its application in spherical phased array synthesis is investigated in this communication. The proposed algorithm combines IGA and IPSO to take advantages of both by means of the grafting principle in botany, where IGA and IPSO are introduced to overcome the drawbacks of GA and PSO. The proposed algorithm is verified by both classical test function and pattern synthesis of a conformal array. These experimental results show that the hybrid algorithm is able to achieve the optimum design for specified design criteria in an effective manner. The accuracy and the robustness of the hybrid algorithm show its potential applications in a wide class of electromagnetic fields. APPENDIX

addition, the design target is to have a scan range of 025 to 25 from broadside and a side-lobe level of 033 dB or lower. The improved algorithms (IGA, IPSO and HIGAPSO) are applied to synthesize the far field radiation patterns. The SGA and PSO are not shown because they are inefficient to perform optimization for this case. Parameters are selected the same for these three algorithms to have a fair comparison. Each run has been conducted with a population of 32 individuals and a maximum iteration of 2000. The rest parameters are selected the same as those in the test function. Fig. 4 shows the normalized absolute ' = 0 plane radiation pattern in dB for three scan directions of (0 ; 0 ), (25 ; 0 ), (25 ; 45 ), respectively. From the results, it can be seen that the optimization results achieved by HIGAPSO satisfy with the requirements while imperfect results are obtained by either IPSO or IGA. Note that the side-lobe levels in other directions are not optimized and the side-lobes in these directions can be higher.

The crossover operator in [15] with its four offspring chromosomes are defined as

~b1 = b11 b21 1 1 1 bn1 = (~as + ~at )=2 ~b4 = b14 b24 1 1 1 bn4 = ((~amax + ~amin )(1 0 w) + (~as + ~at )w) =2 ~b2 = b12 b22 1 1 1 bn2 = ~amax (1 0 w) + max(~as ;~at )w ~b3 = b13 b23 1 1 1 bn3 = ~amin (1 0 w) + min(~as ;~at )w

(19) (20) (21) (22)

where ~as and ~at have the same definition as in Section II-A. Without loss of generality, let’s take the k th variable xk of chromosomes for example. Since w 2 [0; 1], the ranges determined by (20), (21), (22) are s t max s (min((xkmax + xmin + xmin k )=2; (xk + xk )=2);max((xk k )=2; (xk +

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 10, OCTOBER 2010

Fig. 5. A comparison of the optimized array pattern (OAP) by the proposed algorithm and the pattern simulated via HFSS. (a) Scan direction of (0 ; 0 ). (b) Scan direction of (25 ; 0 ). (c) Scan direction of (25 ; 45 ).

max )

and (xkmin ; min(xks ; xtk )), respectively. Actually, some of the search ranges are missed by this crossover operator. Concretely speaking, three cases corresponding to different s t )=2 and (xk + xk )=2 can be relationships between (xkmax + xmin k divided to illustrate this point. )=2, the range a) If (xks + xtk )=2 < (xkmax + xmin k s t s t (min(xk ; xk ); (xk + xk )=2) is missed. s )=2, the range ((xk + b) If (xks + xtk )=2 > (xkmax + xmin k t s t xk )=2; max(xk + xk )) is missed. max + xmin )=2, the range = (xk c) If (xks + xtk )=2 k s t s t (min(xk ; xk ); max(xk + xk )) is missed, except the point (xks + xtk )=2. Therefore, there are certain blind ranges where the offspring can not reach. However, for our proposed crossover, such blind ranges do not exist because the ranges determined by (2) and (3) are ((xks + xtk )=2; max(xks + xtk )) and (min(xks ; xtk ); (xks + xtk )=2), respectively. Therefore, the entire domain can be searched by the offspring.

xtk )=2)),

(max(xk ; xk ); xk s

t

REFERENCES [1] C. Dohmen, J. W. Odendaal, and J. Joubert, “Synthesis of conformal arrays with optimized polarization,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2922–2925, Oct. 2007. [2] L. I. Vaskelainen, “Constrained least-squares optimization in conformal array antenna synthesis,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 859–867, Mar. 2007. [3] K. Wincza, S. Gruszczynski, and K. Sachse, “Conformal four-beam antenna arrays with reduced sidelobes,” Electron. Lett., vol. 44, no. 3, pp. 174–175, 2008. [4] T. E. Morton and K. M. Pasala, “Pattern synthesis of conformal arrays for airborne vehicles,” in Proc. IEEE Aerosp. Conf., 2004, vol. 2, pp. 1030–1038. [5] M. Bucci, A. Capozzoli, and G. D’Elia, “Power pattern synthesis of reconfigurable conformal arrays with near-field constraints,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 132–141, Jan. 2004. [6] P. Knott, “Antenna modeling and pattern synthesis method for conformal antenna arrays,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2003, vol. 1, pp. 800–803. [7] B. h. Wang, L. q. Meng, and Y. Guo, “Pattern synthesis of doublecurved conformal paraboloidal array,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2008, pp. 1–4. [8] P. Paul, F. G. Guimaraes, D. Nair, and D. A. Lowther, “A clonal selection algorithm with varying order finite elements for the optimization of microwave devices,” Microwave Opt. Technol. Lett., vol. 50, no. 5, pp. 1392–1397, May 2008. [9] S. Selleri, M. Mussetta, P. Pirinoli, R. E. Zich, and L. Matekovits, “Differentiated meta-PSO methods for array optimization,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 67–75, Jan. 2008.

[10] F. J. Villegas, “Parallel genetic-algorithm optimization of shaped beam coverage areas using planar 2-D phased arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 6, pp. 1745–1753, Jun. 2007. [11] S. M. Mikki and A. A. Kishk, “Hybrid periodic boundary condition for particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3251–3256, Nov. 2007. [12] J. Robinson, S. Sinton, and Y. Rahmat-Samii, “Particle swarm, genetic algorithm, and their hybrids: Optimization of a profiled corrugated horn antenna,” in Proc. IEEE Int. Symp. on Antennas Propagation, Jun. 16–21, 2002, vol. 1, pp. 314–317. [13] C.-F. Juang, “A hybrid of genetic algorithm and particle swarm optimization for recurrent network design,” IEEE Trans. Syst., Man, Cybern.-Part B: Cybern., vol. 34, no. 2, pp. 997–1006, Apr. 2004. [14] F. Grimaccia, M. Mussetta, and R. E. Zich, “Genetical swarm optimization: Self-adaptive hybrid evolutionary algorithm for electromagnetics,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 781–785, Mar. 2007. [15] F. H. F. Leung, P. K. S. Tam, H. K. Lam, and S. H. Ling, “Tuning of the structure and parameters of a neural network using an improved genetic algorithm,” IEEE Trans. Neural Networks, vol. 14, no. 1, pp. 79–88, Jan. 2003. [16] D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 771–779, Mar. 2004. [17] D. W. Boeringer and D. H. Werner, “Efficiency-constrained particle swarm optimization of a modified Bernstein polynomial for conformal array excitation amplitude synthesis,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2662–2673, Aug. 2005. [18] W. T. Li, S. F. Liu, X. W. Shi, and Y. Q. Hei, “Low-sidelobe pattern synthesis of spherical array using the hybrid genetic algorithm,” Microwave Opt. Technol. Lett., vol. 51, no. 6, pp. 1487–1491, Jun. 2009.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 10, OCTOBER 2010

Enhanced Total-Field/Scattered-Field Technique for Isotropic-Dispersion FDTD Scheme Hyun Kim, Il-Suek Koh, and Jong-Gwan Yook

Abstract—A new total-field/scattered-field (TF/SF) technique is proposed for the isotropic-dispersion finite difference time domain (ID-FDTD) scheme. The proposed technique does not calculate incident fields on the TF/SF boundary by using a one-dimensional (1D) FDTD scheme as is used in the conventional technique or by using a fast Fourier transform (FFT) algorithm such as is used in the analytic field propagation (AFP) method, but proceeds instead by using an analytic expression of the incident field. Hence, the proposed technique does not require the interpolation procedure that must be used to simulate an oblique incident wave in the conventional technique. To verify the effectiveness of the proposed technique, the quality of the plane waves excited by the proposed and conventional techniques are compared and discussed. Also, the accuracy of the proposed method is numerically shown in a scattering problem. Index Terms—FDTD method, isotropic-dispersion finite difference (ID-FD) equation, low dispersion scheme, TF/SF technique.

I. INTRODUCTION The ID-FDTD scheme was proposed to rectify the inherent anisotropic dispersion characteristics of the standard FDTD scheme [1]. The numerical wavenumber of the scheme is nearly isotropic and the discrepancy between the numerical and exact numbers is negligible for both lossless and lossy media [1]–[3]. In the original ID-FDTD scheme [1], the plane wave excitation was implemented by using the conventional TF/SF technique [4]. For the Yee scheme in conjunction with the conventional TF/SF technique, three problems that degrade the quality of the excited plane wave can be observed: 1) the “anisotropic difference” between the numerical phase constants of the 1D and the higher-dimensional standard FDTD schemes, which depends on the propagation directions, 2) the “numerical phase velocity error” of the standard FDTD scheme, whose velocity is different from the exact value, and 3) the “interpolation error” due to the non-collocation between the 1D and higher-dimensional FDTD grids. Since the first and the second problems are caused by the inherent anisotropic dispersion of the standard FDTD scheme, low-dispersion FDTD schemes addressed in [5], can be utilized to reduce the errors. The interpolation error is directly caused by the Yee cell (non-collocation grid); thus, it is very hard to rectify this problem by only using the known low-dispersion scheme. Hence, the AFP method has been proposed to reduce the interpolation error by utilizing the FFT and inverse FFT (IFFT) and replacing the exact velocity with the numerical phase velocity in the desired propagation direction [6], [7]. Thus, the AFP does not require the computation of the 1D Manuscript received July 12, 2009; revised February 17, 2010; accepted April 09, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. This work was supported in part by the Defense Acquisition Program Administration and the Agency for Defense Development under Contract UD070054AD, in part by The Ministry of Knowledge Economy (MKE), Korea, under the Information Technology Research Center (ITRC) support program supervised by the National IT Industry Promotion Agency (NIPA) under Grant NIPA-2010-(C1090-1011-0006). H. Kim and J.-G. Yook are with the Department of Electrical Engineering, Yonsei University, Seoul, Korea. I.-S. Koh is with the Graduate School of Information Technology and Telecommunications, Inha University, Incheon 402-751, Korea (e-mail: [email protected],). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055791

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FDTD method to update the incident field, and as a result, the interpolation procedure can be avoided. However, the AFP has some disadvantages such as more difficult implementation and higher numerical complexity per iteration than the conventional technique. This communication proposes a new TF/SF technique to be used in conjunction with the low-dispersion FDTD schemes, which directly uses the analytic representation of the excitation field on the TF/SF boundary. To validate the effectiveness of the proposed technique, the calculated excitation field is compared to the exact field values, and the error characteristics are investigated in Section II. As an example, a scattering problem is considered to demonstrate the improved accuracy of the proposed TF/SF technique in Section III. II. NEW TF/SF TECHNIQUE FOR ID-FDTD SCHEME As mentioned above, there are three problems in the TF/SF technique: 1) anisotropic phase velocity difference, 2) numerical error in the phase velocity, and 3) interpolation error. The first and second problems can be minimized by using the ID-FDTD scheme, which shows superior isotropic dispersion to other low-dispersion FDTD schemes. Since the numerical wavenumbers of the ID-FDTD scheme vary over the propagation direction, but the fluctuation of the wavenumbers is quite small (!c) = 2:!189 C d n  !t t where Cn2 is the refractive index structure parameter in the range 10010 to 10020 m02=3 [9], !t = V 2=d is the Fresnel frequency,  is the wavelength, d is the path length between the transmitter and the receiver and V (m/s) is the wind velocity. These two asymptotes meet at the corner frequency !c = 1:43 1 V 2=d. The best finite impulse response (FIR) filter, Hs (z ), of a given length which matches the ex-

pected PSD of amplitude scintillation is achieved using the modified Yule-Walker method described in [10]. For LOS links, the scintillation standard deviation can be expressed as [11]

s (t) = 23:17 k7=6 Cn2 (t) d11=6 (5) where , Cn2 and d are as defined in (3) and (4), and k = 2= is the wave-number. In addition to clear-sky fading, scintillation also occurs during rain induced fades where the scintillation standard deviation s ( ) depends on the rain attenuation ( ) (in dB) expressed as [12]

 t

At s (t) = C 1 A(t)5=12

(6)

C is a constant which depends on frequency and antenna size. Reported values of C are 0.039 and 0.056 for 19.77 and 49.5 GHz satellite links, respectively [12]. To the authors best knowledge there are no reported measured C value for terrestrial links. However, increasing of scintil-

lation standard deviation with rain attenuation is also expected in terrestrial links due to increasing atmospheric turbulence during rain. Scintillation time series during clear-sky and rainy conditions can then be simulated by first normalizing the power of the FIR filter s ( ) then multiply the filtered process by (5) and (6), respectively. Fig. 2 shows a model for generating scintillation time series during clear-sky and rainy conditions. Note from Fig. 2 that the output from s ( ) has a Gaussian distribution and s ( ) has a log-normal distribution [13]. This results in ( ) being conditionally Gaussian with log-normally distributed scintillation intensity which is consistent with the MoulsleyVilar model reported in [13].

H z

St

 t

H z

Fig. 3. Typical multipath propagation scenario in BFWA. ab is the LOS path, ac + cb and ad + db are reflected taps from reflection points at c and d, respecand  are the AODs.  and  are the AOAs. tively. 

IV. DYNAMIC VEGETATION EFFECTS The received signal after propagating through vegetation has a diffuse and coherent component. The diffuse component is due to propagation through vegetation and the coherent component results from diffraction from the top and side of vegetation as well as a ground reflected wave [14]. Hence, the received signal envelop can be described by a Nakagami-Rice distribution [15]. The signal fading due to swaying vegetation can be modeled by utilizing a multiple mass-spring system to represent a tree and a turbulent wind model [15]. The time series for the received signal power through vegetation is then obtained as j ( )2 j. For a Nakagami-Rice distributed signal envelope, ( ) can be expressed as the sum of the direct and diffuse signal components as shown below

qt

qt

q(t) = ad exp(j) + af exp j i 0 2 1Li(t) M

Direct

i=1

(7)

Diuse

where the first term in (7) is the contribution of the direct signal component. d and are the amplitude and phase of the direct signal, respectively. The second term in (7) is the contribution of the diffuse component which is the sum of signals scattered from the various parts is the total number of scattering tree components (the of the tree. can be trunk, branches and sub-branches). To reduce complexity, chosen to be equal to 7 which is sufficient to recreate the rich dynamic behavior of the fading from a real tree [15]. f is the amplitude of each scattered signal, i is the phase uniformly distributed within the range [0, 2 ), is the wavelength and 1 i ( ) is the time varying path length difference caused by displacement of the th tree component, see [15] for details on the model. The model has been verified using measurements in [15].

a



M

 



Lt

a

i

M

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Fig. 4. Theoretical model for generating correlated rain attenuation time series experienced by multipath taps. n(t) is a white Gaussian noise, C are coefficients which introduces the appropriate correlation between multipath taps, H (z ) controls the time dynamics of each multipath component as defined in (1), m and  are the mean and standard deviation of ln A(t) (as discussed in Section II). A (t), A (t) and A (t) are the rain attenuations in dB experienced by tap 1, 2 and 3, respectively [16].

V. MULTIPATH PROPAGATION During rain, the taps may be subjected to varying correlated dynamics depending on the spatial correlation of rain attenuation between different paths through rain. In [16] we proposed a statistical multipath model during rain for BFWA systems employed in dense urban area. In the model, the geometrical based single bounce elliptical (GBSBE) channel model [17] was used to estimate channel parameters such as power, delay, angle-of-departure (AOD) and angle-of-arrival (AOA) of each multipath component. The AOD and AOA informations of each multipath component were used to calculate the spatial correlation between multipath components using a spatial correlation model of rain. Fig. 3 shows a multipath scenario with three taps, where ab is the LOS path, ac + cb and ad + db are reflected taps from reflection points at c and d, respectively. The rain attenuation experienced by each multipath component in Fig. 3 can be simulated using the simulator shown in Fig. 4. In the model, n(t) is a white Gaussian noise with zero mean and unite variance. By using a spatial rain attenuation model, and information on the AOD and AOA of each tap, Cij introduces the appropriate correlation between multipath taps, Hr (z ) controls the time dynamics of each multipath component as defined in (1), m and  are the mean and standard deviation of ln A(t) (as discussed in Section II). Note that the rain attenuation Atap2 (t) of tap 2 is the sum of Aac (t) and Acb (t) which are the rain attenuations (in dB) on path ac and cb. Similarly, the rain attenuation Atap3 (t) of tap 3 is the sum of Aad (t) and Adb (t) which are the rain attenuations (in dB) on path ad and db. See [16] for details on the model. Other multipath models for BFWA systems are also reported in, e.g., [18], [19]. The distribution of the short-term received signal envelope after propagating different paths can be described by a Nakagami-Rice distribution [20]. It was found from 38 GHz wideband measurements that the Nakagami-Rice K -factor of the received signal decreases with increasing rain rate [20]

K

= 16:88 0 0:04R

(dB)

(8)

where R is the rain rate in mm/h. Two hypotheses may explain (8). One is due to inhomogeneities in the atmosphere during rain. While the

second is based on the change of the electromagnetic properties of the reflecting surfaces (wet surfaces become better reflectors which results in increasing specular reflected power) [20]. Based on (8), a simulation model of rain attenuation for BFWA is reported in [21]. In addition, using (8), a study on the distribution of the Nakagami-Rice K -factor and the effect of rain on the mulipath behavior of BFWA channels is presented in [22]. Note that (8) is found from measurements at 38 GHz with a specific transmitting (sector horn) and receiving (parabolic reflector) antenna, polarization, site and climate, so further validation data is required for its generic applicability test. VI. COMBINED CHANNEL MODEL According to the environment and weather conditions, the individual dynamic models for rain, scintillation, vegetation and multipath can be combined to capture the dynamic effects experienced in real channels. The objective is to specify the relationship between propagation effects and combine them to construct a comprehensive channel model suitable for FMTs and evaluation of system performance. As discussed in Section III, during rain the scintillation standard deviation increases with increasing rain attenuation. In Section V the dependency of the received multipath signal power on rain rate was discussed. In addition, measurement results reported in [23] shows that wet vegetation attenuates the signal more than dry. Fig. 5 illustrates a channel model for combined propagation degradations. The dashed lines at the side of Fig. 5 indicate that there is a dependency between the propagation effects as discussed above i.e., rain affects the signal amplitude, the scintillation standard deviation and vegetation attenuation. The channel impulse response for the combined model can then be expressed as

h(t; ; t ; r ) =

N

01

n=0

an (t)sn (t)qn (t) n (t) [

 [t 0 t;n (t)] [r 0 r;n

0  (t)] (t)]exp(0j (t)) n

n

(9)

where N is the total number of taps, and is equal to 1, 2 and 3 for good, moderate and bad channel conditions, respectively [19]. This is

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Fig. 5. Combined wideband dynamic channel model.  , , (t), m(t) and u(t) are the delay, phase, magnitude, input and output signals, respectively. a(t), q (t) and s(t) are the time varying rain, vegetation and scintillation effects, respectively. The dashed lines at the side of the figure indicate that rain attenuation influences the signal magnitude (by affecting the K -factor of the received signal), scintillation and vegetation attenuation experienced by each tap. All values are in linear scale.

because the very narrow receiver antenna beamwidth in BFWA system allow only few multipath components to be received with significant power. For each tap number n, n (t) is the tap delay, n (t) is the phase within the range [0, 2 ),  is delta function, t;n (t) is the AOD, r;n (t) is the AOA, n (t) is the signal amplitude including free space path loss and loss in reflection, qn (t) is the time varying vegetation attenuation, sn (t) is the amplitude scintillation and an (t) is the dynamic rain attenuation. Note that lowercase letters are used to indicate that the values are in linear scale. Each received signal component is represented by time delayed taps. These taps are weighted with the experienced degradations (rain, scintillation and vegetation attenuation) as well as the channel multipath power delay profile. The rain attenuation time series of each tap in Fig. 5 can be generated using the simulator shown in Fig. 4. The amplitude scintillations during rain of each tap can be generated using the simulator shown in Fig. 2. The time series of signal fading due to swaying vegetation of each tap can be simulated using (7). In addition, the effect of rain rate on the multipath property of the channel can be accounted for using (8), where the power in the direct, Pd , and multipath, Pmp , component is given by

K Pd = Pr K +1 P Pmp = r K+1

(10) (11)

Pr and K are the received signal power and the Nakagami-Rice K -factor given in (8), respectively (both in linear scale). We can observe from (8) that as the rainfall rate increases the Nakagami-Rice K -factor decreases which in turn results in decreasing and increasing power of the direct and mulitpath component, respectively, see (10) and (11). Depending on the path length traveled through vegetation, rain and turbulence, each tap will have different average vegetation and rain attenuations, and scintillation effects. The movements of the trunk, branches and sub-branches are the same for all multipath taps but the paths traveled through the tree are different. This implies that depending on the angle of incidence to the foliage, taps will be affected by different branches. To summarize, the following steps are performed when setting up the simulator. • Multipath during rain (for mathematical details see [16]): — Use the AOD and AOA information of each tap to calculate the correlation coefficient of rain attenuation, rij , for the different paths;

Fig. 6. Tap attenuations for the combined effects of rain, vegetation and scintillation. R = 28 mm=h, d = 6 km, f = 29 GHz, = 4:5 10 , C = 5:1 10 m , V = 2 m=s, C = 0:056, simulation period = 600 minutes (10 hours) and sampling rate = 20 Hz.

2

2

— Using rij , calculate the corresponding correlation matrix for the Gaussian process of the filter outputs, Y ; — Similarly, using Y , calculate the corresponding correlation matrix for the Gaussian process of the filter inputs, X ; — Obtain the mixing matrix, , by performing Cholesky factorization of X ; — Estimate mi and i for each path component from local measurements or using the ITU-R P.530 [4]; — Obtain the total attenuation due to rain for each path by summing the contribution from the path components. • Scintillation time series during rainy and clear-sky conditions: — Generate white Gaussian noise time series and filter them using a lowpass filter, Hs (z ), which matches the PSD defined in (3) and (4); — For clear-sky condition, multiply the filtered process by (5); — For rainy periods, multiply the filtered process by (6). • Signal fading due to swaying vegetation (for mathematical details see [15]): — Define the number of tree components, M , and the NakagamiRice K -factor; — From a uniform distribution, generate the initial phases, i , of the direct and diffuse components; — From the Nakagami-Rice K -factor calculate the amplitude of the direct, ad , and diffuse, af , component; — Calculate the path length differences, 1Li (t), for i = 1; 2; : : M caused by swaying tree components; — Use (7) to generate time series of signal fading due to swaying vegetation. • The effect of rain on the multipath property of the channel: — Use (8) to introduce rainfall rate dependency to the NakagamiRice K -factor; — Calculate the power of the direct and diffuse component using (10) and (11), respectively.

R

R

R

C

R

VII. NUMERICAL RESULTS AND DISCUSSION An example of simulated combined effects of rain, scintillation and vegetation attenuation for the first three taps is shown in Fig. 6. The fast fading of the multipath taps in Fig. 6 is due to vegetation and scintillation effects. While the large but slow variation of the multipath taps is due to rain attenuation. Figs. 7 and 8 show the complementary cumulative distribution functions (CCDFs) of the bit-error-rate (BER) using BPSK modulation (without error correcting codes) for combinations of

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the propagation effects are combined. In addition, we can also observe from Figs. 7 and 8 that rain attenuation is the dominant propagation impairment which contributes most to the BER of the link. Note also how the BER changes when the rainfall rate increases from 28 mm/h in Fig. 7 to 60 mm/h in Fig. 8. This demonstrates the significant effect of rain attenuation on BFWA systems, and shows also the sensitivity of the model to the choice of simulation parameters. VIII. CONCLUSION

Fig. 7. CCDFs of the BER for the combined effects of rain, vegetation, scintil= ,d ,f , lation and multipath propagation. R : ,C : ,V = ,C : , (10 hours) and .

= 28 mm h = 6 km = 29 GHz = 4 5 2 10 = 5 1 2 10 m = 2 m s = 0 056 simulation time = 600 minutes sampling rate = 20 Hz

In this communication we present a new simulator for system performance evaluation and development of robust BFWA with fade mitigation. The simulator properly combines earlier reported models for rain, scintillation, vegetation and mulitpath propagation by taking into account the dependency between impairments to give a realistic combined channel model. The dynamics of rain attenuation is modeled using the Maseng-Bakken approach. The multipath propagation is modeled using the geometrical based single bounce elliptical channel model. The spatial correlation between multipath taps during rain is incorporated using a space-time rain attenuation model. A lowpass filter which models the PSD of scintillation is used to describe the dynamic characteristics of amplitude scintillation. The signal fading due to swaying vegetation is described by utilizing a multiple mass-spring system to represent a tree and a turbulent wind model. Numerical analyses of the BER under different combinations of propagation impairments were performed. As expected the results show that the worst-case scenario is when rain, scintillation, vegetation, and multipath effects are combined. Furthermore, different examples of BER curves are presented by varying the simulation parameters which in turn shows the sensitivity of the model to the choice of simulation parameters. Future work includes verifying the combined channel simulator using measurements.

REFERENCES

Fig. 8. CCDFs of the BER for the combined effects of rain, vegetation, scintillation and multipath propagation. R = ,d ,f , ,C : ,V = ,C : , : (10 hours) and .

= 60 mm h = 6 km = 29 GHz = 5 1 2 10 m = 2 m s = 0 056 = 4 5 2 10 simulation time = 600 minutes sampling rate = 20 Hz TABLE I : BFWA SYSTEM PARAMETERS (k

= 1 38 10 1

,T

= 290 K) [16]

different propagation impairments. A simulation time of 600 minutes (10 hours) and a sampling rate of 20 Hz were used for all the propagation impairments. Note that different simulation parameters are used in Figs. 7 and 8, see the figure captions. Table I shows typical BFWA system parameters used in the simulations [16]. As expected we can observe from Figs. 7 and 8 that the worst-case scenario is when all

[1] M. Cheffena, L. E. Bråten, and T. Tjelta, “Time dynamic channel model for broadband fixed wireless access systems,” presented at the 3rd Int. COST 280 Workshop, Prague, Czech Republic, Jun. 6–7, 2005. [2] M. Cheffena, L. E. Bråten, T. Tjelta, and T. Ekman, “Time dynamic channel model for broadband fixed wireless access systems,” presented at the IST Mobile and Wireless Summit, Myconos, Greece, Jun. 4–8, 2006. [3] T. Maseng and P. Bakken, “A stochastic dynamic model of rain attenuation,” IEEE Trans. Commun., vol. 29, pp. 660–669, 1981. [4] “Propagation Data and Prediction Methods Required for the Design of Terrestrial Line-of-Sight Systems,” Geneva, 2007, ITU-R P.530-12. [5] M. Cheffena, L. E. Bråten, and T. Ekman, “On the space-time variations of rain attenuation,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1771–1782, 2009. [6] K. H. Craig, Ed., “Propagation Planning Procedures for LMDS,” AC215 CRABS, Deliverable D3P1b Jan. 1999 [Online]. Available: http://www.telenor.no/fou/prosjekter/crabs [7] “Efficient Millimetre Broadband Radio Access for Convergence and Evolution,” EMBRACE [Online]. Available: http://www.telenor.no/fou/prosjekter/embrace/ [8] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York: IEEE Press and Oxford University Press, 1997. [9] J. M. Warnock, T. E. Vanzandt, and J. L. Green, “A statistical model to estimate mean values of parameters of turbulence in the free atmosphere,” presented at the 7th Symp. of Turbulence Diffusion, 1985. [10] B. Friedlander and B. Porat, “The modified Yule-Walker method of ARMA spectral estimation,” IEEE Trans. Aerosp. Electron. Syst., vol. 20, no. 2, pp. 158–173, Mar. 1984. [11] “Radiowave Propagation Effects on Next Generation Fixed-Service Terrestrial Telcommunications Systems,” COST 235, 1996, Final rep., ESBN 92-827-8023-6. [12] E. Matricciani, C. Riva, and M. Mauri, “Scintillation and simultaneous rain attenuation at 49.5 GHz,” presented at the 9th Int. Conf. Antennas Propag., 1995. [13] T. J. Moulsley and E. Vilar, “Experimental and theoretical statistics of microwave amplitude scintillations on satellite down-links,” IEEE Trans. Antennas Propag., vol. AP-30, no. 6, pp. 1099–1106, Nov. 1982. [14] Attenuation in Vegetation Geneva, 2007, ITU-R P.833-6.

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[15] M. Cheffena and T. Ekman, “Dynamic model of signal fading due to swaying vegetation,” EURASIP J. Wireless Commun. Networking. Special Issue on Advances in Propagation Modelling for Wireless Systems, vol. 2009, no. Article ID 306876, doi:10.1155/2009/306876, p. 11, 2009. [16] M. Cheffena and T. Ekman, “Theoretical multipath channel model during rain for BFWA employed in dense urban areas,” presented at the ICSPCS, Gold Coast, Australia, Dec. 15–17, 2008. [17] J. C. Liberti and T. S. Rappaport, “A geometrically based model for line-of-sight multipath radio channels,” in Proc. IEEE Veh. Tech. Conf., 1996, pp. 844–848. [18] P. Soma, L. Cheun, S. Sun, and M. Y. W. Chia, “Propagation measurements and modelling of LMDS radio channel in Singapore,” IEEE Trans. Veh. Techn., vol. 52, no. 3, pp. 595–606, May 2003. [19] P. B. Papazia, G. A. Hufford, and R. J. Achatz, “Study of the local multipoint distribution service radio channel,” IEEE Trans. Broad., vol. 43, pp. 175–184, 1997. [20] H. Xu, T. S. Rappaport, R. J. Boyle, and J. H. Schaffner, “Measurements and models for 38 GHz point-to-multipoint radiowave propagation,” IEEE J. Areas Commun., vol. 18, no. 3, pp. 310–321, Mar. 2000. [21] P. Hou, J. Zhuang, and G. Zhang, “A rain fading simulation model for broadband wireless access channels in millimeter wavebands,” presented at the 51st IEEE VTC Spring, Tokyo, Japan, 2000. [22] A. D. Panagopoulos, K. P. Liolis, and P. G. Cottis, “Rician K-factor distribution in broadband fixed wireless access channels under rain fades,” IEEE Commun. Lett., vol. 11, pp. 301–303, Apr. 2007. [23] I. J. Dilworth and B. L. Ebraly, “Propagation effects due to foliage and building scatter at millimeter wavelengths,” in Proc. Inst. Elect. Eng. Antennas Propag. Conf., Apr. 4–7, 1995, pp. 51–53.

Planar Printed Strip Monopole With a Closely-Coupled Parasitic Shorted Strip for Eight-Band LTE/GSM/UMTS Mobile Phone Fang-Hsien Chu and Kin-Lu Wong

Abstract—A planar printed antenna comprising a driven strip monopole and a parasitic shorted strip, both of comparable length and closely coupled to each other, suitable for eight-band LTE/GSM/UMTS operation in the mobile phone is presented. The proposed antenna is mainly configured along the boundary of the no-ground portion on the system circuit board of the mobile phone to achieve a simple and compact structure. Also, the edge of the no-ground portion facing the system ground plane on the circuit board is not necessarily a straight line, leading to more degrees of freedom in allocating the required no-ground portion on the circuit board for printing the antenna. The driven strip monopole and the parasitic shorted strip both contribute their lowest and higher-order resonant modes to form two wide operating bands centered at about 830 and 2200 MHz to respectively cover the LTE700/GSM850/900 operation (698–960 MHz) and GSM1800/ 1900/UMTS/LTE2300/2500 operation (1710–2690 MHz). Index Terms—Handset antennas, internal handset antennas, mobile antennas, printed monopoles, small antennas.

I. INTRODUCTION Planar strip monopole is suitable to be printed on one surface of the system circuit board of the mobile phone, making it easy to Manuscript received October 31, 2009; revised January 25, 2010; accepted April 06, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. The authors are with the Department of Electrical Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (e-mail: [email protected]. edu.tw; http://antenna.ee.nsysu.edu.tw). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055807

fabricate at low cost and find applications in the slim or thin-profile mobile phone [1]–[10] for its negligible antenna thickness above the system circuit board. Some promising planar strip monopoles for penta-band WWAN (wireless wide area network) operation covering the GSM850 (824–894 MHz), GSM900 (880–960 MHz), GSM1800 (1710–1880 MHz), GSM1900 (1850–1990 MHz) and UMTS (1920–2170 MHz) bands have been reported in the published papers [11]–[14]. To achieve small size yet wideband or multiband operation, the techniques of embedding a chip inductor [11], [12] or using an internal printed distributed inductor [13] in the strip monopole have been demonstrated. The design using a printed distributed inductor to replace the embedded chip inductor facilitates the fabrication of the antenna on the system circuit board. The use of an external matching network is also helpful in achieving improved impedance matching for frequencies over the desired lower band (824–960 MHz) and upper band (1710–2170 MHz) for penta-band WWAN operation [11], although the external matching networks may cause some undesired loss in the input power of the antenna. In these designs, however, the obtained bandwidth cannot cover the recently introduced LTE (long term evolution) operation [15]–[17] in the 700 MHz band (698–787 MHz), 2300 MHz band (2305–2400 MHz) and 2500 MHz band (2500–2690 MHz). The LTE operation is expected to provide better mobile broadband and multimedia services than the existing GSM and UMTS communication systems and will become attractive for the mobile users. Hence, the mobile phone capable of eight-band operation including the LTE700/2300/ 2500, GSM850/900/1800/1900, and UMTS bands will be demanded on the market in the very near future. In this communication we present a novel planar printed strip monopole to cover the desired eight-band LTE/GSM/UMTS operation. The proposed planar monopole has a simple structure which comprises a driven strip monopole and a parasitic shorted strip, both of comparable length and closely coupled to each other. The parasitic shorted strip used in the proposed design is different from the traditional ones that have been applied in the internal mobile device antennas [18]–[21] for bandwidth enhancement, in which the parasitic strip is usually of a much shorter length than the driven element; furthermore, only part of the parasitic strip is gap-coupled to the driven element or vice versa. In the proposed design, both the driven strip monopole and the parasitic shorted strip whose length is slightly longer than that of the driven strip monopole contribute their lowest (0:25) resonant modes to form a wide operating band centered at about 830 MHz for the antenna’s lower band to cover the frequency range of 698–960 MHz. A wide operating band centered at about 2200 MHz for the frequency range of 1710–2690 MHz is also obtained, which is formed by the higher-order resonant modes contributed by both the driven and parasitic shorted strips. That is, the proposed planar monopole can provide two wide lower and upper bands to respectively cover the LTE700/GSM850/900 and GSM1800/1900/UMTS/ LTE2300/2500 operation. In addition, the simple structure of the proposed antenna allows it to easily follow along the boundary of the no-ground portion on the system circuit board of the mobile phone such that the antenna can be implemented with a small printed area. The proposed design is also promising to be implemented in the no-ground portion with a non-straight edge facing the main ground plane on the system circuit board. This can lead to more degrees of freedom in allocating the required no-ground portion for printing the antenna on the system circuit board. Detailed operating principle of the proposed antenna is described in the communication, and the proposed antenna is fabricated and studied. II. PROPOSED ANTENNA Fig. 1 shows the geometry of the proposed antenna having a simple uniplanar structure and comprising only two elements: a driven strip

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Fig. 2. Measured and simulated return loss for the proposed antenna.

frequencies higher than about 800 MHz; the two resonant modes can be incorporated into a wide lower band for the proposed antenna to cover the desired frequency range of 698–960 MHz. In addition, the parasitic shorted strip can contribute its two higher-order resonant modes to occur at about 1700 and 2700 MHz, which can incorporate the higher-order resonant modes of the driven strip monopole occurred at about 2200 and 2600 MHz to form a very wide upper band (bandwidth larger than 1 GHz) to cover the desired frequency range of 1710–2690 MHz. Details of the excited resonant modes will be discussed with the aid of Fig. 3 in the next section. Fur2 ther, note that the extended ground (size 5 2 24 ) at the top edge of the system ground plane can be used to accommodate the associated nearby electronic elements such that some valuable board space can be reclaimed for use, which is attractive for practical applications.

mm

Fig. 1. Geometry of the proposed antenna for eight-band LTE/GSM/UMTS operation in the mobile phone.

monopole (section AC) and a closely-coupled parasitic shorted strip (section BD). A no-ground portion with a non-straight edge facing the system ground plane printed on the system circuit board is allocated to accommodate the proposed antenna. In the study, a 0.8-mm thick FR4 substrate of length 115 mm and width 60 mm is used as the system circuit board, which is further enclosed by a plastic housing fabricated using the plastic plates of thickness 1 mm, relative permittivity 3.0, and conductivity 0.02 S/m as the mobile phone housing. The no-ground 2 2 (15 2 60 less the extended portion has a size of about 780 2 in the system ground plane). The driven ground of size 5 2 24 strip monopole and parasitic shorted strip of the proposed antenna generally follow along the boundary of the no-ground portion to achieve a compact structure. The front end (point A) of the driven strip monopole is the feeding point of the antenna, while the front end (point B) of the parasitic shorted strip is short-circuited to the top edge of the system ground plane through a via-hole in the system circuit board. In-between the driven strip monopole and parasitic shorted strip, there are a first coupling gap (gap1) of 0.5 mm in the front section and a second coupling gap (gap2) of 1.0 mm in the remaining section of length 37 mm t . The use of two coupling gaps leads to more degrees of freedom in adjusting the capacitive coupling between the two strips, which makes it easy to achieve good impedance matching (better than 3:1 VSWR or 6-dB return loss widely used as the mobile phone antenna specification in the practical applications) for frequencies over the desired operating bands. The lengths of the driven strip monopole and parasitic shorted strips are both close to a quarter-wavelength (about 93 mm) at 800 MHz, but are of slightly different lengths. The two strips are expected to contribute their lowest resonant modes, with one at frequencies lower than about 800 MHz and the other one at

mm

()

mm

mm

III. RESULTS AND DISCUSSION The proposed antenna with dimensions given in Fig. 1 was fabricated and tested. Results of the measured and simulated return loss for the fabricated prototype are shown in Fig. 2. The simulated results in this study are obtained using Ansoft HFSS version 11.2 [22]. The measured data generally agree with the simulated results. Two wide operating bands are seen to be excited. The lower band, based on the definition of 3:1 VSWR or 6-dB return loss, shows a wide bandwidth of 305 MHz (665–970 MHz) which allows the antenna to cover the LTE700/GSM850/900 operation. The upper band shows an even wider bandwidth of 1210 MHz (1700–2910 MHz), which covers the GSM1800/1900/UMTS/ LTE2300/2500 operation. Hence, with the obtained wide lower and upper bands, eight-band LTE/GSM/UMTS operation is achieved for the proposed antenna. To analyze the operating principle of the proposed antenna, Fig. 3 shows the comparison of the simulated return loss and input impedance of the proposed antenna and the reference antenna (the corresponding antenna without the parasitic shorted strip). Notice that the driven strip monopole (section AC) in the two antennas tested in the figure is with the same dimensions. In Fig. 3(a), for the reference antenna, the lower band is centered at about 950 MHz and formed by one resonant mode only. The bandwidth of this lower band cannot cover the desired frequency range of 698–960 MHz. Similarly, the upper band of the reference antenna is formed by one resonant mode at about 2200 MHz, whose bandwidth is far from covering the desired frequency range of 1710–2690 MHz. When the parasitic shorted strip is added (the proposed antenna), an additional resonant mode at about 700 MHz (denoted as mode p1 in the figure) is generated; this can be seen more clearly from the input impedance results shown in Fig. 3(b). This new resonant mode incorporates the original one (mode d1 in the figure) contributed by the driven strip monopole to form a wide lower band for the 698–960 MHz operation. Also note that mode d1 shifted down with the adding of the parasitic shorted strip is mainly because the length of

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Fig. 4. Simulated return loss as a function of the end-section length t of the driven strip monopole; other dimensions are the same as given in Fig. 1.

Fig. 5. Simulated return loss as a function of the end-section length d of the parasitic shorted strip; other dimensions are the same as given in Fig. 1. Fig. 3. Comparison of the simulated (a) return loss and (b) input impedance of the proposed antenna and the reference antenna (without the parasitic shorted strip).

the parasitic shorted strip is slightly longer than that of the driven strip monopole and there is also strong coupling between the two coupled strips. Two higher-order resonant modes of the parasitic shorted strip are also excited at about 1700 and 2700 MHz (mode p2 and p3 in the figure) with good impedance matching [also see the input impedance results in Fig. 3(b)]. There are also two higher-order resonant modes at about 2200 and 2600 MHz (mode d2 and d3 in the figure) contributed by the driven strip monopole. Note that the mode at about 2600 MHz is related to the one at about 2900 MHz for the case of no parasitic shorted strip; when the parasitic shorted strip is added, the mode at about 2900 MHz is shifted to lower frequencies at about 2600 MHz. A wide upper band is then formed by these higher-order resonant modes contributed by the parasitic shorted strip and the driven strip monopole to cover the 1710–2690 MHz band operation. A parametric study of the proposed antenna is also conducted. Fig. 4 shows the simulated return loss as a function of the end-section length t of the driven strip monopole. Large effects on both the antenna’s lower and upper bands are observed. With the decreased length t, the resonant mode contributed by the driven strip monopole is greatly affected and is shifted to higher frequencies (see the second resonant mode in the lower band). In addition, the decreased length t also leads to degraded impedance matching for the resonant modes contributed by the parasitic shorted strip. This is largely because the decreased length t decreases the coupling-gap length of gap2, which affects the capacitive coupling between the driven strip monopole and the parasitic shorted strip and hence results in degraded impedance matching for the resonant modes excited through the capacitive coupling.

Fig. 5 shows the effects of the end-section length d of the parasitic shorted strip. Results of the simulated return loss for the length d varied from 8.5 to 12.5 mm are shown. Large effects on both the antenna’s lower and upper bands are also seen. In addition, it is noted that the central or resonant frequencies of the resonant modes contributed by the parasitic shorted strip are varied (see the first mode at about 700 MHz in the lower band and the two modes at the lower edge and upper edge of the upper band). This further confirms the contribution of the parasitic shorted strip in the excited resonant modes for the proposed antenna. Fig. 6 shows the effects of the width w of the extended ground. Results of the simulated return loss for the width w varied from 3 to 7 mm are presented. Small variations in the obtained bandwidth of the upper band are seen. However, when the width w is increased, that is, the extended ground is extended further into the no-ground portion, the obtained bandwidth of the lower band is decreased. For covering the desired operating band of 698–960 MHz, the width w is selected to be 5 mm here. The measured three-dimensional (3-D) total-power radiation patterns for the proposed antenna are plotted in Fig. 7. The antenna is tested in a far-field anechoic chamber [TRC (Training Research Co.) measurement system]. Measured results for frequencies at 740, 925, 1795, 2045 and 2400 MHz are shown. The radiation patterns for frequencies in the lower band generally show dipole-like patterns, and omnidirectional radiation in the azimuthal plane (x 0 y plane) is observed. On the other hand, more variations and nulls in the radiation patterns are seen for frequencies in the upper band. Note that the radiation patterns of the internal mobile phone antenna are generally dependent on the system ground plane of the mobile phone which is also an efficient radiator [23]–[25], especially for frequencies in the lower band. For frequencies in the upper band, since the wavelength is comparable to the length of the system ground plane, surface current nulls are usually

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Fig. 6. Simulated return loss as a function of the width ground; other dimensions are the same as given in Fig. 1.

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w of the extended

Fig. 8. Measured and simulated antenna gain and radiation efficiency for the proposed antenna. (a) The lower band (698–960 MHz). (b) The upper band (1710–2690 MHz). Fig. 7. Measured three-dimensional (3-D) total-power radiation patterns for the proposed antenna.

excited in the system ground plane which leads to the nulls or large variations observed in the radiation patterns at higher frequencies. Fig. 8 presents the measured and simulated antenna gain and radiation efficiency for the proposed antenna. The measured data generally agree with the simulated results. Over the lower band (698–960 MHz) and upper band (1710–2690 MHz) respectively shown in Fig. 8(a) and (b), the measured antenna gain is about 00:4–1:1 dBi and 2.7–4.4 dBi. The measured radiation efficiency is respectively about 53–76% and 52–75% over the lower and upper bands. The radiation efficiency for all the frequencies in the lower and upper bands is all better than 50% for the proposed antenna, which generally is acceptable for practical applications in the modern mobile phones. The SAR (specific absorption rate [26]) results are studied in Fig. 9. The SAR simulation model (SEMCAD [27]) shown in the figure is applied. Note that the mobile phone in the study is with the proposed antenna positioned at the bottom of system circuit board, which has been shown to be a promising arrangement for practical applications of the printed antennas with no back ground plane to achieve decreased SAR values [7], [8], [28], [29]. The simulated SAR values for 1-g head tissue are listed in the table in the figure. The return loss given in the table shows the impedance matching level at the testing frequency. The SAR values are obtained using input power of 24 dBm for the GSM850/900 operation (859, 925 MHz) and 21 dBm for the GSM1800/1900 operation (1795, 1920 MHz), UMTS operation (2045 MHz) and LTE operation (740, 2350, 2595 MHz). The obtained SAR values are all well below the SAR limit of 1.6 W/kg [26], indicating that the proposed antenna is promising for practical mobile phone applications. Fig. 10 shows the comparison of the simulated radiation efficiency and antenna gain of the proposed antenna with and without the plastic

Fig. 9. SAR simulation model (SEMCAD [26]) and the simulated SAR values for 1-g head tissue for the proposed antenna. The return loss given in the table shows the impedance matching level at the testing frequency.

housing. Relatively large effects on the radiation efficiency in the lower band (698–960 MHz) than in the upper band (1710–2690 MHz) are observed. This is largely owing to the larger radiation power absorption by the plastic housing (a lossy material) for the frequencies in the lower band. The plastic housing could also cause some variations on the radiation patterns, especially in the upper band. This results in the antenna gain increase seen in some frequencies when the plastic housing is present. Fig. 11 shows the simulated surface current distributions for the proposed antenna. From the surface current distributions, it can be seen that the parasitic shorted strip is excited at its fundamental mode at 700 MHz and its higher-order modes at 1700 and 2700 MHz. While the driven strip monopole is excited at its fundamental mode at 850 MHz

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Fig. 12. Simulated return loss as a function of the length L of the main ground; other dimensions are the same as given in Fig. 1.

for the desired eight-band LTE/ GSM/UMTS operation have been obtained. Good radiation characteristics for frequencies over the operating bands have also been observed. Further, the proposed antenna can 2 on be easily printed on the small no-ground portion of about 780 the system circuit board of the mobile phone. The planar two-dimensional structure of the proposed antenna makes it very attractive for practical applications in the modern slim mobile phones.

mm

REFERENCES Fig. 10. Simulated antenna gain and radiation efficiency for the proposed antenna with and without the plastic housing.

Fig. 11. Simulated surface current distributions for the proposed antenna.

and its higher-order modes at 2200 and 2600 MHz. Also note that, due to strong coupling through the coupling gap, large surface currents are excited for both the driven strip monopole and parasitic shorted strip, for example, at 2600 and 2700 MHz. Fig. 12 shows the simulated return loss as a function of the length L of the main ground. Large effects on the lower band are observed. This behaviour is similar to those observed for the conventional internal mobile phone antennas that have been reported [23]–[25]. IV. CONCLUSION A planar printed strip monopole with a closely-coupled parasitic shorted strip suitable for LTE/GSM/UMTS operation has been proposed. The antenna has been fabricated and tested. Two wide operating bands for covering the 698–960 MHz and 1710–2690 MHz bands

[1] K. L. Wong, Y. C. Lin, and T. C. Tseng, “Thin internal GSM/DCS patch antenna for a portable mobile terminal,” IEEE Trans. Antennas Propag., vol. 54, pp. 238–242, Jan. 2006. [2] K. L. Wong, Y. C. Lin, and B. Chen, “Internal patch antenna with a thin air-layer substrate for GSM/DCS operation in a PDA phone,” IEEE Trans. Antennas Propag., vol. 55, pp. 1165–1172, Apr. 2007. [3] C. I. Lin and K. L. Wong, “Printed monopole slot antenna for internal multiband mobile phone antenna,” IEEE Trans. Antennas Propag., vol. 55, pp. 3690–3697, Dec. 2007. [4] R. A. Bhatti, Y. T. Im, J. H. Choi, T. D. Manh, and S. O. Park, “Ultrathin planar inverted-F antenna for multistandard handsets,” Microwave Opt. Technol. Lett., vol. 50, pp. 2894–2897, Nov. 2008. [5] R. A. Bhatti and S. O. Park, “Octa-band internal monopole antenna for mobile phone applications,” Electron. Lett., vol. 44, pp. 1447–1448, Dec. 2008. [6] H. Rhyu, J. Byun, F. J. Harackiewicz, M. J. Park, K. Jung, D. Kim, N. Kim, T. Kim, and B. Lee, “Multi-band hybrid antenna for ultra-thin mobile phone antenna,” Electron. Lett., vol. 45, pp. 773–774, Jul. 2009. [7] C. H. Chang and K. L. Wong, “Printed =8-PIFA for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 1373–1381, May 2009. [8] Y. W. Chi and K. L. Wong, “Quarter-wavelength printed loop antenna with an internal printed matching circuit for GSM/DCS/PCS/UMTS operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, Sep. 2009. [9] C. T. Lee and K. L. Wong, “Uniplanar coupled-fed printed PIFA for WWAN/WLAN operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 1250–1257, May 2009. [10] K. L. Wong and W. Y. Chen, “Small-size printed loop antenna for penta-band thin-profile mobile phone application,” Microwave Opt. Technol. Lett., vol. 51, pp. 1512–1517, Jun. 2009. [11] T. W. Kang and K. L. Wong, “Chip-inductor-embedded small-size printed strip monopole for WWAN operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 966–971, Apr. 2009. [12] K. L. Wong and S. C. Chen, “Printed single-strip monopole using a chip inductor for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 58, Jan. 2010. [13] C. H. Chang and K. L. Wong, “Small-size printed monopole with a printed distributed inductor for penta-band WWAN mobile phone application,” Microwave Opt. Technol. Lett., vol. 51, pp. 2903–2908, Dec. 2009. [14] K. L. Wong and T. W. Kang, “GSM850/900/1800/1900/UMTS printed monopole antenna for mobile phone application,” Microwave Opt. Technol. Lett., vol. 50, pp. 3192–3198, Dec. 2008.

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[15] , S. Sesia, I. Toufik, and M. Baker, Eds., LTE, The UMTS Long Term Evolution: From Theory to Practice. New York: Wiley, 2009. [16] J. Cho and K. Kim, “A frequency-reconfigurable multi-port antenna operating over LTE, GSM, DCS, and PCS bands,” presented at the IEEE Antennas Propag. Soc. Int. Symp., Charleston, SC, 2009, Session 317. [17] G. Park, M. Kim, T. Yang, J. Byun, and A. S. Kim, “The compact quadband mobile handset antenna for the LTE700 MIMO application,” presented at the IEEE Antennas Propag. Soc. Int. Symp., Charleston, SC, 2009, Session 307. [18] K. L. Wong, L. C. Chou, and C. M. Su, “Dual-band flat-plate antenna with a shorted parasitic element for laptop applications,” IEEE Trans. Antennas Propag., vol. 53, pp. 539–544, Jan. 2005. [19] M. Ozkar, “Multi-band bent monopole antenna,” U.S. patent 7,405,701 B2, Jul. 29, 2008. [20] Y. X. Guo, M. Y. W. Chia, and Z. N. Chen, “Miniature built-in quadband antennas for mobile handsets,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 30–32, 2003. [21] A. Lehtola, “Internal Multi-band antenna,” U.S. patent 6,476,769 B1, Nov. 5, 2002. [22] Ansoft Corporation HFSS [Online]. Available: http://www.ansoft.com/ products/hf/hfss/ [23] K. L. Wong, Planar Antennas for Wireless Communications. New York: Wiley, 2003. [24] T. Y. Wu and K. L. Wong, “On the impedance bandwidth of a planar inverted-F antenna for mobile handsets,” Microwave Opt. Technol. Lett., vol. 32, pp. 249–251, Feb. 2002. [25] P. Vainikainen, J. Ollikainen, O. Kivekas, and I. Kelander, “Resonator-based analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, pp. 1433–1444, Oct. 2002. [26] Safety Levels With Respect to Human Exposure to Radio-frequency Electromagnetic Field, 3 kHz to 300 GHz, ANSI/IEEE standard C95.1, Apr. 1999. [27] SEMCAD Schmid & Partner Engineering AG (SPEAG) [Online]. Available: http://www.semcad.com [28] C. H. Li, E. Ofli, N. Chavannes, and N. Kuster, “Effects of hand phantom on mobile phone antenna performance,” IEEE Trans. Antennas Propag., vol. 57, pp. 2763–2770, Sep. 2009. [29] C. T. Lee and K. L. Wong, “Internal WWAN clamshell mobile phone antenna using a current trap for reduced groundplane effects,” IEEE Trans. Antennas Propag., vol. 57, pp. 3303–3308, Oct. 2009.

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Bandwidth Enhancement of the Small-Size Internal Laptop Computer Antenna Using a Parasitic Open Slot for Penta-Band WWAN Operation Kin-Lu Wong, Wei-Ji Chen, Liang-Che Chou, and Ming-Ren Hsu

Abstract—By embedding a parasitic open slot in the antenna ground of a small-size internal laptop computer antenna, enhanced bandwidth for the antenna’s lower band to cover the GSM850/900 operation (824–960 MHz) can be achieved. The internal laptop computer antenna in this study is a small-size coupled-fed shorted T-monopole, whose length along the top edge of the display ground of the laptop computer is 43 mm only. The antenna can also provide a wide upper band to cover the GSM1800/1900/UMTS operation (1710–2170 MHz). With the inclusion of the parasitic open slot, the total antenna length along the top edge of the display ground is 48 mm only, and the occupied antenna volume is 10 3 5 48 mm above the top edge of the display ground, which is the smallest for the internal penta-band WWAN laptop computer antenna that have been reported for the present. Detailed results of the proposed parasitic open slot on bandwidth enhancement of the internal WWAN laptop computer antenna are presented. Index Terms—Bandwidth enhancement, laptop computer antennas, mobile antennas, open slot, WWAN antennas.

I. INTRODUCTION In order to provide ubiquitous wireless internet access, the internal antennas for wireless wide area network (WWAN) communications have been demanded in the modern laptop computers. It is required that the internal WWAN laptop computer antennas be with smaller size yet wider bandwidth to cover the penta-band operation of the GSM850 (824–894 MHz),GSM900 (880–960 MHz), GSM1800 (1710–1880 MHz), GSM1900 (1850–1990 MHz) and UMTS (1920–2170 MHz) systems. For the internal WWAN laptop computer antennas that have recently been reported [1]–[7], however, it is required that the length of the antenna along the top edge of the display ground should be at least 60 mm for penta-band WWAN operation. This length requirement is needed for these reported internal antennas to generate the desired operating band at about 900 MHz to cover the GSM850/900 operation. This behavior is owing to the much larger display ground connected to the internal WWAN antenna. The large display ground cannot assist in the generation of a wide lower band at about 900 MHz for the WWAN antenna. This behavior is different from the internal antennas connected to the system ground plane of the mobile phone, which has a much smaller size than the display ground and is generally excited as an efficient radiator in the 900-MHz band [8]. In this communication we present a new bandwidth-enhancement method of using a parasitic open slot to achieve a wide lower band for the internal laptop computer antenna with a small size (antenna length less than 50 mm along the top edge of the display ground) to Manuscript received December 11, 2009; revised February 25, 2010; accepted March 25, 2010. Date of publication July 01, 2010; date of current version October 06, 2010. K.-L. Wong and W.-J. Chen are with the Department of Electrical Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (e-mail: [email protected]; [email protected]; [email protected]. nsysu.edu.tw). L.-C. Chou and M.-R. Hsu are with the Department of High Frequency Business, Yageo Corporation Nantze Branch, Kaohsiung 811, Taiwan (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2010.2055815

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. (a) Geometry of the proposed bandwidth-enhanced internal laptop computer antenna with a parasitic resonant open slot for the penta-band WWAN operation. (b) Dimensions of the metal pattern of the antenna.

cover the GSM850/900 operation and a wide upper band to cover the GSM1800/1900/UMTS operation as well. The parasitic open slot requires no direct feeding and hence does not complicate the structure of the studied small-size internal laptop computer antenna, which is a coupled-fed shorted T-monopole [9] with its length along the top edge of the display ground of the laptop computer 43 mm only in this study. The parasitic open slot is embedded in the ground plane of the shorted T-monopole. It can be excited as a monopole slot or quarter-wavelength slot [10]–[22] at about 900 MHz and contribute a resonant mode to assist in obtaining a wide lower band of at least 824–960 MHz for the antenna to cover the GSM850/900 operation. Notice that the parasitic open slot applied in this study is different from those in the reported monopole slot antennas for the mobile phone or laptop computer applications [10]–[22], in which they all require direct feeding. Details of the proposed parasitic open slot are described, and its detailed effects on bandwidth enhancement of the internal laptop computer antenna are studied. II. PROPOSED ANTENNA WITH A PARASITIC OPEN SLOT Fig. 1(a) shows the geometry of the proposed bandwidth-enhanced internal laptop computer antenna with a parasitic resonant open slot for the penta-band WWAN operation. The antenna is printed on a 0.8-mm thick FR4 substrate of size 48 2 18:5 mm2 and comprises a radiating portion and an antenna ground. The radiating portion is a T-monopole (section CD, DE and DF) short-circuited to the antenna ground through the shorting strip (section CB) and capacitively excited by a feeding strip (section AG). The proposed parasitic open slot has a length (t) of 49.5 mm and a width of 1 mm and is embedded in the antenna ground. Details of the dimensions of the radiating portion and the open-slot-embedded antenna ground are given in Fig. 1(b). Notice that the portion of the antenna ground below the open slot is electrically connected to the display ground through two fixing points (point H, H’). The display

ground in this study is a 0.2-mm thick copper plate of width 200 mm and length 260 mm, which can also be treated as the supporting metal frame of the laptop display. Also, along the top edge of the display ground, the antenna is mounted with a distance of 30 mm to the central line of the display ground. This arrangement is for the practical consideration that the central region is usually reserved to accommodate the lens of the embedded digital camera in many modern laptop computers. The upper part (width 3.5 mm) of the radiating portion is bent such that the total antenna height including the parasitic open slot above the top edge of the display ground is 10 mm only, and the antenna width is 3.5 mm in the direction orthogonal to the laptop display. In this case, the cross-sectional area of the antenna including the parasitic open slot is 10 2 3:5 mm2 , which is promising to be mounted along the narrow spacing between the laptop display and the laptop casing as an internal WWAN antenna [1]–[7]. It should also be noted that the antenna length along the top edge of the display ground is 48 mm only, which is the smallest for the internal penta-band WWAN laptop computer antenna that have been reported for the present. The coupled-fed shorted T-monopole has been studied in [9] for achieving penta-band WWAN operation in the mobile phone. By applying the coupling feed to the shorted T-monopole, two wide operating bands centered at about 900 and 1900 MHz to respectively cover the GSM850/900 and GSM1800/1900/UMTS operation can be obtained for application in the mobile phone with a ground plane of 60 2 100 mm2 . The operating principle of the coupled-fed T-monopole has been extensively studied in [9]. The coupling feed applied here is different from that used in the coupled-fed printed PIFAs (planar inverted-F antenna) for the internal mobile phone antennas [23]–[26], in which the coupling feed comprises a coupling strip and a feeding strip and does not directly couple to the main radiating portion of the antenna. In this study, the left arm (section DF) of the T-monopole is directly coupled by the feeding strip (section AG) and controls the generation of a resonant mode at about 800 MHz and a higher-order resonant mode at about 2100 MHz. The other arm, section DE, contributes a resonant mode at about 1800 MHz which incorporates the one at 2100 MHz to form a wider upper band to cover the desired frequency range of 1710–2170 MHz. For the laptop computer application, the effect of the display ground which has a much larger size (2002260 mm2 here) than the ground plane of the mobile phone is relatively small for the antenna’s upper band. However, for the lower band, the internal antenna with a small size (antenna length less than 50 mm here) is difficult to achieve a wide bandwidth to cover the GSM850/900 operation when applied in the laptop computer. This is the major problem to overcome in obtaining a small-size internal penta-band WWAN laptop computer antenna. In this study, the parasitic open slot embedded in the antenna ground can be excited as a quarter-wavelength resonant structure [10]–[22] to contribute an additional resonant mode to the antenna’s lower band; hence, a wide lower band covering the desired frequency range of 824–960 MHz can be obtained. Also notice that the portion of the antenna ground above the embedded open slot has a width of 2 mm, which is allocated to accommodate the 50- mini coaxial line connected to the internal antenna in the practical applications. The central conductor and outer grounding sheath are respectively connected to point A (the feeding point) and point A’ (the grounding point) as shown in the figure. III. RESULTS AND DISCUSSION Fig. 2 shows the measured and simulated return loss for the proposed antenna with dimensions given in Fig. 1. The measured data are seen to agree with the simulated results obtained using the Ansoft simulation software HFSS version 11.2 [27]. The wide lower band with a measured 3:1 VSWR bandwidth of 185 MHz (790–975 MHz) is obtained,

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Fig. 2. Measured and simulated return loss for the proposed antenna.

Fig. 3. Comparison of the simulated return loss of the proposed antenna and the case without the parasitic open slot.

which covers the GSM850/900 operation. The wide upper band is also obtained. It has a wide bandwidth of 610 MHz (1630–2240 MHz) and covers the GSM1800/1900/UMTS operation. Note that the 3:1 VSWR (6-dB return loss) bandwidth definition is widely used as a specification for the internal WWAN antenna design in the practical mobile devices including the laptop computers and mobile phones. Effects of the parasitic open slot on the bandwidth enhancement of the proposed antenna are clearly seen in Fig. 3, in which the comparison of the simulated return loss of the proposed antenna and the case without the parasitic open slot is shown. The corresponding dimensions of the two antennas in the figure are the same. Results clearly indicate that an additional resonant mode at about 930 MHz is generated owing to the presence of the parasitic open slot. While for the case without the open slot, the excited resonant mode at about 820 MHz shows a narrow bandwidth, which is far from covering the desired frequency range of 824–960 MHz. For the upper band, the first mode is affected and shifted to lower frequencies, while the second mode is very slightly affected. The first mode in the upper band is mainly contributed by section DE as studied in [9] and will be also analyzed in Fig. 6. Fig. 4 shows the simulated distribution of the electric field in the open slot and the surface current on the metal pattern of the antenna. Strong excited electric fields in the open slot are seen at 930 MHz shown in Fig. 4(b). Also, at 930 MHz, the excited surface currents along the open slot are strong and close to symmetric. These two characteristics indicate that the parasitic open slot is excited as a resonant structure at 930 MHz. The observed field and current distributions are also similar to those of the excited traditional half-wavelength slot antenna from its slot center to one of the closed end. That is, the parasitic open slot resonated at 930 MHz is a quarter-wavelength slot or monopole slot as studied in [10]–[22]. While at 820, 1850, and 2100 MHz shown in Figs. 4(a), 4(c), and Fig. 4(d) the field and current distributions indicate that the parasitic open slot is not at resonance as it does at 930 MHz. In Fig. 4(c), results also indicate that section DE is strongly excited, which contributes to the excited resonant mode at 1850 MHz as discussed in Section II Fig. 5(a) shows the simulated return loss as a function of the length t of the parasitic open slot. Other dimensions are the same as given in Fig. 1. It is seen that the second mode in the lower band is shifted to higher fre-

Fig. 4. Simulated distribution of the electric field in the open slot and the sur. (b) f face current on the metal pattern of the antenna. (a) f . (c) f . (d) f .

930 MHz

= 1850 MHz

= 2100 MHz

= 820 MHz

=

quencies when the length t decreases. On the other hand, the central frequency or resonant frequency of the first mode in the lower band is not varied, although some variations in the impedance matching are seen. The observed behavior also confirms that the second mode in the lower band is mainly contributed by the parasitic open slot. Effects of the width w of the parasitic open slot are studied in Fig. 5(b), and results for w varied from 0.5 to 1.5 mm are shown. When the width is smaller (w = 0:5 mm) or larger (w = 1:5 mm), the upper edge frequency of the lower band is decreased. To obtain the desired lower-band and upperband bandwidths, the width w is selected to be 1.0 mm in the study. Fig. 6 shows the comparison of the simulated return loss of the proposed antenna and the case without section DE. It is seen that an additional resonant mode which is the first mode of the upper band of the proposed antenna is generated, owing to the presence of section DE. This indicates that the first mode of the upper band is contributed by section DE. Also, the presence of section DE helps achieve en-

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Fig. 8. Measured antenna gain and radiation efficiency for the proposed antenna. (a) The GSM850/900 bands. (b) The GSM1800/1900/UMTS bands. Fig. 5. Simulated return loss as a function of (a) the length t and (b) the width w of the parasitic open slot; other dimensions are the same as in Fig. 1.

Fig. 9. Simulated return loss for the proposed antenna with and without the keyboard ground. Fig. 6. Comparison of the simulated return loss of the proposed antenna and the case without section DE (strip 2).

Fig. 7. Measured three-dimensional (3-D) total-power radiation patterns for the proposed antenna.

hanced impedance matching of the two resonant modes in the lower band. More detailed effects of the coupled-fed T-monopole have been analyzed in [9]. Fig. 7 shows the measured three-dimensional (3-D) total-power radiation patterns for the proposed antenna. The antenna is tested in a far-field anechoic chamber [ETS- Lindgren measurement system (http://www.ets-lindgren.com)], and the far-field data include the mismatching losses. The radiation patterns at about the central frequencies of the five operating bands are shown. The radiation patterns are quite

different from those shown in [9] for the mobile phone applications, in which the patterns in the 900-MHz band are close to the dipole-like patterns. This is mainly because that the system ground plane of the mobile phone can assist in the mode generation for the 900-MHz band, while the display ground of the laptop computer can not, owing to its much larger size. In Fig. 7, the patterns at 859 and 925 MHz are similar to each other, indicating that stable patterns are obtained in the 900-MHz band. Similar patterns at 1920 and 2045 MHz are also seen, which are smoother than the patterns seen at 1795 MHz. Fig. 8 shows the measured antenna gain and radiation efficiency for the proposed antenna. The results over the lower band for the GSM850/900 operation are shown in Fig. 8(a), while those over the upper band for the GSM1800/1900/UMTS operation are shown in Fig. 8(b). The measured antenna gain is about 0.4–2.3 dBi over the lower band and 1.3–4.1 dBi over the upper band. For the measured radiation efficiency, it is about 58–63% over the lower band and 65–86% over the upper band, which are acceptable for practical applications. Effects of the keyboard ground on the proposed antenna are also studied in Fig. 9. Results of the simulated return loss for the antenna with and without the keyboard ground are shown. The keyboard ground has the same dimensions as those of the display ground 2 (200 2 260 mm ) in Fig. 1. The results show very small variations, when the keyboard ground is present. This is reasonable, since the display ground alone has very large dimensions compared to those of the proposed antenna. Hence, with the presence of the additional keyboard ground, small effects can be expected.

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IV. CONCLUSION The use of a parasitic open slot has been shown to be effective in enhancing the bandwidth of the lower band at about 900 MHz for the small-size WWAN antenna in the laptop computer. The proposed parasitic open slot has been applied to a coupled-fed shorted T-monopole with a small length of 43 mm only, which shows a very narrow lower band far from covering the GSM850/900 operation and a wide upper band to cover the GSM1800/1900/UMTS operation. With the presence of the parasitic open slot, results have shown that a new resonant mode contributed by the open slot has been generated to result in a wide lower band to cover the GSM850/900 operation. The operating principle of the parasitic open slot has been studied in the communication. The radiation characteristics of the proposed antenna with the parasitic open slot have been observed to meet the requirement for practical applications in the laptop computer.

REFERENCES [1] K. L. Wong and L. C. Lee, “Multiband printed monopole slot antenna for WWAN operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 57, pp. 324–330, Feb. 2009. [2] C. T. Lee and K. L. Wong, “Study of a uniplanar printed internal WWAN laptop computer antenna including user’s hand effects,” Microwave Opt. Technol. Lett, vol. 51, pp. 2341–2346, Oct. 2009. [3] K. L. Wong and F. H. Chu, “Internal planar WWAN laptop computer antenna using monopole slot elements,” Microwave Opt. Technol. Lett., vol. 51, pp. 1274–1279, May 2009. [4] K. L. Wong and S. J. Liao, “Uniplanar coupled-fed printed PIFA for WWAN operation in the laptop computer,” Microwave Opt. Technol. Lett., vol. 51, pp. 549–554, Feb. 2009. [5] C. H. Chang and K. L. Wong, “Internal coupled-fed shorted monopole antenna for GSM850/900/1800/1900/UMTS operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 56, pp. 3600–3604, Nov. 2008. [6] X. Wang, W. Chen, and Z. Feng, “Multiband antenna with parasitic branches for laptop applications,” Electron. Lett., vol. 43, pp. 1012–1013, Sep. 2007. [7] C. H. Kuo, K. L. Wong, and F. S. Chang, “Internal GSM/DCS dual-band open-loop antenna for laptop application,” Microwave Opt. Technol. Lett., vol. 49, pp. 680–684, Mar. 2007. [8] P. Vainikainen, J. Ollikainen, O. Kivekas, and I. Kelander, “Resonator—based analysis of the combination of mobile handset antenna and chassis,” IEEE Trans. Antennas Propag., vol. 50, pp. 1433–1444, Oct. 2002. [9] W. Y. Chen and K. L. Wong, “Small-size coupled-fed shorted T-monopole for internal WWAN antenna in the slim mobile phone,” Microwave Opt. Technol. Lett., vol. 52, pp. 257–262, Feb. 2010.

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[10] H. wang, M. Zheng, and S. Q. Zhang, “Monopole slot antenna,” U.S. patent 6,618,020 B2, Sep. 9, 2003. [11] S. K. Sharma, L. Shafai, and N. Jacob, “Investigation of wide-band microstrip slot antenna,” IEEE Trans. Antennas Propag., vol. 52, pp. 865–872, Mar. 2004. [12] S. I. Latif, L. Shafai, and S. K. Sharma, “Bandwidth enhancement and size reduction of microstrip slot antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 994–1002, Mar. 2005. [13] A. P. Zhao and J. Rahola, “Quarter-wavelength wideband slot antenna for 3–5 GHz mobile applications,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 421–424, 2005. [14] R. Bancroft, “Dual Slot Radiator Single Feedpoint Printed Circuit Board Antenna,” U.S. patent 7,129,902 B2, Oct. 31, 2006. [15] P. Lindberg, E. Ojefors, and A. Rydberg, “Wideband slot antenna for low-profile hand-held terminal applications,” in Proc. 36th Eur. Microwave Conf. (EuMC2006), Manchester, U.K., pp. 1698–1701. [16] W. S. Chen and K. Y. Ku, “Broadband design of a small non-symmetric ground =4 open slot antenna,” Microwave J., pp. 110–120, Jan. 2007. [17] C. I. Lin and K. L. Wong, “Printed monopole slot antenna for internal multiband mobile phone antenna,” IEEE Trans. Antennas Propag., vol. 55, pp. 3690–3697, Dec. 2007. [18] C. H. Wu and K. L. Wong, “Hexa-band internal printed slot antenna for mobile phone application,” Microwave Opt. Technol. Lett, vol. 50, pp. 35–38, Jan. 2008. [19] C. I. Lin and K. L. Wong, “Internal hybrid antenna for multiband operation in the mobile phone,” Microwave Opt. Technol. Lett, vol. 50, pp. 38–42, Jan. 2008. [20] C. H. Wu and K. L. Wong, “Internal hybrid loop/monopole slot antenna for quad-band operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 50, pp. 795–801, Mar. 2008. [21] W. S. Chen and K. Y. Ku, “Band-rejected design of the printed open slot antenna for WLAN/WIMAX operation,” IEEE Trans. Antennas Propag., vol. 56, pp. 1163–1169, Apr. 2008. [22] C. I. Lin and K. L. Wong, “Printed monopole slot antenna for penta-band operation in the folder-type mobile phone,” Microwave Opt. Technol. Lett, vol. 50, pp. 2237–2241, Sep. 2008. [23] C. H. Chang and K. L. Wong, “Printed =8-PIFA for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 1373–1381, May 2009. [24] K. L. Wong and C. H. Huang, “Bandwidth-enhanced internal PIFA with a coupling feed for quad-band operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 50, pp. 683–687, Mar. 2008. [25] K. L. Wong and C. H. Huang, “Printed PIFA with a coplanar coupling feed for penta-band operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 50, pp. 3181–3186, Dec. 2008. [26] C. T. Lee and K. L. Wong, “Uniplanar coupled-fed printed PIFA for WWAN/WLAN operation in the mobile phone,” Microwave Opt. Technol. Lett., vol. 51, pp. 1250–1257, May 2009. [27] Ansoft Corporation HFSS [Online]. Available: http://www.dupont. com/mcm/techinfo.html

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Corrections Corrections to “On the use of Nonsingular Kernels in Certain Integral Equations for Thin-Wire Circular-Loop Antennas” George Fikioris, Panagiotis J. Papakanellos, and Hristos T. Anastassiu

Errors were identified in [1]. First, Table I has several numerical errors. It should be replaced by Table I shown here. Also, in the 6th line under (24), replace “59 terms in (24) are required” with “42 terms in (24) are required.” Finally, in the first line of Section II.A, exp(jct=k) should be replaced by exp(jkct). The errors do not affect the discussions in [1].

Index Terms—Galerkin method, integral equations, loop antennas.

ACKNOWLEDGMENT Manuscript received March 30, 2010; accepted April 02, 2010. Date of publication July 08, 2010; date of current version October 06, 2010. G. Fikioris is with the School of Electrical and Computer Engineering, National Technical University, GR 157-73 Zografou, Athens, Greece (e-mail: [email protected]). P. J. Papakanellos is with the Hellenic Air Force Academy, GR 1010 Dekelia, Athens, Greece (e-mail: [email protected]). H. T. Anastassiu is with the Communications and Networks Design and Development Department, Hellenic Aerospace Industry, GR 32009 Schimatari, Viotia, Greece (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2055816

The authors thank Dr. P. Koukoulas and Dr. V. A. Houdzoumis for bringing the errors to their attention.

REFERENCES [1] G. Fikioris, P. J. Papakanellos, and H. T. Anastassiu, “On the use of nonsingular kernels in certain integral equations for thin-wire circular-loop antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 151–157, Jan. 2008.

TABLE I REPLACEMENT FOR TABLE I IN [1]

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 10, OCTOBER 2010

Corrections to “Comparison of Interpolating Functions and Interpolating Points in Full-Wave Multilevel Green’s Function Interpolation Method”

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TABLE II COMPARISON OF ACCURACY BETWEEN DIFFERENT RBFS

Yan Shi and Chi Hou Chan In [1], Table II contained errors, the correct version is shown here. Also, errors appeared in (2) and (3), the correct equations are as follows:

ik r r 9 (*r ; *r ) = e* * r 0 r 0

j

0

j

(2)

0

and

ik r r 9 (*r ; *r ) = e* * 3 r 0 r 0

j

0

j

0

* 0 *r 0 1

ik r

0

:

(3)

REFERENCES [1] Y. Shi and C. H. Chan, “Comparison of interpolating functions and interpolating points in full-wave multilevel Green’s function interpolation method,” IEEE Trans. Antennas Propag., vol. 58, no. 8, pp. 2691–2699, Aug 2010. Manuscript received July 27, 2010; accepted July 27, 2010. Date of current version October 06, 2010. Y. Shi is with the School of Electronic Engineering, Xidian University, Xi’an 710071, Shaanxi, China (e-mail: [email protected]). C. H. Chan is with the State Key Laboratory of Millimeter Waves, City University of Hong Kong, Hong Kong SAR, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2080770

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