IEEE Transactions on Antennas and Propagation [volume 58 number 11]

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NOVEMBER 2010

VOLUME 58

NUMBER 11

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Fundamental Aspects of Near-Field Coupling Small Antennas for Wireless Power Transfer .... ....... J. Lee and S. Nam A Switchable Matching Circuit for Compact Wideband Antenna Designs ...... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... . Y. Li, Z. Zhang, W. Chen, Z. Feng, and M. F. Iskander Electrically Small, Millimeter Wave Dual Band Meta-Resonator Antennas ..... ...... .... ..... I. K. Kim and V. V. Varadan Printed Loop Antenna With a U-Shaped Tuning Element for Hepta-Band Laptop Applications ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ........ C.-W. Chiu and Y.-J. Chi Modified Test Zone Field Compensation for Small-Antenna Measurements .... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ....... J. T. Toivanen, T. A. Laitinen, and P. Vainikainen A Quasi-Planar Conical Antenna With Broad Bandwidth and Omnidirectional Pattern for Ultrawideband Radar Sensor Network Applications .. ........ ......... ......... . ....... ......... ......... ... H. Zhai, S. Tjuatja, J. W. Bredow, and M. Lu A Differential Dual-Polarized Cavity-Backed Microstrip Patch Antenna With Independent Frequency Tuning ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ... C. R. White and G. M. Rebeiz The Backfire-to-Broadside Symmetrical Beam-Scanning Periodic Offset Microstrip Antenna .. ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... Y. Li, Q. Xue, E. K.-N. Yung, and Y. Long On Strongly Truncated Leaky-Wave Antennas Based on Periodically Loaded Transmission Lines ..... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... M. Schühler, R. Wansch, and M. A. Hein Modulated Arm Width (MAW) Spiral: Theory, Modeling, Design and Measurements .. ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ...... W. N. Kefauver, T. P. Cencich, and D. S. Filipovic Arrays Time Domain Characterization of Circularly Polarized Ultrawideband Array ... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... . A. E.-C. Tan, M. Y.-W. Chia, and K. Rambabu A Post-Wall Waveguide Center-Feed Parallel Plate Slot Array Antenna in the Millimeter-Wave Band . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ...... K. Hashimoto, J. Hirokawa, and M. Ando Tracking of Metallic Objects Using a Retro-Reflective Array at 26 GHz ... . J. A. Vitaz, A. M. Buerkle, and K. Sarabandi Adaptive Array Beamforming Using a Combined LMS-LMS Algorithm ...... J. A. Srar, K.-S. Chung, and A. Mansour Adaptive Correction to Array Coefficients Through Dithering and Near-Field Sensing . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ...... R. Janaswamy, D. V. Gupta, and D. H. Schaubert Ultrawideband All-Metal Flared-Notch Array Radiator . ......... ......... ........ ......... ... R. W. Kindt and W. R. Pickles Analyzing Large-Scale Non-Periodic Arrays With Synthetic Basis Functions .. ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ B. Zhang, G. Xiao, J. Mao, and Y. Wang

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(Contents Continued on p. 3441)

(Contents Continued from Front Cover) Electromagnetics Combined Electromagnetic Energy and Momentum Conservation Equation .... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... . O. Zandi, Z. Atlasbaf, and M. S. Abrishamian A Novel Approach for Evaluating Hypersingular and Strongly Singular Surface Integrals in Electromagnetics ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ..... M. S. Tong and W. C. Chew Source Decomposition as a Diakoptic Boundary Condition in FDTD With Reflecting External Regions ....... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... S. Malevsky, E. Heyman, and R. Kastner MAS Pole Location and Effective Spatial Bandwidth of the Scattered Field .... ......... ......... ........ ...... J. E. Richie Complex Materials and Surfaces High Efficiency Wideband Aperture-Coupled Stacked Patch Antennas Assembled Using Millimeter Thick Micromachined Polymer Structures ... ......... ........ ......... ......... .... S. K. Pavuluri, C. Wang, and A. J. Sangster Novel Reconfigurable Defected Ground Structure Resonator on Coplanar Waveguide .. ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ...... H. B. El-Shaarawy, F. Coccetti, R. Plana, M. El-Said, and E. A. Hashish Artificial Impedance Surfaces for Reduced Dispersion in Antenna Feeding Systems ... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ....... G. Goussetis, J. L. Gómez-Tornero, A. P. Feresidis, and N. K. Uzunoglu Multilayered Wideband Absorbers for Oblique Angle of Incidence ...... ....... .. ......... A. Kazemzadeh and A. Karlsson Numerical A Split-Step FDTD Method for 3-D Maxwell’s Equations in General Anisotropic Media ....... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... G. Singh, E. L. Tan, and Z. N. Chen Locally-Conformal FDTD for Anisotropic Conductive Interfaces ........ ........ ......... ..... H. O. Lee and F. L. Teixeira An Auxiliary Differential Equation Method for FDTD Modeling of Wave Propagation in Cole-Cole Dispersive Media .. .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .. I. T. Rekanos and T. G. Papadopoulos Numerically Stable Moment Matching for Linear Systems Parameterized by Polynomials in Multiple Variables With Applications to Finite Element Models of Microwave Structures ...... ....... .. ......... O. Farle and R. Dyczij-Edlinger Propagation Indoor Channel Spectral Statistics, K-Factor and Reverberation Distance ....... ......... ...... Y. Lustmann and D. Porrat Multipath Fading Measurements at 5.8 GHz for Backscatter Tags With Multiple Antennas ..... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .. J. D. Griffin and G. D. Durgin Semi-Deterministic Propagation Model for Subterranean Galleries and Tunnels ........ ......... .. L. Subrt and P. Pechac

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COMMUNICATIONS

Broadband Bowtie Dielectric Resonator Antenna ........ ......... ......... ........ ......... ......... L. Z. Thamae and Z. Wu Miniaturized Dual-Band CPW-Fed Annular Slot Antenna Design With Arc-Shaped Tuning Stub ...... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ...... M.-J. Chiang, T.-F. Hung, J.-Y. Sze, and S.-S. Bor Low Profile Spiral on a Thin Ferrite Ground Plane for 220–500 MHz Operation ........ ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... I. Tzanidis, C.-C. Chen, and J. L. Volakis Bent Two-Leaf Antenna Radiating a Tilted, Linearly Polarized, Wide Beam .... . H. Nakano, Y. Ogino, and J. Yamauchi A Simple Compact Reconfigurable Slot Antenna With a Very Wide Tuning Range .... .. H. Li, J. Xiong, Y. Yu, and S. He A Novel Compact UHF RFID Tag Antenna Designed With Series Connected Open Complementary Split Ring Resonator (OCSRR) Particles ..... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... B. D. Braaten Field Penetration Into a Circular Aperture Pierced by a Long Cylinder .. ........ ...... .... ......... . Y. S. Lee and H. J. Eom An Efficient Asymptotic Extraction Approach for the Green’s Functions of Conformal Antennas in Multilayered Cylindrical Media ...... ........ ......... ......... ........ ......... ......... ........ ... J. Wu, S. K. Khamas, and G. G. Cook A Generalized Asymptotic Extraction Solution for Antennas in Multilayered Spherical Media . ....... .. .... S. K. Khamas Lack of Rotation Invariance in Short-Pulse Communication Between Broadband Circular-Polarization Antennas ...... .. .. ........ ......... ......... ........ ......... ......... ........ H. D. Foltz, J. S. McLean, A. Medina, and J. H. Alvarez Jerkov An Efficient Slope-Deterministic Facet Model for SAR Imagery Simulation of Marine Scene .. ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... H. Chen, M. Zhang, Y.-H. Zhao, and W. Luo Wave Scattering by Dielectric and Lossy Materials Using the Modified Equivalent Current Approximation (MECA) . .. .. ........ ......... ......... ........ ......... ........ J. G. Meana, J. Á. Martínez-Lorenzo, F. Las-Heras, and C. Rappaport Dual-Band Tunable Screen Using Complementary Split Ring Resonators ...... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ... B. Sanz-Izquierdo, E. A. Parker, and J. C. Batchelor Near-Optimal Radiation Patterns for Antenna Diversity . ......... ......... ........ ......... .. D. N. Evans and M. A. Jensen

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CORRECTIONS

Correction to “A Modified Real GA for the Sparse Linear Array Synthesis With Multiple Constraints” ........ ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ..... K. Chen, Z. He, and C. Han

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Is the leading international engineering journal on the general topics of electromagnetics, antennas and wave propagation. The journal is devoted to antennas, including analysis, design, development, measurement, and testing; radiation, propagation, and the interaction of electromagnetic waves with discrete and continuous media; and applications and systems pertinent to antennas, propagation, and sensing, such as applied optics, millimeter- and sub-millimeter-wave techniques, antenna signal processing and control, radio astronomy, and propagation and radiation aspects of terrestrial and space-based communication, including wireless, mobile, satellite, and telecommunications. Author contributions of relevant full length papers and shorter Communications are welcomed. See inside back cover for Editorial Board.

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Digital Object Identifier 10.1109/TAP.2010.2089912

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 11, NOVEMBER 2010

Fundamental Aspects of Near-Field Coupling Small Antennas for Wireless Power Transfer Jaechun Lee, Member, IEEE, and Sangwook Nam, Member, IEEE

Abstract—A physical limitation on the power transfer efficiency between two electrically small antennas in the near-field range is presented. By using a Z-parameter which describes the interaction between two antennas for 10 10 spherical modes in connection with antenna parameters, the maximum power transfer efficiency and the optimum load impedance are shown as functions of the distance between two antennas, the radiation efficiency and the input impedance of the isolated antenna. The theory is verified by a simulation with a small helical antenna, which generates 10 and 10 modes, simultaneously.

TE TM

TE

TM

Index Terms—Antenna efficiency, electromagnetic coupling, near fields.

I. INTRODUCTION IRELESS power transfer through coupled antennas in the near-field range is being widely used in radio frequency identification (RFID) systems. While RFID tags require a small amount of power for an instantaneous process, an increasing number of mobile electric devices are demanding higher amounts of wireless power transfer. The feasibility of this transfer depends on power transfer efficiency, and the results on this topic were reported recently [1]–[3]. As in earlier works, to estimate the power transfer efficiency of near-field coupling antennas, a numerical or analytical method can be used to solve the electromagnetic problem of the specified antennas by varying their positions. However, in view of such a system design, it is desirable that antennas are characterized as a few parameters and a closed-form of formula for the power transfer efficiency is given as a function of them like the Friis transmission formula between antennas in the far-field range. According to that purpose, the spherical mode representation is used to obtain a simple description of the problem, since the antennas used for wireless power transfer are electrically small or spherical modes. and generate predominantly Furthermore, the coupling properties between antennas are obtainable from the interaction theory between spherical modes in space [4]. A similar approach had been studied to investigate

W

Manuscript received October 30, 2008; revised June 18, 2009; accepted May 04, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST). (No. 2010-0018879). J. Lee is with the Samsung Advanced Institute of Technology (SAIT), Yongin 449-712, Korea (e-mail: [email protected]). S. Nam is with the School of Electrical Engineering and INMC, Seoul National University, Seoul 151-742, Korea (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.2010.2071330

the mutual coupling effect between adjacent antennas on the performance of an array antenna system [5], [6]. In this study, a simple and general model of antennas and coupling network among them in wireless power transfer system is presented based on the spherical mode theory. This enable us to evaluate system performance readily with a few parameters of antenna, and it informs us of the necessary parameters of antenna to achieve the power transfer efficiency required by an application. From the formulation, the physical limitation of the power transfer efficiency and the effect of the radiation efficiency on it are examined, and the optimum impedance for maximum power transfer efficiency is found. In most applications, as in RFID systems, the relative orientation, or more generally, the relative polarization of the antennas will not coincide. The formula includes the case of different orientations using identical antennas; The case of different antennas can be treated with a similar method. The characteristics of coupling magnitude, that determines the power transfer efficiency, are analyzed in terms of the coupling coefficients between the spherical modes in parallel or orthogonal directions. The theory is verified with an example of helical antenna. In addition, the optimum frequency of the power transfer efficiency is shown with this actual antenna in example, while an ideal lossless antenna shows the better power transfer efficiency at lower frequency. II. THEORY To obtain coupling network between antennas, first, their transmitting and receiving fields are expressed as spherical waves. Second, each antenna is represented by a scattering parameter between its feed port and spherical waves. Third, space between antennas is described as a Z-parameter among their spherical waves. Finally, antennas and space are cascaded into a two-port parameter between feed ports of two antennas. time convention is assumed for the Throughout this study, field quantities and suppressed. A. Spherical Mode Representation of Antenna Model The electric and magnetic fields outside an antenna can be expressed by spherical modes [8], [9]

0018-926X/$26.00 © 2010 IEEE

(1) (2)

LEE AND NAM: FUNDAMENTAL ASPECTS OF NEAR-FIELD COUPLING SMALL ANTENNAS FOR WIRELESS POWER TRANSFER

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where is the wave impedance and , are the spherical vector wave functions defined in Appendix A. and are the coefficients of and modes, respectively, whose superscripts, and indicate outward and inward traveling directions, respectively. The radiated power is written by Fig. 1. Near-field coupling antennas modeling with T M translation and rotation.

and T E

modes

(3) Each spherical mode can be regarded as propagating in a waveguide of the unit characteristic impedance with its wave coeffior . cient being Between these spherical modes and the feed port of the antenna, a S-parameter can be built [4] (4) Fig. 2. Equivalent network between the input ports of two identical antennas.

where is the reflection coefficient at the feed port. , and are the receiving, transmitting and scattering coefficient matrices, respectively, for the spherical modes. , are the coefficients of the incident and reflected waves at the feed port, respectively. and are the coefficient vectors of the incident , and and reflected spherical modes, which contain , , respectively. Generally, antennas used for wireless power transfer are very small compared with wavelength, and are coupled in the nearfield range. Such antennas predominantly generate the lowestand modes, which order spherical modes, i.e., are represented as and modes, respectively, under properly rotated coordinates as shown in Appendix B. So we consider a small antenna assumed to and modes, having • radiate and receive only no interaction with higher-order modes; • have uniform phase of current distribution; • be reciprocal. The fields generated by the antenna are written in detail in Appendix A. The input impedance of the single antenna is denoted as

is the input current of the antenna and . The phases of the input current and the spherical mode coefficients are same since the current on the antenna is assumed to have a uniform phase and the spherical mode coefficients are determined by [8, Eq. (60)]

where

(8) and are realwhere is the current density and valued functions in free space as defined in Appendix A. Then, from (3) and(7), the relation between the input current and the spherical mode coefficients is given as (9) This can be described as the transformer between antenna feed and spherical mode ports as depicted in Fig. 2. Its S-parameter of the feed port being is given by representation with

(5) and are the input resistances related with the where radiated power into space and the loss in the antenna, respectively. The radiation efficiency is defined by the ratio of the radiated , to the input power, power,

(10) .. .

..

.

.. .

where and , the transmitting coefficients for modes, respectively, are

.. . and

(6) (11) The radiated power is divided into those of modes

and These are related with the radiation efficiency as (7)

(12)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 11, NOVEMBER 2010

B. Space Description as the Relation of Spherical Modes in the Distance Space can be seen as a four-port network between spherical modes centered at each antenna positions as depicted in Fig. 2. This network parameter is obtained from the following relation among spherical modes with different origins and orientations, namely, the Addition theorem in Appendix C [9], [10]

TABLE I CHARACTERISTICS OF THE TRANSLATION COEFFICIENTS IN T IN TERMS OF CYLINDRICAL COMPONENTS AT r < 0:37 WAVELENGTHS

(13) (14) , , , and , are the translation coefficients including rotation in , are different from , in Appendix C. that the -dependent function is instead of , where is the spherical Bessel function of the first kind. and are the spherical Hankel functions of the first and second kind, corresponding to the inward and outward traveling wave, respectively. From (13), the wave coefficients at each , , coordinate can be set as , , where prime indicates origin at Antenna 2 position. By using the voltage and current definition with the unit characteristic impedance [11], Z-parameters are written by where

(15) (16) (17) (18) where . The remaining Z-parameters are obtained from (14) and by using reciprocity as

(19) where

,

are abbreviated as

,

.

C. Whole Coupling Network Between Two Antennas We consider the case of using identical antennas; The case of different antennas can be treated with a similar method. Coupling network between two antennas is described by cascading networks of two antennas and space as depicted in Fig. 2. The whole network between two input ports of antennas can be obtained by conversion of Z-parameter of space in (19) into S-parameter and cascading it between S-parameters of antennas in (10), or by using scattering flow graphs in [4] with the S-parameters of antennas. It was found that two approaches yield

the same result. Since the Z-parameter expression of the result is simpler than S-parameter expression, Z-parameter representation is used being written by (20) where

(21) is the main factor of the maximum power transfer efficiency as will be shown in (29) and increasing its magnitude leads the maximum power transfer efficiency to 1. We can see that is composed of two kinds of the translation coefficients, and , which represent translation between the same kind of modes and different kind of modes, respectively, in (13) and (14). Each translation coefficient is separated into x, y and z-component as shown in (55) in Appendix C. Characteristics of them are rewritten as cylindrical components and summarized in Table I, showing three non-zero components. can be analyzed as a linear combination of three Thus cylindrical components whose magnitude and combining ratio are determined by the position and orientation of Antenna 2, respectively. The resulted Z-parameter is obtained under the assumption and modes generation and interaction. But of only this assumption becomes invalid when two antennas approach very close, because the magnitudes of higher order modes increase too large to neglect in the vicinity of an antenna owing to the faster increase of the higher order Hankel functions in their modes. So the Z-parameter deviates from the formula within a close distance between antennas, which will be shown with an example in Section IV. III. MAXIMUM POWER TRANSFER EFFICIENCY AND OPTIMUM LOAD IMPEDANCE From the Z-parameter, the input impedance of the transmitting antenna is given by (22)

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where is the load impedance of the receiving antenna. The input power is (23) The power delivered to the load is

(24) The power transfer efficiency is defined by

Fig. 3. (a) Legend and (b) 2-dimensional mapping of the orientation of Antenna 2 for the maximum power transfer efficiency pattern in Table II.

(25) The maximum power transfer efficiency can be found where

TABLE II MAXIMUM POWER TRANSFER EFFICIENCY PATTERN WITH RESPECT TO 2-DIMENSIONAL MAPPING OF THE ORIENTATION OF ANTENNA 2 AT r : WAVELENGTHS

=01

(26) This yields the optimum load impedance and the maximum power transfer efficiency as (27) (28)

(29) where . This can be also obtained from the simultaneous conjugate matching condition of a two-port network for the maximum transducer power gain [12, p. 619] as (30) From (29), we can see that the maximum power transfer efficiency depends only on as a function of the relative position and orientation of antennas, their radiation efficiency and the and modes. ratio between In Fig. 4, the maximum power transfer efficiency with for four cases of antenna position and orientation are compared. The cases of (a), (c) and (d) also represent the three cases , and , respecof maximum coupling by tively, in Table I. The result shows that the best performance in a range less than 0.3 wavelengths is obtained by placing the other . antenna with parallel orientation at In Table II, 2-dimensional patterns of the maximum power transfer efficiency with respect to the orientation of Antenna wavelengths and , , are illus2 at trated. A radial and angular position within the circular pattern correspond to and , respectively, of the orientation of Antenna 2 as depicted in Fig. 3(b). Each pattern is separated into the cases of considering only one of three non-zero components in Table I separately in , and compared with the original case where all components are combined in . We can observe that at the position of Antenna 2 at , only -comworks, and away from the position ponent related with , -component related with and -component at

=1 0 ) = (0 0 0)

Fig. 4. The maximum power transfer efficiency with  against the posi (a) ; ; , tion and orientation of the opposite antenna at  ;  ;  (b) = ; ; , (c) = ; = ; , (d) = ; = ; = .

( 2 0 0)

( 4 2 0)

(

( 2 2 2)

related with cross-coupling term, in affect the maxand . imum power transfer efficiency pattern at It is notable that the input reactance of the antenna has no effect on the maximum power transfer efficiency and only shifts the optimum load reactance in (28). So we will see the optimum as a standard. load impedance of the antenna with From (27) and (28), the optimum load impedance corresponding to Fig. 4 is plotted on the Smith chart in Fig. 5 as the optimum load reflection coefficient (31)

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height = 5 cm, diameter = 4 cm, operating frequency = 300 MHz).

Fig. 8. Center-fed helical antennas ( , 4-turns,

wire thickness = 1 mm

Fig. 5. The Smith chart representation of the optimum load impedances correR ,X ). sponding to Fig. 4 (Z

=

=0

Fig. 6. The maximum power transfer efficiencies of antennas with different ; : ; : ; : at  . radiation efficiencies 

(

= 1 0 5 0 1 0 01)

=0

Fig. 9. Normalized Z

of the helical antenna at 

= 0, =2.

IV. EXAMPLE

Fig. 7. The Smith chart representation of the optimum load impedances of antennas with different radiation efficiencies  ; : ; : at  (solid), = (dash) according to the distance between antennas.

2

(

= 1 0 5 0 1)

=0

where the characteristic impedance, , is set as . Fig. 5 exhibits various paths depending on cases, but common direction toward infinite load as the distance between antennas gets closer. So, the larger magnitude of load is proper at a closer distance. The dependence on the radiation efficiency as a crucial factor in the maximum power transfer efficiency is shown , then in Fig. 6. If we use an ideally lossless antenna lowering frequency achieves the higher transfer efficiency at a fixed distance. But since the radiation efficiency of an actual antenna under a given size decreases at the lower frequency, there may exists an optimum frequency as shown in the following example. As the radiation efficiency decreases, more variance is required for the optimum load resistance while less is required for the optimum load reactance as shown in Fig. 7.

To verify the theory, we compare the Z-parameters of an actual antenna with that in (20). A center-fed helical antenna is chosen as illustrated in Fig. 8 with 4 turns, 4 cm diameter, 5 cm height, and a copper wire thickness of 1 mm. The helical antenna consists of open-ended wires and acts both like a dipole and modes and loop antenna. It generates both and resonates by itself at about 295 MHz. We set the operating frequency at 300 MHz for the unit wavelength. Simulation is carried out using commercial software, FEKO, which is based on the method of moments. The input impedance of the single antenna, obtained by simulation, is at 300 MHz. We assume the antenna is tuned to have zero reactance with a series capacitor. The radiation resistance is given from the simulation with a perfect electric conductor as (PEC) wire. The radiation efficiency becomes 0.77 by (6). of the two antennas when close together is computed in Table III with normalization by the input impedance of the single anin (20) becomes invalid in distances tenna. It shows that within 0.1 wavelengths, which is about twice the antenna size. This is because the assumption becomes invalid when two anof the two tennas approach very close as mentioned above. antennas, normalized by the radiation resistance, is depicted in from (20) by the theory. It Fig. 9 in comparison with

LEE AND NAM: FUNDAMENTAL ASPECTS OF NEAR-FIELD COUPLING SMALL ANTENNAS FOR WIRELESS POWER TRANSFER

Fig. 10. The maximum power transfer efficiency of the helical antenna ( ) in four cases. : ,

0 77 frequency = 300 MHz

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= 70%(4) ( = 1)

50%( )

Fig. 13. Distance for the maximum power transfer efficiencies of , , and with the helical antenna and a lossless antenna . 

90%( )

NORMALIZED Z

TABLE III OF THE HELICAL ANTENNA

Fig. 11. The Smith chart representation of the optimum load impedances corresponding to Fig. 10.

Fig. 13 and shows the optimum frequency at around 56 MHz shows a continwhile an ideal lossless antenna with uous increase of the distance at lower frequency. V. CONCLUSION Fig. 12. Radiation efficiency of the helical antenna with different resonant frequency under the same size and corresponding number of turns and thickness of wire.

shows a good agreement in distance of more than 0.1 wavelengths and some deviation within 0.1 wavelengths similar to . Although the Z-parameter shows deviation from the theory when two antennas are very close, at that point, the maximum power transfer efficiency is already converging to 1, and thus does not deviate from the theory as shown in Fig. 10. The optimum load impedance in Fig. 11 also shows the validity of the theory. As mentioned above, to find out the optimum frequency for the maximum power transfer efficiency under the fixed size, the resonant frequency of the antenna is lowered by increasing the number of turns without increasing its size. Under the frequency of 50 MHz the thickness of wire is reduced to avoid the direct contact between wires. The radiation efficiency calculated by simulation in Fig. 12 shows a decrease at lower frequency as expected. From this, the distances required for the maximum power transfer efficiency of 50%, 70% and 90% are found in

To simplify and generalize the wireless power transfer problem, an approximate Z-parameter has been built by relating or mode interaction with the antenna parameters. Using the Z-parameter, we have shown the maximum power transfer efficiency according to the radiation efficiency of the single antenna. The corresponding optimum load impedance has been plotted for reference. Although the Z-parameter shows deviation from the theory when two antennas are very close, the theory provides a helpful measure to estimate the power transfer efficiency of the near-field coupling antennas. APPENDIX A SPHERICAL VECTOR WAVE FUNCTION If we set the electric and magnetic vector potentials outside time dependence as an antenna on the (32) where is the wave number,

,

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, and is the Kronecker delta, then the electric and magnetic fields (33)

where

(34) yield (1) and (2), a linear combination of the spherical vector wave functions [8], [9] (40) is the spherical harmonic function (35)

. and are the rotation angles about the former -axis and the new -axis, respectively, as depicted in Fig. 1. For and modes, we can get from (39) with and

(41) (36) and

in (8) are different from that is replaced by , where ical Bessel function of the first kind. The fields of the antenna generating are written as

and in is the spherand

Applying (41) to(35),(36) and exchange of the original and and angles result in rotated coordinates with

modes

(42) . We can see that arbitrary modes can be represented as a and a in properly rotated coordinates, respectively.

where

(37)

and mode

APPENDIX C ADDITION THEOREM UNDER COORDINATE TRANSLATION AND ROTATION The addition theorem of the spherical vector wave functions under coordinate translation is given as [10, p. 595]

(43) (38)

AND

APPENDIX B MODES UNDER COORDINATE ROTATIONS

The spherical mode under coordinate rotations as depicted in Fig. 1 can be derived from the following formula on the spherical harmonics [9]

(44) where

,

. (45) (46)

(39) where

.

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The general expressions for the translation coefficients, and are complicated, so we consider only the and in [13] as case of

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(57)

REFERENCES (47) (48) . where , modes at The translation coefficients between origin and translated , modes centered at are given by (49) (50) (51) (52) By some manipulations of (43) and (44), the similar form can , modes, in which the translation be built on , modes and coefficients between modes can be written by

(53)

(54) When the translated coordinate is rotated, the coefficients, and in (43) and (44) are replaced by and which are linear combinations of and , and are given from (42) as respectively [9].

[1] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless power transfer via strongly coupled magnetic resonances,” Science, vol. 317, no. 5834, pp. 83–86, Jul. 6, 2007. [2] Y. Kim and H. Ling, “Investigation of coupled mode behaviour of electrically small meander antennas,” Electron. Lett., vol. 43, no. 23, Nov. 2007. [3] A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Efficient wireless non-radiative mid-range energy transfer,” Ann. Phys., vol. 323, no. 1, pp. 34–48, Jan. 2008. [4] J. E. Hansen, Spherical Near-Field Antenna Measurements. London: Peter Peregrinus, 1988, pp. 85–85. [5] W. K. Kahn and W. Wasylkiwskyj, “Coupling, radiation, and scattering by antennas,” in Proc. Symp. on Generalized Networks, Brooklyn, NY, 1966, vol. 16, pp. 83–114. [6] W. Wasylkiwskyj and W. K. Kahn, “Scattering properties and mutual coupling of antennas with prescribed radiation pattern,” IEEE Trans. Antennas Propag., vol. 18, pp. 741–752, Nov. 1970. [7] W. Wasylkiwskyj and W. K. Kahn, “Theory of mutual coupling among minimum-scattering antennas,” IEEE Trans. Antennas Propag., vol. 18, pp. 204–216, Mar. 1970. [8] A. C. Gately, Jr, D. J. R. Stock, and B. R.-S. Cheo, “A network description for antenna problems,” Proc. IEEE, vol. 56, pp. 1181–1193, Jul. 1968. [9] S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math., vol. 19, no. 1, pp. 15–24, 1961. [10] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: IEEE Press, 1995, pp. 595–595. [11] G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, 2nd ed. NJ: Prentice-Hall, 1997, pp. 50–50. [12] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [13] W. C. Chew and Y. M. Wang, “Efficient ways to compute the vector addition theorem,” J. Electromagnet. Waves Appl., vol. 7, no. 5, pp. 651–665, 1993.

Jaechun Lee (S’05–M’08) received the B.S. degree from Soongsil University, Seoul, Korea, in 2000, and the M.S. and Ph.D. degrees from the Seoul National University, in 2002 and 2008, respectively, all in electronics/electrical engineering. He is currently with Samsung Advanced Institute of Technology (SAIT), Yongin, Korea. His research interests include analysis/design of electromagnetic (EM) structures, antennas, and microwave active/passive circuits.

(55) and written by

(56)

Sangwook Nam (S’87–M’88) received the B.S. degree from Seoul National University, Seoul, Korea, in 1981, the M.S. degree from the Korea Advanced Institute of Science and Technology (KAIST), in 1983, and the Ph.D. degree from the University of Texas at Austin, in 1989, all in electronics/electrical engineering. From 1983 to 1986, he was a Researcher with the Gold Star Central Research Laboratory, Seoul, Korea. Since 1990, he has been with Seoul National University, where he is currently a Professor in the School of Electrical Engineering. His research interests include analysis/design of electromagnetic (EM) structures, antennas, and microwave active/passive circuits.

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A Switchable Matching Circuit for Compact Wideband Antenna Designs Yue Li, Zhijun Zhang, Senior Member, IEEE, Wenhua Chen, Member, IEEE, Zhenghe Feng, Senior Member, IEEE, and Magdy F. Iskander, Fellow, IEEE

Abstract—A novel compact wideband antenna, which adopts a multi-state switchable matching circuit, is proposed in this paper. The design of the switchable matching is systematically studied and a two-step iterative procedure is used to obtain optimized values for both the bandwidth division between the various stages and the matching circuit components in each stage. Without loss of generality, a meander line monopole, with dimensions of 80 10 10 3 , is used as the radiator in the proposed wideband antenna design. To validate the proposed new design, a four-state matching circuit controlled by four PIN diodes is fabricated and measured. The reflection coefficient of the prototype antenna is found to be , and the peak gain is higher than better than across the 470–770 MHz bandwidth for the Integrated Services Digital Broadcasting-Terrestrial (ISDB-T) application.

mm

7 3 dB

3 dBi

Index Terms—Iterative gradient searching, PIN diode, switchable matching circuits, wideband matching.

I. INTRODUCTION ITH the rapid development of wireless communication systems, there is a significant interest in providing more digital broadcasting services to small mobile handsets. For example, Integrated Services Digital Broadcasting-Terrestrial (ISDB-T) can deliver television programs or other multimedia content directly to mobile terminals. ISDB-T operates at the frequency range of 470–770 MHz, covering a relatively wide bandwidth of 48.4%. It is always a challenge to install a passive internal antenna in a mobile handset to cover the entire ISDB-T band. This is because the typical dimensions of a mobile handset are quite small in comparison to a quarter of the wavelength which is the antenna dimension needed to effectively transmit in the 470–770 MHz frequency band. Several promising antenna designs have been published in

W

Manuscript received July 08, 2009; revised December 24, 2009; accepted April 18, 2010. Date of publication September 02, 2010; date of current version November 03, 2010. This work was supported in part by the National Basic Research Program of China under Contract 2009CB320205, in part by the National High Technology Research and Development Program of China (863 Program) under Contract 2007AA01Z284, in part by the National Natural Science Foundation of China under Contract 60771009, and in part by the National Science and Technology Major Project of the Ministry of Science and Technology of China 2009ZX03006-008. Y. Li, Z. Zhang, W. Chen, and Z. Feng are with State Key Laboratory on Microwave and Digital Communications, Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). M. F. Iskander is with HCAC, University of Hawaii at Manoa, Honolulu, HI 96822 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.2010.2071345

recent papers including the planer meander sleeve monopole antenna utilized as the radiating element in a mobile handset [1] and the conical monopole loaded with a circular patch on top studied by Zhou, et al. in [2]. These designs, however, occupy unacceptably large space, and, hence, some other techniques need to be investigated to design compact wideband antennas for the ISDB-T applications. Frequency reconfigurable antenna approach represents an alternative method for designing wideband antennas [3]–[8]. This approach has the capacity of switching the operating frequency while maintaining similar radiation pattern and gain throughout the required bands. Examples of these designs include the one described in a study by Kathleen et al. [3], using a varactor to tune the frequency of PIFA. Similar idea is studied by Milosavljevic et al. [4], wherein a varactor is loaded at the end of a PIFA antenna and the operating frequency was tuned by varying the capacitance. This reconfigurable method is also used in another study [5], where a varactor is loaded in a slot antenna at a fixed location to achieve a wide tunability range of the first resonant frequency ( ) and the second resonant frequency ( ) ) that ranges from 1.2 to 1.65. If and a frequency ratio ( different varactor locations are chosen, a wider frequency ratio could be achieved with a range of 1.3 to 2.67 [6]. PIN diodes are also used to change the length of antenna elements [7], adjust locations of feed points [8], and help tune the antennas and achieve impedance matching. Some of these antenna designs actually covered multiple bands including the GSM850 (824–894 MHz), GSM900 (880–960 MHz), DCS (1710–1880 MHz), PCS (1850–1990 MHz), and UMTS (1920–2170 MHz) bands [8]. The circuit used for impedance matching can also be reconfigurable. The antenna impedance can be matched in a relatively wide bandwidth by several states in a switchable matching circuit. The matching circuit designed by Mingo, et al. [9] can generate a great number of impedance values uniformly distributed on the Smith Chart. In this case, the circuit is composed of several lumped capacitances controlled by several PIN diodes. In a switchable matching circuit, several matching components are needed and lumped components are usually more preferred than distributed ones as they provide advantages, such as small footprint, ease of switching, and wide-working frequency range. The insertion losses of lumped components, however, cannot be ignored, but they are often considered acceptable in antennas designed for mobile communication applications. In this paper, a compact wideband monopole antenna with a switchable matching circuit is proposed. In a study by Kang et al. [10], a preliminary switchable matching circuit was proposed, and some preliminary simulations were presented. In

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Fig. 1. 3-D geometry of proposed antenna (unit: mm).

this follow-up manuscript, a matching procedure and a switchable matching circuit topology are systematically described, and a dual-step gradient optimization method is introduced to determine the component values of a matching circuit. Specifically, a four-state switchable matching circuit is designed using four lumped components controlled by four PIN diodes, and each component is reused twice to minimize the complexity of the overall system. A prototype of meander line monopole antenna is fabricated as an example of a switchable system to validate the developed matching technique. The obtained results show that the designed antenna can cover the entire ISDB-T better than when com(470–770 MHz) band with bining the four matching states. The gain and radiation patterns are also measured, and the obtained results are presented in the following sections. The manuscript is arranged as follows. In Section II, the matching technique is studied and the topology of the matching circuit is described. In Section III, a gradient optimization method is introduced to determine values of the matching circuit components. The fabrication of the proposed antenna with a matching circuit is then described and the results of the measured reflection coefficient, gain, and radiation patterns are shown in Section IV. Conclusions are presented in Section V.

Fig. 2. Resonant antenna curve on Smith Chart (470–770 MHz).

II. SWITCHABLE MATCHING CIRCUIT DESIGN A. Compact Meander Line Monopole Without loss of generality, a compact meander line monopole, shown in Fig. 1, is used here to demonstrate the proposed reconfigurable matching technique. The antenna dimensions are (0.125 0.016 0.016 ). The width of 80 10 10 strip is 2 mm. The antenna is installed at the edge of the ground single plane, which is made up of a 2-mm thick 140 80 ). The impedance locus of the anside FR4 board ( tenna is shown in Fig. 2, and the impedance at 770 MHz and 470 MHz are marked by Markers 1 and 2, respectively. This impedance locus is a good representative of most monopole antennas. In a low frequency band, the antenna has a capacitive characteristic, while in a high frequency band, its impedance is inductive. The original reflection coefficient without matching is shown in Fig. 3, and the reflection coefficient bandwidth of is only 100 MHz. B. Matching Circuit Design It is well known that for any given impedance, there is more than one matching circuit, which could be used to achieve good impedance matching [11]. In the example shown in

Fig. 3. Reflection coefficient of proposed antenna without matching.

Fig. 4(a) and (b), all four circuits can be used as a single frequency matching for the impedance at location . Using the left side circuit in Fig. 4(a) as an example, the series capacitor moves the antenna impedance from location along the large anticlockwise curve, while the shunt inductor moves the impedance along the small anticlockwise curve to the matching point. Combining the effect of both components, the impedance at location is successfully matched to 50 . In the proposed procedure, the goal of a reconfigurable matching is to match different portions of the impedance locus, which are equivalent to different frequency bands, by using different circuits. This might sound simple, but in practice the matching must be carefully designed to achieve the desired performance. For example, if the desired frequency range of the antenna is divided into four segments, as shown in Fig. 5, with Band 1 having the lowest frequency band and Band 4 having highest frequency, the total number of possible matching circuits will

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Fig. 5. Division of the whole bandwidth into four regions.

Fig. 4. Four different ways to match the impedance at location X.

be sixteen (i.e., four segments multiplied by four possible circuit elements, see Fig. 4 for each segment). In other words, one way to implement this wideband impedance matching is to utilize four independent circuits, one for each band. Assuming that each matching circuit uses two components, the matching arrangement mentioned above will require eight components and two single-pole four-throw (SP4T) switches. Another way to obtain the required four matching states could be based on an overall and simultaneous optimization of one set of matching components. In this case, when selecting the matching components and the circuit layout, reuse becomes a primary consideration. By using one shunt and one series components, as shown in Fig. 6, it is possible to match all four bands independently. Looking from the antenna side, a series inductor and a shunt inductor can be used to match Band 1, a series inductor and a shunt capacitor for Band 2, a series capacitor and a shunt inductor for Band 3, and a capacitor and a shunt capacitor for Band 4. In Fig. 6, although the matching circuits used to match the four regions have been chosen to have similar topologies, if there are no constraints on the matching circuit design, the result could still be eight different component values. The fol-

Fig. 6. Matching circuits for four bands.

lowing shows that the circuit can be simplified by optimizing the matching circuit as a whole and while covering the required frequency rang. Here, single-pole 2-throw (SP2T) switches are used as the switchable components. Because each SP2T switch can provide two states, switches can then provide a maximum states. Two possible topologies of such matching circuits of are shown in Fig. 7(a) and (b). The maximum numbers of states in these two topologies are 4 and 8, respectively. In the circuit shown in Fig. 7(a), for example, a total of four components are used and each component is active in two states. When the Switch A is in Position 1 and Switch B is in Position 1, and then Band 1 can be matched. The other matching configurations are listed in Table I. III. PARAMETER OPTIMIZATION After deciding on the topology of the matching circuit, this section focuses on describing the procedure for selecting values of the circuit components. There are actually two related optimizations in this process. The first one is on how to divide the entire frequency band into several regions, while the other is on the determination of the values of the components. As may be expected, when the frequency span of a region gets narrower, it is easier to achieve good matching in this region. This, how-

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Fig. 7. Switchable matching circuit (a) 2 SP2T, 4 states and (b) 3 SP2T, 8 states.

TABLE I MATCHING CONFIGURATION OF SWITCH A AND B

Fig. 8. Flow chart for optimization.

ever, makes it more difficult to achieve matching over the entire band, since the process results in a set of well-matched but narrow bands. Naturally, an optimization procedure of multiple iterations is required to achieve an optimized matching over the entire band. As an example, in the circuit shown in Fig. 7(a), all matching components and switches are considered as ideal ones. As a result, the component value vector and frequency band edge vector can be expressed by (1)

(1) In vector , and are the band edges of the overall band, and . In addition, , , with and are sub-band edges, which segment the overall band. The vectors and are variables that need to be optimized. The cost function is given in Formulas (2)–(4). is the meais the matched sured input impedance without matching. is the input impedance in each band, as listed in Table I. sub-band. In this exoptimization goal of return loss in the ample, is set to 10 dB for all four sub-bands. The is the sum of error of all four sub-bands. In each sub-band, the error is calculated by integrating the positive difference between the antenna response and the goal over the specific sub-band. If at one frequency the antenna response is better than the goal, the

error value at that frequency is 0, which means it doesn’t contribute to the overall cost function (2)

(3)

(4) The optimization procedure is illustrated in Fig. 8. It includes the following steps: 1) Random optimization: 100 values of vector are selected randomly between 0 and 30 (nH/pF), as candidates for starting values. Even distribution is selected as the value of

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TABLE II OPTIMIZED VALUE

, i.e.,

, , and . is calculated to find the , whose is the minimum. is 2) Nonlinear gradient optimization I: The optimized found when is fixed; that is, the optimized values of , , , and for the fixed four sub-frequency bands are found. The object function and constraints are illustrated in Formula (5). The index is utilized as the counter of first optimization times while setting Fig. 9. Fabrication of proposed antenna.

(5) is 3) Nonlinear gradient optimization II: The optimized found when is fixed; that is, the optimized ranges of subfrequency bands for four matching circuits are found. The object function and constraints are illustrated as follows:

(6) is ob4) If the calculated error converges, the optimized tained. Otherwise, Step 3 is repeated. The index then increases by one. 5) The process of optimization terminates when it converges . The latest is chosen as the optimized . or in Step 1 is based on The method to find the value of random optimization. The optimization goal in Step 2 and 3 is to find the minimum of constrained nonlinear multivariable function based on the iterative gradient searching. The optimization procedure has been run several times and converges with iteris set to guarantee ations less than 100. As a result, the final convergence. The final results are not sensitive to the starting values. Various initial values, which are randomly distributed between 0 and 30 (nH/pF), are trialed to test the robustness of the optimization procedure and it always converged to the same values. The optimization values of the four components are listed in Table II. The goal of the optimization without any parasitic parameters and insertion loss is to prove the design strategy and find initial values for practical tuning. IV. ANTENNA FABRICATION AND MEASUREMENT RESULTS To demonstrate the validity of the presented matching strategy, a meander line antenna was fabricated and matched. The fabrication of a meander line monopole antenna, which is fed by a 50 coaxial cable, is shown in Fig. 9. The antenna

Fig. 10. Schematic diagram of switchable matching circuit. TABLE III MATCHING CONFIGURATION OF D1-D4

is made of a flat metallic strip, meandered and attached to the outer side of a foam support, with dimensions of 80 10 10 . The 3 V control voltage for the switch is supplied using two AA batteries. The schematic diagram of the four-states of a switchable matching circuit is shown in Fig. 10. The gray area in Fig. 10 is the bias circuit and the clear area is the primary signal path. Four PIN diodes (Philips BAP64-03) are used to control , , , and . The configuration of the four PIN diodes for different matching states is shown in Table III. The bias resistances ( - ) are all 46 , and the inductance (

LI et al.: A SWITCHABLE MATCHING CIRCUIT FOR COMPACT WIDEBAND ANTENNA DESIGNS

Fig. 11. Dimension of switchable matching circuit PCB (unit: mm).

) chokes RF signal from the matching circuit to control the voltage source with a value of 120 nH. RF signal shorting ) are all 470 pF, and the DC block cacapacitances ( pacitance ( and ) are 100 pF each. The entire matching network with voltage bias circuit was printed on a 1-mm thick PCB, as shown in Fig. 11. If the antenna is to 11 12 serve as a transmitting antenna, the power handling capability must be considered when designing the matching circuit. To transmit a 30 dBm signal, a bias voltage of 10 V or more is required. However, the antenna proposed in this paper is supposed to be used as a receiving antenna, so a 3 V bias voltage can already guarantee the positive bias status. , and are difThe values of key components , , ferent from the optimized set-up due to the parasitic parameters introduced by the components and routing from the PCB. Also, the insertion loss of PIN diodes must be involved. Starting from the simulated values of the four components, some iterative tuning procedure must be involved based on the frequency response. The optimizing method is almost identical to the one used in the simulation. Unlike continuous value used in simulation, discrete components with parasitic effect are replaced at and affect the each iteration step. The series components center frequency for four bands, while shunt components and decide the return loss. As a result, a tradeoff between center frequency and return loss in each band must be considered in the components values tuning. The final values of , , , and soldered on the PCB are 12 nH, 18 nH, 4.7 pF, and 2.2 pF. The measured reflection coefficient of the proposed antenna with the matching circuit is shown in Fig. 12. As seen in the figure, the combined bandwidth covers the whole required better than ISDB-T frequency band of 470–770 MHz with . Simulated gain and measured gain in the desired bands are shown in Fig. 13. The measured gain peaks at 2.9 dBi and is , which is the minimum gain limit required better than by the system specification across the whole band. The difference between the straight line and the dashed line is attributed to the insertion loss introduced by the matching components. The average insertion loss is around 2 dB in the ISDB-T band. The

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Fig. 12. Reflection coefficient with matching circuit.

Fig. 13. Measured and simulated gain.

insertion loss is mainly introduced by non-ideal lumped components and PIN diodes. And it can be reduced by adopting high quality components and low insertion loss switches in the matching network. There is a discrepancy at the 495 MHz frequency shown in Fig. 13, where the measured gain is higher than the ideal one. This is due to the error introduced by the chamber measurement. Furthermore, at the low end of the measured frequency range, the chamber uncertainty can be up to 2 dB. It can be observed that the antenna has a below average gain when the shunt matching component is switched to the capacitor, which corresponds to bands of 545–590 MHz and 725–770 MHz. Further investigation and improvement on this phenomenon are needed if better performance is required. The radiation patterns at 495 and 720 MHz are shown in Figs. 14 and 15, respectively. A quasi-omnidirectional pattern is achieved in the - plane. V. CONCLUSION In this paper, a compact frequency reconfigurable monopole antenna with switchable matching circuit is presented. The monopole antenna matching method has been systematically

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Fig. 14. Radiation pattern at 495 MHz.

Fig. 15. Radiation pattern at 720 MHz.

studied and optimized. An iterative gradient optimization method is introduced to divide the band of interest into suitable regions and to determine the four optimized component values that would achieve matching over the entire band. A meander line monopole antenna and a four-state matching circuit are fabricated and measured to validate the proposed technique. Simulation and experimental results show that the antenna can cover the entire ISDB-T (470–770 MHz) frequency band with better than . The gain and radiation patterns of the proposed antenna are also measured. The proposed matching technique can, therefore, successfully be utilized for other applications using different antennas over different frequency bands. REFERENCES [1] B. H. Sun, J. F. Li, T. Zhou, and Q. Z. Liu, “Planar meander sleeve monopole antenna for DVB-H/GSM mobile handsets,” Electron. Lett., vol. 44, no. 8, pp. 508–509, Apr. 2008. [2] S.-G. Zhou, B.-H. Sun, Y.-F. Wei, and Q.-Z. Liu, “Low profile wideband antenna with shaped beams for indoor DVB-H applications,” Electron. Lett., vol. 45, no. 23, pp. 1151–1152, Nov. 2009. [3] K. L. Virga and Y. Rahmat-Samii, “Low-profile enhanced-bandwidth PIFA antennas for wireless communications packaging,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 1879–1888, Oct. 1997.

[4] Z. D. Milosavljevic, “A varactor tuned DVB-H antenna,” in Proc. Int. Conf. on Antenna Technology Small Antennas and Novel Metamaterials, Cambridge, U.K., 2007, pp. 124–127. [5] N. Behdad and K. Sarabandi, “A varactor-tuned dual-band slot antenna,” IEEE Trans. Antennas Propag., vol. 54, pp. 401–408, Feb. 2006. [6] N. Behdad and K. Sarabandi, “Dual-band reconfigurable antenna with a very wide tunability range,” IEEE Trans. Antennas Propag., vol. 54, pp. 409–416, Feb. 2006. [7] D. Peroulis, K. Sarabandi, and L. P. B. Katehi, “Design of reconfigurable slot antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 645–654, Feb. 2005. [8] 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, pp. 1919–1928, Jul. 2007. [9] J. Mingo, A. Valdovinos, A. Crespo, D. Navarro, and P. Garcia, “An RF electronically controlled impedance tuning network design and its application to an antenna input impedance automatic matching system,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 489–497, Feb. 2004. [10] Y. Kang, H. Mi, Z. Zhang, W. Chen, and Z. Feng, “A reconfigurable compact antenna for DVBH application,” in Proc. Int. Conf. on Microwave and Millimeter Wave Technology, Apr. 2008, pp. 1882–1885. [11] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005, ch. 5.

LI et al.: A SWITCHABLE MATCHING CIRCUIT FOR COMPACT WIDEBAND ANTENNA DESIGNS

Yue Li was born in Shenyang, China, in 1984. He received the B.S. degree in telecommunication engineering from the Zhejiang University, Zhejiang, China, in 2007. He is currently working toward the Ph.D. degree at Tsinghua University, Beijing, China. His current research interests include antenna design and theory, particularly in reconfigurable antennas, electrically small antennas and antenna in package.

Zhijun Zhang (M’00–SM’04) received the B.S. and M.S. degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 1992 and 1995, respectively, and the Ph.D. degree from Tsinghua University, Beijing, China, in 1999. In 1999, he was a Postdoctoral Fellow with the Department of Electrical Engineering, University of Utah, Salt Lake City, where he was appointed a Research Assistant Professor in 2001. In May 2002, he was an Assistant Researcher with the University of Hawaii at Manoa, Honolulu. In November 2002, he joined Amphenol T&M Antennas, Vernon Hills, IL, as a Senior Staff Antenna Development Engineer and was then promoted to the position of Antenna Engineer Manager. In 2004, he joined Nokia Inc., San Diego, CA, as a Senior Antenna Design Engineer. In 2006, he joined Apple Inc., Cupertino, CA, as a Senior Antenna Design Engineer and was then promoted to the position of Principal Antenna Engineer. Since August 2007, he has been with the Department of Electronic Engineering, Tsinghua University, Beijing, China, where he is a Professor.

Wenhua Chen (M’07) received the B.S. degree from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2001, and the Ph.D. degree from Tsinghua University, Beijing, China, in 2006. He is currently an Assistant Professor with the State Key Laboratory on Microwave and Digital Communications, Tsinghua University. His research interests include computational electromagnetics, reconfigurable and smart antennas, and high-efficiency power amplifiers. He has authored and coauthored over 30 journal and conference papers.

Zhenghe Feng (SM’92) received the B.S. degree in radio and electronics from Tsinghua University, Beijing, China, in 1970. Since 1970, he has been with Tsinghua University, as an Assistant, Lecturer, Associate Professor, and Full Professor. His main research areas include numerical techniques and computational electromagnetics, RF and microwave circuits and antenna, wireless communications, smart antenna, and spatial temporal signal processing.

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Magdy F. Iskander (F’93) is the Director of the Hawaii Center for Advanced Communications (HCAC), College of Engineering, University of Hawaii at Manoa, in Honolulu (http://hcac.hawaii.edu). He is also a Co-Director of the NSF Industry/University joint Cooperative Research Center between the University of Hawaii, University of Arizona, Arizona State University, and the RPI and the Ohio State University. He was a Professor of Electrical Engineering and the Engineering Clinic Endowed Chair Professor at the University of Utah, Salt Lake City, for 25 years. He was also the Director of the Center of Excellence for Multimedia Education and Technology. In 1986, he established the Engineering Clinic Program at the University of Utah to attract industrial support for projects for undergraduate engineering students and was the Director of this program until joining the University of Hawaii in 2002. From 1997–99 he was a Program Director, in the Electrical and Communication Systems Division at the National Science Foundation. He spent sabbaticals and other short leaves at Polytechnic University of New York; Ecole Superieure D’Electricite, France; UCLA; Harvey Mudd College; Tokyo Institute of Technology; Polytechnic University of Catalunya, Spain; University of Nice-Sophia Antipolis, and at several universities in China including Tsinghua University. He authored a textbook on Electromagnetic Fields and Waves (Prentice Hall, 1992; and Waveland Press, 2001) edited the CAEME Software Books, Vol. I, 1991, and Vol. II, 1994; and edited four other books on Microwave Processing of Materials, all published by the Materials Research Society, 1990–1996. He edited the 1995 and 1996 International Conference on Simulation and Multimedia in Engineering Education Proceedings. He has published over 200 papers in technical journals, has eight patents, and has made numerous presentations in technical conferences. He is the Founding Editor of the journal, Computer Applications in Engineering Education (Wiley). This journal is now in its 18th year, and received the Excellence in Publishing Award, by the Association of the American Publishers, in 1993. His ongoing research contracts include propagation models, low-cost phased array antenna designs, IED and UXO targets detection and classification, HF radar for homeland security applications, and several other research projects sponsored by corporate sponsors including Raytheon, Trex, Motorola, Kyocera Wireless, Corning, Inc., BAE Systems, and L-3 Communications. Dr. Iskander is a Fellow of the IEEE and was a member of the National Research Council Committee on Microwave Processing of Materials. He received the 1985 Curtis W. McGraw ASEE National Research Award, 1991 ASEE George Westinghouse National Education Award, 1992 Richard R. Stoddard Award from the IEEE EMC Society, the 2000 University of Utah Distinguished Teaching Award, and the 2002 Kuhina (Ambassador) Award from the Hawaii Visitors and Convention Bureau. He was a Guest Editor of a Special Issue on Wireless Communications Technology for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION in May 2002, Co-Guest Editor of a Special Issue of the IEICE Journal in Japan (September 2004), and a Co-Guest Editor of a Special Issue on Wireless Communications for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION in 2006. He is the 2002 President of the IEEE Antennas and Propagation Society, Vice President in 2001, and was a member of the IEEE APS AdCom from 1997 to 1999. He was the General Chair of the 2000 IEEE AP-S Symposium and URSI meeting in Salt Lake City, UT, the General Chair of the IEEE Conference on Wireless Communications Technology, in 2003, in Hawaii, the General Chair of the 2005 IEEE/ACES joint conference on Wireless Communications and Applied Computational Electromagnetics, and the General Chair for the 2007 IEEE Antennas and Propagation International Symposium in Honolulu, HI. He was also a Distinguished Lecturer for the IEEE AP-S (1994–97).

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Electrically Small, Millimeter Wave Dual Band Meta-Resonator Antennas In Kwang Kim, Student Member, IEEE, and Vasundara V. Varadan, Senior Member, IEEE

Abstract—A meta-resonator antenna is one in which a metamaterial resonator is the radiating element of the antenna. In this paper, meta-resonator antennas are developed using multilayer low temperature co-fired ceramics techniques. A pair of split-ring resonators (SRR) is used as the radiating element of the antenna. The two SRRs have different resonance frequencies due to the opposite placement of the gaps and the antenna can operate at both frequencies. Other multiband antennas can be designed by adding different metamaterial resonators. No matching network is required since feeding is by inductive/capacitive coupling. The size of the meta-resonator antenna is 10% of a conventional microstrip antenna. The electrical size (ka) of the antenna is 0.386, the bandwidth is 2%, gain is 3.76 dB and efficiency is 71%. An omnidirectional meta-resonator antenna is designed by removing the ground plane and by using a microstrip line as the feed line. The feed line can also serve as a monopole antenna if desired. The radiation pattern, efficiency and gain of the omnidirectional meta-resonator antenna are similar to those of a monopole antenna. Index Terms—Electrically small antenna, low temperature co-fired ceramics (LTCC) process, metamaterial resonator, metamaterials, multiband antenna.

I. INTRODUCTION ITH rapid progress in metamaterial research, many applications are envisaged in the microwave and optical frequency range. Use of metamaterials for antennas is one of the more important applications currently being researched. We can reduce the size of radiators and improve their properties using metamaterial substrates [1], [2]. Split-ring resonators (SRR) (suggested by Pendry [3]) are embedded in the substrate to give rise to magnetic properties to reduce antenna size. The operational frequency of the designed microstrip antenna is usually at a lower frequency than the resonance frequency of the SRR [4]. The SRR has dispersive properties only at the resonance frequency and the resonance frequency of the SRR depends on the geometry and dimensions of the SRR. In [5], a multiband antenna has been designed using various scaled SRRs in the substrate of a microstrip antenna. This approach needs a number of embedded SRRs that requires a large area substrate and ground plane.

W

Manuscript received May 29, 2009; revised October 16, 2009; accepted May 09, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The authors are with the Microwave and Optics Laboratory for Imaging & Characterization, Department of Electrical Engineering, University of Arkansas, Fayetteville, AR 72701 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.2010.2071341

Metamaterial resonators can also be used directly as the radiating element of the antenna [6]–[9]. A nearby feeding structure excites an SRR by inductive/capacitive coupling and the SRR radiates efficiently at the SRR resonance frequency. The operating frequency of the SRR is independent of the resonance frequency of the feeding structure. The electrical size of the at the resonance frequency. The dimensions of SRR is the SRR radiator are compared to a conventional microstrip antenna in Fig. 1. The footprint size reduction achieved using the SRR resonator is 90% (0.4 mm 0.41 mm for the meta-resonator antenna vs 1.65 mm 1 mm for the conventional microstrip antenna). Another advantage of metamaterial resonator antennas is that multiband/wideband antennas can be designed by adding metamaterial resonators that have other resonance frequencies. Because the SRR is active only at its own resonance frequency, the SRRs do not interfere with one another. In conventional multiband antennas, impedance matching networks are required at each. The design of broadband matching networks is complicated and needs a large area. The total size, including the radiating elements and the matching networks, is not electrically small [10]. Multiband antennas with SRR radiators that do not require a matching network may be attractive to antenna engineers. Fabrication of a metamaterial resonator is an important implementation issue because 3-D metallo-dielectric SRR structures are needed for antenna applications. Additionally, the scale of the SRR for millimeter wave applications is less than 1 mm. We have used the low temperature co-fired ceramics (LTCC) process since it is widely used by the MMIC industry. The LTCC process has the desired resolution and 3D connectivity. In the LTCC process, metallized patterns are printed on the ceramic layer and the patterns are 3D connected by via holes filled with metal for multilayer fabrication. LTCC dielectrics are also stable and have very low dielectric loss at millimeter frequencies. In this paper, we present a meta-resonator antenna using a pair of SRRs, one with a gap at the top and the other with a gap at the bottom as shown in Fig. 1(a). The antenna has two operating frequencies due to the orientations of the resonators. Input was provided through a single feeding probe for both frequencies. We also suggest a modified meta-resonator antenna which has an omnidirectional radiation pattern by removing the ground plane and using an extended microstrip line as the feed line. The feeds used for both cases (with and without the ground plane) may be considered as parasitic or inductive/capacitive coupling feeds since there is no direct contact between the feed line and the radiators. Meta-resonator antennas have acceptable performance properties for many applications.

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. (a) Meta-resonator antenna operating at 41 GHz and 52.2 GHz. (b) Conventional microstrip antenna operating at 41 GHz (LTCC dielectric: " : j : ; , SRR and patch: Silver, the thickness of metal line is 0.01 mm).

= 7 4 0 0 026

=1

II. META-RESONATOR ANTENNA DESIGN USING LTCC SPLIT RING RESONATOR A. Meta-Resonator Antennas Design We designed a meta-resonator antenna using a pair of SRRs (as shown in Fig. 1(a)) embedded in multilayer LTCC films. Fig. 2 presents the layout. The SRR is a u-shaped loop with a gap, which is composed of metal lines and two metal filled microvias as shown in Fig. 2(c). Each SRR has a 0.052 mm gap. The radius of the microvia is 0.065 mm. The width of the LTCC dielectric is 3 mm and the thickness is 0.5 mm. A metallized ground plane is placed under the dielectric. We added one layer mm) for a microstrip feed of the dielectric ( line under the ground plane. A microstrip line is printed on the underside of the lower dielectric layer to minimize the effect of the dispersive properties of the SRR. The width of the microstrip line is 0.125 mm to realize a characteristic impedance of 50 . A metal filled microvia, connected to the microstrip line and isolated from the ground plane, inductively feeds power to the pair of SRRs. A time-varying magnetic flux is generated around the feeding microvia (feed probe) and induces currents on the SRRs. The radius of the probe is 0.065 mm and the distance between the center of the probe and the SRR is 0.105 mm. The antenna has two SRRs and gaps have different orientations with respect to each other. The gap of one SRR is close to the ground plane and the gap of the other SRR is closer to the air interface. Fig. 3 presents the charge distributions on the

Fig. 2. (a) Top view of the meta-resonator antenna, (b) side view of the metaresonator antenna, (c), (d) geometry of the SRRs fabricated using LTCC (LTCC dielectric: " : j : ; , SRR, ground plane and feeding structure: Silver, the thickness of metal line is 0.01 mm).

= 7 4 0 0 026

=1

Fig. 3. Charge distributions on the SRRs and images. (a) Gap closer to the ground plane and (b) gap more distant from the ground plane. The SRRs with gap closer to the ground are excited more strongly.

SRRs. An incident -field induces a current on the SRR and the current generates a varying charge density across the gap of the SRR. The image has opposite charge density at corresponding positions. The SRR with gap close to the ground has a strong capacitive coupling with the ground plane and hence the resonance frequency is lower than the other SRR. We can make several resonance frequencies using the same structure simply by varying the capacitive coupling with the ground plane [11].

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Fig. 5. (a) Simulated radiation efficiency of the meta-resonator antenna (71%@41 GHz, 68%@51.6 GHz), (b) radiation pattern at 41 GHz : dB), (c) radiation pattern at 52.2 GHz ( ( : dB).

Max gain = 3 76 42

Fig. 4. (a) Simulated reflection coefficient of the meta-resonator antenna. (b) Reflection coefficient of the meta-resonator antenna using the downward SRR only. (c) Reflection coefficient of the meta-resonator antenna using the upward SRR only. (d) Reflection coefficient of the feeding probe.

The proposed antenna is numerically simulated using a full wave simulator (Ansoft HFSS). Fig. 4(a) shows the reflection coefficient of the proposed antenna. We observe three operating frequencies at 41 GHz, 52.2 GHz and 71.6 GHz. When only the SRR with gap close to the ground is used as the radiator of the meta-resonator antenna, it results in a single resonance frequency at 40.2 GHz as shown in Fig. 4(b). The other SRR antenna by itself operates at 51.6 GHz as shown in Fig. 4(c). The

Max gain =

feeding probe and the ground plane also constitute a monopole antenna at 70.5 GHz as shown in Fig. 4(d). The SRR gives rise to dispersive properties only at the resonance frequency. As can be seen in Fig. 4(a), each SRR operates independently in the presence of the other SRR with very little shift in resonance frequency. This can be seen by comparing the three resonance frequencies in Fig. 4(a) with the individual resonances in Figs. 4(b),(c) and (d). The radiation efficiency defined as the ratio of the radiated power to the incident power is calculated and presented in Fig. 5(a). The efficiency is 71% at 41 GHz, 68% at 52.2 GHz and 83% at 71.4 GHz. Fig. 5(c) presents the radiation pattern at each operating frequency. The maximum gain is 3.8 dB at 41 GHz and 4.2 dB at 52.2 GHz. Both the radiation efficiency and the gain are acceptable for antenna applications in the automotive industry. B. Electrical Size and Chu Limit The wavenumber is 0.8587 rad/mm at 41 GHz and the maximum dimension of the meta-resonator antenna ‘ ’ is 0.45 mm as shown in Fig. 3(b). We can calculate the electrical size of the proposed antenna as follows: (1) The proposed antenna is an electrically small antenna according [12]. The Chu limit to the Wheeler definition since

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Fig. 7. (a) Omnidirectional meta-resonator antenna. No ground plane under SRRs. Microstrip line is extended to SRR for the feeding structure. (b) Top : j : ; , view. (c) Side view. (Substrate: DuPont 951 " SRR, ground plane and feeding structure: Silver, the thickness of metal line is 0.01 mm).

= 7 4 0 0 026

Fig. 6. Comparison between the meta-resonator antenna and the conventional antenna. (a) Reflection coefficient, (b) radiation efficiency, (c) radiation pattern at 41 GHz. The meta-resonator antenna has narrower bandwidth and lower radiation efficiency because of size reduction.

values, and size [13], [14].

The realized bandwidth in Fig. 6(a) is 800 MHz (800 MHz/41 %). The bandwidth of the meta-resonator antenna is GHz less than the Chu limit of 3.5% because the SRR has a high factor and radiates only at the resonance frequency.

, are approximated using the electrical C. Comparison With a Conventional Microstrip Antenna

(2) Using

=1

for the meta-resonator antenna, we obtain (3)

The Chu limit for the bandwidth is given by (4) where is the criterion for the maximum VSWR inside the or a return loss of 10 dB bandwidth of interest. Using we obtain %

(5)

We compared properties of the meta-resonator antenna with those of a conventional microstrip patch antenna which has an operating frequency at 41 GHz. The geometry of the conventional antenna is shown in Fig. 1(b). The free space wavelength at 41 GHz is 7.3 mm. The length of the SRR (0.4 mm) is 0.055 at this frequency. The size of a conventional microstrip patch antenna on the same substrate is 1.65 mm 1 mm at 41 GHz. The electrical size of the microstrip patch antenna is 0.83. The minimum is 2.95 and the maximum bandwidth is 24% by (2) and (3). The conventional microstrip antenna has a bigger radiator and shows a broader bandwidth (7.3%) as shown in Fig. 6(a). The meta-resonator antenna has a narrow and deep minimum in the reflection coefficient because of the reduction of radiator size and the resulting high value. Fig. 6(b) shows the comparison of the radiation efficiency. The efficiency of the microstrip antenna is 91% and is much higher than the meta-resonator antenna. The radiator of the microstrip antenna is on the surface of the substrate and radiates directly into air whereas the

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Fig. 9. Surface charge distribution on the omnidirectional meta-resonator antenna (a) at 46-GHz magnetic resonance (the charge density is of opposite sign at adjacent locations on the two SRRs) and (b) at 51.6-GHz electric resonance (the charge density has the same sign at adjacent locations on the two SRRs).

are acceptable for many applications where a small size is an important criterion. III. OMNIDIRECTIONAL META-RESONATOR ANTENNA

Fig. 8. (a) Simulated reflection coefficient of the monopole antenna, (b) reflection coefficient of the omnidirectional meta-resonator antenna, (c) radiation efficiency of the omnidirectional meta-resonator antenna (76%@46 GHz, 85%@51.6 GHz), (d) radiation pattern of the omnidirectional meta-resonator antenna at 46 GHz ( : dB), (e) radiation pattern of the omnidi: dB). rectional meta-resonator antenna at 51.6 GHz (

Max gain = 2 5

Max gain = 1 8

meta-resonator antenna is embedded in a dielectric. The maximum gain of the meta-resonator antenna is 3.76 dB and the maximum gain of the microstrip antenna is 5.4 dB as shown in Fig. 6(c). The radiation patterns are similar to each other. Both the efficiency and maximum gain of the meta-resonator antenna

Some applications need antennas that have an omnidirectional radiation pattern. We modified the meta-resonator antenna to realize an omnidirectional radiation pattern by removing the ground plane as shown in Fig. 7. A partial ground plane was placed under the substrate just to launch a signal in the microstrip feed line. The width of the microstrip line was 0.125 mm resulting in a 50 line. The microstrip line is extended to the SRRs and the length is 0.9 mm. The feeding structure is arranged such that the induced -field goes through the SRRs. The feeding structure alone operates as a monopole antenna with resonances at 39 GHz and 59 GHz as shown in Fig. 8(a). When we add the pair of SRRs, we observe two more resonances at 46 GHz and 51.6 GHz as shown in Fig. 8(b). Capacitive coupling is absent in this case because there is no ground plane under the SRRs. However, we still observe two resonances. These two resonances are from the magnetic and electric resonances of the two SRRs [15]. A symmetric pair of SRRs may have both magnetic and electric resonance. The two rings have opposite charge distributions of the opposite sign at 46 GHz resulting in a magnetic resonance and identical charge distribution at 51.6 GHz resulting in an electric resonance as shown in Fig. 9. This antenna also has operating frequencies at 38 GHz and 60.4 GHz which are very close to that of the feeding structure (39 GHz and 59 GHz). We optimized the length of the feed line (0.9 mm) to place resonance frequencies of SRRs between the operating frequencies of the feeding structure (38, 60.4 GHz). Further study is required to understand the effect of the resonance of the feeding structure on the resonance frequency of the meta-resonator. Fig. 8(c) shows the radiation efficiency of the proposed antenna. The meta-resonator antenna has 76% efficiency at 46 GHz and 85% at 51.6 GHz. The efficiencies are similar to those of a monopole antenna (83% at 38 GHz, 89% at 60.4 GHz). Fig. 8(d), (e) present radiation patterns at the operating frequencies. We observe omnidirectional patterns similar to that of monopole antennas at both the resonance frequencies. The maximum gain of the SRR radiator

KIM AND VARADAN: ELECTRICALLY SMALL, MILLIMETER WAVE DUAL BAND META-RESONATOR ANTENNAS

is 2.5 dB at 46 GHz and 1.8 dB at 51.6 GHz. The corresponding gain of the monopole antenna is 2.1 dB.

IV. CONCLUSION Meta-resonator antennas in LTCC dielectrics have been studied. We designed a meta-resonator antenna using a pair of SRRs in multilayer ceramic films with a feeding structure. A number of operating frequencies can be realized by embedding different placement of the gaps of the SRRs. Meta-resonator antennas have many advantages. The feeding structure does not contact the resonator and excites the resonator by inductive or parasitic coupling. Hence, electrically small antennas can be designed without considering a matching network. Size reduction is 90% compared to a conventional patch antenna on the same substrate. The bandwidth, gain and efficiency are lower than a conventional microstrip patch antenna but the electrical size is 90% smaller in area. Size may be an important criterion in some antenna applications. An omnidirectional meta-resonator antenna has also been designed by removing the ground plane. A microstrip line serves as a feeding structure and also operates as a monopole antenna away from the resonance frequency of the meta-resonator. We can design multi band and/or wideband antennas easily by proper design of resonators and feeding structures. The efficiency and radiation pattern of the omnidirectional meta-resonator antenna are similar to those of the monopole antenna. In this paper, we have not presented experimental data for a single millimeter wave antenna element. The reason for this is simple. The size of one SRR is 0.4 mm and the total size of a single antenna element is only 3 mm 3 mm. The fabrication of a single small antenna element is very challenging. For measurements, a probe should directly contact the input port. Accurate measurements at millimeter wavelengths are also a challenge and accurate support and fixtures for a single antenna element are simply not possible. Hence, we are considering the fabrication and measurement of an array of such antenna elements using the unit cell antenna described in this paper. This will result in a larger antenna and will permit more accurate mounting and fixturing for measurements. Moreover, most applications at these frequency bands need an array antenna for a high gain radiation pattern. In the next paper, we propose to design, fabricate and characterize a millimeter wave meta-resonator array antenna.

REFERENCES [1] I. K. Kim and V. V. Varadan, “LTCC metamaterial substrates for millimeter-wave applications,” in Proc. IEEE Region 5 Technical Conf., Apr. 2007, pp. 109–112. [2] F. Bilotti, A. Alu, and L. Vegni, “Design of miniaturized metamaterial patch antennas with -negative loading,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1640–1647, Jun. 2008.

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[3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [4] P. M. T. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretykov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1654–1662, Jun. 2006. [5] I. K. Kim and V. V. Varadan, “Millimeter wave dual-band microstrip antennas with metamaterial substrates using the LTCC process,” Metamaterials 2007, pp. 242–245, Oct. 2007. [6] K. B. Alici and E. Ozbay, “Electrically small split ring resonator antennas,” J. Appl. Phys., vol. 101, pp. 083104–, Apr. 2007. [7] V. V. Varadan and I. K. Kim, “Compact, multi band plasmonic resonator antenna,” presented at the Antennas Propag. Society International Symp., Jun. 2009. [8] V. V. Varadan, “Design, fabrication and characterization of electrically small plasmonic resonator antennas,” in Proc. Antenna Applications Symp., Sep. 2009, pp. 431–439. [9] P. Jin and R. W. Ziolkowski, “Low-Q, electrically small, efficient nearfield resonant parasitic antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2548–2563, Sept. 2009. [10] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficient electrically small antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 691–707, Mar. 2008. [11] I. K. Kim and V. V. Varadan, “Microstrip patch antenna on LTCC metamaterial substrates in millimeter wave bands,” presented at the Antennas Propag. Society Int. Symp., Jul. 2008. [12] H. A. Wheeler, “Fundamental limitations of small antennas,” in Proc. IRE, Dec. 1947, pp. 1479–1484. [13] L. J. Chu, “Physical limitations on omni-directional antennas,” J. Appl. Phys., vol. 19, pp. 1163–1175, Dec. 1948. [14] J. S. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, no. 5, May 1996. [15] I. K. Kim and V. V. Varadan, “Electric and magnetic resonances in symmetric pairs of split ring resonators,” J. Appl. Phys., vol. 106, pp. 074504–, Oct. 2009. In Kwang Kim (S’07) received the B.S. degree in electronic engineering from Inha University, Incheon, Korea, in 1997 and the M.S. degree in electrical engineering from Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 1999. He is currently working toward the Ph.D. degree at the University of Arkansas, Fayetteville. From 1999 to 2004, he was a Research Engineer with Hyundai Electronics Inc., Yicheon, Korea, where he was involved in RF system design of CDMA base stations. From 2004 to 2006, he was with SK teletech Co., Ltd., Seoul, Korea, where he was involved in the development of advanced handsets. His research interest includes the design, characterization and applications of metamaterials.

Vasundara V. Varadan (M’82–SM’03) received the Ph.D. degree in physics from the University of Illinois at Chicago, in 1974. Previously, she was with Cornell University, The Ohio State University, and Pennsylvania State University. From 2002 to 2004, she was the Division Director of the Electrical and Communications Systems Division, National Science Foundation (NSF). She is currently the Billingsley Chair and Distinguished Professor of Electrical Engineering with the University of Arkansas, Fayetteville. Her research interests are EM theory and measurements, metamaterials, electrically small antennas, microwave nondestructive evaluation and imaging, smart materials and devices, numerical simulation of wave problems, and embedded sensor systems. Dr. Varadan is a Fellow of the Acoustical Society of America, the Institute of Physics (U.K.), and The International Society for Optical Engineers (SPIE).

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Printed Loop Antenna With a U-Shaped Tuning Element for Hepta-Band Laptop Applications Chien-Wen Chiu, Member, IEEE, and Yu-Jen Chi, Student Member, IEEE

Abstract—A simple printed-loop antenna with wideband characteristics is presented for laptop computer applications. The proposed rectangular loop pattern generates four resonant modes below 4 GHz. The excited modes, two unbalance and two balance, are analyzed and discussed. A novel U-shaped tuning element printed on the back side of the circuit board adjusts the resonant modes to cover GSM850/GSM900/DCS/PCS/UMTS/WLAN and WiMAX bands. Using HFSS software, this study conducted a simulation to optimize the antenna design and fabricate a practical structure to investigate its performance and characteristics. This study also measures various antenna parameters to validate the proposed antenna. Index Terms—Broadband antenna, mobile phone antenna, multiband antenna, portable device applications, printed loop antenna.

I. INTRODUCTION

W

ITH the rapid progress in wireless communication, wireless networking has become a basic but important function for laptop computers. Besides the widely-used wireless local area network (WLAN) standard, the 3.5 G mobile communication system has become extremely popular. The 3.5 G system is based on the high speed downlink package access (HSDPA) protocol, allowing people to access the internet via cellular communication systems. In addition, worldwide interoperability for microwave access (WiMAX) technology, which provides wireless data transmission in a variety of ways, is an alternative for wider coverage than WLAN. To date, only a few internal antennas have been proposed for laptop computers [1]–[8], and even fewer of them can simultaneously cover all of the following communication standards: GSM850/900 (824–960 MHz), DCS (1710–1880 MHz), PCS (1850–1990 MHz), UMTS (1920–2170 MHz), WLAN + Bluetooth (2400–2484 MHz), and WiMAX (2500–2690 MHz). Printed loop antennas have been widely used in laptop computers and mobile phones. Since laptop computers started out as computers and not as personal communication devices, antennas were not considered in the early development stage. As a result, the space allocated for antenna design is usually quite small and narrow. When an internal antenna is embedded into the top edge of an LCD panel, its geometrical configuration must

Manuscript received June 23, 2009; revised May 11, 2010; accepted May 20, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. This work was supported in part by the Taiwan National Science Committee under Grant NSC-96-2221-E-197-001. The authors are with the Department of Electronic Engineering, National Ilan University, Ilan, Taiwan, R.O.C. (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.2010.2071343

be thin or rectangular, with a long length to width ratio, i.e., the practical antenna profile must be nearly planar: thin but long. In the past decade, researchers have proposed some practical antenna designs for laptop applications, some in the WLAN bands and some in the WWAN bands [1]–[5]. Recently, Wong et al., demonstrated a printed monopole slot antenna can be attractive for multiband laptop computer applications. The proposed monopole slot antenna can achieve small size (60 mm 12 mm) [6], [7]. To achieve low prices, a uniplanar inverted-F antenna was proposed for the above application. The uniplanar PIFA which is printed only on a single surface is easier to fabricate [8]. On the other hand, mobile phones, started out as a communication device, are typically equipped with printed antennas, which are planar, easy to fabricate and less expensive [9]–[11]. Therefore, some recent studies have proposed printed loop antennas mounted on top of the handset printed circuit board for mobile phone applications. These antennas can generate operating bands covering both GSM850/900 and DCS/PCS/UMTS standards, and are very promising designs [12], [13]. This paper presents a simple printed-loop antenna with wideband characteristics for laptop applications. The proposed antenna can be easily implemented in a printed circuit board and mounted on the LCD panel of a mini-laptop computer. Its rectangular loop pattern can generate four resonant modes below 4 GHz, two unbalance and two balance modes. The proposed design uses a grounded U-shaped tuning element printed on the back side of the circuit board to adjust the resonant modes to cover GSM850/GSM900/DCS/PCS/UMTS/WLAN and WiMAX bands. The proposed antenna covers sufficient bands to support the desired operation bands. This planar . To antenna requires only a volume of optimize the design of this antenna, this paper performs HFSS simulations and fabricates a practical structure to verify the simulation results. The actual measured antenna parameters are presented to validate the proposed antenna. In contrast to some recent proposed laptop antennas [6]–[8], the current one is double-sided and has slight larger size. However, it covers not only WWAN band but also WLAN and WiMAX bands. II. ANTENNA DESIGN Fig. 1(a) shows the configuration of the proposed antenna. The antenna is printed on a FR4 substrate with a thickness of 0.8 mm and a relative permittivity of 4.4. The antenna is mounted on the top-right corner of a vertical ground plane (of size ), which is the supporting metal frame of a LCD panel. Because the antenna is coated on the double-sided PCB, it measures only . Fig. 1(b) and (c) show the antenna patterns on the front and back sides of the printed circuit board.

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Geometry of the proposed antenna, (a) 3D view, (b) plan view of the front-side, and (c) plan view of the back-side.

Fig. 3. Simulation results for the simple loop pattern.

Fig. 2. Simulation results as a function of different heights.

The U-shape on the back side is a tuning element, which is affixed to the ground plane. The antenna is fed by a 50 mini coaxial cable. The center conductor of the cable launches the signal into the strip end on the left hand-side of the loop pattern. The outer conductor of the cable is connected to the right hand-side of the loop strip that terminates to the ground plane. The proposed antenna has a one-wavelength rectangular loop design. The final design originates from a simple rectangular loop pattern that is illustrated in Fig. 2. The distance between the loop pattern and the vertical ground plane is defined as height .

Since the loop antenna structure is a one-wavelength resonator, the perimeter of the initial loop antenna can be determined by the resonant frequency of the lowest loop-mode [10]–[12]. The total length of the simple loop pattern can simply be calculated. The calculated wavelength is 150 mm at 2 GHz. However, because the loop pattern is printed on the edge of the thin circuit board near the ground plane, the conductor plane and the substrate may influence the electromagnetic field. After some simulations with multiband consideration, the initial-guess perimeter for the loop is designed to around . Fig. 2 shows the influence of varying the height on the reflection coefficient, suggesting that the distance between the bottom of the loop pattern and the vertical ground plane can be used to adjust the input impedance. The best impedance matching occurs at a height , but the height is finally adjusted to 4 mm in taking the overall design consideration. Fig. 3(a) shows the real and imaginary parts of the input impedance, while (b) shows the simulated reflection coefficient for the simple loop antenna with and without the vertical ground plane. The loop antenna is symmetric when there is no system ground plane, and this loop antenna can generate traditional loop modes. The first resonant mode at 0.95 GHz (half-wavelength) has very high values of resistance and reactance (the so-called parallel type, referred to as anti-resonance) [14]. It does not match

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Fig. 4. Simulated vector current distributions. (a) 1.136 GHz; (b) 1.773 GHz; (c) 2.091 GHz; (d) 3.364 GHz.

a 50 transmission line, and therefore does not generate resonance, as Fig. 3(b) shows. The second resonant mode, generated at 1.9 GHz, is a one-wavelength loop mode since the wavelength is about 150 mm. Fig. 3(a) shows that the antenna has smaller values of resistance and reactance (the so-called series type, resonant-mode) [14]. The input impedance allows the matching of 50 transmission lines, and thus the generation of resonance. When the system ground plane is added, the strip end in the right-hand side terminates at the ground plane and the strip end in the left-hand side is fed by a coaxial cable, as Fig. 1 shows. The outer conductor of the coaxial cable is also connected to the grounded strip. The resulting antenna system has a symmetric loop pattern, but an unbalanced feeding scheme. The unbalance mode and balance mode can be simultaneously excited on this antenna system. Therefore, the antenna system generates four resonant modes (series type) below 4 GHz, as Fig. 3(a) and (b) show. The presence of the conducting ground plane shifts the resonant frequencies. The first resonant mode at 1.136 GHz is an unbalance mode, and the mode at 2.091 GHz is its higher mode. The second resonant mode, at 1.773 GHz, is a balance mode, and the mode at 3.364 GHz is its higher mode. To demonstrate that the unbalance mode creates the resonances at 1.136 GHz and 2.091 GHz and that the self-balanced loop mode creates the resonance at 1.773 GHz and 3.364 GHz, Fig. 4 shows the vector current distributions and Fig. 5 shows the surface current densities at 1.136, 1.773, 2.091 and 3.364 GHz, respectively. These results are simulated by HFSS, a commercial electromagnetic simulation tool. The currents shown in Fig. 4(a) and (c) are equal in amplitude, but in the opposite direction since the loop pattern is symmetric from side to side with respect to the line . This behavior and the current density in Fig. 5(a) and (c) imply that the feeding scheme excites the unbalance modes. The current shown in Fig. 4(b) and (d) exhibits a differential behavior in the feeding port. In this case, the currents are equal in amplitude and in the same direction with respect to the line . Obviously, they are one- and two-wavelength loop modes. The balance modes are excited when the electric length of the loop pattern is close to one- or two-wavelength.

Fig. 5. Simulated surface current density of the loop antenna. (a) 1.136 GHz; (b) 1.773 GHz; (c) 2.091 GHz; (d) 3.364 GHz.

On the other hand, the currents along the y-direction in Fig. 4(b) are in the same direction at 1.773 GHz. This characteristics show that the antenna at this mode can also be viewed as a folded dipole [15]. If the effect of ground plane is considered, the half length of the perimeter of the loop (75 mm) is close to the 0.5 wavelength of a dipole antenna at 1.773 GHz. Theoretically, the impedance of a folded dipole is about four times greater than that of an isolated dipole [15]. However, in the present case, there is a substrate and a nearby ground plate below the feeding strip. The upper strip is printed near the edge of substrate, which is more distant from the ground plate than the feeding strip. The current density in Fig. 5(b) on the lower strip is stronger than that on the upper strip. As a result, the current ratio becomes smaller, e.g., , at the resonating frequency 1.773 GHz by HFSS simulation, where

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Fig. 7. Simulated results after adding a grounded-rectangular tuning element. Fig. 6. Simulated reflection coefficient with the size of LCD ground plane as a at 1.136 GHz). parameter (

= 264 mm

and are the total currents on the lower and upper strips, respectively. The step-up impedance ratio is much smaller than that predicted by traditional folded dipole. Finally, some explanations are necessary to account for how the vertical ground plane affects excitations of the unbalance modes and matching of the input impedance. The loop pattern which terminates to the ground plane is fed by an unbalanced feeding port. Due to the presence of the ground plane, the feeding scheme can have not only the balanced but also unbalanced excitation. The half-wavelength mode of a single loop antenna, which is an anti-resonance mode, can be supported by the unbalanced excitation. Since there is a current null for the surface current shown in Fig. 5(a) in this mode, the loop can be viewed as consisting of two arms. The currents on the two arms of the loop are of the same phase. As a result, the arms behave like two parallel bent monopoles mounted on the vertical ground plane. Since the length of one arm is around 70 mm , the resonant frequency of the monopole-like mode is calculated to be about 1.07 GHz (wavelength 280 mm), which corresponds to the resonant frequency of the first resonance mode shown in Fig. 3(b). The resonant frequency of the first unbalance mode is mainly determined by the arm length of the loop pattern but also influenced by the ground plane. The monopole-like mode can also be verified by the radiation patterns discussed in Section IV, where in the x-y plane is basically omni-directional. The large ground plane, which is larger than half a wavelength at the first resonance mode 1.136 GHz, is more like a reflector, not an efficient radiator [3]. Fig. 6 shows the simulated reflection coefficient with the size of LCD ground plane as a parameter. The large ground plane shown in Fig. 1(a) is located nearly on the same plane with the loop. The finding shows that changing the size of the ground plane has no significant influence on the resonance frequency if only the ground plane size is larger than (66 mm). If the ground plane size is less than a quarter-wavelength, the ground plane has significant influence on the impedance matching at the resonance modes. When the size of the ground plane is reduced from one-wavelength to quarter-wavelength, the bandwidth of the first resonance mode gradually decreases from 160 MHz to 110 MHz.

III. BANDWIDTH ENHANCEMENT BY A MATCHING ELEMENT Although the proposed antenna can generate two resonant modes at 1.773 and 2.091 GHz, the bandwidth achieved is not wide enough to cover UMTS, WiFi, and WiMAX operations for the antenna’s upper band. Therefore, this study applied another bandwidth-enhancement technique to achieve better wideband performance. To achieve better bandwidth performance, a parasitic load shown in Fig. 1(c), a grounded tuning element, is used to lower the resonant frequency at 3.364 GHz. The position of this short-circuited element can be decided by investigating the surface current distribution. Since the currents shown in Fig. 5(b) and (d) are strong at the feeding center of the loop antenna, placing the tuning element near this area will dramatically influence the resonant modes. The tuning element, coupled with the antenna, can effectively decrease the fourth-order resonant mode at 3.364 GHz resulting in lower resonant frequencies. Fig. 7 shows the effect of adding a grounded-rectangular tuning element to the back of the printed circuit board. Adding this tuning element broadens the bandwidth from 1.75 GHz to 2.70 GHz. However, the addition of this tuning element may affect the impedance matching at the GSM900 band since the maximum electric field distributions at 1 GHz are located in the center of the loop pattern. In this works, the radiation field that comes from the ground plane is small since the area of the ground plane is large and the induced current on the ground plane is small. The radiation resistance of the input impedance decreases when the tuning element is short-circuited to the ground plane. As a result, the impedance matching deteriorates and the reflection coefficient fails to reach 10 dB. The U-shaped tuning element in Fig. 1(c) is preferred to a rectangular one since it can decrease the impedance matching effect at 1 GHz. Fig. 8 shows a parametric study of the concave depth on the rectangular tuning element. The widest bandwidth at the GSM band occurs when . Impedance matching also improves in the higher bands as the depth increases. Finally, Fig. 8 shows that when the concave depth is 6 mm, the reflection coefficient is close to 10 dB, which meets the requirements and specifications of WLAN and WiMAX applications. Although introducing a U-shaped concavity to the tuning element helps it match the input impedance at the GSM band, the resonant frequency at the GSM band is a little higher than the

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Fig. 8. Simulated results when a concave is introduced to the rectangular tuning element.

Fig. 9. Frequency shifting down as a middle line has been added.

desired operating frequency. Therefore, a horizontal strip line is inserted into the loop pattern to increase the electric length, as Fig. 1 shows. Fig. 9 shows the simulated results with and without inserting the middle line. This figure demonstrates that inserting a middle strip line shifts the resonance to a lower band. However, it also reduces the impedance bandwidth in the higher band from 1500 MHz (1.7 GHz 3.2 GHz) to 1100 MHz (1.7 GHz 2.8 GHz). In addition, the input impedances are and at the second resonance mode for the folded dipole with and without the middle line, respectively. The step-up impedance ratio between the two cases (three lines and two lines) is simulated to be 1.6. The value is different from the traditional prediction of due to the substrate and the ground plane effects. The location where an antenna is attached to on the supporting metal frame is not unique. The antenna can be mounted on the vertical or horizontal edge of an LCD display. According to the location and orientation evaluation at IBM, bandwidths will be wider if the antenna is placed at the top edge of a large ground plane [16]. Fig. 10 shows the location effect of the antenna on the impedance matching. Here, is the distance between the antenna and the top-right edge of the LCD panel. The size of the LCD

Fig. 10. Simulated reflection coefficient as a function of the distance d.

Fig. 11. Effect of varying position P from 34 mm to 35 mm.

Fig. 12. Measured and simulated results for the proposed antenna.

panel which measures 200 mm 160 mm is different from the system ground plane of a mobile phone. The figure shows that the resonant frequency remains almost unchanged but the input impedance will be influenced if the antenna is moved along the top edge. Moving the antenna along the top edge has significant

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Fig. 13. Measured radiation patterns on the horizontal plane. (a) 900 MHz; (b) 1.80 GHz; (c) 2.10 GHz; (d) 2.45 GHz; (e) 2.70 GHz.

effects on the bandwidth, especially in the low GSM band. It indicates that the bandwidth is wider if the antenna is placed near the top-right corner, i.e., . The current distributions shown in Fig. 5(a) have no significant difference on the ground plane but have some change on the loop pattern when the antenna placed near the right corner is compared with that at the top-center of the ground plane. The study finds that better impedance matching is achieved when the loop is placed near the top-right corner than when it is placed at the top center. Finally, Fig. 11 shows the influence of the position of the U-shaped tuning element on antenna performance. The simulated reflection coefficient for position , which varies from 34 mm to 35 mm, shows that the fourth-order resonant frequency can be shifted to the lower band and a wide band operation can be obtained when is 34.5 mm. The position is rather sensitive especially when the tuning element is nearby the feeding port. Since using the coupled tuning element introduces too many loaded effects, the U-shaped tuning element is better than a rectangular shape for obtaining a good impedance match to 50 at high frequencies.

IV. MEASURED RESULTS AND DISCUSSION This study constructed a practical antenna based on the structure shown in Fig. 1. It was tested and measurements were taken using a Network Analyzer E5071B. The measured results were then compared with the simulation results produced by HFSS software. Fig. 12 shows both the experimental results and the simulation results. The solid line in this figure shows the measured results, while the dashed line shows the simulated results. These results exhibit good overall agreement, except for some loss caused by a 200 mm mini-coaxial cable. The bandwidth achieved with a reflection coefficient better than 6 dB is 140 MHz (820–960 MHz) in the GSM band and 1190 MHz (1710–2900 MHz) in the DCS/PCS/UMTS bands. The bandwidth with a reflection coefficient better than 10 dB is sufficient for WLAN and WiMAX applications. The radiation pattern and gain were measured in an anechoic chamber using a Satimo system—SG24 in Taiwan. Fig. 13 shows the measured radiation patterns of the proposed antenna on the horizontal plane (x-y plane). Monopole-like patterns are evident at the unbalance modes, as Fig. 13(a) and (c) shows. A fairly omni-

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TABLE I MEASURED PEAK GAIN, TWO-DIMENSIONAL AVERAGE GAIN, AND RADIATION EFFICIENCY

directional pattern can be observed for , except for 1.8 GHz aland patterns for the folded-dipole though the measured mode at 1.8 GHz are somewhat directional and some variations patterns. The omni-directional and nulls can be found in the feature is perfectly suitable for laptop computers. Table I lists the measured peak gain, two-dimensional average gain in the horizontal plane, and three-dimensional radiation efficiency of the proposed antenna. The antenna gain is approximately 1.12–3.21 dBi. Note that the measured gain results include the loss of a 200 mm mini-coaxial cable. The mini-coaxial cable (I-PEX) used for laptop or netbook computers has a very small diameter, around 1.1 mm, for routing through hinges. Therefore, the cable has more than 3 dB/m loss at the operating bands. This loss effect reduces the antenna gain by about 0.6 dB. Nonetheless, the obtained antenna gain and radiation efficiency are still good for practical applications in laptop computers despite cable loss. V. CONCLUSION This paper proposes a simple printed loop antenna with multiband characteristics that integrate WWAN and WLAN/WiMAX systems. The proposed system consists of a loop pattern and a novel U-shaped tuning element on each side of a printed circuit board. The loop pattern generates four resonating modes below 4 GHz, while the grounded U-shaped tuning element broadens the impedance bandwidth. To validate the proposed design, this study also presents parameter measurements for the reflection coefficient, radiation patterns, antenna gain, and radiation efficiency. The proposed two-dimensional planar-type antenna is compact, wideband, and easy to fabricate, making it very suitable for today’s popular small netbook computers. ACKNOWLEDGMENT The authors are grateful to the National Center for High-performance Computing for computer time and the use of their facilities. They would also like to thank Prof. S. M. Deng at MCU for helpful discussions and the reviewers for their comments and suggestions. REFERENCES [1] 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. [2] D. Liu and B. Gaucher, “A quadband antenna for laptop applications,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Hawaii, Jun. 2007, pp. 128–131. [3] C. H. Kuo, K. L. Wong, and F. S. Chang, “Internal GSM/DCS dualband open-loop antenna for laptop application,” Microw. Opt. Technol. Lett., vol. 49, pp. 680–684, Mar. 2007.

[4] C. Zhang, S. Yang, S. Lee, S. E. Ghazaly, A. E. Fathy, H. K. Pan, and V. K. Nair, “A low profile twin-PIFA laptop reconfigurable multi-band antenna for switchable and fixed services wireless applications,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Hawaii, Jun. 2007, pp. 1209–1212. [5] C. W. Chiu, Y. J. Chi, and S. M. Deng, “An internal multiband antenna for WLAN and WWAN applications,” Microw. Opt. Technol. Lett., vol. 51, pp. 1803–1807, Aug. 2009. [6] 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. [7] K. L. Wong and F. H. Chu, “Internal planar WWAN laptop computer antenna using monopole slot elements,” Microw. Opt. Technol. Lett., vol. 51, pp. 1274–1279, May 2009. [8] K. L. Wong and S. J. Liao, “Uniplanar coupled-fed printed PIFA for WWAN operation in the laptop computer,” Microw. Opt. Technol. Lett., vol. 51, pp. 549–554, Feb. 2009. [9] T. A. Denidni, H. J. Lee, Y. S. Lim, and Q. Rao, “Wide-band high-efficiency printed loop antenna design for wireless communication systems,” IEEE Trans. Vehicular Tech., vol. 54, pp. 873–878, May 2005. [10] Y. W. Chi and K. L. Wong, “Internal compact dual-band printed loop antenna for mobile phone application,” IEEE Trans. Antennas Propag., vol. 55, pp. 1457–1462, May 2007. [11] B. Jung, H. Rhyu, Y. J. Lee, F. J. Harackiewwicz, M. J. Park, and B. Lee, “Internal folded loop antenna with tuning notches for GSM/GPS/ DCS/PCS mobile handset applications,” Microw. Opt. Technol. Lett., vol. 48, pp. 1501–1504, Aug. 2006. [12] K. L. Wong and C. H. Huang, “Printed loop antenna with a perpendicular feed for penta-band mobile phone application,” IEEE Trans. Antennas Propag., vol. 56, pp. 2138–2141, Jul. 2008. [13] Y. W. Chi and K. L. Wong, “Quarter-wavelength printed loop antenna with an internal printed matching circuit for GSM/DCS/PSC/UMTS operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 2541–2547, Sep. 2009. [14] K. D. Katsibas, C. A. Balanis, P. A. Tirkas, and C. R. Birtcher, “Folded loop antenna for mobile hand-held units,” IEEE Trans. Antennas Propag., vol. 46, pp. 260–266, Feb. 1998. [15] C. A. Balanis, Antenna Theory, Analysis and Design, 2nd ed. New York: Wiley, 1997, pp. 458–462. [16] Z. N. Chen, Antennas for Portable Devices. West Sussex, U.K.: Wiley, 2007, pp. 122–124. Chien-Wen Chiu (M’98) was born in Maoli, Taiwan, R.O.C., in 1962. He received the B. S. degree from National Taiwan Normal University, Taipei, in 1984, and the M.S. and Ph.D. degrees in electrical engineering from National Taiwan University, Taipei, in 1990 and 1996, respectively. From 1990 to 1991, he was a member of the Computer and Communication Research Laboratory, Industrial Technology Research Institute, where his primary research focus was the development of the DECT system. From 1991 to 1997, he was with the Department of Electronic Engineering, Jinwen University of Science and Technology, where he was a Lecturer until 1996 when he became an Associate Professor. From 1997 to 2003, he was with the Department of Electronic Engineering, Minghsin University of Science and Technology. In 2003, he joined the Department of Electronic Engineering, National Ilan University, where he is currently an Associate Professor. He is presently the Chairman of the Bachelor Program for the College of Electrical Engineering and Computer Science. His research interests include mobile antenna design, electromagnetic computations, electromagnetic compatibility, and RF ID antenna design. Yu-Jen Chi (S’98) was born in Taipei, Taiwan, R.O.C., in 1985. He received the B.S. and M.S. degrees in electronic engineering from National Ilan University, I-Lan, Taiwan, R.O.C., in 2007 and 2009, respectively. He is currently working toward the Ph.D. degree at National Chiao Tung University, Hsinchu, Taiwan, R.O.C. His main research interests are in multiband antennas for mobile devices, CRLH leaky wave antenna, and metamaterials. Mr. Chi received the Best Poster Award from the International Workshop on Antenna Technology (iWAT 2009).

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Modified Test Zone Field Compensation for Small-Antenna Measurements Juha T. Toivanen, Tommi A. Laitinen, and Pertti Vainikainen, Member, IEEE

Abstract—A method is presented by which full 3-D antenna pattern measurements can be performed without an anechoic measurement environment. The method is a field compensation technique that allows compensating for the effect of arbitrary incident test zone fields. Simulations are performed to demonstrate that the method works reliably in various kinds of surroundings. The applicability range of the method is estimated in terms of the accuracy of the determined far-field antenna pattern with a given uncertainty in individual measurement values. Finally, measurement results are presented that are in line with the simulations and verify the functioning of the method. Index Terms—Antenna measurement, error correction, field compensation, radiation pattern, test zone.

I. INTRODUCTION HE past decades have seen a considerable amount of research on compensation and correction methods for antenna range imperfections. These methods are applied in order to improve the measurement accuracy by reducing the effect of the non-ideal measurement environment on the measurement result. Often, despite the use of anechoic chambers, the reflections from the measurement surroundings constitute an important error source, which can be further minimized through the use of numerical correction techniques. Examples of such techniques are the deconvolution [1], plane-wave synthesis [2], antenna pattern correction [3] [4], virtual array [5], test zone field (TZF) compensation [6], plane wave, pattern subtraction method [7], EAD method [8], MARS technique [9], and IsoFilter technique. Typical limitations in these techniques are the increased measurement time or equipment requirements, limited field models, or restrictions on the level or number of the reflection components. One promising correction method is the TZF compensation technique [6]. In this technique, a complete spherical-wave model is created for the test zone fields, using a minimum amount of extra equipment and measurements. The traditional first-order probe correction technique is then iteratively employed to correct for not only the higher-order azimuthal spherical modes of the probe pattern but also for the undesired

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Manuscript received April 09, 2009; revised November 18, 2009; accepted May 09, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. This work was supported in part by the Academy of Finland, by Nordforsk, The Finnish Foundation for Economic and Technology Sciences, the HPY Research Foundation, The Finnish Society of Electronics Engineers, Walter Ahlström’s Foundation, and in part by Ella and Georg Ehrnrooth’s Foundation. The authors are with the Department of Radio Science and Engineering, SMARAD, Aalto University School of Science and Technology, FI-00076 Aalto, Finland (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.2010.2071335

reflections present in the measurement chamber. The technique is computationally efficient and thus applicable for electrically large antennas. However, the drawback of this technique is that constraints are placed on the amplitude of the reflected signals relative to the direct signal. The application of the technique may thus not be possible in environments with a high reflectivity level. The purpose of this paper is to present a general field compensation method, which, as opposed to [6], can be applied in practically any kind of (stable) electromagnetic environment. This way, one gains the advantage of not needing expensive anechoic measurement facilities. The method is a variation of the technique presented in [6]. It is theoretically exact, assuming negligible multiple reflections between the antenna under test (AUT) and the surroundings; it accounts for all range reflections and interfering signals of constant nature. The idea of this method was briefly introduced in [11]. The method is presented in this paper in a more comprehensive manner, including the theoretical background, computer simulations, and measurements. Application to both singleand multi-probe ranges is discussed in detail. Additionally, an uncertainty analysis is presented that provides information on the applicability of the method under different measurement noise levels. The results of this paper are important in that they present a way to perform antenna pattern measurements without any RF-absorber coating in the measurement facility. The presented method is best suited for measurement of small antennas (e.g., with the radius of the minimum sphere of 2– at maximum), where the computational requirements remain reasonable. It can be applied in both near-field and far-field conditions. II. COMPENSATION METHOD The modified test zone field compensation method is presented for the case where the range probe acts as the transmitter and the antenna placed in the test zone as the receiver. It is, however, applicable also for the opposite case. The method comprises two steps. In the first step (Section II-A), the TZF is determined by measuring a calibration antenna (with known radiation characteristics) in the test zone. In the second step (Section II-B), the knowledge of the TZF is utilized in solving the radiation characteristics of the AUT. It is first assumed that there is only one single-polarized range probe. The generalization to dual- or multi-polarized probes and multi-probe measurement systems is discussed in Section II-C. A. Test Zone Field Measurement The compensation method is based on the spherical-wave theory and the TZF and the antenna radiation characteristics are

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expressed in this context. The first task is to determine the TZF produced by the range probe, with respect to a fixed TZF coordinate system. For this, a calibration antenna, known as the TZF probe, is required. The TZF probe must be larger than the AUTs to be measured, so that it is sensitive to a large-enough number of spherical modes. Alternatively, a small TZF probe can be used provided that it is offset from the test zone origin so that its minimum measurement sphere is larger than the AUT minimum sphere. The radiation characteristics of the TZF probe must be known from a separate calibration measurement. They , are expressed by the antenna transmission coefficients which relate to the spherical wave expansion of the field radiated by an antenna as

, where the integer indices get the values , and . is the so-called truncation number for the spherical-mode series . and is the total number of modes, with Empirical formulas connect the required with the size of the is proportional to , where is the antenna [13], [14]. radius of the minimum sphere enclosing the antenna (in the TZF measurement, is also the radius of the test zone). One can now use (2), (3), and (4) to form a system of linear . With the single-index notation (5), this equations for the system of linear equations is expressed in matrix form as

(1) is the electric field strength in a position given Here, by the spherical coordinates , , and ; is the wave number; is the wave admittance; is the input signal; and are the outgoing spherical vector wave functions. It is well from spherical antenna known, how to determine the measurements [12], and hence it is not discussed here. The TZF probe is measured in different orientations in the test of the TZF probe in the zone. It is necessary to express the fixed TZF coordinate system. Therefore, the transmission coefficients depend on the probe’s orientation in the TZF coordinate system, denoted by the Euler angles , , and . The conversion to the TZF coordinate system is accomplished with the of [12] spherical-wave rotation function

or, in short (6) Here is an -by- matrix such that each row corresponds is the total to a different measurement orientation and number of orientations; is the number of spherical modes, determined by the size of the TZF probe; is a column vector of the TZF; containing the spherical-wave coefficients is a column vector containing the measured signal and is required so that there are at least values. Generally, as much linearly independent rows in the matrix as there are unknown variables. This matrix equation can be solved for the using the Moore-Penrose pseudoinverse [15] (7)

(2) In the formula above, the change in notation signifies the change of coordinate system. Next, the are through the simple converted to receiving coefficients relation

then contain all relevant information on the TZF, The including the range probe signal, the reflected and scattered signals, and other interfering signals. They provide a complete characterization of the fields entering the test zone. B. AUT Measurement

So, now there is a separate set of receiving coefficients for each orientation of the TZF probe, all expressed in the same TZF coordinate system. Using the spherical-wave theory, the signal received by an antenna placed in a test zone can be expressed as [12]

In the AUT measurement, the are used to solve the AUT radiation pattern through a similar procedure that was used in the TZF measurement. In this part, the TZF coordinate system are converted to the AUT is no longer used. Instead, the coordinate system. With the AUT orientation in the TZF coorin the AUT dinate system denoted by , , and , the coordinate system are given as

(4)

(8)

(3)

In this formula, is the received signal, are the receiving are the spherical-wave-excoefficients of the antenna, and pansion coefficients of the TZF. The superscript index 4 means that the coefficients represent the incoming waves. It will be dropped in the following for simplicity of notation. The sphercan be converted into the singleical-mode index triplet index -notation (5)

Now, by placing the AUT in the test zone and measuring the signal for different AUT orientations (different values for , , ), one can use (4) and (8) to form a system of linear equations for the coefficients of the AUT. Again, using the singleindex notation (5), one obtains the matrix equation

TOIVANEN et al.: MODIFIED TEST ZONE FIELD COMPENSATION FOR SMALL-ANTENNA MEASUREMENTS

or (9) is an -by- matrix such that each row corresponds to a is the total number of different measurement orientation and orientations; is the required number of spherical modes, determined by the size of the AUT; is a column vector containing of the AUT and is a column the receiving coefficients vector containing the measured signal values. Again, is required but here both and may be smaller than in the TZF measurement, if the AUT is smaller than the TZF probe. The solution is obtained through the pseudoinverse

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denotes probe number and there are in total where probes. Similarly, the measurement-value vector will contain values measured with every range probe. (14) Naturally, in this case the number of AUT measurement orientations can be reduced, while still obtaining enough linearly independent rows to be able to solve the matrix equation. In the extreme case, a large number of measurement probes are placed all around the AUT so that no AUT rotation at all is required. In this case,

(10)

(15)

This gives the measurement result since, combining (1) and (3), the radiation pattern of the AUT can be calculated from and (11)

(16)

The pseudoinverse gives the least-squares solution to the can be problem through a single matrix multiplication since calculated in advance and applied for all AUTs that fit within the test zone. The measurement point locations can be chosen at will, bearing in mind that the more diversity they exhibit (in terms of polarization and direction) the less sensitive to measurement noise the result will be.

In fact, this is the case with the measurements presented later in this paper in Section V. An added benefit is that in a multi-probe system, the TZF compensation provides also the channel balance calibration (as shown in [11]) since the information about the channel-response differences is included in the TZFs.

C. Multi-Probe Measurement Systems

A virtual antenna measurement environment is created in MatLAB, which allows the testing of the compensation method in a large number of different environments through simulations. The simulations are performed in a virtual measurement range with a single, dual-polarized range probe.

The modified TZF compensation method can be applied also with multi-polarized range probes and multi-probe antenna measurement systems. In this case, the TZF is different for each probe and therefore, the TZF coefficients become a function of the probe . (12) Thus, the TZF measurement and calculations must be repeated for all range probes. In practice, the signals corresponding to every range probe can be recorded in one scan with the TZF probe, provided that the measurement setup does not involve movement or rotation of the range probes during the measurement. Intuitively, one needs less measurement orientations for the AUT, when using a multi-probe system, compared to a singleprobe system. This is also the case with the presented method. When it is applied for a multi-probe range, the number of rows matrix in (9) is multiplied by the number of probes, in the because, as discussed above, each probe produces a different TZF. The matrix then gets the form

(13)

III. SIMULATIONS

A. Simulation Environment The simulation consists of a range probe, TZF probe, AUT, and several reflectors, which are modeled by short dipoles. The number, location, amplitude and polarization of the reflectors are varied. The AUT is a mobile terminal antenna model with a maximum directivity of 4.7 dBi [16]. A dual-ridged horn antenna model (directivity 9.5 dBi) is used as the range probe and also as the TZF probe. The number of reflections in the simulations is between 1 and 10 and the amplitude varies between 20 dB and 0 dB compared to the range probe signal in the test zone. The simulation proceeds through the following steps: 1) Definition of the radiation characteristics of all antennas; 2) Definition of the reflector locations and signals; 3) Calculation of the true TZF; 4) Virtual measurement of the TZF using the TZF probe; 5) Addition of noise to the measurement values; 6) Calculation of the TZF from the noisy measurement values; 7) Virtual measurement of the AUT; 8) Addition of noise to the measurement values; 9) Application of the compensation using the TZF calculated from the noisy measurement values; 10) Calculation of the measured AUT radiation pattern and comparison with the true pattern.

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TABLE I MEAN ERROR IN TOTAL RADIATED POWER [dB] (C = corrected, U = uncorrected)

Fig. 1. Antenna pattern measurement accuracy for different noise levels (gray = 1 re ection, black = 10 re ections).

The measurement noise, which is added to all measurement values, has a Gaussian distribution. The number of spherical . An oversampling factor modes used for the AUT is of approximately 2 is used in all virtual antenna measurements, meaning the number of measurement points is approximately twice the number of modes. B. Simulation Results The results of this analysis are presented in terms of the uncertainty in the total radiated power (TRP) and the directivity of the AUT, as derived from the antenna measurements. For each , referenced to simulation run, the error in the directivity the true antenna pattern, is calculated for the full space angle. The error values are normalized to the maximum directivity, i.e., (17) The root-mean-square (RMS) value of these directivity errors provides an average figure of merit for the accuracy of an antenna pattern measurement. When the measurement noise is non-existent, the directivity and TRP errors are zero, meaning the method is theoretically exact. Of course, a realistic measurement scenario will include a certain level of noise and other sources of uncertainty. Fig. 1 shows the RMS directivity errors as a function of the maximum reflection amplitude (dBs relative to the direct range probe signal in the test zone), for different numbers of reflectors and different measurement noise levels. The graphs show a sliding average i.e., the RMS error values of all simulations within a 5-dB window are averaged to give a point on the graph. For readability, graphs are shown only for cases with 1 and 10 reflectors. The number or amplitude of the reflections does not have a large influence on the overall uncertainty. The slight tilt of the graphs originates from the matrix inversion; the condition number of the matrix increases slightly (on average) with reflector amplitude and amount, resulting in higher uncertainty.

Only the measurement noise level has a large impact on the results. This overall noise level can be thought of as representing a number of different sources of uncertainty, such as positioning errors, multiple reflections, measurement instrument noise etc. It can be seen from Fig. 1 that Gaussian noise with 40 dB standard deviation (relative to the maximum received signal) causes on average an RMS directivity error of 20 dB and a 10-dB rise in the noise level causes a 5-dB rise in the directivity error. Even with 20 dB noise level, the pattern measurement accuracy can still be sufficient for, e.g., mobile terminal antenna measurements, where the highest accuracy is typically not required. In Table I, the mean errors (in dBs) of the TRP values are shown for each noise level. The values are for the case with 10 reflectors and are shown as a function of the maximum reflection . It can be seen that without correction the error amplitude rises steadily with the reflection amplitude, as expected. The correction eliminates most of the error in all cases, regardless of the noise level. However, the higher the noise level, the more residual error still remains after the correction. IV. EXPERIMENTAL VALIDATION—SINGLE-PROBE SYSTEM The presented method is verified experimentally by measurements with a single-probe measurement system installed in a standard laboratory room without anechoic properties. This kind of a measurement environment produces reflected fields with high amplitude and thus enables proper testing of the presented compensation method. A. Test Zone Field Measurement The room, in which the measurements were performed, contained many metal objects and reflecting surfaces relatively close to the test zone. A single, dual-polarized range probe was installed in the room in approximately 3-m distance from the test zone. For both polarizations of the range probe, all signals entering the test zone were characterized with the TZF measurement. The TZF measurement set-up is shown in Fig. 2, where the TZF probe can be seen in the middle installed on a rotating arm, which enabled a full 360-degree azimuth scan . Elevation scan and the change of measurement powere made possible by the probe holder design, larization which incorporated rotary joints for this purpose. A wideband, Vivaldi-type printed-circuit-board antenna element was used both as the range probe and as the TZF probe. The circular absorber sheet seen in Fig. 2 was used in order to reduce the back-lobe radiation of the TZF probe. The required number of measurement points in the TZF measurement is determined by the number of significant spherical

TOIVANEN et al.: MODIFIED TEST ZONE FIELD COMPENSATION FOR SMALL-ANTENNA MEASUREMENTS

Fig. 2. Test zone field measurement with a calibration antenna.

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Fig. 4. Measurement of AUT1 in a laboratory room with no RF-absorber coating on the walls.

of the TZF probe at

Fig. 5. Directivity of AUT1 with and without TZF compensation compared to the reference pattern.

modes of the TZF probe pattern, which is proportional to the square of the radius of the minimum measurement sphere of the probe. The mode series must be truncated appropriately. If the selected truncation number is too low, one runs into aliasing problems and if it is too high, the computational burden of the problem increases unnecessarily. That being said, the result is not extremely sensitive to the selection of the truncation number. of the TZF probe must be known from a separate The measurement so its radiation characteristics were measured at the DTU-ESA Spherical Near-Field Antenna Test Facility, in the Technical University of Denmark [17]. Fig. 3 shows the spherical-mode spectrum of the TZF probe at 1.8 GHz. The spherical mode number in the horizontal axis corresponds to the -notation (5). It can be seen from Fig. 3 that there exist . After this, the relatively high modes up to about was modes attenuate rapidly. A truncation number of used throughout the measurement frequency range of 1–3 GHz, modes. giving The number of measurement points should be somewhat higher than the number of modes. In the TZF measurement, a 20-degree step was chosen for both and and the mea-

surement was performed in two polarizations, giving 292 , the measurement points in total. Therefore, with oversampling factor was approximately 1.8.

Fig. 3. The normalized transmission coefficients 1.8 GHz.

B. AUT Measurement The test antenna (AUT1), a wideband horn antenna, is shown in Fig. 4. Its dimensions are . Since AUT1 is similar in size to the TZF probe, a truncation number was used also in the AUT measurement. Also, the of number of measurement points in the AUT measurement was the same as in the TZF measurement, with a 20-degree grid in both and . The measurement was performed in the range of 1–3 GHz with 100-MHz spacing. C. Measurement Results Based on the measurement data, the TZF compensation was applied as presented in Section II. The directivity of the AUT was calculated as a function of direction and compared with that provided by a reference measurement, which was performed at the DTU-ESA facility. A pattern cut at 2 GHz showing this comparison is presented in Fig. 5. Also the pattern measured without

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Fig. 7. Diversity antenna structure as AUT.

V. EXPERIMENTAL VALIDATION—MULTI-PROBE SYSTEM Fig. 6. Average directivity error as a function of the frequency with and without the TZF compensation.

TZF compensation is shown and, as can be seen, it is severely distorted due to the high reflectivity level of the measurement room. The performance of the TZF compensation is good down to about 15 dB from the pattern maximum. Below this level, the measurement errors start to have a large effect and the compensation accuracy degrades. The RMS directivity errors, calculated according to (17) asto be equal to the DTU-ESA measurement value, suming are presented in Fig. 6. Without TZF compensation, the errors are in the range of 4 to 10 dB. Error levels this high can be expected, since the level of the strongest signal reflections in the measurement room was 5 to 10 dB compared to the direct range probe signal, depending on the frequency. Obviously the room is a very poor environment for antenna measurements. Fig. 5 and Fig. 6 show that the application of the TZF compensation improves the results considerably. The average directivity error level after compensation is approximately 15 dB. There is no significant frequency dependence except that compensation accuracy is slightly limited at the lower end of the frequency range by the back-lobe radiation of the TZF probe. D. Uncertainty Analysis To estimate the measurement uncertainty in the TZF measurement, a control measurement was performed at regular intervals in a predetermined TZF probe position. The standard deviation of these values was approximately 40 dB relative to the maximum signal. This uncertainty is very low and, based on the simulation results in Section III-B, is not the limiting factor in the compensation accuracy. There were more significant error sources in the measurements such as the rotating antenna support structure, which interferes with the TZF (and AUT) measurement since it is not stationary. Also, due to the back-lobe radiation of the TZF probe, the mounting flanges in the DTU-ESA calibration measurement and the actual TZF measurement affected the pattern of the TZF probe in a different manner. Especially this was a problem at lower frequencies, where the level of the back-lobe radiation of the probe was higher. It is likely that these errors can be mitigated with better TZF probe and antenna holder designs.

Tests of the presented method were also performed by measurements with a multi-probe range at the Department of Radio Science and Engineering in the Aalto University School of Science and Technology. This multi-probe range presents a suitable environment for the testing of the method, because the probes effectively act as a source of reflections in the measurement. A. Test Zone Field Measurement The measurements were performed with the multi-probe system presented in [18] and [19]. This system consists of 32 dual-polarized probes that are placed uniformly on a spherical surface. Uniform placement means here that the probe locations are those of the vertices of the concentric icosahedron and dodecahedron. The test zone is located in the centre of the sphere. The TZF probe, measurement grid, and other parameters are the same that were used in the single-probe measurements. For every orientation of the TZF probe, a full measurement was performed, with each range probe at a time measuring the signal transmitted by the TZF probe (due to reciprocity, this is the same as considering the measurement probes as transmitters and the TZF probe as the receiver). This way, enough data was gathered for the characterization of the TZF corresponding to every measurement probe. The total amount of data is quite large, but the obtained data is valid as long as there are no significant changes in the system or in the measurement environment, and it can be used for all AUTs that fit within the characterized test zone. The scanning of the TZF probe was performed manually. B. AUT Measurement Three AUTs were used in the measurements. AUT2 is a simple sleeve dipole antenna, operating at 1.8 GHz. AUT3 and AUT4 are the two ports (left- and right-side elements, respectively) of the mobile terminal diversity antenna structure shown in Fig. 7. The model of AUT3 was used in the simulations in Section III. The dimensions of the whole antenna structure are and it operates at 1.6 GHz. All AUTs are relatively small in wavelengths. This means their spherical-mode spectrum does not extend very far and the used TZF probe is certainly large enough in comparison so that the required modes can be resolved. Due to the small size of the AUTs and the relatively small number of measurement probes,

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TABLE II NORMALIZED DIRECTIVITY ERRORS FOR MEASUREMENTS WITH THE MODIFIED TZF COMPENSATION

Fig. 9. AUT3 pattern cut, showing the DTU-ESA reference measurement and the multi-probe measurement with the TZF compensation.

Fig. 8. E plane (dashed) and H plane (solid) pattern cuts for AUT2 (sleeve dipole). The multi-probe measurement with the TZF compensation is compared to the DTU-ESA reference measurement.

a truncation number of was used in the measurements. The AUTs were measured by placing them in the centre of the test zone and recording the signal received by each measurement probe. The AUTs were measured in a single orientation. Due to the three-dimensional measurement probe configuration, sufficient measurement data was obtained without AUT rotation. C. Measurement Results Again, the TZF compensation was applied and the results calculated. Since the AUTs were not rotated during the measurements, the matrix and vector in (9) were of the form in (15) and (16), respectively. The directivity values of the AUT were calculated as a function of direction and compared to the DTU-ESA reference measurement. The RMS directivity errors, to be equal to the calculated according to (17) assuming DTU-ESA measurement value, are presented in Table II. The values are in the range of 7 to 10 dB for all AUTs. The error in TRP was also calculated and it was less than 0.4 dB for all AUTs. Fig. 8 shows the E plane and H plane pattern cuts for the sleeve dipole (AUT2). In the H plane, the maximum difference between the DTU-ESA reference measurement and the measurement with the TZF compensation is approximately 0.7 dB. These differences are mainly caused by the errors in the TZF measurement (see uncertainty analysis in the next section). In the E plane, the pattern measured with the TZF compensation is smoother than the reference pattern. This is because the truncation number (and the number of measurement points) in the multi-probe measurement was smaller than in the DTU-ESA reference measurement. Thus, some of the pattern variation is averaged out in the multi-probe measurement.

A representative pattern cut for AUT3 is shown in Fig. 9, with a similar comparison between the multi-probe measurement and the DTU-ESA reference measurement. The accuracy of this measurement is of the same order as with AUT2, although on average the result is slightly better, as can be seen from Table II. D. Uncertainty Analysis As in the single-probe measurements, a control measurement was performed at regular intervals in a predetermined TZF probe position to estimate the measurement uncertainty in the TZF measurement. The standard deviation of these values was approximately 25 dB relative to the maximum signal. This uncertainty is considerably higher than in the single-probe measurements and is mainly due to positioning errors because of the manual scanning. In addition, there are other error sources as mentioned in Section IV-D. The effect of the support structure of the TZF probe is now different since it is not present during the AUT measurement (AUT rotation was not required in the measurements). Consequently, it seems realistic to assume that, when comparing the measurements with the simulation results, the overall measurement uncertainty in the measurements was comparable to the 20 dB noise level in the simulations. The reflectivity level in the measurement chamber has been earlier determined to be in the order of 20 dB. The reflections mostly originate from the probe antennas. With these parameters, the simulation results in Fig. 1 and Table I are in fairly good agreement with the measurement results. VI. DISCUSSION It is worth attention to discuss the differences of the modified TZF compensation presented in this paper, as compared with the original TZF compensation presented in [6]. The essential difference between the two methods is the employed calculation technique. The modified TZF compensation uses a matrix-inverse solution method instead of the more traditional Fouriertransform-based method used in the original TZF compensation. Several advantages are gained with the use of the matrix-inverse solution method. First, the solution is obtained directly;

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no iterations are required. Second, there are no limitations on the amplitude of the range reflections. Since the original TZF compensation is an iterative technique, the level of the range reflections must be lower than the range probe field for the iterative process to converge, whereas the modified TZF compensation, as presented here, can be used in a measurement chamber without any anechoic properties, i.e., without RF absorbers. Although for the measurements presented in this paper, the reflected fields were smaller in amplitude than the range probe field, in theory there are no restrictions in the method on the level of the reflected signals relative to the probe signal, because the probe signal has no special status in the method. As usual, however, the method does not compensate for multiple reflections between the AUT and the probes. Third, it is not necessary to use a measurement point grid that is regular in and . Instead, the measurement point locations can be chosen, e.g., to provide a uniform distribution over a spherical surface with no point clustering near the poles. This was also the case with the multi-probe measurements presented in this paper. Actually, it is necessary to use the matrix inversion only in the second step of the modified TZF compensation, i.e., the AUT measurement, to realize the benefits discussed above. The TZF coefficients can be found in another way, e.g., as presented in (first-order) probe as the TZF [20], by using a so-called probe. However, the matrix-inverse solution method can be used also in the first step for the probe correction. This general probe correction method, although not often used, is known [21] and provides one additional advantage, namely the possibility to use a general (non-first-order) probe as the TZF probe. This makes it easier to find a suitable, wideband TZF probe for characterization of the TZF over a wide frequency band. In this paper, the matrix-inverse solution method is used in both the TZF measurement and the AUT measurement. The downside of this solution method is that it is computationally demanding and potentially more sensitive to errors in the measurement data due to the matrix-inverse operation. The number of required spherical modes depends on the size of the test zone, which must be large enough to enclose the AUTs to be tested. Hence, as the electrical size of the AUT increases, the number of spherical modes required for the characterization of the AUT and test zone fields expands rapidly. This directly influences the size of the matrix and the computational requirements in the matrix inversion. In the general case, the size of , the matrix, that is required to be inverted, is where is the number of measured probe signals. This matrix size sets the upper limit for the acceptable size of the AUT in wavelengths, for which the method is computationally applior greater, there are more than cable. For example, for 10 million elements in the matrix, and the method can hardly be used. For such AUTs, the original TZF compensation method could still be used. The computational complexity of the modi, whereas in the original TZF fied TZF compensation is [22], [6]. compensation it is In summary, a compensation method has been presented, which enables small-antenna pattern measurements in surroundings with an arbitrarily high reflectivity level, provided that the amplitude of multiple reflections is small compared to the primary reflections. The accuracy of the method is

mainly determined by the uncertainties in the TZF measurement. Potential applications include measurements in normal, non-anechoic rooms and also multi-probe measurement systems, where the neighboring probes are an unavoidable source of reflections. The functioning of the method has been confirmed with measurement results. ACKNOWLEDGMENT M. Mustonen and J. Ilvonen are thanked for providing the antenna-model data for the simulations. S. Pivnenko is acknowledged for performing the measurements at the DTU-ESA Spherical Near-Field Antenna Test Facility and L. Nyberg and P. Rummukainen for helping with the measurements at Aalto University. REFERENCES [1] J. C. Bennett and A. Griziotis, “Removal of environmental effects from antenna radiation patterns by deconvolution processing,” in Proc. Inst. Elect. Eng. Conf., 1983, pp. 224–228, Pub. 219, Pt. 1. [2] J. F. R. Pereira, A. P. Anderson, and J. C. Bennett, “New procedure for near-field measurements without anechoic requirements,” Inst. Elect. Eng. Proc. Microw. Opt. Antennas, vol. 131, no. 6, pp. 351–358, 1983. [3] J. Appel-Hansen, “Reflectivity level of radio anechoic chambers,” IEEE Trans. Antennas Propag., vol. 21, no. 4, pp. 490–498, Jul. 1973. [4] J. van Norel and V. J. Vokurka, “Novel APC-methods for accurate pattern determination,” in Proc. Antenna Measurement Techniques Assoc. Symp., 1993, pp. 385–389. [5] W. D. Burnside and I. J. Gupta, “A method to reduce signal errors in antenna pattern measurements,” IEEE Trans. Antennas Propag., vol. 42, no. 3, pp. 399–405, Mar. 1994. [6] D. N. Black and E. B. Joy, “Test zone field compensation,” IEEE Trans. Antennas Propag., vol. 43, no. 4, pp. 362–368, Apr. 1995. [7] D. A. Leatherwood and E. B. Joy, “Plane wave, pattern subtraction, range compensation,” IEEE Trans. Antennas Propag., vol. 49, no. 12, pp. 1843–1851, Dec. 2001. [8] S. A. Goodman and I. J. Gupta, “A method to correct measurement errors in far-field antenna ranges,” presented at the Antenna Measurement Techniques Assoc. Symp., 2007, Paper A07-0068. [9] A. C. Newell and G. Hindman, “Scattering reduction in spherical nearfield measurements,” presented at the 2008 IEEE AP-S Int. Symp., San Diego, CA, Paper 439.3. [10] D. W. Hess, “The IsoFilter technique: Isolating an individual radiator from spherical near-field data measured in a contaminated environment,” in Proc. 28th Antenna Measurement Techniques Assoc. Symp., Austin, Texas, 2006, pp. 289–295. [11] J. T. Toivanen, T. A. Laitinen, S. Pivnenko, and L. Nyberg, “Calibration of multi-probe antenna measurement system using test zone field compensation,” in Proc. 3rd Eur. Conf. Antennas Propag., Berlin, Germany, 2009, pp. 2916–2920. [12] J. E. Hansen, Spherical Near-Field Antenna Measurements. London, UK: Peter Peregrinus Ltd., 1988. [13] F. Jensen and A. Frandsen, “On the number of modes in spherical wave expansions,” presented at the Antenna Measurement Techniques Assoc. Symp., Atlanta, Georgia, 2004, Paper PID-105. [14] T. Laitinen, “Spherical Wave Expansion-Based Measurement Procedures for Radiated Fields,” Lic. Sc. thesis, Dept. Electrical and Communications Eng., Helsinki Univ. of Tech., Espoo, Finland, 2000. [15] T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation. Englewood Cliffs, NJ: Prentice Hall, 2000. [16] M. Mustonen, “Multi-Element Antennas for Future Mobile Terminals,” Lic. Sc. thesis, Dept. Radio Science and Eng., Helsinki Univ. of Tech., Espoo, Finland, 2008. [17] DTU-ESA Spherical Near-Field Antenna Test Facility, Specifications [Online]. Available: http://www.dtu.dk/centre/ems/English/research/facilities/facility_specifications.aspx [18] T. A. Laitinen, J. Toivanen, C. Icheln, and P. Vainikainen, “Spherical measurement system for determination of complex radiation patterns of mobile terminals,” Electron. Lett., vol. 40, no. 22, pp. 1392–1394, 2004. [19] J. Toivanen, T. A. Laitinen, C. Icheln, and P. Vainikainen, “Spherical wideband measurement system for mobile terminal antennas,” in Proc. 2nd IASTED Int. Conf. Antennas, Radar and Wave Propagation, Banff, Canada, 2005, pp. 360–365.

TOIVANEN et al.: MODIFIED TEST ZONE FIELD COMPENSATION FOR SMALL-ANTENNA MEASUREMENTS

[20] R. C. Wittmann, “Spherical near-field scanning: Determining the incident field near a rotatable probe,” in Antennas Propag. Symp. Dig., 1990, pp. 224–227. [21] F. Jensen, “On the probe compensation for near-field measurements on a sphere,” Archiv fuer Elektronik und Uebertragungstechnik, vol. 29, pp. 305–308, Jul. 1975. [22] F. H. Larsen, “Probe-Corrected Spherical Near-Field Antenna Measurements,” Lic. Sc. thesis, Electromagn. Inst., Tech. Univ. of Denmark, Lyngby, Denmark, 1980. Juha T. Toivanen was born in Kaavi, Finland, in 1980. He received the MSc (Tech) degree in radio engineering from the Helsinki University of Technology, Espoo, Finland, in 2005. From 2006 to 2009, he worked in the telecommunications industry. Currently he is working as a Researcher in the Department of Radio Science and Engineering, Aalto University, in the area of measurement methods for mobile terminal antenna testing.

Tommi A. Laitinen was born in Pihtipudas, Finland, on March 19, 1972. He received the Master of Science in Technology, the Licentiate of Science in Technology, and the Doctor of Science in Technology degrees in electrical engineering from Helsinki University of Technology (TKK), Helsinki, Finland, in 1998, 2000, and 2005, respectively. He joined the Radio Laboratory at TKK as a Master’s thesis student in 1997, and continued there as a doctoral student. His major research interests at TKK were small antenna measurements. From 2003 until the end of 2006, he was with the Technical University of Denmark

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(DTU), as a Postdoctoral Researcher and Assistant Professor. His research interests at DTU were spherical near-field antenna measurements. During these years, he mainly contributed to the development of an accurate antenna pattern characterization procedure for the DTU-ESA Spherical Near-Field Antenna Test Facility based on spherical near-field antenna measurements with a high-order probe. Since the beginning of 2007, he has been with the Radio Laboratory at TKK and later, from the beginning of 2010, with its successor Aalto University, in the Department of Radio Science and Engineering as Senior Researcher. While still carrying on with the research on spherical near-field antenna measurements, he now works also with small antenna measurements and sensor applications. His other duties include occasional teaching on master’s and postgraduate courses at Aalto University. He is the author or coauthor of approximately 50 journal or conference papers. Dr. Laitinen is the recipient of the IEEE Antennas and Propagation Society’s 2009 R. W. P. King Award for one of his papers.

Pertti Vainikainen (M’91) 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) 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) at TKK, and a Visiting Professor in 2000 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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A Quasi-Planar Conical Antenna With Broad Bandwidth and Omnidirectional Pattern for Ultrawideband Radar Sensor Network Applications Huiqing Zhai, Saibun Tjuatja, Senior Member, IEEE, Jonathan W. Bredow, Senior Member, IEEE, and Mingyu Lu, Senior Member, IEEE

Abstract—A quasi-planar conical antenna with broad bandwidth (more than 10:1) and omnidirectional radiation pattern has been designed. In order to achieve mechanical stability, conical radiator is realized through thin metallic coating over a cavity etched into a dielectric slab. Due to dielectric loading, the conical antenna in this paper is quasi-planar, low-cost, light, mechanically robust, and easy to fabricate, integrate, and re-configure. A simulation tool based on the Method of Moments is developed to analyze the quasi-planar antenna. With simulation results as the guideline, the dielectric loading’s material and geometries are optimized to attain wideband input impedance and omnidirectional radiation pattern. Three antenna prototypes are fabricated by using high density polyurethane foam as the dielectric loading material. Simulation and measurement data show excellent agreements. Input impedance bandwidths of all the three prototypes are greater than 10:1; and all of them show omnidirectional patterns in the azimuth plane. These quasi-planar conical antennas can be readily applied to ultrawideband communication and radar applications. In this paper, an ultrawideband radar sensor network testbed is constructed using one of the three prototype designs. Target localization is successfully demonstrated by five radar nodes with the aid of grid based location estimation algorithm. Index Terms—Conical antennas, dielectric loading, method of moments (MoM), omnidirectional, wideband.

I. INTRODUCTION N recent years, enormous research interests have been attracted to ultrawideband radar sensor network, in which low-cost radar nodes are deployed over unattended regions for security/surveillance monitoring [1]. At each radar node, the antenna is required to be wideband, omnidirectional in azimuth plane, low-cost, and mechanically robust. Conical antennas are nice candidates for ultrawideband sensor network, since it satisfies the first two requirements (wideband and omnidirectional) by nature [2]–[4]. Nevertheless, their mechanical drawbacks prevent conical antennas from being directly applied to radar sensor network. Due to their three-dimensional configurations, conical antennas are bulky and difficult to fabricate, integrate, and re-configure. Moreover, since conventional conical antennas comprise of free-standing metal, they are typically heavy in

I

Manuscript received January 28, 2010; revised April 21, 2010; accepted May 11, 2010. Date of publication September 02, 2010; date of current version November 03, 2010. This work was supported in part by the U.S. AFRL under Contact (FA 8650-07-2-5061). The authors are with the University of Texas at Arlington, Arlington, TX 76019 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.2010.2072899

order to achieve sufficient mechanical stability. Several remedies have been proposed to improve conical antennas’ mechanical performance. In [5]–[7], parasitic rings and supporting pins serve to stabilize the conical radiators; and, wire grid structures are exploited in [8], [9] to replace the solid conical surfaces. The techniques in [5]–[9] still involve three-dimensional configurations and free-standing metal, thus do not resolve the conical antennas’ mechanical deficiencies fundamentally. In the past few years, a lot of efforts were reported on planar monopoles, which are two-dimensional counterparts of the conical antennas [10]–[12]. Since planar monopoles are printed onto circuit boards, they are light and easy for fabrication and integration. However, planar monopoles are not perfect either. Although planar, they are not conformal to the mounting surface hence also belong to three-dimensional antennas. In addition, radiation of planar monopoles unavoidably deviates from omnidirectional pattern in the azimuth plane [10]–[12]. In essence, planar monopoles rely on dielectric material’s support to achieve mechanical stability; and dielectric loading can actually be applied by various means. In [13], [14], part of the conical radiators are covered by dielectric material to improve antenna performance; but, such partial coverages do not make the entire antenna mechanically stable. Analyses and simulation of interactions between dielectric loadings and conical antennas are conducted in [15]–[17]. Resistive loading for conical antennas, which is investigated in [18], does not constitute the optimal solution as it reduces the antennas’ efficiency. In [19], a radome-like dielectric cover is used to reinforce conical antennas. Since the antenna configuration in [19] requires integrating the conical part and cover part, it is difficult to fabricate. In [20], [21], magnetic dielectric materials are proposed to support conical antennas while maintaining wideband input impedance characteristics; apparently, the usage of magnetic material is not the most practical. A conical antenna loaded with non-magnetic SPS material is accomplished in [22]. Unfortunately, the dielectric loading degrades the electrical performances of the original conical radiators: satisfactory input impedance and radiation only (in pattern are shown in frequency band other words, a 3:1 bandwidth, which is far narrower than those of typical conical antennas [23]). In summary, no prior research has completely resolved the conicalantennas’ mechanical drawbacks without significantly demoting their electrical characteristics. In this paper, a novel dielectric-loaded conical antenna is designed. Specifically, conical radiators are realized by thin metallic coating over cavities etched into non-magnetic dielectric slabs; and the resultant antennas are termed quasi-planar conical antennas. The novel dielectric loading technique

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ZHAI et al.: A QUASI-PLANAR CONICAL ANTENNA WITH BROAD BANDWIDTH AND OMNIDIRECTIONAL PATTERN

fundamentally improves the mechanical characteristics of conventional conical antennas: it enables the quasi-planar conical antennas to be low-cost, light, mechanically robust, and easy to fabricate and integrate. Furthermore, re-configuration of the quasi-planar conical antennas can be straightforwardly accomplished through changing the cavity shapes (both input impedance and radiation pattern of conical antennas can be improved by re-shaping conical radiators, as demonstrated by extensive previous literatures [23]–[26] as well as this paper). Because the dielectric slab has revolutionary symmetry, the quasi-planar conical antennas retain omnidirectional radiation pattern in the azimuth plane. A full-wave simulation tool based on the method of moments (MoM) is developed to analyze and optimize our quasi-planar antennas. The simulation results suggest that, materials with low dielectric constant (less than 4) and low loss should be picked for dielectric loading in order to maintain wideband input impedance. Three antenna prototypes are fabricated by using high density polyurethane foam as the dielectric loading material. The first prototype comprises of regular mono-cone geometry; and the other two are re-configured from the regular conical shape to achieve different band coverages and elevation radiation patterns. Simulation and measurement data show excellent agreements. Input impedance bandwidths of all the three prototypes are greater than 10:1. Furthermore, one of the three prototype designs is taken advantage of to construct an ultrawideband radar sensor network testbed in the anechoic chamber of the University of Texas at Arlington. Target localization is successfully demonstrated by five sensor nodes with the aid of available grid based location estimation algorithm. It is therefore concluded that the quasi-planar conical antennas designed in this paper can be applied to real-world radar sensing applications. In fact, they can be extended to ultrawideband communications [27], [28] with little modification. This paper is organized as follows. Antenna design and fullwave simulation are described in Section II. In Section III, the antenna’s performance is demonstrated by extensive simulation and measurement results. And finally, conclusions of this study are drawn in Section IV. II. ANTENNA DESIGN AND ANALYSIS The proposed quasi-planar conical antenna is illustrated in Fig. 1(a). A cylindrical dielectric slab is made of homogeneous and permeability , where material with permittivity and are the permittivity and permeability of free space, respectively. The bottom side of the dielectric slab is coated by metal and behaves as the ground plane. A conical cavity is etched into the slab, and a metallic layer is coated on the cavity wall. The metallic cone and the ground plane jointly form a mono-cone radiator. Photo of a fabricated antenna prototype is shown in Fig. 1(b). If any planar circuits reside on the other side of the ground plane, they can be connected to the antenna through a via hole. In our prototypes, the antenna is fed by a co-axial connector, with its outer and inner conductors connected to the ground plane and the cone tip respectively. The mono-cone geometry in Fig. 1 can be re-configured by simply changing the cavity’s shape; two re-configured prototypes are presented in Section III. If the dielectric slab is removed, the configuration in Fig. 1 reduces to a conventional conical antenna [2]. Presence of the dielectric slab provides mechanical support to the cone radi-

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Fig. 1. The proposed quasi-planar conical antenna. (a) Antenna geometry and (b) photo of one antenna prototype.

ator thus makes the conical antenna quasi-planar, mechanically stable, and easy to fabricate, integrate, and re-configure. The major research topic of this paper is how to design the dielectric loading so that the cone radiator’s nice electrical characteristics (including wide bandwidth and omnidirectional pattern) are not deteriorated. Apparently, the quasi-planar conical antenna in Fig. 1 retains omnidirectional pattern because of its revolutionary symmetry. Therefore, the focus in the below is on maintenance of wide input impedance bandwidth. The quasi-planar conical antenna in Fig. 1 shares the same radiation mechanism as conventional conical antennas [2]. Since the feed is located at the center of a revolutionarily symmetric structure, spherical transverse electromagnetic (TEM) wave is launched in the dielectric material. The cone and the ground plane can be considered constituting a transmission line with characteristic impedance [2] (1) where denotes the angle of the conical cavity (Fig. 1(a)). If the cone’s size is semi-infinite, there only exists an outgoing TEM wave. In realistic antenna configurations, the outgoing TEM wave is reflected and scattered due to truncation of the cone. The reflection and scattering attenuate with the increase of frequency; consequently, the antenna approaches a semi-infinitely long transmission line when frequency is high enough. For conventional mono-cone antennas, the antenna’s input impedance is close to the value in (1) when the cone size (Fig. 1) is greater than quarter wavelength typically. Compared to the conventional conical antennas, introduction of dielectric material brings up a few complications. (a) The wavelength within the dielectric material is shorter than that in the air. As a result, the electrical length of the transmission line is enlarged due to the dielectric loading, which is helpful to the low frequency input impedance performance. ), in (1) is (b) Without dielectric loading (i.e., when frequency-independent. However, a dielectric material’s electrical properties usually vary with respect to the frequency (in (1), the dielectric constant ’s dependence on frequency is implicit). Such variations may deteriorate the antenna’s wideband feature if not carefully taken into account. (c) The dielectric-air interface results in more reflection and scattering of the outgoing TEM wave, making the antenna less frequency-independent.

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(d) The dielectric material forms a cavity, which resembles that underneath microstrip patch antennas [2]. This cavity tends to store energy, hence would reduce the antenna’s bandwidth. (e) Dielectric loss of the loading material consumes energy hence would reduce the antenna’s efficiency. Consequently, both material and geometry of the dielectric loading are critical for designing the quasi-conical antenna. In order to precisely model the interaction between conical radiator and dielectric loading, a Maxwell’s equations solver based on MoM is developed as a simulation tool for the body-of-revolution configuration in Fig. 1. Simulation results indicate that, it is possible to maintain the cone radiator’s wideband input impedance as long as the dielectric material’s relative permittivity is less than 4 (as shown by simulation results in Section III). Our simulation tool is described in the rest of this section, which follows two techniques: body-of-revolution MoM in [29] and generalized Poggio-Miller-ChangHarrington-Wu-Tsai (PMCHWT) scheme in [30]. Due to revolutionary symmetry, geometry of the quasi-planar conical antenna can be fully represented by revolving a generating curve (which resides in the plane) with respect to the axis, as shown in Fig. 2(a). Two regions are separated by the generating curve: air region and dielectric region. Part of the generating curve is covered by metal (solid lines in Fig. 2(a)) and the other part is dielectric-air interface (dashed lines in Fig. 2(a)). Two equivalent problems are constructed in Fig. 2(b) and (c) respectively, to facilitate the MoM development. In Fig. 2(b), there is air in the whole space. The exterior fields are the same as those in Fig. 2(a); and the interior fields are zero. Equivalent surface electric and magnetic currents over the generating curve are deand , respectively. In Fig. 2(c), the whole space is noted . The filled with the dielectric material interior fields are the same as those in Fig. 2(a), and the exterior fields are zero. In the interior equivalence problem, equivalent electric and magnetic currents over the generating curve are and , respectively. In the exterior equivalence denoted are radiproblem (Fig. 2(b)), electromagnetic fields ated by and in free space; fields in Fig. 2(c) and together with the co-axial feed in are produced by . Electric fields a homogeneous space with material and share similar mathematical expressions; and so do magand . Formulations of , , , and netic fields are given below.

Fig. 2. Illustration of MoM analysis for the quasi-planar conical antenna. (a) Original problem. (b) Exterior equivalence problem. (c) Interior equivalence problem.

(3)

(2)

, is the angular In (2) and (3), , time frequency, is the operating frequency, dependence is suppressed, , are fields due to the co-axial feed, and . In order to discretize the unknown equivalent currents, nodes are picked over the generating curve with roughly equispaced locations expressed in cylindrical coordinate , . Next, the system as generating curve is approximated by line segments: , as depicted in Fig. 3. Because of revolutionary symmetry in the quasi-planar conical

ZHAI et al.: A QUASI-PLANAR CONICAL ANTENNA WITH BROAD BANDWIDTH AND OMNIDIRECTIONAL PATTERN

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where

(9) (10) (11)

(12) Fig. 3. Illustration of linear interpolation function in MoM.

antenna, it is not difficult to show that the magnetic field only has component and the electric field has no component. has no As a consequence, equivalent electric current component and equivalent magnetic current has component only. Then, the equivalent currents can be expressed using linear interpolation as

, , , with dimensions

, ,

,

,

, and

are all row vectors

(13) (14) (15)

(4) (16) (5)

and matrices , , all have dimensions

,

,

,

,

, and

,

, are unknown coefficients, the two types where , , of linear interpolation basis functions, see (6)–(7) at the bottom is the revolutionary projection of point of the page, onto plane, , and . Next, the electric fields in (2) are tested by and the magnetic fields (17)

in (3) are tested by . After the substitution of (4) and (5), the following matrix equations are yielded.

(18)

(8)

(6) elsewhere

(7) elsewhere

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(19)

through dividing the feed voltage by the total current at feed point. Typically, the simulated input impedance converges (with fluctuation less than 10%) when the spacing among nodes in , where and is the Fig. 3 is smaller than wavelength in free space. III. NUMERICAL AND EXPERIMENTAL RESULTS

(20) Evaluation of the matrix elements in (17)–(20) is detailed in and [23], [29]. Two methods are adopted for calculating in (13) and (14): one is the magnetic frill current model in [31] and the other is the delta gap model. MoM results are almost indistinguishable from these two methods, which was observed , , , and are in [15] as well. The entries in vectors not required explicitly in the MoM solver. Finally, boundary conditions in Fig. 2(a) are incorporated into matrix (8), by following the generalized PMCHWT procedure in [30]. Specifically, there are two types of boundary conditions in Fig. 2(a). (i) Over the part of generating curve that is covered by metal, tangential electric fields are zeros and tangential magnetic fields from the two sides are independent of each other. and is As a result, over this metal part, independent of . (ii) Over the part of generating curve that is dielectric-air interface, both tangential electric fields and tangential magnetic fields are continuous across the interface. As a reand sult, over this dielectric-air interface part, . Obviously, not all the elements of vector are independent unknowns. Based on the above boundary conditions, the “non-redundant” unknowns are picked from to form a non-redundant vector with dimension . Also, connectivity matrix with is constructed to link and as dimension (21) The matrix equation (8) can then be rearranged as (22) , which is Equation (22) relies on the fact that guaranteed by the two aforementioned boundary conditions. can be readily solved from (22) using direct matrix Vector inversion methods like LU decomposition. In the simulations rarely goes greater of our quasi-planar conical antennas, than 1,000. Therefore, the cost associated with direct matrix inversion is trivial. is solved, total current at the feed point is After vector calculated as with and (note that, ” does not make the total current zero because of the “ ” term in (6)). Then, the antenna’s input impedance is found “

In this section, the quasi-planar conical antenna design in Section II is verified by some numerical and experimental results. This section is divided into three subsections. Section III-A investigates the impacts of dielectric loading onto conical radiators through numerical simulations. In Section III-B, simulation and measurement results of three fabricated antenna prototypes are presented. One of the three prototype designs is exploited to construct a testbed for ultrawideband radar sensor network. Configuration and target localization results related to this testbed are elucidated in Section III-C. In this section, it is always assumed that the antenna is to be matched to impedance 50 . A. Simulation Results for Dielectric Materials’ Impact onto Conical Antennas In this subsection, the quasi-planar mono-cone antenna geometry in Fig. 1 is simulated by the MoM solver developed in Section II, with various dielectric materials. Figs. 4 and 5 show the simulated voltage standing wave ratios (VSWRs) at the antenna feed. In Fig. 4, the dielectric constant of the loading material is assumed to be 1, 2, 3, and 4, respectively; and the . When cone angle is changed by through (1) with , the antenna reverts to a conventional conical antenna. ) Its input impedance is matched (with criterion when is larger than quarter wavelength. In practice, high end of the matching frequency band is restricted by the feed strucis greater than 1 ture. It is observed in Fig. 4 that, when but stays a small value, the antenna maintains the wideband beis, the more ripples there are in havior. However, the larger the VSWR curves. When is as large as 4, the antenna starts losing the wideband feature: some ripples in the VSWR curve go beyond 2. It is concluded that, when is reasonably small (less than 3, to be safe), the quasi-planar conical antenna’s input within a wide frequency band. impedance maintains close to Indeed, large value of is not desirable for the antenna fabresults in small value of . rication either. From (1), large has sharp variations when its argument Since function is small, a tiny error in the antenna fabrication could produce a large error in the input impedance. The data plotted in Fig. 5 are similar to those in Fig. 4. However in Fig. 5, the cone angle is fixed while changes. Specifically, is determined by asin (1). The three curves in Fig. 5 correspond suming to three loading materials with , 2, and 3, respectively. Fig. 5 indicates that, even when the cone angle is not matched to the dielectric material, the quasi-planar conical antenna retains wideband input impedance behavior. Based upon this observation, the wide-band characteristics of our quasi-planar conical antennas are not very sensitive to the variation of . In other words, the value of does not have to be exactly known during design/fabrication; and ’s dependence on frequency can be tolerated as long as it is not too strong. As a result, a wide range

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Fig. 6. Geometries of the three fabricated antenna prototypes (unit: mm). (a) Prototype A, (b) Prototype B, (c) Prototype C.

Fig. 4. Simulated input impedance results for various dielectric constants (with cone angle changed).

Fig. 7. Simulated and measured input impedance results for Prototype A.

Fig. 5. Simulated input impedance results for various dielectric constants (with cone angle fixed).

of materials can be chosen for the proposed quasi-planar conical antenna. Obviously, low conductivity is preferred for the loading material to minimize the dielectric loss. B. Simulation and Measurement Results for Three Antenna Prototypes Numerical results in Section III-A establish clear guidelines for the choice of loading materials. (i) Its dielectric constant should stay below 3, over the entire frequency band of concern. (ii) It should have low conductivity over the entire frequency band. (iii) It had better be light, low-cost, and have sufficient mechanical strength. (iv) Because cavities are to be etched into the dielectric slab, it is preferable that the material can be processed using traditional machine tools. Based on these guidelines, we choose high density polyurethane foam, which is a cheap, light, and easy-to-cut hard foam, as the loading material. A crude measurement using HP 85070A probe kit together with network analyzer

reveal that, dielectric constant of the high density polyurethane foam is between 1.3 and 1.8 in the frequency band of our concern (1 GHz–20 GHz). Then, three antenna prototypes are designed, fabricated, and measured. Geometries of the three prototypes are illustrated in Fig. 6(a)–(c), respectively. Prototype A is a regular mono-cone. Prototype B follows the design ” in [26]; the cone radiator is deformed to a “ structure. Prototype C is a further deformation from Prototype B. In Prototype C, the ground plane is bended; hence it is taller but thinner than Prototype B. As a result, Prototype C is approaching a bi-cone configuration. Extensive simulations were carried out to arrive at these three designs. The simulations show that, it is better to overestimate for Prototype A while should be underestimated for Prototype B and Prototype is designed to be 36.2 (corresponding C. As a result, to ) for Prototype A but 42.3 (corresponding to ) for Prototypes B and C. Simulation and measurement results for the three antenna prototypes are given in Figs. 7–12. In Fig. 7, simulated and measured VSWR results for Prototype A match one another very well. The VSWR stays below 2 from 2 GHz to 18 GHz. Our measurement stops at 18 GHz due to instrumentation limit; but from the trend in Fig. 7, the antenna is expected to keep matched beyond 18 GHz. The simulated and measured antenna gain results in E-plane for Prototype A are compared with each other in Fig. 8, at four frequencies: 2 GHz, 8 GHz, 14 GHz, and 18 GHz. The patterns are a little bit upward-looking, as required

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Fig. 8. Simulated and measured antenna gain results for Prototype A. (a) f = 2 GHz, (b) f = 8 GHz, (c) f = 14 GHz, (d) f = 18 GHz.

Fig. 10. Simulated and measured antenna gain results for Prototype B. (a) f = 2 GHz, (b) f = 8 GHz, (c) f = 14 GHz, (d) f = 18 GHz.

Fig. 9. Simulated and measured input impedance results for Prototype B.

Fig. 11. Simulated and measured input impedance results for Prototype C.

by radar sensor network applications (the radar sensors are usually installed on the ground and they must look upward for target detection). The H-plane patterns are almost perfectly omnidirectional hence are not plotted. It is observed that, the simulated data nicely predict the measured patterns, although the exact value of is unknown. Furthermore, conductivity is assumed to be zero in the simulation; but the simulated gain values are not substantially smaller than the measured gain values. It means that dielectric loss due to our dielectric loading is negligible. Simulated and measured VSWR results for Prototype B are plotted in Fig. 9. Prototype B’s matching band starts with 1.5 GHz, which is lower than that of Prototype A. This phenomenon is consistent with the research finding in [26]: deformation ” structure reduces the anfrom mono-cone to “ tenna size. Such a deformation not only re-configures the input impedance behavior but also tilts the E-plane radiation pattern

a little downward, as visualized from the measured and simulated gain data in Fig. 10. As expected, the further geometry deformation in Prototype C results in more re-configuration. The two VSWR curves for Prototype C in Fig. 11 are below 2 when frequency is above 1.1 GHz. In other words, to reach the same frequency band, size of Prototype C is about half as that of Prototype A. Also as expected, the E-plane radiation patterns in plane. As a matter of Fig. 12 are shifted more toward the fact, if the ground plane is re-shaped to be identical to the “ ” radiator, bi-conical dipole configuration is arrived at and the E-plane radiation would have its main beams plane. located in the Overall, simulation and measurement results in Figs. 7–12 exhibit nice agreements. Especially, matching frequency bands ) of the three antenna prototypes (with criterion are precisely predicted by the simulation tool. Discrepancies

ZHAI et al.: A QUASI-PLANAR CONICAL ANTENNA WITH BROAD BANDWIDTH AND OMNIDIRECTIONAL PATTERN

Fig. 12. Simulated and measured antenna gain results for Prototype C. (a) f = 2 GHz, (b) f = 8 GHz, (c) f = 14 GHz, (d) f = 18 GHz.

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Fig. 13. Radar sensor network testbed for target localization. (a) Testbed setup and measurement result from one node (without target), (b) testbed setup and measurement result from one node (with target), (c) illustration of measurement and processing related to one sensor node.

between the simulation and measurement results are largely attributed to two factors. First, is a fixed value in simulation while it has frequency dependence in reality. Second, the fine geometrical details at the feed point are not rigorously accounted for in the MoM simulation. C. A Radar Sensor Network Testbed and Target Localization Results In this subsection, Prototype B discussed in Section III-B is taken advantage of to construct a testbed for ultrawideband radar sensor network in the anechoic chamber of the University of Texas at Arlington. The testbed is shown by two photos in Figs. 13(a) and (b). A few sensor nodes are deployed around the target, which is an aluminum sphere in our experiment. As elucidated in Fig. 13(c), each sensor node consists of three components: an omnidirectional wideband antenna, an ultrawideband transceiver (including transmitter TX and receiver RX), and post-processing module. In our testbed, five identical quasiplanar conical antennas are built as the sensor node antennas; all of them follow the design of Prototype B in Section III-B. A vector network analyzer is adopted to emulate the ultrawideband transceivers. And, post-processing is carried out at a personal computer. In our experiment, target detection measurement is repeated at every sensor node. The vector network analyzer operates in frequency stepping mode over a wide frequency range. At each frequency, a continuous wave is transmitted from the conical anis recorded as the echo signal. When tenna and the complex information at all the frequencies is collected, a windowing funcis applied to the spectrum and both the transmitted tion and echo signals are synthesized into narrow impulses through inverse Fourier transform (IFT). In fact, the vector network analyzer has a built-in synthesis software to transform frequency

Fig. 14. Results of target localization using grid based location estimation algorithm.

domain data to time domain. In Fig. 13(a) and (b), the time waveforms on the vector network analyzer’s screen domain are shown when the target is absent and present, respectively. When there is no target, the impulse at time zero corresponds to the direct reflection from the conical antenna (Fig. 13(a)). As expected, with the presence of target, an echo impulse appears after a certain time delay (Fig. 13(b)). In Fig. 13(b), the time delay 2.89 ns is resulted from a round trip with distance about . In our post-processing, this 0.4 m and velocity time delay is precisely determined through correlation between the transmitted and echo impulses. Then, distances between the target and all the sensor nodes can be calculated, and the target’s location can be found through triangulation. In this paper, it is assumed that there only exists one target, which greatly relieves the processing complexity. In Fig. 14, one localization result is presented using grid based location estimation algorithm [32]. As illustrated in Fig. 14, five sensor nodes are deployed around the target. In grid based location estimation, which is a graph-based algorithm, each sensor’s measurement is represented by a trace centered at the sensor’s location. When multiple

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traces intersect, the trace’s brightness is enhanced and eventually the brightest spot is picked as the target’s location. Apparently, the localization algorithm’s output in Fig. 14 matches the physical location of the target in Fig. 14. The feasibility of applying the quasi-planar conical antennas to radar sensor network applications is therefore demonstrated. Radar sensor network processing is not the focus of this paper. More sophisticated processing techniques, which address realistic concerns like multiple targets, measurement uncertainties, and targets’ sizes, are currently under investigation. IV. CONCLUSIONS A quasi-planar conical antenna with broad bandwidth (more than 10:1) and omnidirectional radiation pattern is designed in this paper. The usage of dielectric loading fundamentally improves the mechanical characteristics of conventional conical antennas: it makes the quasi-planar conical antenna low-cost, light, mechanically robust, and easy to fabricate, integrate, and re-configure. A full-wave simulation tool based on MoM is developed to analyze the quasi-planar antenna. Simulation results indicate that, it is possible to design the dielectric loading such that wideband input impedance and omnidirectional radiation pattern of conventional conical antennas are maintained. Three antenna prototypes are fabricated by using high density polyurethane foam as the dielectric loading material. Simulation and measurement data show excellent agreements. Input impedance bandwidths of all the three prototypes are greater than 10:1; and all of them show omnidirectional patterns in the azimuth plane. An ultrawideband radar sensor network testbed is constructed using one of the three prototype designs. Target localization is successfully demonstrated by five radar nodes with the aid of grid based location estimation algorithm. ACKNOWLEDGMENT The authors would like to acknowledge Texas Advanced Computing Center (TACC) for granting access to its computational facilities. REFERENCES [1] G. Shingu, K. Takizawa, and T. Ikegami, “Human body detection using MIMO-UWB radar sensor network in an indoor environment,” presented at the 9th Int. Conf. on Parallel and Distributed Computing, Applications and Technologies, Dunedin, New Zealand, Dec. 2008. [2] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York: Wiley-Interscience, 2005. [3] C. H. Papas and R. King, “Input impedance of wide-angle conical antennas fed by a coaxial line,” Proc. IRE, vol. 37, no. 11, pp. 1269–1271, Nov. 1949. [4] S. S. Sandler and R. W. P. King, “Compact conical antennas for wideband coverage,” IEEE Trans. Antennas Propag., vol. 42, pp. 436–439, Mar. 1994. [5] H. Nakano, H. Iwaoka, K. Morishita, and J. Yamauchi, “A wideband low-profile antenna composed of a conducting body of revolution and a shorted parasitic ring,” IEEE Trans. Antennas Propag., vol. 56, pp. 1187–1192, Apr. 2008. [6] Y. K. Yu and J. Li, “Analysis of electrically small size conical antennas,” Progr. Electromagn. Res. Lett., vol. 1, pp. 85–92, 2008. [7] J. Ma, Y.-Z. Yin, S.-G. Zhou, and L.-Y. Zhao, “Design of a new wideband low-profile conical antenna,” Microw. Opt. Technol. Lett., vol. 51, no. 11, pp. 2620–2623, Nov. 2009. [8] K.-H. Kim, J.-U. Kim, and S.-O. Park, “An ultrawide-band double discone antenna with the tapered cylindrical wires,” IEEE Trans. Antennas Propag., vol. 53, pp. 3403–3406, Oct. 2005.

[9] S. Palud, F. Colombel, M. Himdi, and C. L. Meins, “Compact multioctave conical antenna,” Electron. Lett., vol. 44, no. 11, pp. 659–661, May 2008. [10] J. Jung, W. Choi, and J. Choi, “A small wideband microstrip-fed monopole antenna,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 703–7705, Oct. 2005. [11] S. Gupta, M. Ramesh, and A. T. Kalghatgi, “Ultra wideband embedded planar inverted conical antenna,” Microw. Opt. Technol. Lett., vol. 48, no. 12, pp. 2465–2468, Dec. 2006. [12] S. N. Khan, J. Xiong, and S. He, “Low profile and small size frequency notched planar monopole antenna from 3.5 to 23.64 GHz,” Microw. Opt. Technol. Lett., vol. 50, no. 1, pp. 235–236, Jan. 2008. [13] F. Hoshi, S. Sugawara, and T. Minewaki, “A dielectric-loaded broadband antenna with frequency-insensitive E-plane radiation pattern,” presented at the IEEE Antennas Propag. Society Int. Symp., Honolulu, HI, Jun. 2007. [14] S. Palud, F. Colombel, M. Himdi, and C. L. Meins, “A novel broadband eighth-wave conical antenna,” IEEE Trans. Antennas Propag., vol. 56, pp. 2112–2116, Jul. 2008. [15] W. Huang, A. A. Kishk, and A. W. Glisson, “Analysis of a thick monopole antenna loaded with dielectric material,” Archiv fur Elektronik und Ubertragungstechnik, vol. 48, no. 4, pp. 177–183, Jul. 1994. [16] F. Nusseibeh and R. Bansal, “Transient response of a wide-angle cone with dielectric loading,” Radio Sci., vol. 31, no. 5, pp. 1047–1052, Sep. –Oct. 1996. [17] D. Y. Xia, H. Zhang, F. Z. Geng, and Q. Y. Zhang, “A study of sphereloaded and dielectric-covered mono-conical antenna for broadband application,” presented at the IEEE Antennas Propag. Society Int. Symp., Albuquerque, NM, Jul. 2006. [18] J. G. Maloney and G. S. Smith, “Optimization of a conical antenna for pulse radiation: An efficient design using resistive loading,” IEEE Trans. Antennas Propag., vol. 41, pp. 940–947, Jul. 1993. [19] S. Inoue, H. Abe, M. Tokuda, and S. Ishigami, “Radiation characteristic of mono-conical antenna for wideband electromagnetic field generation,” presented at the 20th Int. Zurich Symp. on Electromagnetic Compatibility, Zurich, Switzerland, Jan. 2009. [20] G. B. Gentili, M. Cerretelli, and L. Cecchi, “Coated conical antennas for automotive application,” J. Electromagn. Waves Applicat., vol. 18, no. 1, pp. 85–97, 2004. [21] Y. Y. Kyi, L. Jianying, and G. Y. Beng, “Study of broadband small size conical antennas,” presented at the IEEE Antennas Propag. Society Int. Symp., Albuquerque, NM, Jul. 2006. [22] S. Kuroda, H. Asai, and T. Yamaura, “A study of dielectric covered mono-conical antenna for broadband and small-sized application,” presented at the IEEE Antennas Propag. Society Int. Symp., Columbus, OH, Jun. 2003. [23] X. Liang and M. C. Y. Wah, “Low-profile broadband omnidirectional monopole antenna,” Microw. Opt. Technol. Lett., vol. 25, no. 2, pp. 135–138, Apr. 2000. [24] T. Taniguchi and T. Kobayashi, “An omnidirectional and low-VSWR antenna for the FCC-approved UWB frequency band,” presented at the IEEE Antennas Propag. Society Int. Symp., Columbus, OH, Jun. 2003. [25] I. H. Choi, S. S. Choi, J. K. Park, H. W. Song, and H. S. An, “Design of a compact rectangular mono-cone antenna for UWB applications,” Microw. Opt. Technol. Lett., vol. 49, no. 6, pp. 1320–1323, Jun. 2007. [26] J. L. McDonald and D. S. Filipovic, “On the bandwidth of monocone antennas,” IEEE Trans. Antennas Propag., vol. 56, pp. 1196–1201, Apr. 2008. [27] F. Demmerle and W. Wiesbeck, “A biconical multibeam antenna for space-division multiple access,” IEEE Trans. Antennas Propag., vol. 46, pp. 782–787, Jun. 1998. [28] S. Yiqiong, S. Aditya, and C. L. Law, “Design and time domain characterization of UWB conical antennas,” presented at the IEEE/Sarnoff Symp. on Advances in Wired and Wireless Communication, Princeton, NJ, Apr. 2005. [29] H. Kawakami and G. Sato, “Broad-band characteristics of rotationally symmetric antennas and thin wire constructs,” IEEE Trans. Antennas Propag., vol. 35, pp. 26–32, Jan. 1987. [30] L. N. Medgyesi-Mitschang, J. M. Putnam, and M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Society Amer. A, vol. 11, no. 4, pp. 1383–1398, 1994. [31] L. L. Tsai, “A numerical solution for the near and far fields of an annular ring of magnetic current,” IEEE Trans. Antennas Propag., vol. 20, pp. 569–576, Sep. 1972. [32] C. Fretzagias and M. Papadopouli, “Cooperative location-sensing for wireless networks,” presented at the 2nd IEEE Annual Conf. on Pervasive Computing and Communications, Orlando, FL, Mar. 2004.

ZHAI et al.: A QUASI-PLANAR CONICAL ANTENNA WITH BROAD BANDWIDTH AND OMNIDIRECTIONAL PATTERN

Huiqing Zhai was born in Jilin province, China. He received the B.S., M.S., and Ph.D. degrees in electromagnetic fields and microwave technology from Xidian University, Xi’an, China, in 2000, 2003, and 2004, respectively. In 2004, he joined the School of Electrical Engineering, Xidian University. From April 2005 to March 2008, he worked at Tohoku University, Sendai, Japan, as a Research Fellow. Currently, he works at the University of Texas at Arlington, as a Postdoctoral Research Associate. His primary research interests include computational electromagnetics, microwave remote sensing, and microwave circuits and antennas for wireless communication. Dr. Zhai was awarded the Japan Society for Promotion of Science (JSPS) Research Fellowship from April 2006 to March 2008. He won the Best Paper Award and Zen’iti Kiyasu Award in the Institute of Electronics Information and Communication Engineers (IEICE) of Japan in 2008.

Saibun Tjuatja (SM’03) received the BSEE (magna cum laude) degree from the University of Texas at Arlington (UTA), the MSEE degree from Purdue University, and Ph.D. in electrical engineering from UTA, in 1987, 1988, and 1992, respectively. He is currently an Associate Professor and Director of Wave Scattering Research Center, Department of Electrical Engineering, UTA, where he has been a faculty member since 1993. He served as the Associate Chairman of the UTA Department of Electrical Engineering from 1999 to 2002. His current research interests include wave propagation and scattering in random media, subsurface sensing, radar signal processing, and neural interfacing. Dr. Tjuatja is a member of IEEE, Tau Beta Pi, and Sigma Chi, and a Fellow of the Electromagnetic Academy.

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Jonathan W. Bredow (M’85–SM’05) received the B.S. degree in electrical engineering from Kansas State University, in 1977, the M.S. degree in biomedical engineering from Iowa State University, in 1980, and the Ph.D. degree in electrical engineering from the University of Kansas, in 1989. He joined the faculty of the Department of Electrical Engineering, University of Texas at Arlington (UTA), in 1989, and is now Chair and Professor in the department. He served for many years as Director of the anechoic measurement facility at UTA. He has authored or coauthored a number of papers on microwave measurements and on wireless systems for characterization of wireless channels, as well as low power radars for study of phenomenology in environmental remote sensing. An important aspect of this has been antenna selection/development depending on frequency, bandwidth, polarization, medium (such as air or soil) and range to target in these studies. His current research interests include multi-antenna and multi-sensor instrumentation for wide area remote sensing retrieval.

Mingyu Lu (M’03–SM’08) received the B.S. and M.S. degrees in electrical engineering from Tsinghua University, Beijing, China, in 1995 and 1997, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 2002. He was a Research Assistant at the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, from 1997 to 2002. From 2002 to 2005, he was a Postdoctoral Research Associate at the Electromagnetics Laboratory, University of Illinois at Urbana-Champaign. He joined the faculty of the Department of Electrical Engineering, University of Texas at Arlington, as an Assistant Professor in 2005. His current research interests include radar systems, antenna design, computational electromagnetics, and microwave remote sensing. Dr. Lu was the recipient of the first prize award in the student paper competition of IEEE AP-S International Symposium, Boston, MA in 2001. He received Outstanding Service Award from IEEE Fort Worth Chapter in 2008.

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A Differential Dual-Polarized Cavity-Backed Microstrip Patch Antenna With Independent Frequency Tuning Carson R. White, Member, IEEE, and Gabriel M. Rebeiz, Fellow, IEEE

Abstract—A dual-polarized cavity-backed microstrip patch antenna has been demonstrated with independent tuning of both polarizations from 0.6 GHz to 1.0 GHz using varactor diodes (1.2–5.4 pF). The 10-dB impedance bandwidth varies from 0.9% to 1.8% over that range. Differential coaxial feeds and the placement of the tuning varactors in the appropriate symmetry planes are employed to achieve a differential-mode port-to-port isolation of 28 dB over the tuning range. Accordingly, the 20 dB at broadside. The antenna is cross-polarization level is 100 100 6 4 mm3 and is fabricated on a dielectric substrate ( = 2 2). The feed point and matching circuit for an impedance match to 50 are determined using the transmission line model. This model underestimates the sensitivity of the resonance frequency to the loading capacitance, but leads to a good initial design. To the authors’ knowledge, this is the first dual-polarized antenna with independent tuning over a 1.7:1 bandwidth and single-sided radiation. Index Terms—Cavity-backed antennas, dual-polarized antennas, microstrip antennas, reconfigurable antennas.

I. INTRODUCTION N the constant push achieve higher data rates and higher levels of flexibility in mobile communication systems, the antenna design cannot be ignored. Software Defined Radios, for example, can be reconfigured to communicate using many different protocols, which may be at different frequencies and or polarizations. The instantaneous bandwidth of efficient passive antennas is limited as they become small with respect to the wavelength [1], and frequency tuning can be a solution when small efficient antennas are required to cover a large frequency range. In addition, tunable narrow-band antennas provide frequency selectivity, relaxing the requirements of the receive filters. Applications in multipath environments may take advantage of polarization diversity in order to maximize signal strength, and satellite applications often require circular polarization. A dual-polarized slot-ring antenna with independent tuning of both polarizations over a 1.7:1 bandwidth and double-sided

I

Manuscript received November 21, 2009; revised March 30, 2010; accepted April 20, 2010. Date of publication September 02, 2010; date of current version November 03, 2010. This work was supported in part by the UCSD Center for Wireless Communications. C. R. White is with HRL Laboratories, LLC., Malibu, CA 90265 USA (e-mail: [email protected]). G. M. Rebeiz is with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093 USA. 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.2010.2071364

radiation has been demonstrated by the authors [2]. Single-sided radiation is necessary for most wireless applications, and the square microstrip patch antenna is a good candidate for a dualpolarized design with single-sided radiation. Frequency agility [3]–[5] and dual-polarized operation [6] have been studied in detail, but a literature search produces few dual-polarized antennas with independent tuning. The cavity-backed microstrip patch [7], [8] is a microstrip patch antenna in a cavity that is slightly larger than the patch. The cavity-backed design is not printed on a grounded dielectric slab, and therefore, has less mutual coupling in an array setting than the standard design. The depth of the cavity may be chosen such that the patch is in the same plane as the ground plane, which is very convenient for a varactor-tuned antenna. In this case, the radiation and impedance characteristics are very similar to those of the standard design [8]. In this paper, a dual-polarized cavity-backed microstrip patch antenna with differential probe-feeds is presented with independent tuning from 0.6 GHz to 1.0 GHz and differentialmode isolation 28 dB using four varactor diodes. The antenna geometry, design, and full-wave simulations are presented in Section II, and measured results are presented in Section III. The design principles and methodology described in this paper can also be directly applied to the standard microstrip patch antenna. II. DESIGN A. Realization of Dual-Polarized Independent Tuning A dual-polarized antenna with independent tuning of the resonant frequency of both polarizations and high port-to-port isolation can be achieved under the following conditions: 1) The antenna has two planes of symmetry. 2) The modes responsible for the two polarizations are orthogonal due to their even-odd symmetry with respect has to the two symmetry planes (e.g. mode and symmetries, and mode has and symmetries, where and symmetry means that the and planes are a perfect electric conducting (PEC) and perfect magnetic conducting (PMC) symmetry planes, respectively) (see Fig. 1). and have the same symmetry 3) The antenna feeds for and , respectively. The simplest case is when the as is in the PEC symmetry plane of , and vice feed for . versa for the feed of 4) The tuning elements for are in the PEC symmetry plane and vice versa. of

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The square cavity-backed patch antenna with differential probe feeds and varactor diodes in line with the feeds satisfies all of the above conditions. Condition 2 can be illustrated using the cavity model for the microstrip patch antenna, in which the resonant modes are approximated by assuming that it is a rectangular cavity with PEC top and bottom walls and PMC side walls. The fundamental -polarized mode, the mode, has 2 planes of symmetry; the plane is a PMC plane is a PEC symmetry plane. symmetry plane, and the mode is -polarized and it has PEC and Likewise, the and planes, PMC symmetry with respect to the respectively. Dual-polarized operation is typically achieved by feeding the - and -polarized modes along the and planes, respectively, using either coaxial probes or microstrip lines. If the patch antenna is fed in a single-ended manner, Condition 3 is not satisfied and the - and -polarized feeds may couple to higher-order modes, which increase the cross-polarized radiation, as well as coupling the two feeds together, reducing the and planes [9], isolation. If fed differentially in the [10], however, the - and -polarized feeds enforce PEC symand planes, respectively, and the isometries about lation is theoretically infinite. When the patch is not loaded, Condition 2 is satisfied by and modes. As tuning elements change the the resonance frequencies and mode distributions, orthogonality—and therefore, port-to-port isolation—is preserved due to the even-odd symmetry of the modes. , , can be tuned without The resonance frequency of , , if the tuning elchanging the resonance frequency of are placed in the PEC symmetry plane of ements for because they are short circuited from the perspective of . In , can be tuned without changing . the same manner, B. Antenna Geometry and Fabrication The geometry of the dual-polarized tunable cavity-backed microstrip patch antenna is shown in Figs. 1 and 2. The patch , and the horizontal and vertical polarizais tions ( -pol and -pol, respectively) are fed by differential probe-feeds at ports 1 and 2, respectively, where each probe is connected to an SMA connector soldered to the bottom of the cavity. The four probes are placed 16 mm from the center along the and axes. A square gap with inner and outer widths of 3.5 and 4.7 mm, respectively, surrounds the probe feeds. This type of gap was used by Hall [11] to compensate for the probe-feed inductance in thick substrates. In this work, however, it allows the antenna to be impedance matched over the entire tuning range by placing surface-mount 5.8-nH lumped inductors (Coilcraft 0603CS-5N6X_L [12], 5.8 nH at 900 MHz) in series with the antenna at each feed while directly connecting the coaxial feed to the patch. , and is filled with The cavity is , ) [13] dielectric maRogers 5880 ( terial. The sidewalls of the cavity wrap around to the top producing a 2-mm-wide ring of metal which aids in the fabrication. It can also be used to conformally mount the antenna to a ground plane. This leaves a 6-mm-wide slot surrounding the patch.

Fig. 1. Geometry of the dual-polarized cavity-backed patch antenna with independent tuning. RF-short-circuiting capacitors are placed across the diagonal biasing slots every 11 mm, as shown in the insets. (a) Side view; (b) top view.

Fig. 2. Top-view photograph of the realized antenna.

Varactor diodes and are placed across the radiating and are placed slots along the axis and tune the -pol; along the -axis and tune the -pol. Pads are extended from both the patch and the cavity walls to mount the varactor diodes; the

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Fig. 3. Transmission-line model of the differentially-fed varactor-tuned patch antenna with mutual coupling analysis based on [17]. (a) Two probe feeds; (b) odd mode.

pads are 2 mm wide and the gap is 1 mm. The diodes are Skyworks SMV1405-074 silicon abrupt-junction common-cathode pairs [14] that are connected in parallel to achieve a capacitance range of 1.2–5.4 pF from 0–30 V with an equivalent series resistance (ESR) of . This configuration was chosen in order to achieve a higher varactor than using a single diode with higher junction capacitance [15]. The varactor diodes are reverse biased by applying a DC voltage to the patch relative to the cavity. Independent tuning with high isolation requires that

where is the bias voltage across varactor . Therefore, the patch is separated into quadrants with 0.5-mm-wide biasing gaps. Each of these quadrants is DC-connected to Ports , , , and through the matching inductors, , and the feed probes, and therefore, independent bias voltages can be applied to each quadrant using external bias tees. Although each varactor can be biased independently to compensate for varactor mismatch, the horizontal and vertical quadrants are connected with jumpers (Fig. 1(b)), reducing the complexity by enforcing and . The current on the microstrip patch from the and modes is significant over the whole width of the patch, but goes to zero at the respective E-plane edges. Therefore, the biasing gaps must be RF-short-circuited everywhere, except perhaps at the corners. In order to achieve this, five 47-pF capacitors (AVX SQCS, 0603 package, reactance at 600 MHz) [16] are placed across the slot at an interval of 11 mm starting 11 mm from the corners. C. Circuit Model and Full-Wave Simulations The microstrip patch antenna can be simulated accurately using full-wave simulation methods; however, considerable insight is gained using the transmission-line model. The tuning

characteristics of the capacitively loaded patch are first investigated using the transmission line model, leading to an initial design. The design is then evaluated using a full-wave Method of Moments (MoM) simulation. 1) Transmission-Line Model: The patch is assumed to be a transmission-line section terminated by radiating slots with ad(Fig. 3(a)). The admittance of the mittance slots is calculated considering the mutual coupling admittance, , between the two radiating slots, following Pues et al. [17], and the via-hole inductance is calculated following [18]. When the tuning capacitors are symmetric and the antenna is fed differentially, and the oddmode equivalent circuit is shown in Fig. 3(b). The effective slot increases linearly with frequency conductance, at 0.5–1.0 GHz; the effective slot susceptance, , is capacitive and also increases linearly with frequency. The input impedance is calculated as (1) where feed, and is given by

is the antenna admittance at the probe

(2) is the characteristic admittance of the transmission line, (3) is the total admittance at the slot,

, and (4)

is the admittance of the varactor diode with capacitance and parasitic series resistance when , where and . Resonance occurs at the frequency, , where (5)

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Fig. 4. Calculated resonance frequency of the 84 84 mm patch antenna on a 6.35-mm-thick substrate with  = 2:2 (transmission line model). Fig. 6. Calculated antenna conductance, G , for different values of R and ` for the 84 84 mm microstrip patch (transmission line model).

2

Fig. 5. Calculated radiation efficiency,  , versus resonance frequency for the 84 84 mm microstrip patch for different values of R (transmission line model).

2

As varies from 0–15 pF, varies from 1.15–0.62 GHz (Fig. 4). The radiation efficiency (assuming lossless transmission to the total power lines) is the ratio of the power absorbed in absorbed at the slot: (6) The efficiency is plotted in Fig. 5 for and varying from 0–15 pF. Most varactor diodes have parasitic series resistance between 0.5 and 0.9 for these capacitance values, and to our knowledge, diodes with have not been realized. However, 0.2 can be achieved using MEMS switched capacitors [19]. for the diodes in the fabricated prototype is 0.55 . While the antenna is tuned to resonate at different frequencies, the resistance must also be matched to 50 over the tuning , then range. Assuming that (7) If , then (7) reduces to , as in [20]. This assumption is not valid for the varactor-tuned antenna, however, and the behavior of the denominator in (7) defor feed lopends on the location of the feed. Fig. 6 shows and 16 mm and for different values of . For cations is nearly constant for the high frequencies, ideal varactors, but decreases at the low frequencies when . When and , the antenna is matched to 50 over the tuning range, but these resistance values are unrealistic in most cases. Therefore, the antenna is matched using a series inductor like the cavity-backed slot antenna in [21]. This inductor approximates an inductive admittance inverter, which

2

Fig. 7. Reflection coefficient of the 84 84 mm microstrip patch predicted by the transmission-line model as C varies from 0–15 pF (L = 5:8 nH).

scales an admittance, , as . The fact that an inductor is the matching element is convenient because the probe feeds are inductive. The reflection coefficient is plotted in Fig. 7 for as C varies from 0–15 pF. The transmission-line model predicts one octave of tuning; however, it will be shown by full-wave simulations that the transmission-line model underestimates the sensitivity of the microstrip patch antenna to capacitive loading. results in a better impedance match, but 5.8 Also, reducing nH is plotted in order to compare with full-wave simulations. 2) Full-Wave Simulations: The tunable cavity-backed microstrip patch antenna has been simulated in IE3D (MoM) [22]. A finite dielectric, meshed at 40 cells per wavelength, is used to simulate the cavity-backed patch as shown in Fig. 1, and copper is assumed for all metal. Probe ports are used at ports , , , and , and horizontal localized ports are used in place of the varactor diodes. An infinite ground plane is used for the bottom of the cavity, both to reduce the computational complexity, and to simulate the case where the antenna placed on a large conducting object. The biasing gaps and the gaps surrounding the probe feeds (see Fig. 1) are neglected, and the matching inductors are included as lumped elements in series with all four input ports in IE3D. The 4-port S-parameters are then calculated by loading the appropriate ports with the varactor-diode equivalent circuit. The differential and common mode S-parameters are found by assuming that the antenna is connected to an ideal 180 hybrid. The differential-to-differential-mode S-parameters are calculated as (8a) (8b)

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Fig. 8. Simulated reflection coefficient of the cavity-backed patch antenna as all loading capacitors vary from 1.2–5.4 pF in 0.84-pF steps (L = 5:8 nH).

Fig. 10. Comparison of measured S and S of the cavity-backed patch antenna with either capacitors or conductive-adhesive copper tape RF-short-circuiting the biasing gaps for V = V = f0; 1; 2; 4; 8; 30g.

(8c) (8d) where the subscripts and refer to and respectively. The differential-to-common-mode S-parameters are calculated as (9a) (9b) (9c) (9d) The simulation predicts tuning from about 1.0 GHz to 0.6 GHz for a capacitance range of 1.2–5.4 pF with (Fig. 8). The simulated 10-dB impedance bandwidth varies from about 1.8-1% over the tuning range. As the loading caand become independent, the two polarizations pacitors is dependent on but not resonate at different frequencies; , and vice versa. The simulated differential-differential-mode and ) and differential-common-mode isolation ( coupling ( , , , and ) are 40 dB over the tuning range, as expected due to symmetry. In order to validate the use of microstrip patch antenna models to model the cavity-backed patch antenna (ground plane and the patch in the same plane), a microstrip patch antenna with , an infinite 6.35-mm-thick substrate with , , and a differential feed with was simulated in IE3D. The impedance characteristics (not shown) are very similar to those of the cavity-backed patch; the mi15 dB reflection coefficient crostrip patch is matched to and tunes from 1.05 down to 0.65 GHz as the loading capaci. tance changes from 1 to 6 pF with The under-estimation of the sensitivity of the resonance frequency to capacitive loading when using the transmission line model was also observed in [23] and [24], and can be explained as follows: the transmission-line model assumes no variation of the field or the current along the radiating slot, i.e. that the slot admittance is distributed along its length. When a single varactor diode is placed across each slot, current flows along the slot edge to meet the reactive boundary condition at the location of the varactor—adding inductance in series with the varactor and increasing its loading effect. Although the model underestimates the tuning sensitivity, the difference in the resonant input resistance is small enough that the transmission-line model pro-

Fig. 9. Setup to measure the S-parameters at ports 1 and 1 . There are five more combinations necessary to measure the 4-port S-parameters.

vides good initial values for the feed location and matching inductance. III. MEASURED DATA AND DISCUSSION A. S-Parameters The 4-port S-parameters of the antenna were measured using a two-port vector network analyzer; the two-port S-parameters were meaof at all six combinations of ports sured—the unused ports were terminated in 50 —as shown in Fig. 9. The source power was set to 30 dBm in order to ensure that the diodes were in the small signal region, and the antenna was placed on a foam block about 1 m from any scatterers. All measurements shown in this paper are of the antenna with no ground plane. However, the S-parameters were also measured ground plane, and are nearly identical. Ason a suming that the antenna is connected to an ideal 180 hybrid, the differential and common mode S-parameters are found by (8) and (9), respectively. In order to determine the effect of the RF-short-circuiting capacitors on the antenna, the S-parameters were measured with five capacitors per biasing slot, and also with copper tape shortcircuiting the biasing slot (Fig. 10). There is very good agreement between the two cases. Two other cases (not shown) were investigated: 1) three capacitors per biasing slot and 2) placing insulating-adhesive copper tape over the biasing slot; in both cases, the impedance match was degraded. All remaining measurements are made with five 47-pF capacitors across each biasing gap.

WHITE AND REBEIZ: A DIFFERENTIAL DUAL-POLARIZED CAVITY-BACKED MICROSTRIP PATCH ANTENNA

Fig. 11. Measured S (solid line) and S V f ; ; backed patch antenna for V

=

(dashed line) of the cavity-

= 0 1 2; 4; 8; 30g.

Both polarizations tune from 0.6 GHz to 1.0 GHz with 17 dB reflection. for the lower bias voltages, at the higher bias voltages. It is clear that there and is some asymmetry in the antenna that is causing the difference in resonance frequency at the lower frequencies. The fact that this occurs identically for the 5-capacitor case and the conductive-adhesive copper tape case eliminates asymmetry in the bias and follow capacitors as the cause. Furthermore, nearly identical paths on the Smith Chart (Fig. 11). Therefore, the effects of inductance variations between the matching inductors (which shift the curves toward the top of the Smith Chart) are minor. The most likely reason for the asymmetry is varactor mismatch, which not only causes the two polarizations to resonate at different frequencies, but also couples the differential and common modes together. The 4-port S-parameters were measured for all combinaand in {0, 0.5, 1, 2, 4, 8, 16, 30}, and the tions of differential-to-differential mode S-parameters are shown in . As varies from 0–30 V, varies Fig. 12 for from 0.6–1.0 GHz while remains constant and the isolation, , remains better than 30 dB. The differential-to-differential S-parameters of all tuning states are summarized in Figs. 13 is independent of , and is independent of and 14. over the entire tuning range (Fig. 13(a)). The 10-dB impedance bandwidths show little variation with the cross-polarized bias voltage, and vary from 0.9% at 0.6 GHz to 1.8% at 1 GHz (Fig. 13(b)). The variations that are present are caused by coupling to the common modes, as shown below. The in-band differential-to-differential mode isolation—defined as the within the 10-dB reciprocal of the maximum value of impedance bandwidths of both polarizations—is 28 dB over the entire tuning range (Fig. 14). It was expected that the highest values would occur when , and therefore, the antenna was re-measured with both polarizations tuned to the same frequency over the 650–980 MHz range. In this case, the differential-to-differential and differential-to-common mode isolations are still higher than 29 dB and 22 dB, respectively.

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Fig. 12. Measured differential-to-differential mode S-parameters of the cavity.S

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The Backfire-to-Broadside Symmetrical Beam-Scanning Periodic Offset Microstrip Antenna Yuanxin Li, Member, IEEE, Quan Xue, Senior Member, IEEE, Edward Kai-Ning Yung, Senior Member, IEEE, and Yunliang Long, Senior Member, IEEE

Abstract—A backfire-to-broadside beam-scanning antenna with a periodic structure is described. The proposed antenna consists of the offset double-sided parallel-strip lines (DSPSL). The characteristics of the offset DSPSL have been studied, and then a series of offset DSPSL have been applied as the periodic antenna with a backfire-to-broadside symmetrical beam scanning capability. Both in - plane and - plane, main beams symmetrically steer from the backfire direction to the broadside direction with the increase of the operating frequency. The fractional impedance bandwidth has been broadened to nearly 50% by using the offset DSPSL structure. Index Terms—Backfire to broadside, double-sided parallel-strip lines (DSPSL), microstrip leaky wave antenna, offset.

I. INTRODUCTION

HE microstrip leaky wave antenna (LWA) has been used in many applications for its attractive properties such as low profile, compact structure, and easy to be analyzed by the multi-mode cavity model or the other numerical techniques [1]. The main lobes of the most microstrip LWA designs scan in the forward direction [2]. There are several researches focus on the dual-beam scanning microstrip LWA in the -plane ( - plane) [3]. However the performance of the microstrip LWA is limited because of the limitation of ratio of available highest and lowest impedance [4]. The double-sided parallel-strip lines (DSPSL) is a good candidate to overcome this problem. Besides, since the DSPSL is a balance structure, more freedom can be provided in the microwave circuit design. The design of DSPSL is simply related to the design of the conventional microstrip line [5]. The DSPSL with offsetting strips can increase the characteristic impedance dramatically while keeping width of the strips the same [6]. In the past, the most researches of the DSPSL focused on the microwave circuit design such as the ultra-wide band balun and the low-pass filter [7], [8]. On the other hand, several microstrip antenna designs with backward-to-forward beam-scanning capability have been

T

Manuscript received September 26, 2009; revised February 04, 2010; accepted April 30, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. This work was supported in part by the Natural Science Foundation of China under Grant 60901028 and in part by the NSFCGuangdong under Grants U0635003 and U0935002. Y. Li and Y. Long are with the Department of Electronics and Communication Engineering, Sun Yat-sen University, Guangzhou, China (e-mail: [email protected]). Q. Xue and E. K.-N. Yung are with the State Key Laboratory of Millimeter Waves, City University of Hong Kong, Kowloon, Hong Kong SAR, China. Digital Object Identifier 10.1109/TAP.2010.2071352

studied for a long time [9]. The conventional periodic LWA operated as a linearly polarized traveling-wave antenna with the main lobe points to either the backward or the forward direction due to the positive or negative phase constant [10]–[12]. The conventional periodic LWA consisted of a rectangular waveguide which loaded with a periodic array of holes in the narrow wall of the waveguide, the constructions are so complex [9]. In our previous works, a periodic half-width microstrip LWA was presented, the main lobe scanned from 149 to 28 in H-plane ( - plane) [13]. Compared with the conventional periodic LWA, this new design had the advantage of simple and compact structure. The backward-to-forward beam-scanning capability could be achieved by the application of the composite right/left-handed (CRLH) metamaterial as LWA [14]. A CRLH metamaterial exhibited the phase constant varied from negative to positive values due to the negative index of refraction [15], [16]. At first, a LWA with a backward to forward capability used the CRLH structure with dominant mode was introduced in [17]. On the other hand, the fix-frequency beam-scanning LWA has been reported [18]. In [19], the beam electronically steered over a wide range in both the forward and backward directions with a simple optimization algorithm. There were several researches about the 2–D CRLH microstrip structure [20], [21]. An antenna with only backward radiation was reported in [22]. In simulated results of the above researches, the main lobes begin to scan when the normalized phase constant increases , which means the calculated main lobes should steer from up from 180 (measured from the direction). However, in most of the the main lobes just steer up from experimental results, and the main beam scans from backward to forward only in one plane. In this paper, a backfire-to-broadside symmetrical beam-scanning periodic offset microstrip antenna is presented. The proposed antenna is printed on a substrate, and consists of a series of offset DSPSL. The effect of the periodic offset between the two parallel patches has been shown. The radiation patterns and the bandwidth of this novel microstrip antenna have been studied, too.

II. THE OFFSET DOUBLE-SIDED PARALLEL-STRIP LINES In general, the width of ground plane is at least 5 times wider than the microstrip line in the conventional microwave circuit design as shown in Fig. 2(a) [1]. The characteristics impedance

0018-926X/$26.00 © 2010 IEEE

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W = 14:8 mm, L = 150 mm, T = 8:5 mm, l = 15 mm, l = 10 mm,

s = 2:2 mm h = 0:8 mm " = 2:65

Fig. 1. The backfire-to-broadside beam-scanning periodic offset microstrip antenna. ( , , ) (a) 3-D view, (b) Top view.

Fig. 2. The side view of different microstrip line structure: (a)The conventional microstrip line, (b) The double-sided parallel-strip lines (DSPSL), (c) The offset DSPSL.

of the conventional microstrip line is mainly determined by the width of the microstrip line, and is given by

such as high characteristics impedance, low transmission loss, and good electromagnetic isolation [7]. In Fig. 2(b), when the width of the microstrip line and the finite ground plane are the same, this finite ground microstrip line is called as the double-sided parallel-strip lines (DSPSL). The offset of the top and bottom patches in the DSPSL makes the characteristics increase effectively. As shown in Fig. 2(c), the impedance offset of the DSPSL increases the distance between two parallel patches. The coupling between top and bottom patches reduces, which results in the decreased of distributed shunt capacitance [8]. The and then increasing the characteristics impedance could be increased though high characteristics impedance extending the offset distance of the DSPSL. The characteristics is controlled not only by varying the width of impedance two parallel microstrip lines, but also by changing the offset distance in the offset DSPSL. Several methods to simulate the characteristics impedance of the microstrip transmission line have been reported in [1], [23]. A simple method is given to calculate the of DSPSL with the modified ratio of the patch’s width to the thickness of the substrate, that is

(2)

(1)

where is the effective dielectric constant of the substrate. is The suitable expression of

where and are the distributed shunt inductance and shunt capacitance. On the other hand, the finite ground microstrip line has been applied in microwave integrated circuit for its special features,

(3)

LI et al.: THE BACKFIRE-TO-BROADSIDE SYMMETRICAL BEAM-SCANNING PERIODIC OFFSET MICROSTRIP ANTENNA

Fig. 4. The simulated normalized phase constant method.

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=k

calculated by FDTD

extends, which due to the coupling between two offset DSPSL reduces. Fig. 3. The simulated characteristics impedance Z of the offset DSPSL.

III. DESIGN OF THE PERIODIC OFFSET MICROSTRIP ANTENNA

where is the relative dielectric constant of the substrate, is the modified ratio. For the conventional microstrip lines

(4) for the offset or non-offset DSPSL

(5) where is the width of the patch, is the offset distance, and is the thickness of the substrate in the DSPSL. of the conventional miThe characteristics impedance crostrip line, the DSPSL, and the offset DSPSL with different construction are shown in Fig. 3. The solid lines with different symbols are calculated by (2), and in the line with the dotted lines gotten by Zeland IE3D. When the offset distance increases, the characteristics impedance significantly raises at . At the same ratio of the microstrip the same ratio line’s width to the space between two parallel-strip lines, the of offset DSPSL is about 5 times greater than the of conventional microstrip line and the non offset DSPSL when . On the other hand, the the offset distance increases to of the offset DSPSL drops from 168 to 25 when the increases from 6 to 14, comparing with the of the ratio conventional microstrip line just decreases from about 29 to of the offset DSPSL could be changed in a large 14 . The region. Moreover, the offset DSPSL broadens the bandwidth of the microwave circuit; due to the DSPSL is a balance structure. of the offset The simulated normalized phase constant DSPSL is illustrated in Fig. 4. The normalized phase constant is calculated by the finite difference time domain (FDTD) method. The phase constant decreases when the offset distance

Fig. 1(a) and (b) show the 3-D view and top view layout of the proposed backward beam-scanning antenna. The side view of each uniform radiating period is shown in Fig. 2(c). In this periodic offset microstrip antenna, there are the top and bottom patches on the two sides of a substrate board. The proposed antenna consists of ten offset double-sided parallel-strip lines (DSPSL) periodically. In the top patch, it seems that the periodic radiating strips are on the alternate sides of a transmission line. As we know, the power could leak away as the space wave from the periodic waveguide structure, which has been applied as the antenna and been called as the periodic LWA. The periodic LWA has the advantage of the main lobe scanning in backward direction compared with the uniform LWA. The propagation characteristics of the periodic LWA are characterized by , where the complex propagation constant is the phase constant and is the attenuation constant. The normalized attenuation constant is a measurement of the power leaks away per unit length along the antenna. The determined the direction of the main lobe. phase constant of the The complex propagation constant wave in the periodic LWA is given by [9]

(6) where is the wave number of the fundamental Floquet wave in the uniform structure, is the period, is the order of the space harmonics. The space harmonics in the uni, and the foundational mode in the form structure is just uniform structure is a slow wave where in the frequency range of operation. The periodic construction creates a guide wave that consists of an infinite number of the space harmonics (Floquet waves). The periodic antenna working in either ) or the bound the leaky region (fast-wave region, ) determines the charregion (slow-wave region, acter of the single space harmonic. When the periodic LWA is . So the in the operating region, the space harmonics is

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normalized phase constant and the main lobe’s direction the periodic LWA are expressed as follows:

of

(7) (8) where is the angle of the maximum of the lobe (measured from direction) and is the free-space wave number. The angle of the main lobe is defined by the phase constant with varying of the operating frequency. In the proposed off-set microstrip antenna, ten offset DSPSL are applied as the uniform radiating periods of this antenna. The of each single offset DSPSL with the proper phase constant patch’s width and the offset distance has been calculated by the FDTD method in Fig. 4. The offset DSPSL increases the space between two parallel patches in each radiating period, and of the then affects the uniform normalized phase constant uniform propagation constant. With the appropriate structure of radiating period, the period and space harmonics , the periodic construction makes the slow wave radiate out along the . In the period structure, the radiating edge when surface wave propagates away from the feed, but the power leaks , which deterin the backward direction with mines the radiating angle of the space wave. As a result, the energy radiates in the backward scanning beam. According to the when an appropriresults shown in Fig. 4 and (7), ated period is designed and space harmonics . With the of the periodic offset increase of the operating frequency, microstrip antenna varies from negative to zero values, and then the main lobe of the periodic antenna scans from backfire direcis tion to the broadside direction. The attenuation constant the same for all space harmonics, and its value is the same of the attenuation constant of the fundamental Floquet harmonic. The top and bottom patches are parallel printed on two sides of a substrate board with the relative dielectric constant of , the dielectric loss tangent of and the thick. Two parallel patches are ness of long and wide. Two patches have 10 DSPSL periods. The dimensions of the offset DSPSL are shown as follow: , , and the top strip offsets from the bottom strip. The ratio of the offset distance to the patch’s width is . The ratio of the radiating period’s , the width to the space between two patches is ratio of the offset distance to the space between two patches is . From Fig. 3, the characteristics impedance of this offset DSPSL radiating period is almost 50 . So in the feeding is the width of the transmission line with circuit, . 50 , and the length of the feeding line is IV. EXPERIMENTAL RESULTS As shown in Fig. 1, the offset microstrip antenna consists of two patches on the different side of the substrate. The bottom patch looks like a mirror image of the opposite patch. The omnidirectional radiation patterns are achieved by this parallel-strip for structure. The width of the bottom patch is less than the good omnidirectional performance [4], [24]. The proposed

Fig. 5. The simulating 3-D radiation patterns by GHz. (a) 3.4 GHz (b) 5.2 GHz.

HFSS at 3.4 GHz and 5.2

yz

Fig. 6. The measured - plane radiation patterns at 3.4 GHz, 4.4 GHz and 5.2 GHz.

antenna radiates in two symmetrical beams in - plane and plane. The total scanning angle of this offset microstrip antenna is about 80 in each plane. From the simulating radiation patterns shown in Fig. 5, the radiation patterns change from backward end-fire radiation at 3.4 GHz to nearly omnidirectional radiation at 5.2 GHz. Fig. 6 plots the measured backward two-beam scanning radifor the proposed periodic ation patterns in - plane offset antenna at three frequencies. As previously said, the main lobe elevation steers from the backfire direction by increasing the frequency of operation in - plane. When the antenna works at . Notes that at 4.4 GHz 3.4 GHz, the main lobe directs at and 5.2 GHz the measured scanning angle for the upper beams are and , meanwhile the scanning angle for the and . At the same time, the lower beams are main lobe scans in azimuth from backward - plane to for the left beam, and for the right beam when the operating frequency changes from 3.4 GHz to 5.2 GHz as demonstrated in Fig. 7. The measurement results agree closely with the simulating results. Fig. 8 indicates that this periodic offset antenna has a wide .A impedance bandwidth by the measured -parameter

LI et al.: THE BACKFIRE-TO-BROADSIDE SYMMETRICAL BEAM-SCANNING PERIODIC OFFSET MICROSTRIP ANTENNA

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Fig. 9. The comparison of the measured and calculated attenuation constant  =k and the normalized phase constant =k of the backfire-to-broadside symmetrical beam-scanning periodic offset microstrip antenna.

Fig. 7. The measured x-z plane radiation patterns at 3.4 GHz, 4.4 GHz and 5.2 GHz.

per unit length along the microstrip patch [1], are also shown in Fig. 9. The measured normalized attenuation constant is [2] (10) is the measured half-power beamwidth of where the upper main lobe in - plane. According to Fig. 9, the beamwidth of main lobe is linearly related to . The attenuation constant drops to 0 when the phase constant is near to 0, which means it is difficult to obtain a main lobe in the broadside direction. V. CONCLUSION

Fig. 8. The S and the gain of the backfire-to-broadside symmetrical beamscanning periodic offset microstrip antenna.

wide impedance bandwidth over 3.36 GHz to 5.46 GHz, defined , is 2.1 GHz or about 47.6% referenced to by the the center frequency at 4.41 GHz. The comparison of the experimental and calculated normalized propagation constant is illustrated in Fig. 9. The measured is obtained by phase constant

A backfire-to-broadside symmetrical beam-scanning periodic offset microstrip antenna is presented in this paper. The proposed antenna consists of the offset double-sided parallel-strip lines periodically. The mirror of the opposite parallel less than patches makes the normalized phase constant zero, which results in the backward beam-scanning. The main lobe symmetrically steers from the backfire direction to the broadside direction with the varying of the operating frequency both in - plane and - plane. The total scanning angle of this offset microstrip antenna is about 80 in each plane. The wide impedance bandwidth of 47.6% for is obtained over 3.36 GHz to 5.46 GHz by using the offset double-sided parallel-strip structure. The configuration of this antenna is simple and low cost. This new type of antenna may be used in wireless application such as cruise control or traffic management.

(9) REFERENCES where is the measured angle of the maximum of the upper main lobe in - plane, elevating from end-firing direction ( direction). As shown in Fig. 6, the main lobe steers from backfire direction to broadside direction continually with the increase of operating frequency in - plane. The measured results are close to the calculated results obtained by (7) and FDTD method, with and space harmonics the period . The similar change of below 4 GHz could also be found as shown in Fig. 4. The measured normalized attenuation , which is a measure of the power leaked away constant

[1] A. A. Oliner, “Leaky waves: Basic properties and applications,” Proc. APMC’97, vol. 1, pp. 397–400, Dec. 1997. [2] A. A. Oliner and K. S. Lee, “Microstrip leaky wave strip antennas,” in Proc. Antennas and Propagation Society Int. Symp., 1986, pp. 443–446. [3] Y. X. Li, Q. Xue, E. K.-N. Yung, and Y. L. Long, “Dual-beam steering microstrip leakywave antenna with fixed operating frequency,” IEEE Trans. Antennas Propag., vol. 56, pp. 248–252, Jan. 2010. [4] J. R. James, P. S. Hall, and C. Wood, Microstrip Antenna Theory and Design. London: Peter Peregrinus, 1981. [5] P. H. Rao, M. R. Ranjith, and L. Naragani, “Offset fed broadband suspended plate antenna,” IEEE Trans. Antennas Propag., vol. 53, pp. 3839–3842, Nov. 2005.

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[6] S. G. Kim and K. Chang, “Ultrawide-band transitions and new microwave components using double-sided parallel-strip lines,” IEEE Trans. Microwave Theory Tech., vol. 52, pp. 2148–2152, Sep. 2004. [7] S. Sun and L. Zhu, “Stop band-enhanced and size-miniaturized low-pass filters using high-impedance property of offset finite-ground microstrip line,” IEEE Trans. Microwave Theory Tech., vol. 53, pp. 2844–2850, Sep. 2005. [8] J. X. Chen, J. L. Li, and Q. Xue, “Lowpass filter using offset double-sided parallel-strip lines,” IEE Electron. Lett., vol. 41, no. 24, pp. 1336–1337, Nov. 2005. [9] D. R. Jackson and A. A. Oliner, “Leaky-wave antennas,” in Modern Antenna Handbook, C. A. Balanis, Ed. Hoboken, NJ: Wiley, 2008. [10] J. D. Kraus, “A backward angle-fire array antenna,” IEEE Trans. Antennas Propag., vol. 12, pp. 48–50, Jan. 1964. [11] J. N. Hines and J. R. Upson, A Wide Aperture Tapered-depth Scanning Antenna Ohio State Univ. Research Foundation, Columbus, OH, 1957, Rep. 667-7. [12] R. C. Honey, “A flush-mounted leaky wave antenna with predictable patterns,” IRE Trans. Antennas Propag., vol. 7, no. 10, pp. 320–329, Oct. 1959. [13] Y. X. Li, Q. Xue, E. K.-N. Yung, and Y. L. Long, “The periodic halfwidth microstrip leaky-wave antenna with a backward to forward scanning capability,” IEEE Trans. Antennas Propag., vol. 58, pp. 963–966, Mar. 2010. [14] C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-wave and resonant antennas,” IEEE Antenna Propag. Mag., vol. 50, no. 5, pp. 25–39, Oct. 2008. [15] S. Paulotto, P. Baccarelli, F. Frezza, and D. R. Jackson, “Full-Wave modal dispersion analysis and broadside optimization for a class of microstrip CRLH leaky-wave antennas,” IEEE Trans. Microwave Theory Tech., vol. 56, pt. Part 1, pp. 2826–2837, Dec. 2008. [16] G. Lovat, P. Burghignoli, and D. R. Jackson, “Fundamental properties and optimization of broadside radiation from uniform leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 1442–1452, May 2006. [17] L. Liu, C. Caloz, and T. Itoh, “Dominant mode leaky-wave antenna with backfire-to-endfire scanning capability,” IEE Electron. Lett., vol. 38, no. 23, pp. 1414–1416, Nov. 2002. [18] S. J. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microwave Theory Tech., vol. 52, pp. 2678–2690, Dec. 2004. [19] D. F. Sievenpiper, “Forward and backward leaky wave radiation with large effective aperture from an electronically tunable textured surface,” IEEE Trans. Antennas Propag., vol. 53, pt. 1, pp. 236–247, Jan. 2005. [20] A. Grbic and G. V. Eleftheriades, “Periodic analysis of a 2-D negative refractive index transmission line structure,” IEEE Trans. Antennas Propag., vol. 51, pt. 1, pp. 2604–2611, Oct. 2003. [21] C. Caloz and T. Itoh, “Array factor approach of leaky-wave antennas and application to 1-D/2-D composite right/left-handed (CRLH) structures,” IEEE Microwave Wireless. Compon. Lett., vol. 14, no. 6, pp. 274–276, Jun. 2004. [22] A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. App. Phys., vol. 92, no. 10, pp. 5930–5935, Nov. 2002. [23] D. M. Pozar, Microwave Engineering, 3rd ed. Hoboken, NJ: Wiley, 2005. [24] R. Bancroft and B. Bateman, “An omnidirectional planar microstrip antenna,” IEEE Trans. Antennas Propag., vol. 52, pp. 3151–3154, Nov. 2004.

Yuanxin Li (M’08) was born in GuangZhou, China. He received the B.S. and Ph.D. degrees from the Sun Yat-sen University, China, in 2001, and 2006, respectively. From 2006 to 2008, he was a Senior Research Assistant with the Wireless Communications Research Centre, City University of Hong Kong, Hong Kong SAR. In 2008, he joined the Department of Electronics and Communication Engineering, Sun Yat-sen University, China, where he is currently a Lecturer. His research interests include antennas design and wireless communication applications.

Quan Xue (M’02–SM’04) was born in Xichang, Sichuan Province, China. He 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, China, in 1988, 1990, and 1993, respectively. In 1993, he joined 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, Kowloon, Hong Kong, where he is currently an Associate Professor and the Director of Applied Electromagnetics Laboratory. Since May 2004, he has been the Principal Technological Specialist of the State Integrated Circuit Design Base, Chengdu, Sichuan Province, China. He has authored or coauthored over 120 internationally refereed journal papers. His current research interests include antennas, smart antenna arrays, AIAs, power amplifier linearization, microwave filters, millimeter-wave components and subsystems, and microwave monolithic integrated circuits (MMICs) RF integrated circuits (RFICs). Dr. Xue was awarded Distinguished Academic Staff status for his contribution to the development of millimeter-wave components and subsystems. He co-supervised two IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) best student contest papers (third place 2003, and first place 2004). Edward Kai-Ning Yung (M’85–SM’85) was born in Hong Kong. He received the B.S., M.S., and Ph.D. degrees from the University of Mississippi, Columbia, in 1972, 1974, and 1977, respectively. He was briefly with the Electromagnetic Laboratory, University of Illinois at Urbana-Champaign. He returned to Hong Kong in 1978 and began his teaching career with Hong Kong Polytechnic. He joined the newly established City University of Hong Kong in 1984 and was instrumental in setting up a new department. He became a full Professor in 1989. In 1994, he received one of the first two personal chairs in the university. He is Founding Director of the Wireless Communications Research Center, formerly known as the Telecommunications Research Center. He remains active in research in microwave devices and antenna designs for wireless communications. He is the Principle Investigator of many projects. He is the author of more than 450 papers, including 270 in refereed journals. He is also active in applied research, consultancy, and other technology transfers. Prof. Yung is a Fellow of the Chinese Institution of Electronics, the Institute of Electrical Engineers, and the Hong Kong Institution of Engineers and a member of the Electromagnetics Academy. He has received many awards in applied research, including the Grand Prize in the Texas Instrument Design Championship and the Silver Medal in the Chinese International Invention Exposition. He is listed in Who’s Who in the World and Who’s Who in Science and Engineering in the World. Yunliang Long (M’01–SM’02) was born in Chongqing, China. He received the B.Sc., M.Eng., and Ph.D. degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 1983, 1989, and 1992, respectively. From 1992 to 1994, he was a Postdoctoral Research Fellow, then employed as an Associate Professor, with the Department of Electronics, Sun Yat-Sen University, Guangzhou, China. From 1998 to 1999, he was a Visiting Scholar in IHF, RWTH University of Aachen, Germany. From 2000 to 2001, he was a Research Fellow with the Department of Electronics Engineering, City University of Hong Kong, China. Currently, he is a full Professor and the Head of the Department of Electronics and Communication Engineering, Sun Yat-Sen University, China. He has authored and coauthored over 130 academic papers. His research interests include antennas and propagation theory, EM theory in inhomogeneous lossy medium, computational electromagnetics, and wireless communication applications. Prof. Long is a member of the Committee of Microwave Society of CIE, and on the editorial board of the Chinese Journal of Radio Science. He is Vice Chairman of Guangzhou Electronic Industrial Association. His name is listed in Who’s Who in the World.

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On Strongly Truncated Leaky-Wave Antennas Based on Periodically Loaded Transmission Lines Mario Schühler, Rainer Wansch, Member, IEEE, and Matthias A. Hein, Senior Member, IEEE

Abstract—Leaky-wave radiation of strongly truncated periodic structures is investigated. Starting with the analysis of an infinite structure with one-dimensional periodicity, the implications of a truncation on the far-field radiation characteristic are pointed out. In particular, the influence on the half-power beamwidth and the direction of maximum radiation is examined. The theoretical results are verified by the design of a transmission-line based leakywave antenna. The antenna consists of a microstrip line loaded periodically by inductive short-circuited stubs. The direction of maximum radiation and the half-power beamwidth are determined by the dispersion of the infinite structure and the total length of the antenna, respectively, both parameters representing degrees of freedom for the design. It is also shown how the total efficiency of strongly truncated antennas can be enhanced considerably. Index Terms—Leaky-wave antennas, microstrip antennas, periodic structures.

I. INTRODUCTION HE nature of leaky-wave radiation has been investigated extensively for several decades, see [1]–[5], and it is still of interest to today’s antenna engineers. Well-known examples exploiting leaky-wave radiation are antennas based on slotted waveguides [2], [6]. While a wave propagates along such a waveguide, it emits part of its energy through the slot(s) of specifically shaped apertures into the surrounding medium [6]. The direction of radiation is determined by the propagation constant inside the waveguide. Providing an appropriate excitation, leaky-wave radiation can also be observed at dielectric gratings [7] and dielectric image lines [8], for instance, or at parallel-plate waveguides consisting of a plate periodically structured on one side, often called “partially reflective surface” [9], [10]. Characteristic for the types of leaky-wave antennas mentioned is the frequency scanning capability, i.e., the direction of maximum radiation is frequency dependent. To analyze a leaky-wave structure in terms of its dispersion characteristic, an infinite structure is assumed normally. In practice, however, the structure has to be truncated. A typical leaky-wave antenna measures about 20 wavelengths in length, for instance [11]. For this case, the radiation characteristic of the finite structure differs slightly from that of an infinite one. Reducing the length to below 10 wavelengths, however, the

T

Manuscript received September 21, 2009; revised March 22, 2010; accepted April 18, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. M. Schühler and M. A. Hein are with the Institute for Information Technology, Ilmenau University of Technology, 98684 Ilmenau, Germany (e-mail: [email protected]). R. Wansch is with the RF and Microwave Department, Fraunhofer Institute for Integrated Circuits IIS, 91058 Erlangen, Germany. Digital Object Identifier 10.1109/TAP.2010.2071353

radiation characteristic is affected considerably. In the space domain, the truncation of the leaky structure can be considered a windowing of the field distribution along the infinite structure [12], [13], corresponding to a convolution in the spectral domain. Strongly truncated leaky-wave structures display beamwidths and directivities comparable to patch antennas or dipoles over ground. The remaining advantage of truncated leaky-wave antennas over such implementations is the possibility to align the main beam towards the desired direction, exploiting the dispersive behavior of the leaky structure. The effect of the truncation on the half-power beamwidth was already investigated [11], [14]. Under the assumption that the truncation has no influence on the direction of maximum radiation, which holds for sufficiently long structures, these approaches yield reliable results. In contrast, we illustrate the implications of a strong truncation on the direction of maximum radiation as well as the half-power beamwidth, setting both parameters in relation to the dispersion characteristic of the infinite structure and the length of the finite structure. We then determine how the dispersion of the infinite structure has to be engineered and how long the structure has to be in order to obtain the direction of maximum radiation and the half-power beamwidth desired. For the sake of simplicity and analytical description, we assume an ideal exponential field distribution along the structure, instead of analyzing the problem by a full-wave solution. To verify the theoretical results, a transmission-line based leaky-wave antenna has been designed, implemented and measured. The antenna consists of a microstrip line loaded periodically by short-circuited stubs that act inductively; it is a simplified implementation of the antenna proposed in [15], [16], which consists of balanced composite right-handed/left-handed transmission line sections. The implementation offers several advantages such as the simple analysis and fabrication as well as a low profile. Furthermore, since the line is fed in its fundamental mode, it does not require a special feeding structure [17] as would be necessary for leaky-wave radiation from higher-order modes on microstrip lines [18]. The drawback of the bare antenna implemented here is its low efficiency, resulting from the small leakage factor of the periodically loaded microstrip line and the strong truncation. Using an appropriate feeding network, however, considerably improves the efficiency, although at the expense of the impedance bandwidth. Strongly truncated leaky-wave antennas may therefore be interesting for several applications as long as bandwidth is not an issue. II. LEAKY-WAVE RADIATION FROM STRUCTURES WITH ONE-DIMENSIONAL PERIODICITY We first focus on the nature of leaky-wave radiation from structures offering a periodicity in a single direction, say . We

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assume that the field distribution along the periodic structure consists of a tangential component only, independent of , the transverse in-plane direction, and that can be represented by the . scalar function A. (Semi)-Infinite Structures of the wave equation in a periodic medium The solution with period is given by the sum over an infinite set of harmonic functions, called space harmonics, each of them representing a plane wave [19], [20]. The space harmonics differ from each other by their wavenumbers (1) is the complex wavenumber of the where fundamental harmonic. In the spectral domain two regions can be distinguished: a , and a fast-wave region, slow-wave region, where , with the wavenumber of free-space propwhere agation. For antennas as considered here, the slow-wave region is of minor interest, because this region includes those propagating space harmonics that are tightly bound to the periodic structure and do not radiate. Inside the fast-wave region, however, we find space harmonics that may couple to leaky waves. Due to the leakage of power, leaky waves are characterized by a complex wavenumber [4], [5], whose imaginary part is called “leakage factor”. Let us assume that the periodic structure supports only one and for leaky space harmonic with . Then, the only part of the scalar field distribution that contributes to the energy transfer into the far field is given by if otherwise

The far-field pattern given by (4) was derived assuming that only one space harmonic is leaky, as mentioned above. The remaining harmonics were expected to be bound to the structure, i.e., they are located outside the visible range. But, strictly speaking, this assumption holds only for an infinite structure. For the semi-infinite structure, the space harmonics in the invisible range can, indeed, contribute to the far field, because, in contrast to the infinite structure, the spectrum is a continuous function of . The spectra of each space harmonic therefore interfere with each other, which is pronounced particularly for . On the other hand, sufficiently large attenuation constants the impact of the space harmonics in the invisible range on the far-field characteristic is of minor interest for structures of periods much smaller than , as in the context of the present paper, because the space harmonics are separated far apart from each other, as implied by (1). Therefore, (4) describes the radiation characteristics accurately. B. Finite Structures—Radiation Pattern Let us now consider structures of finite size. Again, we assume that only one leaky space harmonic exists. In mathematical terms, by truncating the periodic structure, the field distribu, the window function [13]. tion is weighted by a function With if if otherwise

(6)

where represents the window width normalized to , the field along the structure that participates in the radiation process can be written as

(2) (7)

where we assumed that the structure is limited to the region , i.e., to a semi-infinite structure, excited at . In (2), represents the coordinate normalised to the free-space the amplitude of the leaky harmonic, and wavelength , . Applying the Fourier transform [21, p. 32] with respect to yields the spectral representation of [3]

assuming that the structure is excited at (cf. (2)). The of is given by the convolution of spectrum and the instrument function , which follows from [23]. For the spectrum the Fourier transform of we eventually arrive at (8)

(3) In the same manner as the complex wavenumber , the Fourier is also normalised to . In the spectral domain, variable we therefore find the visible region for or, if we switch to the angular domain [22], for . With (3), we obtain the (power) far-field pattern from (cf. [11]) (4)

Comparing (8) with (3), we find that the spectrum of the field distribtution of the truncated structure is equal to the spectrum for the semi-infinite structure, weighted by a complex factor.1 With (3) and (8), we obtain for the (power) far-field pattern (9) with

where the direction of maximum radiation, i.e., the peak angle, follows from [5, Eq. (29)] (5)

1(8) can also be interpreted as a sum of two terms, where one term is equal to the spectrum for the semi-infinite structure.

SCHÜHLER et al.: ON STRONGLY TRUNCATED LEAKY-WAVE ANTENNAS BASED ON PERIODICALLY LOADED TRANSMISSION LINES

Fig. 1. Far-field radiation characteristic (9) of truncated periodic structures of different sizes (parameters ~ and F (10)), supporting a single leaky space harmonic, with K = 1= 2 and K = 0:01.

p

Fig. 1 shows the directivity following from the far-field patand , where tern (9) for the directivity was calculated in accordance with [5, Eq. (45)]. , the far-field charFor sufficiently wide windows, e.g., acteristic resembles that of the infinite structure, because, according to the ratio [11]

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Fig. 2. Direction of maximum radiation ^ obtained for a truncated periodic structure in comparison with ^, the direction for an infinite structure. Parameter is the length of the structure ~. TABLE I DIRECTIVITY OF A TRUNCATED STRUCTURE AT  = 50 WITH AND WITHOUT COMPENSATION OF THE MAIN BEAM SHIFT

(10) a major part of the energy carried by the traveling wave has come off the structure at .—This behavior can be also con.—Reducing cluded from (9) by taking the limit of for , however, the radiation pattern displays additional side lobes, which are caused by the instrument function. Furthermore, due , the main beam is broadened, to the convolution with corresponding to a reduced directivity. This is particularly pro, where also a shift of the main beam towards nounced for can be observed. In practice, the window length is usu, i.e., at least 90 percent ally chosen in accordance with of the power carried by the leaky wave is radiated [11, p. 11-7]. In our example, this corresponds to a minimal window length of . According to (7), we notice that reflections at the boundary were not accounted for. Obviously, if reflections were present, a standing wave would appear as a superposition of two waves that travel into opposite directions, leading to a disturbed far-field pattern. A remedy of this problem is to terminate the structure with a matched load. As demonstrated in Section III, the matching condition can be accounted for in the design procedure, hence, an analysis of the implications of reflections is unneccesary at this point. C. Finite Structures—Beam Shift and 3-dB Beamwidth Let us now consider the actual direction of the main beam. For the (semi)-infinite periodic structure, the direction of maximum radiation is determined by (5). For the truncated structure, we have to look for the maximum of the far-field radiation pattern given by (9). As this problem cannot be solved in closed-form,2 we have investigated (9) numerically [24]. The results are disfor played in Fig. 2, where the actual main beam direction 2Even

an approximate solution is large and unpractical.

the truncated structure is plotted as a function of , the direction of maximum radiation for the (semi)-infinite structure (cf. [25]). , scales approxiFor sufficiently small , i.e., mately linearly with , where the slope depends on . In acfor cordance with the shift of the main beam towards narrow windows, the slope decreases for decreasing widths of , for instance, the slope is about 2/3, the window. For the slope is approximately unity, i.e., the while for main beam direction expected for the (semi)-infinite structure coincides with the actual main beam direction for the truncated structure. The shift of the main beam, particularly pronounced for strongly truncated structures, can be compensated for by modifying the dispersion characteristic of the (semi)-infinite is structure. For instance, if a main beam direction of , then, as follows from Fig. 2, the (semi)-indesired for at the finite structure has to be modified such that frequency of interest. The compensation, however, is possible , the slope of only for sufficiently small . For decreases for increasing , and eventually approaches a . For maximum value inside the visible region example, cannot exceed a value of 55.4 for (see Fig. 2). According to Table I, we notice that the need for a compensation of the beam shift depends on the directivity at a certain observation angle, since a large beam shift does not necessarily , for instance, the imply a large loss in directivity. For , the improvedirectivity can be improved by 1.1 dB. For ment would be 0.7 dB only, and for , a compensation would not show an improvement at all. Among the influence on the direction of the main beam, the truncation of the periodic structure leads to a broadened main beam, since the effective radiating aperture is reduced.

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Fig. 4. Model of an infinitesimal section of a differential transmission line loaded periodically by shunt inductances, assuming a period much smaller than  . The series resistance per unit length R represents the leakage.

1

Fig. 3. Half-power beamwidth  as a function of the expected main beam direction  given by (5) (solid lines). The curves were obtained numerically from the far-field pattern (9) evaluated for several widths of the window function  . The dashed lines are isolines, each of them indicates a certain constant actual main beam direction  .

^

^

~

The effect can be quantified by the half-power, or 3-dB, beamwidth . cannot be given in closed form. However, Unfortunately for periodic structures supporting only one leaky space harmonic and of periodicity much smaller than , the approximate solution for uniform leaky-wave antennas can be applied [17], which reads [11] (11)

Another approximation can be found in [14]. But both solutions were found under the assumption that the main beam is not shifted towards as a consequence of the truncation. Therefore, they are only applicable for sufficiently large struc, where . For smaller structures, the actual tures main beam direction has to be determined first. , we studied the power far-field pattern numerTo evaluate ically by computing (9) with respect to the co-elevation for the parameters and . The results are displayed in Fig. 3, where is plotted as a function of for and variable. , increases steadily, as expected from (11). For For also displays a monotonically smaller structures, however, decreasing region for , which can neither be predicted by (11) nor the approximate expression derived in [14]. (8) with the This effect is caused by the weighting of , required for the description of the far-field compoterm nents [21]. Caused by the weighting as well, approaches a constant value, as reaches 90 . In addition to the beamwidth, Fig. 3 illustrates the variation of the actual main beam direction with and by means of isolines. Thus, we can draw conclusions about both and

for varying values of and . But even the other way round for , Fig. 4 given desired main beam direction and beamwidth tells us how suitable values for and have to be engineered. and a beamwidth of For instance, if an antenna with is desired, the periodic structure has to be limited to a and, corresponding to , a phase length of about is required. Unfortunately, constant of as can be concluded from Fig. 2 and Fig. 3, we cannot always . For find appropriate values of and for each pair of and instance, it is impossible to manufacture a truncated leaky-wave , antenna with a 20 wide main beam pointing towards since the size of the aperture needs to be small enough in order to achieve the sufficiently broad beam but, at the same time, the aperture needs to be large enough in order to obtain maximum radiation into the desired direction. In this respect, Fig. 2 and Fig. 3 provide not only a simple way of designing an antenna, but also help to decide whether or not a particular characteristic can be achieved at all. III. LEAKY-WAVE ANTENNAS BASED ON PERIODICALLY LOADED TRANSMISSION LINES We now focus on the implementation of truncated leaky-wave antennas with one-dimensional periodicity based on periodically loaded transmission lines. Recalling the condition for or, in terms of the phase leaky-wave radiation, namely , where denotes the phase velocity of velocity, the leaky structure periodic in , and the free-space phase velocity. The phase velocity of a lossless transmission line is given by [26] (12) the series inductance and the shunt capacitance with per unit length . Since , an ordinary transmission line supports only slow waves. In order to satisfy the radiation has to be increased, either by reducing , condition, , or both. Means of reducing the reactive elements per unit length by periodic loading of the transmission line with series capacitances and shunt inductances were already proposed in several publications, e.g., in [15] and [16]. We restrict our considerations to transmission lines loaded periodically by shunt inductances. The analysis of the dual counterpart is straightforward. While the derivations hold for transmission lines in general, we should keep in mind that uncovered lines are essential in order to allow for energy transfer into the surrounding medium.

SCHÜHLER et al.: ON STRONGLY TRUNCATED LEAKY-WAVE ANTENNAS BASED ON PERIODICALLY LOADED TRANSMISSION LINES

For a period , the periodically loaded transmission line can be analyzed by transmission line theory [17]. An infinitesimal element of the line is depicted in Fig. 4. Beside the series inductance and the shunt capacitance, both per unit length, the , which model includes a shunt inductance denoted by represents the periodic loading, and a series resistance per unit , which represents the power leakage of the wave length [27], [28]. A. Lossless Line: The characteristic impedance is given by (13) with the series impedance per unit length (14) and the shunt admittance per unit length

(15) According to (15), the inductive loading can be modeled by an . The effective, frequency dependent shunt capacitance , which we can characteristic impedance of the loaded line , the impedance of the express in terms of unloaded line, becomes frequency dependent, too: (16)

would be chosen equal to the system On most occasions, for optimum power matching. reference impedance in (16) and in Essentially, the frequency response of (18) can be separated into two regions. For , is becomes inductive, leading to an imaginary negative, i.e., and wavenumber . The line, hence, impedance displays a stop-band, which corresponds to the first stop-band according to the dispersion relation (32) (see Appendix). For , and are real, since shows capacitive behavior. Consequently, the line offers a pass-band in this frequency region, which, in turn, is associated with the first passrepresents the lower band following from (32). Hence, cutoff frequency of the loaded line. However, whereas (32) provides a description of the entire band structure, the transmission line model here is only accurate at sufficiently low frequencies, . where There is another important difference between (32) and (18): While (32) inherently characterizes all space harmonics, function, (18) as implied by the ambiguity of the describes only the fundamental harmonic supported by the transmission line; higher-order harmonics are not considered as a consequence of the homogenization of the line. Yet, in terms of the radiation characteristic, the transmission-line approach is applicable, because the influence of higher-order harmonics , as on the far-field characteristics is negligible for mentioned in Section II. B. Lossy Line: To determine the characteristic impedance in (14) has to be modified as

for

,

(21) With (15) and (21), we obtain from (13)

Here, denotes the resonant frequency of the shunt admit. The propagation constant of the loaded transmission tance line can be determined from [26, p. 584, Eq. (5)] (17)

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(22)

For the wavenumber (17), we find

Using (14) and (15), we obtain (18) (23) in terms of the propagation constant of the unloaded line Solving for the shunt inductance yields

.

(19)

can be derived from the paramwhere eters of the unloaded line. The characteristic impedance of the by unloaded line can now be written as a function of rearranging (18) and (16) (20)

and are weighted Thus, in contrast to (16) and (18), both by a complex factor depending on the ratio . In terms of the design we have to realize that if we choose a in order to achieve a desired phase constant , particular will also be affected. Futhermore, the attenuation constant the loaded line possibly cannot be matched to arbitrary by . mere adjustment of Even though (22) and (23) describe the parameters of the , we concentrate on loaded line for the general case the special case , which turns out to be realistic, can as pointed out below. Then, the term

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be approximated by the first two terms of its Maclaurin series, leading to (24) and (25) We observe that , or even . This condition is essential in order to allow for a main beam to point away [29]. Otherwise, for , from broadside, i.e., the traveling wave decays rapidly, leading to a small effective . aperture with

Fig. 5. Sketch of the optimized unit cell a) and the dispersion characteristic obtained from simulation and calculation using (18) b). The dispersions for freespace propagation and for the unloaded microstrip are shown for information.

IV. DESIGN OF TRANSMISSION-LINE LEAKY-WAVE ANTENNAS—AN EXAMPLE In this section, we design a leaky-wave antenna based on a periodically loaded transmission line. For a frequency of , the antenna shall display a main beam with a 3-dB width of with its maximum directed towards . The antenna is based on a microstrip line, since it is, on the one hand, uncovered, essential for radiating applications, and, on the other hand, easy to fabricate. We used “RO4003C” as the substrate, 1.524 mm in height, with a relative permittivity of 3.55 and a dielectric loss factor of 0.0027 [30]. A. Unit Cell Design and Dispersion Characteristic Before the unit cell can be designed, the (normalized) phase constant and the overall length of the antenna, i.e., the width of the window , have to be determined. From Fig. 3 we find and , respectively. We chose a pe. The antenna therefore comprises 27 unit cells riod of and the approximate expressions (19) and (20) are applicable. . Replacing by , First, we have to evaluate denotes the effective relative permittivity of the unwhere loaded microstrip line, we obtain from (20) (26) and , as well as , are functions of , the width of the unloaded microstrip line. Both quantities are related to one another and, thus, cannot be determined independently. Consequently, we have to find a value for that satisfies (26), with known and set to 50 in our case. Accurate models and empirical expressions describing the impedance and the dispersion of microstrip lines were derived in [31], [32]. Since these expressions cannot be solved explicitly for , a solution must , be found numerically [24]. We eventually obtain and . resulting in We are now able to determine from (19). Replacing by and by , the shunt inductance follows from (27)

Fig. 6. Photograph of the fabricated antenna. The SMA jacks were mounted on the bottom side to minimize their influence on the measurement results. The microstrip is aligned with the x axis.

In our example, amounts approximately to 3.30 nH. A sketch of the unit cell is depicted in Fig. 5(a), including the dimensions optimized by simulations [33]. To obtain the desired phase constant, the width of the line had to be slightly modified as the simulation of the entire structure revealed. The shunt inductance is implemented as a short-circuited microstrip line of proper length, where the inductance caused by the via has to be accounted for [34]. The via diameter and the width of the short-circuited line were chosen to allow for a simple manufacturing process. The dispersion characteristic obtained from the simulation of the scattering parameters of the unit cell is shown in Fig. 5(b), along with the approximate dispersion (18). The curves are found to agree well with each other. As expected from Section III, the structure displays a high-pass characteristic, where the lower cutoff frequency is equal to the resonant frequency of (15). From simulation we achieved a phase constant and an attenuation constant , which, in accordance with (25), corresponds to a series resistance per . unit length of B. Scattering Parameters and Radiation Characteristics After designing the unit cell, the leaky-wave antenna was fabricated and its scattering parameters and radiation characteristics were measured. A photograph of the antenna is depicted in Fig. 6. The antenna is fed by 50 microstrip lines. The board measures approximately 267 mm 75 mm. The scattering parameters measured as a function of frequency are shown in Fig. 7. At the design frequency , the forward transmission is 1.0 dB. That is, in view of the small reflection coefficient of , most of the input power is transmitted to the output port and, hence, cannot contribute to the far field.

SCHÜHLER et al.: ON STRONGLY TRUNCATED LEAKY-WAVE ANTENNAS BASED ON PERIODICALLY LOADED TRANSMISSION LINES

Fig. 7. Frequency response of the magnitude of the input reflection coefficient s and the forward transmission coefficient s of the fabricated antenna. The diagram shows the parameters obtained after deembedding the antenna; influences of the SMA connectors were eliminated.

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Fig. 9. Concept of the antenna feed including the feedback network and the coupler a) and the corresponding matrix signal flow graph using a four-port coupler b). ~b is the vector of source waves.

the derivation of (9) in Section II. To illustrate the scanning capability of the antenna, the far-field patterns measured at 2.13, 2.23, 2.33, and 2.43 GHz are portrayed in Fig. 8(b). Due to the truncation, a peak angle of 55.0 cannot be exceeded, in accordance with expectation. The maximum directivity of the antenna at 2.33 GHz is 6.8 dBi. Notice that, in accordance with [5, Eq. (29)], the directivity was determined for the far-field cut measured over at for comparison with theory. Due to the low total antenna efficiency obtained from the scattering parameters measurement, the maximum gain was expected below 0.2 dBi. C. Efficiency Improvement With Feed Network

Fig. 8. Measured and calculated far-field patterns as a function of  for zero azimuth ( = 0 ). Only the upper half space is illustrated for convenience. The calculated patterns (dashed lines) in a) were obtained from (9) and (4), respectively. The patterns in b) were obtained from measurements. All measurements were performed for a distance of 5.2 m between the source antenna and the antenna under test.

Consequently, we achieved a rather small antenna efficiency of in total. Fig. 8(a) illustrates the far-field pattern (related to the total electric field) measured at 2.33 GHz (dashed line) in compar(dotted line); ison with the pattern obtained from (9) for the pattern of the semi-infinite structure given by (4) (solid line) is shown for information. Theory and measurement are found in good agreement, in particular around the main beam, whose and which exhibits a 3-dB width maximum points to of 28.5 . Because of the good agreement, we conclude that the tangential field components along the aperture are primarily responsible for the main lobe. In terms of the side lobes, however, normal field components, resulting from the -directed current flow along the vias, come into play, which were neglected for

To increase the antenna efficiency and, thus, the gain, a feed network was implemented. Since a great amount of the power fed to the antenna input reaches the output terminal, the idea is to feed the output signal back to the antenna input; Fig. 9(a) illustrates the concept. The network can be separated into two parts: the feedback network provides the actual signal feedback, while the coupler superimposes the input and the output signal appropriately. Both parts are assumed passive and reciprocal. To avoid a mismatch at the input port of the entire system, the input ports of the coupler have to be decoupled. This is also crucial in terms of the radiation characteristic of the antenna: Coupling of the input signal to the feedback path would lead to a leaky wave along the antenna that travels into the direction opposite to the desired one, causing a distortion of the far-field pattern. We therefore chose a rat-race coupler, which consists of two pairs of decoupled ports [35]. For the general case of an amplitude imbalanced coupling, the scattering matrix of the coupler, matched to a given reference impedance, reads

(28)

where , , , and are 2 2 block matrices . and According to (28), ports “1” and “4” as well as ports “2” and “3” are isolated. Suppose that, for instance, port “1” constitutes the system input, then port “3” has to be connected to the antenna input, while the feedback signal is fed to port “4”. The corresponding matrix signal flow graph is depicted in Fig. 9(b),

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Fig. 10. Photograph of the feed network mounted on the antenna’s back side. The feedback lines were spline-interpolated.

where denotes the scattering matrix that results from the cascade of the antenna and the feedback network. The scattering matrix of the entire system follows from (29) where denotes the 2 2 identity matrix. The optimum port , with the 2 2 efficiency of the system is achieved for zero matrix. The feed network therefore has to ensure that power fed to port “1” is neither reflected at port “1” nor transmitted to port “2”. Assuming that the cascade of antenna and feedback network is matched perfectly, we can write for its scattering matrix

Substituting into (29) and using (28), can be expressed in terms of two parameters, namely and , and with its main diagonal elements equal to zero. Since the matching condition is a priori satisfied, only the transmission coefficient magnitude has to be minimized, leading to the condition (30) Thus, is equal to the conjugate complex of the transmission of the coupler. coefficient We can summarize the principal design steps for the feed network as follows. by simulation or measurement. 1) Determination of 2) Determination of the required phase delay of the feedback , network according to the condition . 3) Design of the feedback network. 4) Determination of . from (30). 5) Calculation of 6) Design of the coupler and implementation of the whole feed network. Following these steps, we have designed a feed network tailored to the manufactured antenna; a photograph of the network is shown Fig. 10. The feedback network was implemented using microstrip lines, having an electrical length of 108.2 , in order to satisfy the phase condition in step 2). The characteristic impedances of the coupler branches, which follow from

Fig. 11. Measurement results for the antenna with feed network: forward transand input reflection coefficient s a) and the efficienmission coefficient s cies b) all as a function of frequency. The total efficiency of the bare antenna  is shown for comparison.

are and , providing a power split (4.0 dB). As the antenna, the network was ratio of fabricated on a 1.524 mm thick “RO4003C” substrate [30]. Fig. 11(a) shows the scattering parameters obtained from the measurement of the entire system, i.e., including the antenna and the feed network. At 2.33 GHz, the forward transmisis 14.8 dB and the input reflection is sion 19.1 dB, that is, the system ports can be considered decoupled and matched. Consequently, the antenna efficiency was increased considerably, as portrayed by the measurement data in Fig. 11(b). At 2.33 GHz, we obtained a total efficiency of and a radiation efficiency , where denotes the ratio between the of power radiated and the power accepted by the antenna. was determined by comparing the far-field measurement results with those achieved for an antenna of known gain [36, p. 869]. A radiation efficiency of at least 0.5 is achieved over a band. width of V. SUMMARY This paper focuses on the design of leaky-wave antennas based on periodically loaded transmission lines. Starting with the investigation of the radiation characteristics of one-dimensional periodic structures supporting leaky waves, we analyze the dispersion characteristic of a transmission line loaded by shunt inductances by approximate expressions. The theoretical findings are applied to an example design. The first part of the paper deals with the radiation from leaky structures periodic in one direction. The implications of the truncation of the periodic structure on its radiation characteristics are analyzed. The truncation can be considered an additional degree of freedom, allowing for the control of the beamwidth of the main lobe. The shift of the radiation maximum towards , also caused by the truncation, can be compensated for by appropriate adjustment of the dispersion characteristic of the periodic structure. The concept of loading transmission lines periodically provides a simple way of implementing leaky-wave antennas. Though we discussed only the loading by shunt inductances in detail in terms of its dispersion characteristic and its characteristic impedance, the analysis is also applicable to lines loaded either by series capacitances or both.

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Floquet’s theorem, for a symmetric two-port, the dispersion relation reads [19] (31) For the problem considered here, we eventually obtain (cf. [37]) (32)

Fig. 12. Unit cell of a differential transmission line loaded periodically by shunt inductances.

To verify the theoretical results in practice, a leaky-wave antenna was manufactured. The antenna is based on a microstrip line loaded periodically by short-circuited shunt stubs. The 3-dB beamwidth and the direction of maximum radiation define the total antenna length and the required phase constant of the (infinite) structure. With these parameters given, the characteristic impedance of the unloaded line and the shunt inductance can be determined. Indeed this permits a simple design process, but there is one problem: the low leakage factor. As shown in the example, most part of the power fed to the antenna passes right through it, leading to a rather low efficiency—the antenna is therefore inappropriate for the majority of practical applications. In order to improve the efficiency, we presented the design of a network that feeds the portion of power not radiated back towards the antenna input. At the center frequency, the efficiency can be enhanced considerably, but at the expense of bandwidth, making such antennas suitable for narrow-band applications. Related to the radiation characteristic of truncated leakywave structures, an important point, left out of the discussion here, is the level of the side lobes, which are caused by the side lobes of the instrument function. For instance, the antenna implemented displays a side lobe level of about 7 dB with respect to the main lobe level in the measurement (cf. Fig. 8)—a rather high value not acceptable for some applications. To reduce the side lobe level, the amplitude of the aperture distribution has to be tapered [7], [11], leading to a variation of the leakage factor with the position along the leaky structure. Mathematically, this can be expressed by modifying the definition of the window function in (6) accordingly—provided that remains constant along the structure, what may be difficult to achieve in practice. Despite the fact that the main beam becomes broader due to the reduced effective aperture, qualitatively, for a modified window the effects of the truncation on the radiation characteristic of leaky structures are similar to the results presented here for the uniform window. Quantitatively, however, the implications have to be studied yet. APPENDIX A. Dispersion of a Transmission Line Loaded Periodically by Inductances For the sake of completeness, in addition to the approximate analysis in Section III we discuss briefly the general case of arbitrary periods . The unit cell, which is now not necessarily infinitesimal, is illustrated in Fig. 12. It consists of a transmission line section of electrical length and characteristic impedance loaded by a shunt inductance at its center point. To determine the dispersion relation of the periodically parameters can be used. Applying loaded line, the

thus, the dispersion of the unloaded line, , is extended by an additional term. Under the assumption of a small leakage factor, all quantities on the right-hand side in (32) can be considered real. Hence, we can distinguish two cases if if

(33)

is either real or imaginary. The first case represents that is, the pass-bands, the second case the stop-bands of the loaded transmission line. ACKNOWLEDGMENT The authors would like to thank their colleagues C. Volmer, M. Huhn, M. Zocher, and H. Adel for providing scientific and technical assistance. REFERENCES [1] N. Marcuvitz, “On field representations in terms of leaky modes or eigenmodes,” IRE Trans. Antennas Propag., vol. 4, no. 3, pp. 192–194, 1956. [2] L. Goldstone and A. Oliner, “Leaky-wave antennas I: Rectangular waveguides,” IRE Trans. Antennas Propag., vol. 7, no. 4, pp. 307–319, 1959. [3] A. Hessel, “On the influence of complex poles on the radiation pattern of leaky-wave antennas,” IRE Trans. Antennas Propag., vol. 10, no. 5, pp. 646–647, 1962. [4] T. Tamir and A. Oliner, “Guided complex waves part 1: Fields at an interface,” Proc. Inst. Elect. Eng., vol. 110, no. 2, pp. 310–324, 1963. [5] T. Tamir and A. Oliner, “Guided complex waves part 2: Relation to radiation patterns,” Proc. Inst. Elect. Eng., vol. 110, no. 2, pp. 325–334, 1963. [6] J. Hines, V. Rumsey, and C. Walter, “Traveling-wave slot antennas,” Proc. IRE, vol. 41, no. 11, pp. 1624–1631, Nov. 1953. [7] F. K. Schwering and S.-T. Peng, “Design of dielectric grating antennas for millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 31, no. 2, pp. 199–209, Feb. 1983. [8] K. Solbach and B. Adelseck, “Dielectric image line leaky wave antenna for broadside radiation,” Electron. Lett., vol. 19, no. 16, pp. 640–641, Aug. 1983. [9] G. Trentini, “Partially reflecting sheet arrays,” IRE Trans. Antennas Propag., vol. 4, no. 4, pp. 666–671, Oct. 1956. [10] A. Feresidis and J. Vardaxoglou, “High gain planar antenna using optimised partially reflective surfaces,” Inst. Elect. Eng. Proc. Microw., Antennas Propag., vol. 148, no. 6, pp. 345–350, Dec. 2001. [11] A. A. Oliner and D. R. Jackson, “Leaky-wave antennas,” in Antenna Engineering Handbook, 4th ed. New York: McGraw-Hill, 2007, ch. 11. [12] P. García-Müller and A. Roederer, “A physical optics based plane wave spectrum approach to the analysis of finite planar antennas,” IEEE Trans. Antennas Propag., vol. 40, no. 8, pp. 906–911, Aug. 1992. [13] A. Sutinjo, M. Okoniewski, and R. Johnston, “Beam-splitting condition in a broadside symmetric leaky-wave antenna of finite length,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 609–612, 2008. [14] I. J. Bahl and K. C. Gupta, “A leaky-wave antenna using an artificial dielectric medium,” IEEE Trans. Antennas Propag., vol. 22, no. 1, pp. 119–122, Jan. 1974. [15] L. Liu, C. Caloz, and T. Itoh, “Dominant mode leaky-wave antenna with backfire-to-endfire scanning capability,” Electron. Lett., vol. 38, no. 23, pp. 1414–1416, 2002.

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[16] A. Lai, C. Caloz, and T. Itoh, “Composite right/left-handed transmission line metamaterials,” IEEE Microw. Mag., vol. 5, no. 3, pp. 34–50, 2004. [17] C. Caloz and T. Itoh, Electromagnetic Metamaterials. New York: Wiley, 2006. [18] A. Oliner and K. Lee, “The nature of the leakage from higher modes on microstrip line,” in Microwave Symp. Digest, MTT-S Int., Jun. 1986, vol. 86, no. 1, pp. 57–60. [19] L. Brillouin, Wave Propagation in Periodic Structures, 2nd ed. New York: Dover, 1953. [20] C. Elachi, “Waves in active and passive periodic structures: A review,” Proc. IEEE, vol. 64, no. 12, pp. 1666–1698, Dec. 1976. [21] S. Drabowitch, A. Papiernik, H. D. Griffiths, J. Encinas, and B. L. Smith, Modern Antennas, 2nd ed. Germany: Springer, 2005. [22] H. Booker and P. Clemmow, “The concept of an angular spectrum of plane waves, and its relation to that of polar diagram and aperture distribution,” Proc. Inst. Elect. Eng., vol. 97, pp. 11–17, 1950. [23] A. Papoulis, The Fourier Intregral and Its Applications. New York: McGraw-Hill, 1962. [24] Wolfram Mathematica, 6th ed. Wolfram Research, Inc., 100 Trade Center Drive. Champaign, IL, 61820. [25] M. Schühler, R. Wansch, and M. A. Hein, “Experimental study of the radiation characteristics of a finite periodic structure excited by a dipole,” in Proc. EuCAP’2009, Berlin, Germany, Mar. 23–27, 2009, pp. 3055–3059. [26] K. Simonyi, Theoretische Elektrotechnik, 9th ed. Berlin: VEB Deutscher Verlag der Wissenschaften, 1989. [27] P. Burghignoli, G. Lovat, and D. Jackson, “Analysis and optimization of leaky-wave radiation at broadside from a class of 1-D periodic structures,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2593–2604, Sep. 2006. [28] S. Paulotto, P. Baccarelli, F. Frezza, and D. Jackson, “Full-wave modal dispersion analysis and broadside optimization for a class of microstrip crlh leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2826–2837, Dec. 2008. [29] G. Lovat, P. Burghignoli, and D. Jackson, “Fundamental properties and optimization of broadside radiation from uniform leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1442–1452, 2006. [30] High Frequency Laminates Rogers Corporation. Rogers, CT, 06263. [31] M. Kirschning and R. Jansen, “Accurate model for effective dielectric constant of microstrip with validity up to millimetre-wave frequencies,” Electro. Lett., vol. 18, no. 6, pp. 272–273, 1982. [32] R. Jansen and M. Kirschning, “Arguments and an accurate mathematical model for the power current formulation of microstrip characteristic impedance,” Arch. für Elektronik und Übertragungstechnik, vol. 37, 1983. [33] ADS Momentum 2008 Agilent Technologies. Santa Clara, CA, 95051. [34] M. Goldfarb and R. Pucel, “Modeling via hole grounds in microstrip,” IEEE Microw. Guided Wave Lett., vol. 1, no. 6, pp. 135–137, June 1991. [35] J. Malherbe, Microwave Transmission Line Couplers. Norwood: Artech House, 1988. [36] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [37] A. Harvey, “Periodic and guiding structures at microwave frequencies,” IRE Trans. Microw. Tech., vol. 8, no. 1, pp. 30–61, Jan. 1960.

Mario Schühler received the Dipl.-Ing. degree in computational and electrical engineering from Ilmenau University of Technology, Ilmenau, Germany, in 2005, where he is currently working towards the Ph.D. degree. Before he joined the RF and Microwave Research Laboratory at Ilmenau University of Technology in 2006, he was with the RF and Microwave Department at Fraunhofer Institute for Integrated Circuits IIS, Erlangen, Germany. His research interests include development of RF and microwave circuits and antennas.

Rainer Wansch (M’03) received the Dipl.-Ing. degree in electrical engineering from the FriedrichAlexander-University of Erlangen-Nuremberg, in 1995 with a focus on RF and microwave design and digital signal processing. Since 1996, he has been with Fraunhofer IIS, working on antennas. He leads the antenna group inside the RF and Microwave Department since 2001 and is responsible for the design and measurement of antennas. As a member of the AMTA he published papers about the special measurement challenges of small antennas and of antennas for localization systems. He also published some papers on specific antenna design for satellite reception, integrated antennas, GNSS and RFID antennas.

Matthias A. Hein (M’06–SM’06) received the diploma and doctoral degrees (with honors) from the Bergische Universität Wuppertal, Germany, in 1987 and 1992, respectively. While at the Bergische Universität Wuppertal, he was involved in the development of superconductors for microwave applications, e.g., in mobile communications and satellite systems. In 1999, he received a British Senior Research Fellowship of the EPSRC at the University of Birmingham, Birmingham, U.K. From 1998 until 2001, he headed an interdisciplinary research group of applied physics and electrical engineering, focusing on the microwave engineering of passive microwave devices. Meanwhile, he has authored and coauthored about 310 publications or proceedings, provided more than 30 invited talks and several tutorials at international conferences. He has supervised about 55 Diploma and Doctoral students and acts as a referee for high-ranking scientific journals and international funding agencies. In 2002, he joined Ilmenau University of Technology as the Head of the RF and Microwave Research Laboratory, where he presently leads the Institute for Information Technology, comprising eight professors and about 100 scientific researchers. His current research interests concern device and system aspects of antennas, high-efficiency amplifiers, and sensors for various applications (more information under http://www.tu-ilmenau.de/init).

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Modulated Arm Width (MAW) Spiral: Theory, Modeling, Design and Measurements W. Neill Kefauver, Senior Member, IEEE, Thomas P. Cencich, Senior Member, IEEE, and Dejan S. Filipovic, Senior Member, IEEE

Abstract—The design and performance is discussed of equiangular four-arm modulated arm width (MAW) spiral antennas. Effects of expansion factor, modulation ratio, termination and feedpoint region on cross-polarization, pattern symmetry, gain, and impedance for both forward and reverse mode 1 operation are studied. The bandstop region is characterized using the transmission line theory and full-wave modeling. Three composite ideal beamforming measurements are developed and are shown to be in excellent agreement with theoretical predictions. Obtained results indicate that the expansion factor should be less than 1.5 and the modulation ratio needs to be higher than 4 to achieve high purity radiation patterns in both modes of operation. Beamformer design, effects of the dielectric loading, and non-ideal cavity-backing are not discussed. Index Terms—Log periodic antennas, multimode transmission lines, spectral domain analysis, spiral antennas.

I. INTRODUCTION

T

HE robust sensing of unknown signals usually requires the capability to detect orthogonal polarizations over wide bandwidths by an antenna sensor occupying a small area [1]. A very small number of antenna designs provide this capability, and to a degree they each have inherent design shortcomings to tolerate. This paper discusses a dual-polarized four-arm modulated arm width (MAW) spiral antenna invented by Paul Ingerson [2]–[5]. Although developed in the early 1970s with various modifications patented up to the current time [6]–[9], the MAW spiral is still not well understood and there is a void in open literature related to this unique antenna structure. This work attempts to fill this void by addressing the theory, modeling and design, the most critical aperture parameters, as well as the measurements of MAW spirals. The development of spirals in the 1950s showed that a planar, multi-octave wideband antenna capable to simultaneously respond to many different signals and thus consolidate many Manuscript received October 03, 2009; revised March 09, 2010; accepted May 11, 2010. Date of publication September 02, 2010; date of current version November 03, 2010. This work was sponsored in part by the Office of Naval Research under Grant #N00014-07-1-1161. W. N. Kefauver is with Lockheed Martin, Waterton, CO 80127 USA and also with the Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, CO 80309 USA (e-mail: w.neill.kefauver@lmco. com). T. P. Cencich is with Lockheed Martin, Waterton, CO 80127 USA (e-mail: [email protected]). D. S. Filipovic is with the Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, CO 80309 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.2010.2071357

sensors into a small aerodynamic design was achievable. Unfortunately conventional spirals do not have ability to detect all polarizations efficiently in a single antenna. While two spirals with reverse rotation give rise to a dual polarization capability, this solution still requires twice as much area. Additionally, the detection of different polarizations being spatially separated, usually by several wavelengths, can add a requirement for ambiguity resolution to a direction finding system. For moderate bandwidths, the orthogonal polarization can be realized by adding feedlines to the spiral outside and exciting the spiral inwards as the antenna has the opposite wrapping sense [10]. This approach requires separate beamformers and is limited in for a four arm spiral). To achieve frequency bandwidth ( independent-like performance, both polarizations must be fed from the antenna center. A common embodiment addressing this issue is a log-periodic sinuous antenna [11]. Spiral antennas can support the radiation of different modes with the total number of balanced, frequency independent modes being one less than the number of arms [12]. Thus, a two-arm spiral only controls one mode, referred to as mode 1, sum mode, or broadside mode. With the addition of more arms, control and excitation of the higher order modes are available through the use of mode-forming (beamformer) networks. These modes are polarized same as the sum mode; however, they are characterized by a deep (theoretically infinite) null at broadside and other features of lesser importance for the herein presented discussion [1], [5], [12]–[14]. In order for the higher modes to radiate efficiently, the antenna needs to have a large enough aperture. Typically the spiral must have a diameter of to radiate mode . For spirals too small to radiate all modes available based on the number of arms, an interesting phenomenon occurs. Specifically, the generation of reverse polarization in a lower mode results as the currents reflect from the arm terminations. This phenomenon led to the development of a MAW spiral with at least three arms to control both right ( ) and left ( ) hand polarized beams. Four arm designs are investigated here since their mode-forming networks can be easily implemented with 90 and 180 hybrids. The early work on MAW spirals introduced a basic geometry and a description of the modulated transmission line for band rejection [2], [3]. Pattern measurements of a MAW spiral, sinuous and planar log-periodics are given in [4] however; no information is provided on MAW spiral designs and how structural parameters affect its performance. Later patents introduce various modifications to the resonant sections including use of additional arms [6], rejection enhancement chokes [7], tapering the sections [8], and use of lowpass circuits to suppress higher

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order modes [9]. While the reasoning behind these modifications makes some sense, there is no published documentation that any of these actually improves the performance of a baseline MAW spiral. In this paper we discuss theory, modeling, design and measurements of MAW spirals. The most important geometry features are investigated in greater detail including modulation ), feedpoint region and ratio ( ), expansion factor ( arm termination. These structural parameters are then used to demonstrate how to improve the performance of a MAW spiral including impedance matching, gain, and beam purity characterized by axial ratio and omnidirectionality. Commercial tools based on method of moments (MoM) [15] and finite-element method (FEM) [16] are employed and some guidelines for their use in studying MAW spirals are provided. It is important to note that our focus is only on the most critical antenna aperture parameters, thus the beamformer design and its effects, dielectric loading, and non-ideal cavity backing are not considered and will be demonstrated in future works. Herein utilized antenna (cavity) backing for measurements is a multi-layer absorber panel separated from the antenna aperture by a 1.5” thick unloaded foam layer. This arrangement corresponds well to the simulated model of a free-standing MAW spiral antenna. The paper is organized as follows. • In Section II, the theory of operation is discussed. This includes radiating modes, geometrical parameters and their relationship to antenna layout, active region, termination effects, and application of transmission line theory to describe bandstop regions. • In Section III, the full-wave modeling peculiarities including the effect of mesh asymmetries and a method for mitigation thereof are demonstrated. • In Section IV, the design of the MAW spiral is discussed with care being taken to refine each part of the element – feed structure, active region, bandstop and final termination. Broadside gains, pattern symmetry, and wide-angle cross-polarization rejection for forward and reverse modes ( ) are evaluated. The terms forward and reverse are used in lieu of the relationship between the antenna polarization and its wrapping sense. Specifically, the forward mode is polarized corresponding to the MAW wrapping sense as with regular spirals. The reverse mode has the opposite polarization. It is shown that an expansion lower than 1.5 and modulation greater than 4 are needed to achieve high quality dual-polarized performance. • In Section V, three composite measurements are proposed and used to experimentally verify full-wave modeling and previously discussed conclusions. II. THEORYOF OPERATION A. Layout of the MAW Spiral Geometry Layouts of four different four-arm equiangular self-complementary MAW spirals are shown in Fig. 1. In the figure, the geometry is varied by expansion factor and modulation ratio often referred to as expansion and modulation, respectively.

Fig. 1. Layout of planar MAW spirals with expansion and modulation of: 3 and 2 (a), 3 and 8 (b), 1.25 and 2 (c), 1.25 and 8 (d).

The modulation ratio is defined as the ratio of arm width at the transition between the low and high impedance sections. The modulation ratio is constant throughout the spiral resulting in a self-complementary structure globally, even though local regions are not highly symmetric implying the impedance will not be constant. Expansion factor correlates to radial growth per full cycle. The growth rate, which is the exponential rate of increase in radians, is another term often used to describe growth of a spiral. In a fundamental equation describing the equiangular spiral (1), the growth rate and expansion factor are denoted as and , respectively (1) where is the radius of a point on the spiral, is the starting radius, and is the progressive growth angle in the spiral’s plane. The forward mode for the broadside (or -axis) pointing to the observer is right-hand circularly polarized. The reverse mode corresponds to the opposite wrapping sense, thus it is left-hand circularly polarized. B. Spiral Antenna Contribution to the MAW Spiral Spirals, depending on the phasing of their arms generate different radiation modes. The modes generated being the standard set of harmonic functions in [12]. If one cycle of phase progression is applied over -arms such that the phase on arm is (and the arms’ wrap is consistent with phase clocking), the pattern is a simple sum pattern, or mode 1. In other words, if a four-arm spiral is fed with 90 sequential phasing, then at the active region the phasing between neighboring arms will be zero since the expansion of the spiral will have lost quarter of a wavelength due to rotation at the circumference of one wavelength. The sum mode is the only spiral mode that can , denotes elevation angle) and have gain at broadside ( for many applications it is the only one desired from a spiral.. For a two-arm spiral a frequency independent method to get both

KEFAUVER et al.: MAW SPIRAL: THEORY, MODELING, DESIGN AND MEASUREMENTS

polarizations does not exist since the phasing is the same regardless of polarization and the wrap of the spiral determines the polarization sense. Only two modes are available and the in-phase feeding would result in a differential mode if it would even radiate (typically impedance is very high). ) is counter If the wrapping of the spiral with arms ( to the center-fed feed phasing progression with one cycle, the spiral will generate opposite polarization if the circumference , where is the mode 1 cutoff waveis smaller than length. If the spiral circumference is comparable to or greater , a higher-order co-polarized difference pattern than will radiate. This pattern will have cycles of phase rotationally in the far-field. More importantly, the active region for mode will be at a circumference consistent with the the radiated mode. In the case of a four-arm spiral, this is mode 3. Because of this property, if a low-pass circuit element is dering of radiation, being signed and placed before the the signal wavelength, the bandwidth of the dual polarization spiral can be extended almost indefinitely. Without a low-pass circuit, the switch between co-polarized sum and cross-polarized difference operation mode limits the useable dual polariza, where , tion bandwidth of the spiral to and is the speed of light. To make the spiral both frequency independent and polarization diverse the spiral must be modified to reflect before it can efficiently radiate the undesired difference mode. To properly achieve this result at least two drastically different quarter-wave impedance sections are cascaded to create significant reflection before the spiral’s difference active region for the reverse phasing condition. For a four-arm MAW spiral, these sections which is intentionally will be placed at a circumference of outside the mode 1 active region. The sum pattern will radiate at a circumference of and the difference pattern (consistent with reverse single cycle phasing) will be suppressed by the circumference. The feed struccurrents not reaching the ture (beamformer) must support the two symmetric impedance modes (both positive and negative for forward and reverse operations). The MAW spiral is typically fed with a coaxial cluster mated to a beamformer which provides proper phasing between arms. As with traditional spirals, the limiting bandwidth features are the feedpoint radius (upper end) and antenna circumference (lower end).

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Fig. 2. Reflection and transmission of bandstop section modeled at 2 circumference. T is transmission through the bandstop, 0 is reflection from the bandstop, m is the modulation of the spiral arm.

complementary structure including dielectric loading as computed below (3)

Here, is the mode of operation, is the effective dielectric constant of the coplanar self-complementary structure, is the self-complementary modal impedance deand rived for mode of an -arm complementary antenna and can be computed as [14] (4) denotes where, is the free space impedance (377 ), and free space. The absolute value of the denominator insures positive values ). For a 4-arm for the reverse operation modes ( , the rejection of occurs MAW spiral fed in ring. With two quarter-wave sections in the bandstop at the region, the reflection coefficient can be computed as in (5). Note and fed that the impedances for the rejected are the same (see (4)) (5)

C. Bandstop Design and MAW Spiral Parameters The effectiveness of the bandstop can be estimated by utilizing transmission line theory to determine its rejection. The phasing between adjacent arms at the circumference bandstop region is approximately 180 . Thus, the characteristic impedance of coplanar strip sections for arbitrary mode can be computed as [13] (2) where, is the non-complementary impedance for mode with an arbitrary ratio of metal to non-metal, is the quasi-TEM non-complementary impedance for the transis the quasi TEM self-commission line of the antenna, is the impedance for a self plementary impedance, and

where, are the characteristic impedances of the first (high or low impedance) and second (low or high impedance) complementary sections, and is the reflection coefficient of the bandstop. Unfortunately, increasing the modulation does not scale directly to impedance and to achieve a reflection of 0.97 or suprequires a 30:1 modulation ratio. pression of mode 3 to The reflection and transmission coefficient responses of these lines computed using FEM code Ansoft HFSS [16] are shown in Fig. 2. For usable expansion factors, only one circuit will be near the resonant length. If the expansion factor is extremely low then the spiral may have significant ohmic losses. However, the bandstop performance will improve since the rejection of the

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bandpass will be increased by the cascaded response of multiple bandstops (more bandstops in the reflection region with a lower expansion factor). The practical limit for slow expansion with high modulation allows no more than 2 bandstop sections per arm in the bandstop region. For a MAW spiral with an expansion factor of 1.5 or effectively 9 turns per arm for the antennas evaluated, the result is 1.5 bandstops. The expansion factor needs to be reduced to 1.25 to get two bandstops in the band rejection region. The 1.25 expansion factor combined with 8:1 modulation with a 10:1 bandwidth gives a requirement of 1000:1 aspect ratio between the smallest and largest details in etching. In practice, it is difficult to etch these fine structures (line widths and gaps). Note that the use of the coplanar transmission line model for the circuit is consistent with the operation of the antenna, where each arm is feed with a different but radially symmetric potential from the adjacent arms. Several papers address the case where the adjacent conductors have opposite polarity. In [17] the expected impedance of a coplanar stripline is generalized for planar -fold structures. This case is for straight transmission lines where the phasing between arms is constant on a support structure similar to the MAW spirals considered here. Validated finite element simulations [18] for the case of a four-arm spiral need to be modified by multiplying the impedance with to generate the nominal antenna impedance of mode . The stability of the solution over frequency is excellent. However, the difficulty in creating a good bandstop with one or two sections remains.

III. MODELING The modeling of a MAW spiral is challenging due to a combination of fine and coarse geometric details, and the necessity to operate over multiple octaves while containing many narrowband rejection regions. Use of dielectrics, cavity backing, and any practical balun implementation makes modeling even more difficult. The first objective is to ensure the mesh provides numerically stable and accurate results. To do this, several examples are run with the emphasis placed on the four fold symmetry of the mesh. The MAW spiral geometry ensures that the far-field must be rotationally symmetric when ideal phasing is applied. This is important because the theoretical broadside cross-polarization must be zero for a symmetric geometry and ideal beamformer. The nominal case, shown in Fig. 3, uses a mesh trianat the highest frequency. gular side length of less than However, the geometry discretization results in slightly different triangular elements on antenna arms. When the symmetric mesh is enforced by the rotation of a single, already meshed arm, the broadside cross-polarization drops to the value set by the here). Fig. 3 demonstrates computer’s accuracy (about the effect of mesh asymmetry associated with an implemented meshing algorithm in [15]. The use of the automated meshing tools does not enforce mesh symmetry even though it will impose finer mesh around the feed region. As seen from Fig. 3, the antenna needs to have the symmetry imposed by using the same mesh on all arms that is obtained by rotating a single meshed arm. This technique is

Fig. 3. Effects of mesh asymmetry on broadside gain for MAW spiral with the growth of 2 and modulation of 4 that is fed with wire probes. “Nominal” refers to the automatic mesh by FEKO, “Symmetric” refers to the enforced mesh symmetry.

easily done in the MOM solver since it does not refine the mesh during the solution process. Use of partial differential equation based solvers for accurate modeling of MAW spirals is more challenging as the implementation of 3-D mesh symmetries is not easy. Experiments conducted with FEM code [16] with continuous mesh refinement and increased order of expansion did not significantly improve the accuracy. In MOM solver, the MAW spiral was excited with different excitation techniques including edge, wire and coaxial feeds. There were no appreciable differences in impedance or far-field performance. In herein presented results, the wire port was used. The fabricated antenna was fed using a coaxial bundle [5] with the center conductor directly soldered to each arm. No attempts to use matching baluns were undertaken. IV. DESIGN The geometry of the equiangular self-complementary MAW spiral has three control parameters: expansion factor, modulation ratio, and arm count. Here, we investigate four-arm structures. The simulations are performed using the MoM code FEKO [15]. Increasing the modulation and decreasing the expansion factor improves the overall pattern symmetry and polarization purity. The drawbacks include increased impedance modulation and ohmic loss. In addition, the antenna performance is dependent on a design of four operational regions: balanced feedline, active region, bandstop section and termination. Typically, the feed lines should be optimized for accepting input to a 50 unbalanced coaxial line. However, more complex feed line geometries could be utilized such as the Dyson infinite balun [19] or a non-coaxial transmission line [20]. For mode 1 operation the ratio of metal to non-metal should approach 10:1 for the feed line region. This feed line section needs to be a significant portion of the longest wave) to improve the match across the length of operation ( operation band. Getting this feed line length inside a circumfor 10:1 bandwidth (modulation dimension ference of to eliminate mode 3 operation) requires a low expansion factor. In the active region, the expansion factor needs to be low to

KEFAUVER et al.: MAW SPIRAL: THEORY, MODELING, DESIGN AND MEASUREMENTS

Fig. 4. Simulated broadside gain for different expansion factors. MAW spiral is terminated by the low-impedance termination (45 low Z in Fig. 5), and its modulation is set at 8.

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Fig. 6. Simulated results for antenna input resistance (top) and reactance (bottom) variation over frequency. Modulation ( ) of 8 has a quarter turn matching section, modulations of 2 and 4 have full turn transmission line launches.

m

TABLE I COMPUTED AVERAGE BROADSIDE GAIN (DBIC) FOR DIFFERENT MODULATION AND EXPANSION FACTORS (NO OHMIC LOSSES INCLUDED). G IS GAIN IN FORWARD OPERATION AND G IS GAIN IN REVERSE OPERATION

TABLE II COMPUTED AVERAGE RELATIVE CROSS-POLARIZATION AT 30 . X IS CROSS-POLARIZATION IN FORWARD OPERATION AND X IS CROSS-POLARIZATION IN REVERSE OPERATION

Fig. 5. Simulated reverse operation broadside gain near cutoff – for different termination lengths. Z is the quasi-static line impedance for the partial section, as described in the Section II.C.

reduce the gain ripple. Fig. 4 shows the decrease from 1.5 dB to 0.75 dB in gain ripple as the expansion factor decreases. The bandstop region needs to have minimal reflection when it is part of the active region at lower operating frequencies. When used as the bandstop section, the rejection should be a perfect open or short to prevent further propagation of a traveling wave and undesired radiation of other modes. Even a good rejection leads to increased WoW (the ratio expressed in dB of the peak to minima variation in the co-polarized gain at a constant elevation angle for all azimuth angles at a specific frequency; WoW is abbreviation for “Wobble on Wave”) and a high cross-polarization at angles greater than 30 where the higher order modes can have significant power. WoW is an important parameter for direction finding systems and its reduction is critical for improving the accuracy thereof. The MAW spiral should not be terminated with a resistive load since the dual polarization operation is then compromised at near twice the cutoff frequency. Investigation of the length

of the terminating section indicates the reverse gain drop-out at about can be reduced if the antenna ends in a low-impedance section. This phenomenon is depicted in Fig. 5, where the last section was varied from 45 (an eighth rotation or at ) to a full low-impedance section. As seen, when the MAW spiral termination became a low impedance transmission line the large gain drop-out was significantly reduced. Reverse gain changes significantly with the terminating section due to the interaction between the bandstop and the open end of the MAW spiral. If the termination is high impedance then the phase between the two reflections quickly changes near from destructive to constructive and creates a discontinuity in gain. Once the reflection of the bandstop is shifted off resonance, the smoothness of the gain improves until the section becomes low impedance. At this point, the gain gradually degrades back to the high impedance resonance. The phasing between the bandstop region and the open at the end of the MAW spiral will always vary some because the difference in delay. Tuning for the removal of the resonance is the last step after choosing an expansion factor reasonable for achieving low loss, an accurate etch, and acceptable pattern performance. Note, the better the bandstop, the more the termination reflection is reduced.

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TABLE III COMPUTED AVERAGE RELATIVE CROSS-POLARIZATION AT 60 (NEGATIVE VALUES EXCEED NOMINAL CO-POL). X IS CROSS-POLARIZATION IN FORWARD OPERATION AND X IS CROSS-POLARIZATION IN REVERSE OPERATION

The selected MAW spiral designs are expected to have a reasonable match to 100 when using a matching transmission line section at the feedpoint as described in the previous section. Although the coax is 50 , the antenna impedance is based on the mode, whereas the coax impedance is the arm referenced to ground. Since the MAW spiral is self-complementary beyond the feedpoint, fairly constant real impedance should be obtained. This is shown in Fig. 6 for expansion factor 2.5 and different modulation ratios. The model predicts impedance mostly varying between 100 and 150 . Below twice the cutoff frequency, the reflection of the termination causes excessive ringing making this design an unlikely candidate to minia. The turize since it only has constant impedance above impedance ripple increases with higher modulation and the period decreases slowly relative to frequency. Other methods to improve the match to conventional coax through the feedpoint are possible but not investigated as part of this study. Reactance increases due to the reflection at the end of the transmission line. The effects of expansion factor and modulation ratio on dual-polarized performance are demonstrated next. These results are obtained by averaging computed to avoid variation in the terperformance parameters above mination effects for different geometries. Dielectric effects are not considered for design tradeoffs, however, the presented results are still valid for antennas built on thin low-permittivity substrates. ) The quality of dual circularly polarized operation (mode is first evaluated using broadside gain. Ideally, a MAW spiral will have the same gain in both polarizations. As Table I shows, the expansion factor must be fairly low for this limit to be approached, unless a high modulation is used. However, a high modulation increases mismatch loss. While the improvement in forward gain for smaller expansion factors is less than 0.5 dB (reaching a maximum of 5.5 dBic), the reverse operation gain or the expansion factor exceeds 5 dBic if the modulation is is less than 1.5 and modulation is greater than 4. The average values are obtained as arithmetic mean of a specific parameter , ) for all computed frequency points above . Data ( , are excluded to eliminate the effects of impoints below perfect termination and thus represent the antenna performance only over its frequency independent bandwidth. Polarization purity is investigated next. In a model, broadside cross-polarization is always suppressed unless imposing beamformer imbalances. Thus, the cross-polarization performance investigation is off-axis at 30 and 60 . The polarization purity degrades when off-axis angle increases and the reverse mode operation is always substantially worse than forward mode (see

Tables II and III). Note that the axial ratio and cross-polarization ( in dB) are related as

(6) Thus, the average axial ratio at 30 in the reverse mode of operation for a MAW spiral with expansion factor and modulation cross-pol). ratio of 1.5 and 4, respectively, is 2.2 dB ( At 60 , the axial ratio deteriorates to about 5.7 dB ( cross-pol). Higher modulation improves the cross-polarization under all conditions, with the reverse operation approaching forward for an expansion factor of 1.25. Note that this topology corresponds to 16 wraps and thus increased ohmic loss and tighter tolerances. In accomplishing this cross-polarization reduction, only slight performance degradation is predicted in forward operation. A slow expansion improves the cross-polarization by 5 dB over an expansion of 3. The recommendation is to use low expansion factors to achieve similar cross-polarization for both modes. The final parameter investigated is the pattern symmetry expressed as WoW. WoW arises from modes radiated by the currents flowing in the same direction as the desired mode. For a four-arm MAW spiral this is predominately mode 5. Figs. 7 and 8 show the average WoW at 30 and 60 for various expansions and modulations. As seen, WoW converges faster than the cross-polarization performance, but still requires a modulation of 4 or better to achieve wide-angle beam symmetry for both modes of operation. Presented results show that the expansion factor should be less than 1.5 and the modulation needs to be higher than 4 to achieve good pattern stability for dual-polarized operation. V. MEASUREMENTS A test article having two modifications of a traditional MAW spiral given in [2] is fabricated and shown in Fig. 9. The modifications include moving the modulation outside the active region at the highest desired operational frequency and increasing metal to slot ratio in the center to lower nominal antenna impedance. A 13 cm diameter antenna was printed on a 0.5 mm thick Rodgers 6002 dielectric with permeability of 2.94 and loss tangent of 0.0012 at 10 GHz. An expansion factor of 3 was used to allow an accurate etching of the 4:1 bandstop. The termination transmission line sections were matched to a full eighth rotation (45 in the inset of Fig. 5). A 2.18 mm (0.086”) coax feed cluster [5] was used to feed the antenna. The chosen geometry does not have the best performance among

KEFAUVER et al.: MAW SPIRAL: THEORY, MODELING, DESIGN AND MEASUREMENTS

M = +1

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M = 01

Fig. 7. Computed forward ( ) and reverse ( ) average WoW at 30 for different MAW spiral expansion factors and modulation ratios.

M = +1) and reverse (M = 01) = 30 ,  = 0 .

Fig. 10. Measured gain for the forward ( modes by the ideal beamformer methods at 

M = +1

M = 01

Fig. 8. Computed forward ( ) and reverse ( ) average WoW at 60 for different MAW spiral expansion factors and modulation ratios.

Fig. 9. Photograph of a fabricated MAW spiral with expansion of 3, modulation of 4 and non-complementary center region.

evaluated antennas, however, it enables radiation of undesired , , and 9 to a level that the far-field pattern modes would have significant contamination and modeling validation becomes more thorough. Measurements are processed within the limits of the phase matching of the coaxial cluster feed. Three methods are used to predict the composite pattern from a single arm measurement. The first is to measure each arm independently and combine

the results with an ideal beamformer. This approach is prone to fabrication uncertainty due to phase matching and connectorization. This uncertainty can be reduced by inspection of measurements since each arm should match the response of the other arms except for a physical clocking of the arm. One advantage of this processing method is that the modes are still available that a perfect beamformer would terminate to a load (so that if mode 1 is excited, the data will still have any pattern information from all modes). The second method uses a single arm measurement and post processes data by rotating a particular arm and synthesizing the overall response by summing all properly clocked arm rotations. This processing approach results in an fold symmetric total data set. This method reduces fabrication errors since all arms are assumed identical. Any asymmetry or misalignment of the antenna will however increase pattern errors from prediction. Determining the misalignment is more difficult from measurement because often the components of the patterns due to multiple mode operation cannot be isolated from alignment. Alignment errors also cause increase in adjacent modes due to phasing modulation of the alignment error. The last approach uses measured single arm data to calculate the mode spectrum by Fourier transform. While this method easily determines the contamination arising from undesired modes, it filters the components that cause pattern WoW and axial ratio. Special care must be exercised to pass the modes that the ideal beamformer would also pass. When done correctly all methods should provide similar results for all modes of operation if the antenna is well built and aligned. Measured results for all three methods are shown in Fig. 10. , (rotate), and , for They are denoted as the first, second, and third method, respectively. Off-axis gain was used since the rotation of arms and Fourier at . As seen, there is a transform become equivalent at strong consistency between different measurement processing methods which implies the errors in antenna fabrication and range alignment are small. In addition, the modal contamina) since only a slight tion is small for the selected angle (

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M = +1

Fig. 11. Dielectric loading effects on WoW at 60 using measurements and simulations denoted in superscript as “ ” and “ ,” respectively. Subscripts denote: “ ” for all arms excited, “1” for one arm excited, “ ” for free-standing MAW computed using free-space Greens function, and “6002” for Rogers dielectric 6002 computed using MLGF.

n

m

n

M = +1

c

M = 01

Fig. 13. Forward ( ) and reverse ( ) WoW at 60 obtained using modeling (MLGF), and single arm measurements.

fs

M = 01

Fig. 12. Forward ( ) and reverse ( ) WoW at 30 obtained using modeling (MLGF), and single arm measurements.

performance change was observed between the first and the third method. Fig. 11 shows that the WoW is affected as the effective elecwhich octrical thickness of the dielectric increases past . Below this frequency, curs for this specific article at the difference between the method of moments computations using multi-level (MLGF) and free-space (MOM) Green functions are relatively small relative to other unknowns (proving our premise of design tradeoffs not requiring dielectric). This result implies the parametric evaluations done on the free space design apply if the dielectric is kept thin and low-loss. Note that the adverse effects of dielectric loading require the use of thin, low permittivity substrates.The model prediction is consistent with the measured results as shown in Fig. 11 where the pattern of only one measured arm is used. The inclusion of additional arms in the final pattern prediction increases the overall errors due to both fabrication and range measurement. As shown in Fig. 12, the full-wave model predicts negligible while the measureWoW for the forward operation below ments show a highly varying WoW with the average value about 0.25 dB higher than simulations. It is likely that the chamber multipath is the cause for this discrepancy, but otherwise the

Fig. 14. Cross-polarization for reverse mode at 30 and 60 obtained using modeling (MLGF) and single arm measurements.

correlation is excellent. Results are similar at larger angles as shown in Fig. 13. is expected due The degradation in WoW above to the modification in the feed region. Therefore, without a full bandstop, mode 3 is generated by the same beamformer as mode . Thus mode 3 can radiate providing a large WoW at 60 while causing a much smaller increase at 30 since mode 3 rolls off at least 5 dB at 60 . Note peaks at 45 while mode are rejected by the beamformer. that modes The cross-polarization predictions in Fig. 14 also track the measurements well. Obviously, an optimized design would reduce the observed high axial ratio by using higher modulation and slower expansion. The degradation in performance of is due to the gain, WoW and cross-polarization at the the MAW spiral termination. The antenna has an imperfect reflection when it cannot radiate efficiently and the reflection causes the observed high cross-polarization. With lower expansion factors the reflection decreases due to improved radiation efficiency. However, a significant narrow band resonance can still occur if parameters are not carefully tuned. Similarly, as with log-periodic antennas, gain dropouts occur particularly where the termination (finite size) effects interfere with the active region. Computed and measured -parameters for a 50 reference are shown in Fig. 15. Due to the symmetry of antenna

KEFAUVER et al.: MAW SPIRAL: THEORY, MODELING, DESIGN AND MEASUREMENTS

Fig. 15. Computed (c superscript in legend) and measured (m superscript in legend) S-parameters using HFSS [16].

feed . Good agreement between measurements and models insure that the presented modal impedances are computed correctly. VI. CONCLUSION This paper investigated dual polarized operation of a MAW spiral as a function of expansion rate, modulation ratio, feed and bandstop regions. The MAW spirals under these conditions show good pattern purity for expansion factors less than 1.5 and modulation greater than 4. Other parameters including heavier dielectric loading, beamformer design and its imperfections, and cavity-backing have not been considered. Three different ideal beamformer composite measurements are devised for studying the antenna element. Good agreement with full-wave predictions validates the presented results and conclusions.

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[12] R. G. Corzine and J. A. Mosko, Four Arm Spiral Antennas. Norwood, MA: Artech House, 1990. [13] J. A. Huffman and T. P. Cencich, “Modal impedances of planar, noncomplementary, N-fold symmetric antenna structures,” IEEE Antennas Propag. Mag., vol. 47, no. 1, pp. 110–116, Feb. 2005. [14] G. A. Deschamps, “Impedance properties of complementary multiterminal planar structures,” IRE Trans. Antennas Propag., vol. AP-7, pp. 371–378, Dec. 1959. [15] FEKO User’s Manual Suite 5.4 EM Software & Systems-S.A. (Pty) Ltd., 2008. [16] High Frequency Structure Simulation (HFSS)11 ed. Pittsburgh, PA, Ansoft LLC., 2008. [17] E. Yamashita and K. Atsuki, “Analysis of microstrip-like transmission lines for non-uniform discretization of integral equation,” IEEE Trans. Microw. Theory Tech., vol. MTT-24, pp. 195–200, 1976. [18] J. S. McLean and T. Itoh, “Analysis of a new configuration of coplanar stripline,” IEEE Trans. Microw. Theory Tech., vol. MTT 40, pp. 772–774, 1992. [19] J. D. Dyson, “The equiangular spiral antenna,” IRE Trans. Antennas Propag., vol. 7, no. 2, pp. 181–187, Apr. 1959. [20] T. P. Cencich, J. A. Huffman, and D. Walcher, “Simultaneous mode matching feedling,” U.S. Patent 6,549,175, Apr. 15, 2003.

W. Neill Kefauver (S’82–M’84–SM’08) received the B.S.E.E. and M.S.E.E. degrees from Virginia Polytechnic Institute and State, Blacksburg, in 1983 and 1984, respectively. He is currently working toward the Ph.D. degree at the University of Colorado, at Boulder. He works professionally in the area of antenna measurement and design for Lockheed Martin, Waterton, CO, as a Senior Staff Engineer. Over the past 25 years he has supported numerous products including the Magellan, MGS, MRO interplanetary space probes, and the Kepler Observatory. In addition, he has provided results to be input into the Atlas, Orion, and Targets programs and full characterizations of antennas for GEOSAT GFO, Dawn, and New Skies, as well as several other military programs. He is the co-inventor of three patents. Mr. Kefauver received the NASA Bravery Award in 1986.

ACKNOWLEDGMENT The authors would also like to acknowledge the internal support of L. Martin. REFERENCES [1] S. E. Lipsky, Microwave Passive Microwave Direction Finding. New York: SciTech Publishing, 2004. [2] P. G. Ingerson, “Modulated arm width (MAW) log-spiral antennas,” presented at the Antenna Applications Symp., Monticello, IL, 1970. [3] P. G. Ingerson, “Modulated arm width spiral antenna,” U. S. Patent 3,681,772, Aug. 1, 1972. [4] P. G. Ingerson, P. M. Ingerson, and D. C. Senior, “Comparison of sinuous and MAW spiral antennas for wideband, dual polarized multi-arm applications,” presented at the Antennas Application Symp., Monticello, IL, 1991. [5] D. S. Filipovic and T. P. Cencich Sr., “Frequency independent antennas,” in Antenna Engineering Handbook, 4th ed. New York: McGraw Hill, 2007, ch. 13. [6] C. Walter, “Multi-mode dual circularly polarized spiral antenna,” U.S. Patent 5,451,973, Sep. 19, 1995. [7] G. Andrews, “Broadband center fed spiral antenna,” U.S. Patent 4,243,993, Jan. 6, 1981. [8] G. Andrews, “Broadband spiral antenna with tapered armwidth modulation,” U.S. Patent 4,605,934, Aug. 12, 1986. [9] R. M. Honda, “Constant beamwidth spiral antenna,” U.S. Patent 4,725,848, Feb. 16, 1988. [10] S. Kuo, “Multiple polarization spiral antenna,” U.S. Patent 3,562,756, Feb. 9, 1971. [11] R. H. DuHamel, “Dual polarized sinuous antennas,” U.S. Patent 4658262, Apr. 14, 1987.

Thomas P. Cencich (M’99–SM’08) received the B.S.E.E. degree from the University of Colorado, Boulder, in 1983. In 1983, he joined Lockheed Martin (formerly, Martin Marietta) in Littleton, CO, and as a Senior Staff Engineer has been involved with antenna analysis, design and testing. His primary research interests are in broadband and multimode antennas, where he holds several patents.

Dejan S. Filipovic (S’97–M’02–SM’08) received the Dipl. Eng. degree in electrical engineering from the University of Niˇs, Serbia, in 1994, and the M.S.E.E. and Ph.D. degrees from The University of Michigan at Ann Arbor, in 1999 and 2002, respectively. He is currently an Associate Professor at the University of Colorado at Boulder. His research interests are antenna theory and design, modeling and design of passive millimeter-wave components and systems, as well as computational and applied electromagnetics. Mr. Filipovic was the recipient of the Nikola Tesla Award and Provost’s Faculty Achievement Award. He and his students were co-recipients of the Best Paper Award presented at the IEEE Antennas and Propagation Society (AP-S)/ URSI and Antenna Application Symposium conferences.

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Time Domain Characterization of Circularly Polarized Ultrawideband Array Adrian Eng-Choon Tan, Michael Yan-Wah Chia, Member, IEEE, and Karumudi Rambabu

Abstract—The transmitted signal by a circularly polarized ultrawideband (CP-UWB) array is a short pulse of circularly rotating E-field vector in time domain. A mathematical model is derived for a transmitted pulse radiated by a CP-UWB array antenna. To validate the model, an array of four sequentially rotated ridged horns is constructed, and the transmitted pulse is measured in time domain. The measurement is done with a linearly polarized antenna in far-field, for different rotational angles. The E-field vector transmitted by the CP-UWB array is constructed with the measured time-domain pulses. The measured pulse is compared with the derived pulse for validation. Both measured and derived pulses are compared for various parameters such as the CP-UWB array’s sense of polarization, axial ratio, gain and phase difference between orthogonal signals. Index Terms—Antenna array, circular polarization, time-domain electromagnetics, ultrawideband antenna.

I. INTRODUCTION IRCULARLY polarized signals have interesting applications in radars, especially in the ultrawideband (UWB) radars to enhance the performance. These signals are used in applications such as interferometers, mine detection [1], radar warning receiver antennas [2], phased array elements and polarimeters [3]. In our UWB human tracking radar development, we found that tracking a human target is challenging, due to the fact that the human body is a complex target [4] thus the reflected signal is highly time-variable. Furthermore, the target is a non fixed frame—shifts its position, waves its hands, walks, etc. The radar cross section (RCS) of the human target fluctuates significantly with time owing to constructive and destructive interferences of the scattered signals from multiple points of the human body (Swerling effect [4]). To our knowledge the RCS of a human body, for different poses, has not been measured in the 3–10 GHz band. On the other hand, measurements of the RCS of a car at 24.5–28.8 GHz (UWB Ka Band) show fluctuations of more than 20 dB for both vertical and horizontal polarizations [5]. These measurements, for the case of fixed frame targets, give an indication of the expected fluctuations in RCS of the non fixed frame targets such as the human body. By transmitting the signal in two orthogonal polarizations a circularly polarized ultrawideband (CP-UWB)

C

Manuscript received January 19, 2010; revised April 27, 2010; accepted May 11, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The authors are with the University of Alberta, Canada; and the Institute for Infocomm Research, Agency for Science, Technology and Research (A*STAR), Singapore (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.2010.2071369

radar promises improved performance in tracking complex targets (e.g., human) by reducing the RCS fluctuations of the target [4]. In the literature, various types of antennas have been used for transmitting and receiving circularly polarized UWB signals. In [6], it is reported that a coaxial cavity antenna can be used to receive circularly polarized signals. In [7], Archimedean and equiangular slot spiral antennas, which exhibit circular-polarized radiation, are designed for UWB communications. In [8], polarization effects on wave scattered by the targets are analyzed, so that land mines can be differentiated from environmental clutter, in ground penetrating radar. Recently, there has been a growing interest in circularly polarized UWB radars at Ka Band in the automotive industry for obstacle detection and tracking. Many wideband antenna designs that are capable of transmitting and receiving CP-UWB signals are reported in patents [9]–[13]. In [14], the radiated field of an infinitesimal dipole collocated with an infinitesimal loop are derived, and are shown to define circular polarization for UWB pulse transmission. One of the challenges in constructing the CP-UWB array is to generate UWB pulses that are coincident in time and are timeorthogonal, and feed the spatially orthogonal antennas without introducing relative time-delays. The existing method to overcome this difficulty is to delay the nominal radiated in the nominal polarization (0 deg. signal) by a quarter period of its centre frequency [15]–[17]. In [15], the above said method is used in chiral polarization technique (wherein the space orthogonal signals are only partially time-coincident) for UWB with two spatially orthogonal dipoles. In [16], [17] the transmitted waveforms for spatially orthogonal dipoles and resistively loaded dipoles are derived to evaluate the quality of the CP-UWB signal transmission. In our previous works [18], [19], elliptic and circular polarized UWB systems were investigated, in boresight direction, with an array of horns arranged in sequential rotation [20], [21] configuration. In this paper, we propose a mathematical model for the transmitted pulse radiated by a CP-UWB array. The significance of this model is it predicts the changes in the pulse shape accurately at boresight and off-boresight angles. The model can be applied to different UWB antennas [22] and arrays [23] in time-domain, if the aperture field distribution of the antenna is known. The mathematical derivations presented in this paper have been used to model the beamforming and angle estimation using Monopulse radar principles [24]. In Section II, the transmitted pulse radiated by a sequentially rotated UWB array is derived using time-domain transfer functions [24]–[26] of the array elements. To validate the model, a CP-UWB array is constructed, and the transmitted pulse is measured with a linearly polarized receiver at different polarization planes in

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, and ; is where and are the lengths and the speed of light in free space; in heights of the -th antenna aperture, as defined in Fig. 1. (1) is the delay term that depends on the position of the -th antenna in the array, and can be derived as (3) , and . is the position of the -th antenna in the -plane, as defined in Fig. 1. Let the -th antenna (Fig. 1) be excited with source pulse . We can express the transmitted pulse of the array as

where

N

Fig. 1. Diagram illustrates N antenna apertures ( = 3) of different lengths ( ), heights ( ) and polarization directions (^ ) are located at different lo= ( ), in the -plane. The antennas are radiating in the cations, + -axis, which is pointing out of the page.

 z

p

p ;p

xy

a

Section III. In Section IV, the transmitted E-field vector of the CP-UWB array is constructed from the measurements, and is compared with modeled E-field vector as verification for the proposed theory. The transmitted pulse is also characterized in terms of polarization sense, axial ratio, circular polarization gain, and phase difference between orthogonal signals. II. TRANSFER FUNCTIONOF THE CP-UWB ARRAY A. General Transfer Function of the UWB Array Refs. [28], [29] have suggested that the transfer function of an antenna is an important consideration in the UWB radio system. In [24], [26], [27], it has been shown that the time-domain transfer function of an antenna can be used to model the transmission and reception characteristics of the antenna. Based on this principle, we can derive the time-domain transfer function of an array of aperture antennas, as shown in Fig. 1, as

(1) where “ ” denotes convolution, describes the time-domain transmission characteristics of the array in a particular direction at far-field. With reference to the , and (pointing out of the page) axes in Fig. 1: and are the angles subtending from the and -axes respectively. is the time-domain transfer function in receiving mode, and is the polarization direction of the -th antenna. Time derivative is applied to because in transmitting mode the aperture antenna performs time derivative to the input signal [22], [25]. Assuming that the aperture field distribution of the -th an, and the principle of separation of variables tenna is can be applied to the aperture function, i.e., ; can be derived [24], [26], [27] as

(2)

(4) Equation (4) describes the transmitted pulse as an E-field and vector, and as a function of and . At boresight, will be reduced to Dirac’s delta functions. Hence, the radiated E-field (4) in the boresight will be the vector sum of input pulses [18], [19]. The significance of (4) is evident in off-boresight angles, where the transmitted pulse will be affected by the geometry of the array and the radiation characteristics of the array elements. The effect of the array and its radiating elements on the transmitted pulse, i.e., shape and width, can be estimated using (4). The transmitted pulse depends on the number of array elements and the size of the radiating aperture. Increased element aperture results in wider transmitted pulse at off-boresight angles. Transmitted pulse also depends on the inter-element spacing in the array. The pulse becomes wider for increased inter-element spacing at off-boresight angles. Another very important aspect of the transmitted pulse is the constructive and destructive addition of the pulses that are radiated from individual elements of the array in a particular direction. This will lead to changes in pulse shape. Large inter-element spacing may lead to multiple pulses at off boresight angles. Furthermore, (4) can be used as an array factor for the case of narrowband sinusoidal signals. This will be a validation for (4). In the case of narrowband source signal, the time-domain transfer function, , provides amplitude scaling and phase shifting to the narrowband signal. The time-derivative operation provides a time-orthogonal version of the signal (observed as 90 phase-shift in time-harmonic signals); while the array factor is the summation of the radiation components of all array elements. B. Time-Domain Transfer Function of CP-UWB Array To validate the proposed analysis technique, we consider the CP-UWB array that is constructed with four broadband horn antennas, as shown in Fig. 2. Fig. 3 shows the polarity of the electric field in the aperture of the array elements. The array consists of four similar horns arranged in a sequential rotation manner, with the centres of each horn located at the vertices of a square of length 2 s. The dimensions of the horn are and . The polarization directions, , of the horns are oriented as shown in

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Fig. 2. Circularly polarized ultrawideband (CP-UWB) antenna array constructed with four sequentially rotated broadband horn antennas.

Fig. 4. Circuit to generate orthogonal UWB source pulses (s , s , s and s ) in time-domain, to be fed to the CP-UWB array in Fig. 2.

and must be It should be noted that the source signals time-orthogonal and time coincident for (5) to yield circularly polarized radiated pulses. III. TRANSMITTED CP-UWB SIGNAL A. Feed Circuit for CP-UWB Array Fig. 3. Dimensions of the CP-UWB array in xy -plane that will be used to validate the time-domain transfer function.

Fig. 3. The values of the array parameters are: , and . To derive the transmitted pulse radiated from the array, we first compute using (2), then use (1) to compute the transfer function of the array. The transmitted waveform in free space can be computed using (4). To achieve circularly polarized rato (Fig. 2) to be fed diation, it requires that the antennas with time-orthogonal pulses, i.e., and are fed with positive and negative amplitude versions of the nominal radiated pulse and are fed with the in the nominal polarization, while positive and negative amplitude versions of the time-orthogonal pulse in the spatially orthogonal polarization. The shapes of the time orthogonal pulses have been shown in Section III of the paper. and . The aperture Hence, we can express field distribution of is the same as , while the aperture field distribution of is the same as , hence, ; . is opposite to , while the polarity Thirdly, the polarity of of is opposite to . Applying the above simplifications, the transmitted pulse of the CP-UWB array shown in Fig. 2 can be derived as

To feed the antennas to , orthogonal UWB source pulses and ) are generated using the circuit shown in ( , , Fig. 4. The circuit consists of a step generator which generates a periodic 60 ps step function, cascaded with two stages of impulse forming networks (IFNs). The IFNs perform time derivative on the input step, generating a Gaussian monocycle pulse at the output. The output of the last IFN is cascaded with a Butterworth filter (4th order) to band limit the signal to a bandwidth of 3–5 GHz. The filter output is split into 0 deg. and 180 deg. components with a broadband 180 deg. hybrid coupler. The coupler outputs are fed to two identical power splitters and 90 deg. phase shifters [30] that generate the desired orthogonal pulses , , and . These pulses are then connected to antennas , , and respectively, with coaxial wires of equal electrical lengths. Time-of-arrival of the pulses is measured at the to , to verify the time-delay differinput of the antennas ences of the pulses. The measured maximum delay between the pulses is less than 10 ps (2% of pulse width). Theoretical expressions for the source signals can be derived as (8) (9) (10)

(5) where

and

(11)

are expressed as (6) (7)

and pulse-width are parameters where the amplitude that model the Gaussian monocycle. The envelope response of and centre frequency are parameters that the filter

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Fig. 6. Measurement setup that is used to record the transmitted pulse by the CP-UWB array.

Fig. 5. Plots compare between the measured (line) and theoretical (dashes) s , s , s and s signals.

model the Butterworth filter. The Butterworth filter is designed (4 GHz) and as a 4th order filter with

) and diagonal planes— , . Let and be the theoretical transmitted pulse in the above-mentioned azimuth and diagonal planes respectively,

(15)

(12) where , , and (1.5 GHz). The filter is implemented by microstrip line shorted-stubs, and fabricated in Rogers 4003 substrate. Equations (8)–(12) are verified by measurement and shown in Fig. 5. Fig. 5 compares the derived , , and pulses with the measured pulses, which shows a good agreement.

(16) where (17)

B. Estimation of the Transmitted Pulse by the CP-UWB Array To derive the transmitted pulse from the CP-UWB array, we first need to determine the aperture field distribution of the broadband ridged horn elements in the array. It is known that the field intensity in the aperture of the horn increases at the region of the ridges [24]. Hence, the aperture field distribution of the horn can be approximated as a higher order cosine term [24], [32]. To determine the exact shape of the aperture field distribution, the radiation pattern is measured at various in both E and H planes. Use these angles within measurements iteratively to finalize the field distribution of the antenna aperture. Using the above said method, the aperture field distribution of the ridged horn has been estimated as (13) (14) To derive the theoretical transmitted pulse from CP-UWB array, first compute using (13) and (14). Using the above simplification, and can be represented in terms of and . Substitute , , and into (6) and (7). Equations (6) and (7) are the components of (5). The estimated transmitted pulse from the CP-UWB , array is verified in the principal (azimuth—

(18) (19)

IV. MEASUREMENTS, RESULTS AND DISCUSSIONS A. Measurement Setup The transmitted pulse by the CP-UWB array is measured in an anechoic chamber. The position and orientation of the prototype array antenna and the receiving horn are shown in Fig. 6. The receiving horn is linearly polarized and is placed in far-field, i.e., 5.0 m from the CP-UWB array. To measure the transmitted pulse, the rotational angle (polarization direction, Fig. 6) of the to 180 deg.. Furthermore, receiving horn is rotated from the CP-UWB array is rotated in the azimuth plane so that pulse at various angles can be recorded with a sampling oscilloscope at the output of the receiving horn. Three sets of measurements are made: First, the transmitted is measured at different rotapulse at boresight tional angles of the receiving horn. Second, the transmitted pulse ., } is in the azimuth plane { measured for vertical and horizontal polarizations. Third, the

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Fig. 7. Measured input pulse to antenna p (solid line) is compared with transmitted pulse at boresight direction (dashed line) when the receiver polarization ^ direction (i.e., rotational angle, or tilt = 0 deg:). is at a

Fig. 9. Measured (dotted line) and theoretical (solid line) received signal at boresight direction with different rotational angles (tilt).

Fig. 8. Measured (dotted line) and theoretical (solid line) normalized E-field vector transmitted by the CP-UWB array at boresight.

Fig. 10. Measured axial ratio for 0 deg. (solid line) and 45 deg. (dashed line) rotational angles and measured phase difference between 0/90 deg. signals (solid line) and 45/135 deg. signals (dashed line) for frequency 2–6 GHz.

transmitted pulse in the diagonal plane { , } is measured for vertical and horizontal polarizations. The measured pulses are compared with the estimated pulses to validate the proposed theory. Fig. 7 compares the input pulse to antenna (solid line) with the transmitted pulse at boresight when the receiver horn is oriented along direction (dashed line). This comparison clearly shows that the transmitted pulse (dashed line) is the time derivative of the source pulse (solid line), and the effect of the group delay. However, effect of the group delay is not significant. B. Boresight: Fig. 8 compares the measured (dotted line) and theoretical (solid line) normalized E-field vector transmitted by the CP-UWB array. Fig. 8 verifies the accuracy of the proposed model, and shows that the transmitted pulse is right-hand circularly polarized. The slight differences between measurement and theory are attributed to two factors: First, non-ideal input pulse generated by non-ideal circuit, which is evident in Fig. 5; Second, the nonlinear gain response of the ridged horn in the operating frequency band. Fig. 9 compares the measured (solid line) and theoretical (dotted line) received pulses at rotational angles of 0, 45, 90

and 135 deg. respectively. The axial ratio of the CP-UWB array is computed as the ratio of the Fourier transform of the received pulses in orthogonal directions, as shown in Fig. 10. The bandwidth of the measured axial ratio (solid line and dotted line) of the CP-UWB transmitter is 2.3–5.1 GHz , while the measured phase difference (solid line and dotted line) between orthogonal pulses are between 70 deg. to 110 deg. Ideally, it is expected to be 90 deg. Through this method, it is possible to measure the polarization sense, axial ratio and phase difference between orthogonal pulses, thus enabling unique determination of the polarization characteristics in the Poincaré sphere [33]. C. Azimuth Plane: Fig. 11 compares the measured (dotted line) and theoretical (solid line) normalized E-field vector transmitted by CP-UWB , , 30, 45 and 60 deg.}. The comarray at { parison shows that, except for slight deviations, the measured E-field vector corresponds to the estimated pulse. The slight deviations are attributed to the factors explained in Section III-B. The spectrum of the radiated pulse in nominal and orthog, onal planes are plotted in Fig. 12 for various angles ( 15, 20 and 30 deg.). Comparison between the nominal

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Fig. 13. Measured (solid line)/theoretical (dashed line) gain patterns, and axial ratio (solid line) of the CP-UWB transmitter at centre frequency (4 GHz) for the azimuth plane { :,  :; : }.

= 0 deg

= [090 deg 90 deg ]

Fig. 11. Measured (dotted line) and theoretical (solid line) normalized E-field vector transmitted by the CP-UWB array at different angles in the azimuth plane :,  , 30, 45 and 60 deg.}. {

= 0 deg

= 10

Fig. 14. Measured (dotted line) and theoretical (solid line) normalized E-field vector transmitted by the CP-UWB array at different angles in the diagonal plane , 30, 45 and 60 deg.}. :,  {

= 45 deg

Fig. 12. Measured radiated spectrum at the nominal (solid line) and orthogonal (dashed line) rotation planes are compared with theoretical radiated spectrum (dotted line) at various angles in the azimuth plane ( , 15, 20, 30 deg.).

=0

plane spectrum (solid line) and the orthogonal plane spectrum (dashed line) shows good circular polarization characteristics. For illustration, theoretical pulse spectrum is also plotted in Fig. 12 as dotted line, showing the spectral nulls at various frequencies due to the destructive additions of the pulses at different azimuth angles. Fig. 13 compares the measured (solid line) and theoretical (dotted line) gain, in normalized dBic, in the azimuth plane. Both the measured and theoretical gains show that the 3 dB beamwidth of the transmitter is 16 deg. Fig. 13 also plots the measured axial ratio (solid line) of the transmitter in the azimuth plane. Measurements have shown that the CP-UWB array mainwithin in the azimuth tained an axial ratio of plane.

= 10

D. Diagonal Plane: Fig. 14 compares the measured (dotted line) and theoretical (solid line) normalized E-field vector transmitted by CP-UWB , , 30, 45 and 60 deg.}. Fig. 15 array at { compares the measured (solid line) and theoretical (dotted line) gain, in normalized dBic, in the diagonal plane. Both the measured and theoretical gains show that the 3 dB beamwidth of the transmitter is 22 deg. Fig. 15 also plots the measured axial ratio (solid line) of the transmitter in the azimuth plane. Measurements have shown that the CP-UWB array maintained an within in the diagonal plane. axial ratio of

V. CONCLUSION In this paper, we have presented a mathematical model for the transmitted pulse radiated by a CP-UWB array. The proposed modeling has been validated by a prototype, sequentially

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Fig. 15. Measured (solid line)/theoretical (dashed line) gain patterns, and axial ratio (solid line) of the CP-UWB transmitter at centre frequency (4 GHz) for the :,  :; : }. diagonal plane {

= 45 deg

= [090 deg 90 deg ]

rotated, CP-UWB array. The normalized E-field vector transmitted by the CP-UWB array is discussed. The polarization sense, axial ratio, time orthogonality between the input signals, and gain of the transmitted pulse are also presented. Good agreement is shown between the theory and measurements. REFERENCES [1] S. R. Cloude and C. Thornhill, “Polarimetric radar interferometry: A new sensor for vehicle based mine detection,” in UWB, Short-pulse Electro-magnetics 5. New York: Kluwer Academic, 2000, pp. 519–526. [2] T. R. Holzheimer, “An implementation of a 0.5 to 2.0 GHz circular 360 degree direction finding antenna,” in Proc. Antenna Applications Symp., Sep. 1999, pp. 374–404. [3] S. V. Nesteruk and M. B. Protsenko, “Antenna device for analysis of space-time vector signals,” UWB Ultrashort Impulse Signals, pp. 361–362, Sep. 2006. [4] M. I. Skolnik, Introduction to Radar Systems, 3rd ed. New York: McGrawHill, 2001, ch. 4. [5] H. Osaki, T. Nishide, and T. Kobayashi, “Measurement of ultra wide-band radar cross sections of an automobile at Ka band using circular polarization,” IEICE Trans. Fundamentals, vol. E91, no. 11, pp. 3190–3196, Nov. 2008. [6] T. Holzheimer, “The low dispersion coaxial cavity as an ultra wideband antenna,” Proc. IEEE UWBST, pp. 333–336. [7] Y. Zhang and A. K. Brown, “Archimedean and equiangular slot spiral antennas for UWB communications,” in Proc. 36th Eur. Microwave Conf., UK, Sep. 2006, pp. 1578–1581. [8] W. M. Boerner and S. R. Cloude, “An introduction to polarisation effects in wave scattering and their applications in target classification,” in UWB, Short-pulse Electro-magnetics 5. New York: Kluwer Academic, 2000, pp. 493–500. [9] M. C. Wicks and P. V. Etten, “Orthogonally polarized quadraphase electromagnetic radiator,” U.S. Patent 5068671, 1991. [10] J. T. Apostolos, “Dual polarization Vivaldi notch/meander line loaded antenna,” U.S. Patent 6824154, 2005. [11] H. G. Schantz, “Chiral polarization ultrawideband slot antenna,” U.S. Patent 7391383, 2008. [12] V. Shtrom, W. Kish, and B. Barron, “Antennas With Polarization Diversity,” U.S. Patent 7498996, 2009. [13] P. V. Etten and M. C. Wicks, “Polarization diverse phase dispersionless broadband antenna,” in US Statutory Invention Regis. H1877. Washington, DC: USA, 2000. [14] K. Siwiak and Y. Bahreini, Radiowave Propagation and Antennas for Personal Communications, 3rd ed. Norwood, MA: Artech House, 2007, pp. 16–20.

[15] M. A. Barnes and L. W. Fullerton, “Chiral and dual polarization techniques for an ultra-wideband communication system,” U.S. Patent 5764696, 1998. [16] E. L. Mokole, A. K. Choudhury, and S. N. Samaddar, “Transient radiation from thin, half-wave, orthogonal dipoles,” Radio Sci., vol. 33, no. 2, pp. 219–229, 1998. [17] A. K. Choudhury, “Polarization characteristics of resistively loaded orthogonal dipoles excited by ultrawideband signals,” presented at the IEEE UWBST, 2002. [18] M. Y.-W. Chia and A. E.-C. Tan, “Transmissions and polarizations from pulse based UWB antenna and array,” in Proc. EuCAP, Nov. 2007, pp. 1–5. [19] A. E.-C. Tan, M. Y.-W. Chia, and K.-W. Khoo, “Circularly polarized ultrawideband array and transmission,” presented at the APMC, Hong Kong, Dec. 16–20, 2008. [20] T. Teshirogi, M. Tanaka, and W. Chujo, “Wideband circularly polarized array antenna with sequential rotations and phase shift of elements,” in Proc. ISAP, Aug. 1985, pp. 117–120. [21] P. S. Hall, J. S. Dahele, and J. R. James, “Design principles of sequentially fed, wide bandwidth, circularly polarized microstrip antennas,” IEE Proc., vol. 136, Pt. H, no. 5, Oct. 1989. [22] K. Rambabu, A. E.-C. Tan, K. K.-M. Chan, and M. Y.-W. Chia, “Estimation of antenna effect on ultra-wideband pulse shape in transmission and reception,” IEEE Trans. EMC, vol. 51, no. 3, pp. 604–610, Aug. 2009. [23] A. E.-C. Tan, M. Y.-W. Chia, and K. Rambabu, “Design of ultra-wideband monopulse receiver,” IEEE Trans. MTT, vol. 54, no. 11, pp. 3821–3827, Nov. 2006. [24] A. E.-C. Tan and M. Y.-W. Chia, “Beamforming and monopulse technique on sequentially-fed circularly-polarized ultra-wideband radar array,” presented at the IMS, Boston, Jun. 7–12, 2009. [25] M. Kanda, “Time-domain sensors and radiators,” in Time-Domain Measurements in Electromagnetics, E. K. Miller, Ed. New York: Van Nostrand Reinhold, 1986, ch. 5. [26] H. D. Griffiths and A. L. Cullen, “Sidelobe response of antennas to short pulses part I: Theory,” in IEE Proc. Microwaves, Antennas and Propagation, Aug. 2002, vol. 149, pp. 189–193. [27] M. G. M. Hussain, “Principles of space-time array processing for ultrawide-band impulse radar and radio communications,” IEEE Trans. Veh. Tech., vol. 51, no. 3, pp. 393–403, May 2002. [28] E. G. Farr and C. E. Baum, “Extending the definitions of antenna gain and radiation pattern into time domain,” in Sensor and Simulation Notes, Note 350. Wright-Patterson AFB, OH: Air Force Research Lab., 1992. [29] X. Qing and Z. Chen, “Transfer function measurement for UWB antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 20–25, 2004, pp. 2532–2535. [30] L. Bian et al., “Wideband circularly-polarized patch antenna,” IEEE Trans. AP, vol. 54, no. 9, pp. 2682–2686, Sep. 2006. [31] K. L. Walton and V. C. Sundberg, “Broadband ridged horn design,” Microw. J., pp. 96–101, Mar. 1964. [32] K. Rambabu, A. E.-C. Tan, K. K.-M. Chan, M. Y.-W. Chia, and S.-W. Leong, “Study of antenna effect on UWB pulse shape in transmission and reception,” presented at the ISAP, Singapore, Nov. 2006. [33] IEEE Standard Test Procedure for Antennas, IEEE 149–1979, 1979.

Adrian Eng-Choon Tan received the B.Eng. and Ph.D. degrees from the National University of Singapore (NUS), Singapore, in 2002 and 2008, respectively. In 2008–2009, he worked at the Institute for Infocomm Research I R as a Research Engineer. Currently, he is a Postdoctoral Fellow in the University of Alberta, Canada. His research interests include microwave circuits, time-domain analysis and ultrawideband transceiver systems Mr. Tan was a recipient of the Agency for Science, Technology and Research (A-STAR) Graduate Scholarship.

(

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TAN et al.: TIME DOMAIN CHARACTERIZATION OF CIRCULARLY POLARIZED ULTRAWIDEBAND ARRAY

Michael Yan-Wah Chia (M’94) was born in Singapore. He received the B.Sc. (1st class honors) and Ph.D. degrees from Loughborough University, U.K. He joined the Center for Wireless Communications (CWC), Singapore, in 1994 as a Member of Technical Staff. He became the Communications Division Director in the Institute for Infocomm Research (I R) of Agency for Science, Technology and Research, (A*STAR) in 2003. In 2007, he was appointed as the Program Director of Power Aware Wireless Sensor Networks. Since April 2010, he is holding the appointment of Director of Industry for I R, leading a team of program managers at I R. He is also holding an adjunct position in the National University of Singapore. He has held appointments in several technical/advisory committees in industry and national government bodies, such as Infocomm Development Authority, IDA and SPRING. To date, he has contributed to more than 153 publications in international journals and conferences. He has at least 12 patents granted. He has also led several major wireless research programs in Singapore, particularly in the areas of RFID and UWB. Recently, he is leading a large research program on terahertz by the Science and Engineering Research Council (SERC), Singapore. He has secured many large projects funded by industry such as BOSCH, EADS, IBM, etc. His team has also been contributed to the IBM Business Partner Program for silicon design. His main research interests are UWB, terahertz, beam-steering, wireless broadband, RFID, antenna, transceiver, RFIC, amplifier linearization and communication and radar system architecture. Prof. Chia was a recipient of Overseas Research Student (ORS) Awards from U.K. Universities and Research Studentship Award from British Aerospace, U.K. He has been an active member of organizing committees in various international conferences and was the program co-chair of IEEE International Workshop of Antenna Technology (IWAT) 2005. He was also invited to be a keynote speaker in IEEE International Conference of UWB 2005. He has been the General chair of ICUWB 2007. He is also been a member of the Executive Committee of ICUWB since 2007. He is a member of the editorial board of the Transactions on Microwave Theory and Techniques.

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Karumudi Rambabu received the Ph.D. degree in electrical and computer engineering from the University of Victoria, Victoria, BC, Canada, in 2005. From July 2005 to January 2007, he was a Research Member at the Institute for Infocomm Research (I R), Singapore. Since February 2007, he has been an Assistant Professor with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. His current research interests include design and development of miniaturized microwave and millimeter-wave components and systems. He is also involved in ultrawideband radar systems for medical and security applications. He has authored or coauthored more than 50 journal and conference papers. He is an Associate Editor for the International Journal of Electronics and Communications. Dr. Rambabu received the Andy Farquharson Award for excellence in graduate student teaching from the University of Victoria in 2003 and the Governor General’s Gold Medal for Ph.D. research in 2005.

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A Post-Wall Waveguide Center-Feed Parallel Plate Slot Array Antenna in the Millimeter-Wave Band Koh Hashimoto, Student Member, IEEE, Jiro Hirokawa, Senior Member, IEEE, and Makoto Ando, Fellow, IEEE

Abstract—A center-feed structure using cross-junctions is designed for a post-wall waveguide-fed parallel plate slot array antenna. The length of the radiating waveguide in the center-feed antenna is half of that of a conventional end-feed antenna, and the center-feed antenna is expected to enhance bandwidth depending on the long line effect. The symmetrical structure of the antenna can keep the main beam at boresight as the frequency is changed. The cross-junction is analogous to that previously proposed for a single-mode waveguide array, but it excites plane TEM waves in both directions in the oversized parallel plate. The authors have designed and fabricated a slot array with uniform aperture distribution at 61.25 GHz. The bandwidth for 1-dB gain reduction is 1.7 times as wide as that of a conventional end-feed type antenna. The sidelobe level in the E-plane increases to 7 8 dB because of the slot-free area over the center-feed waveguide, which is suppressed to 11 1 dB by introducing a 8 3 dB amplitude tapered distribution in the slot design of the array. Index Terms—Millimeter-wave antennas, parallel plate waveguides, post-wall waveguides, slot antennas.

Fig. 1. Post-wall waveguide-fed parallel plate slot array.

I. INTRODUCTION ARALLEL plate slot array antennas [1]–[5] are attractive candidates for high efficiency, mass-produced planar array antennas in millimeter-wave applications because the waveguide has low transmission losses. The first implementation of the post-wall waveguide technique to a parallel plate slot array was reported in [4]. A post-wall waveguide, also called “substrate integrated waveguide (SIW)” [6], [7] or “laminated waveguide” [8], is realized by making holes and plating the walls in a dielectric substrate. The post-wall waveguide can be manufactured by conventional print circuit board (PCB) fabrication process at low cost. The efficiency characteristics depending on the antenna size were investigated in the 60- and 76-GHz bands. The efficiency remains at 55–60% in the 21–33 dBi range in the 60 GHz band, and about 40–50% efficiency is realized in the 76 GHz band. Post-wall waveguide-fed parallel plate slot arrays can achieve higher efficiencies than planar line-based antennas especially in the high-gain range. Fig. 1 shows a conventional end-feed type of the post-wall waveguide-fed parallel plate slot array. The feed waveguide is placed at the end of the parallel plate waveguide. The feed waveguide has a series of coupling windows, and the parallel plate

P

Manuscript received September 30, 2009; revised May 05, 2010; accepted May 05, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The authors are with the Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2071356

Fig. 2. Bandwidth of post-wall waveguide-fed parallel plate slot arrays as a function of antenna gain.

waveguide also has a series of slot pairs. The antenna is a kind of series-fed array antenna so that the frequency bandwidth becomes narrower due to the long line effect as the antenna size becomes larger. Fig. 2 shows the bandwidth for 1-dB gain reduction of the post-wall waveguide-fed parallel plate slot array as a function of the antenna gain. The end-feed type has a further disadvantage: the main beam direction is slanted, as shown in Fig. 1, when the frequency is shifted from the design frequency because the guided wavelength changes in the parallel plate waveguide. The center-feed structure using cross-junctions for the parallel plate slot array as shown in Fig. 3 was proposed by the authors in [9]. The feed waveguide is placed at the center of the parallel plate waveguide in a single layer, and two subarrays of slot pairs are arranged on both sides of the feed waveguide. The concept of the center-feed structure using cross-junctions was originally proposed to enhance the frequency bandwidth of a single-layer single-mode waveguide slot array [10].

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Fig. 4. Post-wall center-feed waveguide.

Fig. 3. Post-wall waveguide center-fed parallel plate slot array.

The center-feed structure consists of series of unit cross-junctions. The length of the radiating waveguide in the center-feed antenna is halved in comparison with that of an end-feed antenna, so the center-feed antenna can be expected to have a wider bandwidth; besides, the symmetrical structure of the antenna assures the main beam direction at the boresight irrespective of the frequency. This concept has now been adopted in the post-wall waveguide-fed parallel plate slot array. In contrast with the previous feed for the single-mode waveguide arrays, this new feed acts as a plane wave generator in the oversized parallel plate waveguides. In the design of a unit cross-junction, periodic boundary conditions are imposed for the radiating waveguide assuming periodicity in the large and oversized width of the parallel plate waveguide. The post-wall center-feed waveguide is designed so that the amplitude and the phase of the divided waves at the coupling windows are uniform at 61.25 GHz. The transmission phase advance of the cross-junctions is large and the spacing between adjacent cross-junctions is also large for the in-phase division; to eliminate the generation of higher modes or the obliquely propagating plane waves, the width of the feed waveguide is carefully determined. The center-feed antenna can overcome the defects in the endfeed antenna. However, it has as another disadvantage, a slotfree area at the center of the aperture, and this causes higher sidelobes in the E-plane. The sidelobes are reduced in this article by adopting an amplitude tapered distribution in the slot pair array. This causes the reduction of the antenna gain. The performances of the center-feed antenna with tapered-excited distribution and the end-feed antenna are compared in Section VI. II. STRUCTURE Fig. 4 shows the post-wall center-feed waveguide. The feed waveguide is an array of the cross-junctions shown in Fig. 5. A cross-junction has two coupling windows and four posts in front of these. The posts are arranged symmetrically with respect to the axis of the feed waveguide. Post pairs #1 and #2 are those close to Ports 1 and 2. The width of the feed waveguide and and . A TE wave is that of the radiating waveguide are incident from Port 1. Reflection to Port 1, transmission to Port 2, and division to Port 3 are controlled by the posts arranged at

Fig. 5. A unit cross-junction.

the center of the junction. The design parameters are only and for post pair #1 and and for post pair #2, taking symmetry into account. Post pair #2 is determined to control the transmission to Port 2, while post pair #1 is used to suppress the reflections to Port 1. Then, the remaining power is divided to Ports 3 and 4 equally according to the structural symmetry. The cross-junctions are placed with the spacing of one guided wavelength in the feed waveguide to be excited in phase. The feed waveguide is fed by a rectangular waveguide through an aperture cut out on the back of the antenna. For the boresight in phase radiation, one of two subarrays must have a half wavelength longer distance from the feed waveguide. III. DESIGN Fig. 6 shows the model for the analysis of the cross-junction that excites oversized waveguides. Taking into account the periodicity of the field in the transverse direction in the parallel plate waveguide, two periodic boundaries are imposed as narrow walls. This may be compared to the conductor walls in the cross-junction for a single-mode waveguide array. Only half of the cross-junction is analyzed for symmetry. This is done by introducing a perfect magnetic conductor at the center of the feed waveguide. To reduce the computational load in the method of moments analysis, the walls consisting of posts are replaced with a perfect electric conductor (PEC) solid wall with the thickness of the post diameter and the post-wall waveguide is replaced with a solid-wall waveguide with the same guided wavelength. The field equivalence theorem is applied and the area of the analysis is divided into three parts (regions) by placing perfect electric conductor walls with and on the two apertures unknown magnetic currents of the coupling windows as shown in Fig. 6(b). The three regions are the feed waveguide (Region I), the wall-thickness (Region II), and the periodic boundary wall waveguide (Region III). All the posts are replaced with unknown electric currents , and

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Fig. 6. Model for the analysis of the cross-junction: (a) Parameters, (b) regional divisions.

assumed on the surface , and , respectively. The continuity condition for the tangential component of the magnetic field on the window apertures ( and ) and the null condition for the tangential electric field on the PEC post surfaces , and ) result in a set of integral equations for the ( , and . The integral unknown currents equations are shown in [4]. The scattering matrix of the junction can be calculated from the solutions to these integral equations , and . The cross-junctions were designed to be at 61.25 GHz. controls the transmission to Port 2. and are given to minimize the reflection to Port 1. The width of the feed waveguide is 3.0 mm and that of the radiating waveguide is 4.8 mm. The relative permittivity of the substrate is 2.17. The post-wall feed waveguide is composed of posts with a diameter of 0.5 mm. From a conventional manufacturing concern, the aspect ratio, the substrate thickness over the post diameter, is 1 to 3. The diameter of the regular posts is chosen as 0.5 mm for a 1.2-mm thick substrate according to conventional rules. However, when the 0.5-mm post diameter was used for reflection-canceling post pairs, the frequency characteristics of the reflection became too narrow, and the diameter of the posts arranged at the center of the junctions is slightly thinner, 0.3 mm. Fig. 7 shows the results of the analysis. Both the amplitude and the phase of the transmission to Port 2, (a) and (b), are mainly dependent on because the spacing of 2 effectively acts as the width of the coupling window to Port 2. For larger amplitudes to Port 2, the phase becomes small. The phase of the division to Port 3, (c), depends on . The parameters and are specified to suppress the reflection for each and set. The radiating waveguides of the slotted waveguide array in [10] are excited at the same amplitude with adjacent waveguides out-of-phase by a series of cross-junctions, which are spaced in the half guided wavelength in the feed waveguide. In the parallel plate slot array, all the cross-junctions arranged at intervals

Fig. 7. Results of the unit analysis: (a) Amplitude of the transmission to Port 2, (b) phase of the transmission to Port 2, (c) phase of the division to Port 3.

of the guided wavelength in the feed waveguide divide the incident waves in phase to excite the quasi-TEM mode in the oversized waveguide. The junctions are numbered from termination to the input port. For uniform excitation, the required divided because each cross-junction power for a junction # is has two output ports. The phase-matching condition between , is shown in [4]. The width adjacent junctions, # and # of the feed waveguide of the center-feed single layer slotted (wavelength in the dielectric). In waveguide array is the post-wall center-feed parallel plate slot array, the transmission phase to Port 2 of the cross-junctions is 30 degrees larger than the single layer slotted waveguide array. Therefore, the to decrease width of the feed waveguide is widened to the transmission phase. But this is insufficient to realize the in-phase division. The spacing of the cross-junction, which is one guided wavelength in principle, was widened to 1.2 guided wavelengths. Only half of the feed waveguide was designed due to the symmetry with respect to the feed point, and the design had a cascade of eight cross-junctions as half of the feed waveguide. The post positions of the junctions from #2 to #8 can be determined recursively from the results of the unit design as shown in Fig. 7(a)–(c). Junction #1 is a matching cross-junction that is terminated by shorting posts, and it was designed separately because it has a different structure. The incident power into the matching junction is split equally into Ports 3 and 4, because the transmission to Port 2 is negligible and the reflection to Port 1 is well suppressed. Then, only the phase-matching condition needs to be considered. Fig. 8 shows the post position for each junction. When the junction gets closer to the termination, the post positions and become small, corresponding to a narrower coupling window to Port 2, and becomes larger,

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Fig. 8. Parameters of each junction.

Fig. 9. Feeding structure.

corresponding to a larger coupling window to Port 3, so that the transmission power to Port 2 becomes smaller. Fig. 9 shows the structure of the input aperture. The feed waveguide is fed by a rectangular waveguide attached on the back of the antenna through the input aperture. A 0.3-mm diameter post is placed near the aperture to suppress reflection. The post-wall waveguide is replaced by the PEC wall waveguide with an equal guided wavelength. The size and the position of the aperture and the post position are determined so that reflecdB over a 2.5-GHz bandwidth. tion is suppressed to below

Fig. 10. Model antennas: (a) Center-feed antenna (uniform and tapered distribution), (b) end-feed antenna (uniform distribution).

IV. MODEL ANTENNA Fig. 10(a) shows a picture of a model antenna. The feed waveguide and the slot array are designed to obtain a uniform aperture distribution at 61.25 GHz. The design procedure of the slot array is described in [3]. The substrate is made of PTFE; the dielectric constant is 2.17, the height of the substrate is 1.2 (wavelength in mm; the aperture is 77 mm 88 mm ( ); and the antenna has a slot-free area at its free space) center, with the width of the slot-free area 13 mm . For later comparison, the end-feed antenna with almost the same size (80 mm by 85 mm) is also fabricated as in Fig. 10(b). V. EXPERIMENTAL RESULTS Fig. 11 shows the measured frequency characteristic of the overall reflection at the input aperture. The overall reflection at dB over a 1.6-GHz the input port is suppressed to below bandwidth. The reflection in the end-feed antenna is added to this figure as a reference. Fig. 12 shows the two-dimensional aperture field distribution obtained from near-field measurement in (a) amplitude and (b)

Fig. 11. Frequency characteristics of reflection.

phase at 61.25 GHz. The amplitude is weak over the slot-free area because there are no slots. The amplitude deviation over the aperture is less than 6 dB except at the slot-free area, which does not result in a serious gain reduction. The phase is almost completely uniform so that the phase difference is less than 30 degrees. Fig. 13 shows the frequency characteristics of the antenna gain obtained from far-field measurements in an anechoic chamber. The peak of the measured gain is 33.1 dBi at 61.5 GHz. The efficiency is 56.8 % for the aperture size (77 mm 88 mm) including the slot-free area. The measured gain of the

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Fig. 13. Frequency characteristics of the gain and the bandwidth of 1-dB gain reduction. Bandwidth for the gain at 31.7 dB is also indicated.

Fig. 14. Radiation patterns of the uniformly excited antenna.

Fig. 12. Aperture distribution of the uniformly excited antenna: (a) Amplitude, (b) phase.

end-feed antenna shown in Fig. 10(b) is also included in Fig. 13 for comparison. The aperture size of the end-feed antenna is 80 mm by 85 mm including both the aperture with slots and the feed waveguide, which is almost the same size as the center-feed antenna. The peak gain of the end-feed antenna is 32.7 dBi, which is lower than that of the center-feed antenna by 0.4 dB. The difference in gain is small but is due to larger effective aperture for the center-feed array, while it may be partly due to the additional effects such as fabrication errors, loss in the substrate, etc. The bandwidth for a 1-dB gain reduction is 1.3 GHz in the center-feed antenna and 0.75 GHz in the end-feed type. This is 1.7 times as wide as that of the conventional end-feed type due to the reduction in the long line effect. The relative bandwidth of the center-feed antenna is added in Fig. 2, and the bandwidth enhancement of the center-feed antenna in comparison with the conventional end-feed antennas is confirmed. Fig. 14 shows the radiation patterns obtained from far-field measurements in an anechoic chamber at 61.25 GHz. The sidedB and dB, lobe levels in the E- and H-planes are respectively. The sidelobes in the E-plane are larger due to the slot-free area at the center of the antenna. Fig. 15 shows the measured frequency characteristic of the beam direction. The main beam direction of the center-feed antenna shifts only 0.3 degrees over a 4-GHz bandwidth, while that of the conventional end-feed antenna shifts 7.9 degrees. The beam shift in the end-feed antenna mainly comes from

Fig. 15. Beam directions.

the changes of the guided wavelength in the parallel plate waveguide. VI. SIDELOBE SUPPRESSION IN THE E-PLANE The center-feed antenna has a slot-free area without slots at the aperture center, making the sidelobes higher in the E-plane. The sidelobe level is roughly estimated using two sets of equally spaced discrete point sources with the slot-free area between them. The relationship between the sidelobe levels in the E-plane and the width of the slot-free area is shown in Fig. 16. To suppress the sidelobes, the width of the slot-free area must be as narrow as possible. At the same time, the slot subarrays need to be a proper distance from the feed waveguide is to avoid mutual coupling. From these considerations, a lower limit for the width of the slot-free area. Therefore, the aperture illumination of the slot array was modified to suppress the sidelobes in the E-plane. In the first attempt to achieve this, a tapered distribution in the array of the radiating slot pairs was adopted, and a 61.25-GHz model antenna was designed. The antenna size and the dielectric substrate parameters are the

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Fig. 16. Sidelobe level in the E-plane as a function of the width of the slot-free area. Fig. 18. Radiation patterns in the E-plane.

0.3 GHz. If the bandwidth is defined with gain higher than 31.7 dB, which is 1 dB lower than the peak gain of the reference end-feed antenna, that of the tapered-excited antenna is 0.55 GHz, which is narrower than that of the end-feed antenna by 0.2 GHz. Therefore, it is noted that the end-feed antenna has better performances with respect to both the gain and the bandwidth if the gain value and a reasonable sidelobe level (for example, dB) is the main concern. less than VII. CONCLUSION Fig. 17. Aperture distribution of the tapered-excited antenna.

same as those in the uniformly excited antenna in Section 4. The Taylor distribution of the dB sidelobe level and an with a slot-free area at its center was adopted; the edge level is dB in comparison with the maximum. The parameter is related to the number of sidelobes with nearly constant level in Taylor pattern. The design procedure of the slot array for the tapered distribution developed in [5] was adopted. The measured frequency dependence of the reflection of the tapered-excited antenna is added in Fig. 11. The reflection is slightly larger for the tapered-excited antenna than for the uniformly excited one possibly due to etching errors. The feed waveguides are common for the two, but the reason is not clear at this moment. Fig. 17 shows the measured two-dimensional aperture field distribution of amplitude at 61.25 GHz. There is about dB amplitude taper in the y-direction perpendicular to the feed waveguide. Fig. 18 shows the measured radiation patterns in the E-plane at 61.25 GHz. The sidelobe level in the E-plane of the tapereddB while that of the uniformly excited excited antenna is dB. The sidelobe level in the tapered antenna antenna is as considered here is still somewhat higher than the dB in uniform excitation without a slot-free area. The frequency dependence of the gain of the tapered-excited antenna is added in Fig. 13. The peak of the measured gain is 32.0 dBi at 61.4 GHz, and the efficiency for this aperture size is 47.4 %. The sidelobe is suppressed at the cost of gain reduction of 1.1 dB. The bandwidth for 1-dB gain reduction is 1.0 GHz, which is narrower than that of the uniformly excited antenna by

A center-feed structure using cross-junctions was adopted to a post-wall waveguide-fed parallel plate slot array to obtain a wider bandwidth. The post-wall center-feed waveguide was designed so that the amplitude and the phase of the divided waves are uniform at 61.25 GHz. In the design of the unit cross-junction, the average of the transmission phase to Port 2 in the cross-junctions is larger (72 degrees) than with the conventional design for single-mode slot arrays. To excite quasi-TEM waves in the oversized parallel plate, the spacing between the adjacent cross-junctions was 1.2 guided wavelengths to realize the in-phase division. A uniformly excited antenna was manufactured based on these considerations, and the overall reflection at the input port dB over a 1.6-GHz bandwidth. The is suppressed below bandwidth with a gain reduction of 1 dB is 1.7 times as wide as that of conventional end-feed type antenna. The sidelobe level in the E-plane increases to dB due to the slot-free area associated with the center-feed waveguide; the sidelobe level is suppressed to dB by introducing a dB amplitude tapered distribution in the array of the radiation slot pairs, at the cost of gain reduction and the narrowed bandwidth. ACKNOWLEDGMENT The authors wish to thank Y. Arai and N. Uchida of Taisei Co. Ltd. for the fabrication of the model antennas. REFERENCES [1] J. P. Quintez and D. G. Dudley, “Slots in a parallel plate waveguide,” Radio Sci., vol. 11, no. 8–9, pp. 713–724, Aug.–Sept. 1976. [2] H. A. Auda, “Quasistatic characteristics of slotted parallel-plate waveguide,” IEE Proc., pt. H, vol. 135, pp. 256–262, Aug. 1988.

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[3] J. Hirokawa, M. Ando, and N. Goto, “Waveguide-fed parallel plate slot array antenna,” IEEE Trans. Antennas Propag., vol. 40, no. 2, pp. 218–223, Feb. 1992. [4] J. Hirokawa and M. Ando, “Single-layer feed waveguide consisting of posts for plane TEM wave excitation in parallel plates,” IEEE Trans. Antennas Propag., vol. 46, no. 5, pp. 625–630, May 1998. [5] J. Hirokawa and M. Ando, “Sidelobe suppression in 76-GHz postwall waveguide-fed parallel-plate slot arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 11, pp. 1727–1732, Nov. 2000. [6] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wirel. Compon. Lett., vol. 11, no. 2, pp. 68–70, Feb. 2001. [7] J. F. Xu, W. Hong, P. Chen, and K. Wu, “Design and implementation of low sidelobe substrate integrated waveguide longitudinal slot array antennas,” IET Microw. Antennas Propag., vol. 3, no. 5, pp. 790–797, Aug. 2009. [8] H. Uchimura, T. Takenoshita, and M. Fujii, “Development of a laminated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2438–2443, Dec. 1998. [9] K. Hashimoto, T. Kai, J. Hirokawa, and M. Ando, “A center-feed postwall waveguide parallel plate slot array,” in Proc. IEEE AP-S URSI Int. Symp., Jul. 2006, pp. 3043–3046. [10] S. H. Park, J. Hirokawa, and M. Ando, “Design of a multiple-way power divider for center-feed single-layer waveguide arrays,” in Proc. IEEE AP-S Int. Symp., Jun. 2003, vol. 2, pp. 1165–1168.

Koh Hashimoto (S’06) was born in Tokyo, Japan, on January 28, 1982, and received B.S. and M.S. degrees in electrical and electronic engineering from Tokyo Institute of Technology, Tokyo, Japan, in 2005 and 2007, respectively, where he is currently working toward the D.E. degree. His main interest is planar array antennas.

Jiro Hirokawa (S’89–M’90–SM’03) was born in Tokyo, Japan, on May 8, 1965, and received B.S., M.S., and D.E. degrees in electrical and electronic engineering from Tokyo Institute of Technology (Tokyo Tech), Tokyo, Japan, in 1988, 1990, and 1994, respectively. He was a Research Associate from 1990 to 1996 and is currently an Associate Professor at Tokyo Tech. From 1994 to 1995, he was with the antenna group of Chalmers University of Technology, Gothenburg, Sweden, as a Postdoctoral Fellow. His research area has been in slotted waveguide array antennas. Dr. Hirokawa received an IEEE AP-S Tokyo Chapter Young Engineer Award in 1991, a Young Engineer Award from IEICE in 1996, a Tokyo Tech Award for Challenging Research in 2003, a Young Scientists’ Prize from the Minister of Education, Cultures, Sports, Science and Technology in Japan in 2005, and a Best Paper Award from IEICE Communication Society in 2007. He is a Member of IEICE, Japan.

Makoto Ando (SM’01–F’03) was born in Hokkaido, Japan, on February 16, 1952, and received the B.S., M.S., and D.E. degrees in electrical engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1974, 1976, and 1979, respectively. From 1979 to 1983, he worked at Yokosuka Electrical Communication Laboratory, NTT, and was engaged in development of antennas for satellite communication. From 1983 to 1985, he was a Research Associate at Tokyo Institute of Technology, where he is currently a Professor. His main interests have been high-frequency diffraction theory such as physical optics and geometrical theory of diffraction. His research also covers the design of reflector antennas and waveguide planar arrays for DBS and VSAT. Most recently, his interests include the design of high-gain millimeter-wave antennas and their systems. Dr. Ando served as the Chairman of ISAP (International Symposium on Antennas and Propagation) in 2007, the Technical Program Co-Chair for the 2003 IEEE Topical conference on wireless communication technology, the Chair of 2004 URSI International Symposium on Electromagnetic Theory, and the Co-Chair of 2005 IEEE ACES International Conference on Wireless Communications and Applied Computational Electromagnetics. He served as the Guest Editor-in-Chief of several special issues in the IEICE Transactions on Communications and on Electronics, Radio Science and IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He received the Young Engineers’ Award of IEICE Japan in 1981, the Achievement Award and the Paper Award from IEICE Japan in 1993. He also received the 5th Telecom Systems Award in 1990, the 8th Inoue Prize for Science in 1992 and the Meritorious Award on Radio, the Minister of Public Management, Home Affairs, Posts and Telecommunications and the Chairman of the Broad of ARIB in 2004. He serves as the Chair of Commission B of URSI, 2002-2005, the member of Administrative Committee of IEEE Antennas and Propagation Society 2004–2006 and also the member of Scientific Council for Antenna Centre of Excellence - ACE in EU’s 6’th framework program for research a network of excellence since 2004-2007. He was the President of Electronics Society of IEICE in 2006 and is the 2009 President of IEEE Antennas and Propagation Society. He is the member of IEE Japan, and is the Fellow, IEICE Japan, and IEEE.

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Tracking of Metallic Objects Using a Retro-Reflective Array at 26 GHz Jacquelyn A. Vitaz, Student Member, IEEE, Amelia M. Buerkle, Member, IEEE, and Kamal Sarabandi, Fellow, IEEE

Abstract—The detection and tracking of targets in highly cluttered environments often poses a difficult engineering challenge as the strong clutter backscatter contained in the scene makes it difficult to distinguish the target of interest. A design concept to aide in standoff detection and tracking of large metallic objects in a warehouse setting is presented. This includes the design and implementation of an active retro-reflective array system that is able to determine the precise location and identification of an object. To accomplish this, a unique series fed grounded coplanar waveguide patch antenna is designed and implemented with minimal cross coupling among elements. This compact array has a relatively large radar cross-section (RCS) while maintaining the desired retro-reflectivity. Additionally, a tilted beam is incorporated in the linear series-fed array to isolate the large RCS of the planar array structure at boresight from the desired modulated, retro-reflective RCS. This method for enhanced detection is implemented at 26 GHz. The incorporation of the high-speed PIN switches into the array structure provide the tag with a unique identification. Measurement results for the modulated RCS from such an array in a cluttered environment are presented. Index Terms—GCPW series array, long range RFID, retro-directive array, retro-reflector, Van Atta array.

I. INTRODUCTION ARGET tracking can generally be accomplished with the use of retro-reflectors which are able to return an incident signal back in the same direction with a radar cross-section (RCS) that is independent of angle of arrival for a large range of incidence angles. In low clutter environments, corner reflectors adequately accomplish this task, however, they are large, three-dimensional structures and require significant space. Additionally, corner reflectors’ purely passive nature does not allow for external modulation and, as such, they cannot be employed in situations where range gating alone is insufficient for target identification. The Van Atta retro-reflector [1], first proposed in 1959, proves to be a viable substitution for corner reflectors. The retro-reflective properties of these arrays have since become very useful for a large number of applications, from satellite communication [2]

T

Manuscript received September 13, 2009; revised April 01, 2010; accepted April 30, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. J. A. Vitaz is with the Electrical Engineering Department, University of Michigan Ann Arbor, MI 48109 USA on leave from Raytheon Corporation, Andover, MA 01810 USA (e-mail: [email protected]). A. M. Buerkle and K. Sarabandi are with the Electrical Engineering Department, University of Michigan, Ann Arbor, MI 48109 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.2010.2071350

to polarimetric active radar calibrator (PARC) development [3]. In both of these cases, horn antennas are used as the reflective elements and are connected by coaxial cables, ultimately yielding a cumbersome structure. Planar antennas and printed transmission lines are used to implement a variety of Van Atta retro-reflectors mainly for wireless communications and vehicle tracking applications. In [4], a planar, aperture coupled microstrip Van Atta retro-reflective array at 10 GHz is presented, but the feed network employed for the individual radiating elements prohibits operation on metallic surfaces. In [5], proximity-coupled dual-ring antennas are combined in sub-arrays to arrive at an 8 16 element retro-directive array at 35 GHz. However, the feeding structure of this array makes the integration of active components difficult. In [6], a 2 8 active Van Atta microstrip array operating at 24 GHz is discussed. Eight amplifiers are integrated into the microstrip feed lines but the result does not yield significant gain and the dominance of the passive flat plate backscatter makes it difficult to distinguish the Van Atta response. The current application requires the use of a planar retro-reflective structure which is capable of providing large RCS and signal modulation for unique tag identification to aide in the standoff detection and tracking of objects in a cluttered environment. The application is similar to RFID but differs in that much longer range of operation is needed and operation on metallic objects is required. To achieve small dimensions, an RF tag operating in the frequency range from 25.5–26.5 GHz is proposed. Linear, series-fed GCPW patch arrays are employed as the radiating elements. Using array elements with high gain (e.g., series fed patch array) allows maximization of RCS using the minimum number of active components for modulation or amplification, if needed. Additionally, the unique orientation of the series-fed arrays provides beam squinting, allowing for ease of target identification and distinguishing the response from that of the array ground plane [7], [8]. This paper details the design of a planar Van Atta array that is retro-reflective in one plane. The overall retro-reflector methodology is first introduced followed by a description of the array and feed network design. Good agreement is achieved between the measurement and simulation results. Modulation is then described along with the overall system development. This paper concludes with measurement results. II. RETRO-REFLECTIVE ARRAY OVERVIEW A. Van Atta Array Concept The RF tag is based on a planar Van Atta retro-reflector. In a Van Atta array configuration, individual linear elements are connected in pairs, depending on their relative distance from the

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Fig. 1. Phase relationships of the radiating elements for the Van Atta configuration both in the transmit and receive states.

Fig. 3. Grounded coplanar waveguide feed lines fabricated on 20 mil Rogers RT/duroid 6002 along exploded view of fabricated GCPW feed line.

B. GCPW Line Design

Fig. 2. Individual linear arrays are connected in pairs, depending on their relative distance from the array center with lines of equivalent phase length. The intra-element and intra-array spacings are noted in the figure along with the plane of retro-directivity of the reflective structure.

array center, with lines of equivalent phase length such that the incident energy reradiates back in the direction of incidence [9]. To understand its operation, consider a plane wave incident on a linear array as shown in Fig. 1. This incident wave induces cur, rents on each element with a certain phase delay, where is the wavenumber, is the angle of incidence, and is the intra-element spacing. After the energy travels through and , the interconnecting lines of identical phase length, the phase relationship is reversed such that the energy reradiates back in the direction of the source. For a one dimensional Van Atta array, the elements are connected as shown in Fig. 1. Elements 1 and 4 are equidistant from the array center and hence are connected together; similarly with elements 2 and 3. The cyclic nature of phase relationships implies that the lines are not required to be exactly the same length, rather only differ by a multiple of wavelength. It will later be shown that the difference in the total path lengths is restricted by the bandwidth of operation. Maximizing the RCS for a given source radar footprint allows greater ease in target distinction and identification. For this reason, maximizing RCS is the primary driving force for the majority of the design decisions. For the current design, linear series-fed patch arrays are used as the radiating elements of the Van Atta retro-reflector, as illustrated in Fig. 2, to increase the RCS and provide a narrow beam in the y-direction. In general, the Van Atta operates over a wider range of incidence angles than a typical corner reflector; here the only limitation is the directivity of the individual radiating elements.

Due to the high frequency of operation of this design, to minimize the radiation and substrate loss and allow for maximum energy transfer, GCPW lines are chosen for the feed network. These GCPW lines consist of a coplanar waveguide (CPW) over a metal backed substrate. The two semi-infinite ground plates of CPW are connected to the metallic groundplane of the substrate by two rows of metallic via holes. These GCPW lines have the additional benefits of ease of fabrication with both shunt and surface mount devices, reduced radiation loss and reduced cross talk effects between adjacent lines. Grounded coplanar waveguide lines are used for both the connection of Van Atta array elements and for the lines connecting the patches internal to the series arrays. Fig. 3 shows a picture of the fabricated interconnecting feed lines. In order to achieve the required retro-reflectivity discussed in Section II.A, the lines must be of such length to provide identical phase delay relative to one another. The bend geometry is identical in each line in order to eliminate this variable from the design process. Straight segments are altered to achieve identical phase delays within a multiple of a wavelength. Fig. 4 presents the results using one port reflection coefficient measurements and demonstrates the identical phase at 26.2 GHz. It should be noted that the measured insertion loss of the GCPW feed lines is approximately 1–2.5 dB. This loss is significantly less than a comparable microstrip design (approximately 4 dB, as fabricated and measured). The feed lines are important to the overall operation of the retro-reflective array in terms of both performance efficiency and bandwidth of operation. The frequency range over which acceptable phase error is maintained ultimately determines the operational bandwidth for this application. For the current application, the maximum tolerable phase variation is chosen to be 90 for a 4% (1 GHz) bandwidth. Therefore, the length difference between the longest and the shortest lines cannot exceed 6.25 at the design frequency. This will effectively limit the overall dimension and routing options for the lines. C. Series Fed GCPW Patch Array Design The series-fed array configuration offers unique advantages over other antenna array configurations [10]. The feed line

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Fig. 6. Photograph of 8

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2 1 series array as designed and fabricated.

Fig. 4. Phase measurements of the feed network lines. Note identical phase at 26.2 GHz. Results presented using one port reflection coefficient measurements, where a second port was left open.

Fig. 7. Simulated and measurement results of the reflection coefficient for the 8 1 series array with associated transition.

2

Fig. 5. Layout of GCPW patch antenna.

lengths are inherently minimized, reducing losses through radiation and dissipation. Using these constituent arrays also uses the minimum number of active components, thus increasing the RCS of the overall structure by reducing the number of switches and other associated circuitry required for the active design. The series array, from an equivalent circuit perspective, is the combination of parallel resonant circuits, each of which represents a single patch element. Since the elements are spaced a wavelength apart at resonance, their reactances theoretically cancel, leaving the parallel combination of input resistances. In order to achieve 50 input impedance to the 8 1 series array, the input impedance of each resonator is designed to be at 26 GHz. The addition of multiple series of resonant circuits significantly reduces the bandwidth of the single resonant element. Additionally, matching is highly susceptible to small variations in the lengths of the lines that separate the radiating elements. The individual coplanar patch elements consist of a patch antenna closely surrounded by a ground plane and fed with a GCPW line, as depicted in Fig. 5 [11]. This arrangement reduces radiation into the substrate and cross coupling among adjacent array-elements. For the development of the series array, a single patch is first designed using patch design principles and then verified using full wave electromagnetic field simulation tools (HFSS).

Individual patch elements are then placed into an 8 1 array configuration and simulated in HFSS. A photograph of the fabricated array is shown in Fig. 6. The simulated and measured reflection coefficient is compared in Fig. 7 and it is observed to be less than 10 dB over the band of interest. The high degree of sensitivity to the series line lengths is believed to be the main cause of discrepancy between the measured and simulated results. Simulated and measured gain measurements for the 8 1 series array over the band of interest are shown in Fig. 8. The gain is fairly consistent at high, low and mid-band. An overall beam squint of approximately 11 is noted in the elevation plane at the center frequency of 26 GHz. This is the result of the traveling wave associated with the series feed geometry. Ultimately, the squint is desirable at this point in the design as it allows for the RCS of the array to be distinguished from that of the flat metal ground plane which is a unique feature to the overall design. III. PASSIVE RETRO-REFLECTIVE ARRAY RESULTS The entire 8 8 Van Atta retro-reflector is designed and fabricated. The feed lines are integrated with the 8 1 series arrays and the resulting structure is pictured in Fig. 9. This design is entirely passive and measurement of the RCS magnitude and the RCS response versus angle is used to quantify the overall array performance. The antenna is measured at a distance of 14 meters in the University of Michigan anechoic chamber using a radar fashioned from a network analyzer, two horn antennas, and an amplifier. The return signal is measured at various azimuth and elevation

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Fig. 8. Simulated and measurement results of normalized gain for the 8 1 series array with associated transition. The results are shown for 25.5 GHz, 26 GHz and 26.5 GHz, the low, middle, and high ends of the operating band.

Fig. 9. Photograph of the fabricated passive 8

2 8 Van Atta array.

angles. The method of substitution is used to determine the total RCS of the Van Atta retro-reflector. As seen in Fig. 10, a relatively high RCS response is observed for the array. The measured RCS response for a flat metallic plate is included for comparison. The lack of dependence on azimuth angle of the RCS demonstrates the retro-reflective capabilities of the fabricated array. The absolute RCS of the passive retro-reflector is consistent with its predicted value resulting from the gain of the individual radiating elements and the losses of the associated feed network lines.

Fig. 10. Time domain measurement results comparing the RCS of the passive-retro-reflector to that of a flat plate. Both time domain measurement results correspond to the entire operating frequency range (25.5–26.5 GHz).

Fig. 11. Photograph of the PIN diode switch and its associated bias circuitry.

switch requires about 20 mA of current at 1 V bias voltage. The switches operate up to 26.5 GHz, the upper edge of the desired band of the array. The placement of the switches within the feed lines themselves is also important with regard to the overall operation of the retro-reflective array. When the array is in the off position, it is important that short circuits are observed at the array input terminals. This minimizes the RCS of the individual radiating elements allowing for a maximum on-to-off RCS ratio [13]. V. ACTIVE RETRO-REFLECTIVE ARRAY RESULTS

IV. MODULATION OF THE RETRO-REFLECTED SIGNAL As mentioned before, modulation helps distinguish the target from background clutter and different targets from one another. The retro-reflected signal can relay information from the target to the source by using amplitude modulation in the interconnecting lines [12]. For this configuration, a switch is placed within the line connecting each antenna pair which guides the signal to either a matched load to the transmit antenna. This allows the retro-reflected signal to be turned either on or off, effectively modulating the backscattered RCS. In this design, amplitude modulation is accomplished using MaCOM MA4SW110 PIN diode switches. Four switches are placed in the lines interconnecting the series array elements; an example switch and bias network are shown in Fig. 11. Each

A. RCS Measurements The active 8 8 retro-reflective structure is designed and fabricated. This design incorporates the switches and associated bias networks into the array feed lines. A photograph is shown in Fig. 12. The measurement setup is identical to that of the passive array. A battery pack including a switch to manually change the array into its on or off state is fastened to the back of the retro-reflector. RCS variation with angle along with the change in RCS from on to off states is used to quantify array performance. As seen in Fig. 13, a relatively high RCS response is observed in the on state; its lack of dependence on azimuth angle demonstrates the retro-reflective capability of the array. The on RCS is the same as that of the entirely passive array in Fig. 9 with

VITAZ et al.: TRACKING OF METALLIC OBJECTS USING A RETRO-REFLECTIVE ARRAY AT 26 GHz

Fig. 12. Photograph of fabricated 8

2 8 active retro-reflective array.

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Fig. 14. The measurement setup for the active retro-reflective tag in high clutter environment. An assortment of trihedral corner reflectors were placed near the tag to determine its ability to distinguish itself from other targets not of interest.

Fig. 15. Photograph of the switch driver circuit board as fabricated.

Fig. 13. RCS measurements of the active Van Atta array with switches and battery pack in the anechoic chamber. Comparing the on and off performance.

additional loss resulting from the switches and their associated bias networks. A significant increase in RCS is noted from the on position compared to the off. It is noted that the placement of the switches in the off state leads to more angular dependence in the form of periodic peaks in the RCS response. Signals reflected from the short-circuited switches propagate back to the series antenna elements in phase for specific azimuth angles. Further testing using copper tape to change the location of the short circuits in the lines resolved this issue and, in future designs, the location of the modulating components is staggered. B. Performance in High Clutter Environment The performance of this tag is then tested in a high clutter environment outside of the anechoic chamber. To simulate this clutter, an assortment indoor objects and a number of trihedral corner reflectors are placed in different ranges but still in close proximity to the retro-reflective tag. The radar is set up in the same manner as the chamber measurement. The tag is then placed atop a pedestal located next to a metal cabinet as shown in Fig. 14.

Fig. 16. Measurement results illustrating the magnitude of the retro-reflected return from the active tag in a high clutter environment for both the on and off states. (a) Return signal from off array. (b) Return signal from on array.

To achieve portability for a standalone tag, a driver circuit is required to provide power and control the current and bias voltage delivered to the switches. A photograph of the driver circuit board layout is shown in Fig. 15. A MaCOM driver chip (MADRMA0002) and 7555 clock chip at a frequency of approximately 1.5 Hz is used in initial testing. Eventually, the on/off sequence will be determined by a more complex m-sequence pseudo-noise (PN) code generator circuit. The m-sequence will be a code that uniquely identifies each retro-reflective array tag. The results are shown in Fig. 16. The additional clutter is displayed on the network analyzer screen along with the target of interest. A 5 dB change in RCS is noted when the array is in the off state compared to when it is in the on state. For enhanced detection, subtraction of two successive measurements can be performed to cancel out all stationary targets.

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VI. CONCLUSION

Amelia Buerkle (S’03–M’07) received the B.Sc. degree in electrical engineering from The Cooper Union, New York, in May 2002, and the M.S. and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, in April 2004 and May 2007. Currently, she is a Postdoctoral Researcher with the Radiation Laboratory, University of Michigan, Ann Arbor. Ms. Buerkle is a recipient of an NSF Graduate Research Fellowship and the 2005 URSI Young Scien-

In conclusion, a low cost, retro-reflective, long range RFID tag is presented. Its performance is demonstrated with existing radar hardware. An implementation of a planar retro-reflective tag at 26 GHz using grounded coplanar waveguide linear series arrays is demonstrated in both simulation and measurement. REFERENCES [1] L. C. Van Atta, “Electromagnetic Reflector,” U.S. patent 2 908 002, Oct. 6, 1963. [2] E. Gruenberg and C. Johnson, “Satellite communications relay system using a retrodirective space antenna,” IEEE Trans. Antennas Propag., vol. 12, no. 2, pp. 215–223, Mar. 1964. [3] M. Fujita, “Development of a retrodirective parc for alos/palsar calibration,” IEEE Trans. Geosci. Remote Sensing, vol. 41, no. 10, pp. 2177–2186, Oct. 2003. [4] Y.-J. Ren and K. Chang, “A broadband Van Atta retrodirective array for ka-band applications,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2007, pp. 1441–1444. [5] S.-J. Chung and K. Chang, “A retrodirective microstrip antenna array,” IEEE Trans. Antennas Propag., vol. 46, no. 12, pp. 1802–1809, Dec. 1998. [6] T.-J. Hong and S.-J. Chung, “24 GHz active retrodirective antenna array,” Electron. Lett., vol. 35, no. 21, pp. 1785–1786, Oct. 1999. [7] J. Vitaz, A. Buerkle, and K. Sarabandi, “A 26 Ghz retro-reflective array for long-range rfid applications,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2009, pp. 1–4. [8] J. Vitaz, A. Buerkle, and K. Sarabandi, “Tracking of metallic targets using a retro-reflective array at 26 Ghz,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2009, pp. 1–4. [9] E. Sharp and M. Diab, “Van Atta reflector array,” IRE Trans. Antennas Propag., vol. 8, no. 4, pp. 436–438, Jul. 1960. [10] T. Metzler, “Microstrip series arrays,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 174–178, Jan. 1981. [11] K. Li, C. Cheng, T. Matsui, and M. Izutsu, “Coplanar patch antennas: Principle, simulation and experiment,” in Proc. IEEE Antennas and Propagation Society Int. Symp, 2001, vol. 3, pp. 402–405. [12] S.-J. Chung, T.-C. Chou, and Y.-N. Chin, “A novel card-type transponder designed using retrodirective antenna array,” in IEEE Int. Microwave Symp. Digest, 2001, vol. 2, pp. 1123–1126. [13] D. Pozar, “Radiation and scattering from a microstrip patch on a uniaxial substrate,” IEEE Trans. Antennas Propag., vol. 35, no. 6, pp. 613–621, Jun. 1987.

Jacquelyn A. Vitaz (S’08) received the B.S. degrees in electrical engineering and physics from Lehigh University, Bethlehem, PA, in May 2001 and January 2002, respectively, and the M.S. degree in electrical engineering from the University of Michigan Ann Arbor, in May 2005, where she is currently working toward the Ph.D. degree. She has worked with Raytheon Corporation Andover, MA, since 2001 as both a power design and antenna and microwave Engineer. Ms. Vitaz has been the recipient of Raytheon’s Advanced Studies Scholarship both in 2003 and 2007 for continuing graduate study.

tist Award.

Kamal Sarabandi (S’87–M’90–SM’92–F’00) received the B.S. degree in electrical engineering from Sharif University of Technology, in 1980, the M.S. degrees in electrical engineering and mathematics, both in 1986, and the Ph.D. degree in electrical engineering in 1989 from The University of Michigan, Ann Arbor. He is the Rufus S. Teesdale Professor of Engineering and Director in the Radiation Laboratory, Department of Electrical Engineering and Computer Science, University of Michigan. His research areas of interest include microwave and millimeter-wave radar remote sensing, metamaterials, electromagnetic wave propagation, and antenna miniaturization. He has 25 years of experience with wave propagation in random media, communication channel modeling, microwave sensors, and radar systems and is leading a large research group including two research scientists, and 14 Ph.D. students. He has graduated 35 Ph.D. and supervised numerous postdoctoral students. He has served as the Principal Investigator on many projects sponsored by NASA, JPL, ARO, ONR, ARL, NSF, DARPA and a larger number of industries. Currently he is leading the Center for Microelectronics and Sensors funded by the Army Research Laboratory under the Micro-Autonomous Systems and Technology (MAST) Collaborative Technology Alliance (CTA) program. He has published many book chapters and more than 180 papers in refereed journals on miniaturized and on-chip antennas, metamaterials, electromagnetic scattering, wireless channel modeling, random media modeling, microwave measurement techniques, radar calibration, inverse scattering problems, and microwave sensors. He has also had more than 420 papers and invited presentations in many national and international conferences and symposia on similar subjects. Dr. Sarabandi is a member of NASA Advisory Council appointed by the NASA Administrator. He is serving as a vice president of the IEEE Geoscience and Remote Sensing Society (GRSS) and is serving on the Editorial Board of The IEEE Proceedings, and served as Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE Sensors Journal. He is a member of Commissions F and D of URSI. Dr. Sarabandi was the recipient of the Henry Russel Award from the Regent of The University of Michigan. In 1999 he received a GAAC Distinguished Lecturer Award from the German Federal Ministry for Education, Science, and Technology. He was also a recipient of the 1996 EECS Department Teaching Excellence Award and a 2004 College of Engineering Research Excellence Award. In 2005 he received the IEEE GRSS Distinguished Achievement Award and the University of Michigan Faculty Recognition Award. He also received the best paper Award at the 2006 Army Science Conference. In 2008 he was awarded a Humboldt Research Award from The Alexander von Humboldt Foundation of Germany and received the best paper award at the IEEE Geoscience and Remote Sensing Symposium. He was also awarded the 2010 Distinguished Faculty Achievement Award from the University of Michigan. In the past several years, joint papers presented by his students at a number of international symposia (IEEE APS’95,’97,’00,’01,’03,’05,’06,’07; IEEE IGARSS’99,’02,’07; IEEE IMS’01, USNC URSI’04,’05,’06,’10, AMTA’06, URSI GA 2008) have received best paper awards.

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Adaptive Array Beamforming Using a Combined LMS-LMS Algorithm Jalal Abdulsayed Srar, Kah-Seng Chung, and Ali Mansour

Abstract—A new adaptive algorithm, called least mean squareleast mean square (LLMS) algorithm, which employs an array image factor, , sandwiched in between two least mean square (LMS) algorithm sections, is proposed for different applications of array beamforming. It can operate with either prescribed or adaptive . The convergence of LLMS algorithm is analyzed for two different operation modes; namely with external reference or self-referencing. The range of step size values for stable operation has been established. Unlike earlier LMS algorithm based techniques, the proposed algorithm derives its overall error signal by feeding back the error signal from the second LMS algorithm stage (LMS2 ) to combine with that of the first LMS algorithm section (LMS1 ). Computer simulation results show that LLMS algorithm is superior in convergence performance over earlier LMS based algorithms, and is quite insensitive to variations in input signal-to-noise ratio and actual step size values used. Furthermore, LLMS algorithm remains stable even when its reference signal is corrupted by additive white Gaussian noise (AWGN). In addition, the proposed LLMS algorithm is robust when operating in the presence of Rayleigh fading. Finally, the fidelity of the signal at the output of an LLMS algorithm beamformer is demonstrated by means of the resultant values of error vector magnitude (EVM) and scatter plots. Index Terms—Adaptive array beamforming, error vector magnitude (EVM), least mean square-least mean square (LLMS) and least mean square (LMS) algorithms, Rayleigh fading.

I. INTRODUCTION

I

N recent years, adaptive or smart antennas have become a key component for various wireless applications, such as radar, sonar and cellular mobile communications [1] including worldwide interoperability for microwave access (WiMAX) [2]. They lead to an increase in the detection range of radar and sonar systems, and the capacity of mobile radio communication systems. These antennas are used as spatial filters for receiving the desired signals coming from specific direction or directions while minimizing the reception of unwanted signals emanating from other directions. Beamforming is central to all antenna arrays, and a summary of beamforming techniques is presented in [3]. An overview of signal processing techniques used for adaptive antenna array beamforming is described in [4]. Manuscript received November 04, 2009; revised March 28, 2010; accepted May 03, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. J. A. Srar is with Department of Electrical and Electronic Engineering, 7th October University, Misurata, Libya (e-mail: [email protected]). K.-S. Chung and A. Mansour are with the Department of Electrical and Computer Engineering, Curtin University of Technology, Perth, WA, Australia (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.2010.2071361

Because of its simplicity and robustness, the least mean square (LMS) algorithm has become one of the most popular adaptive signal processing techniques adopted in many applications including antenna array beamforming. Moreover, there is always a tradeoff between the speed of convergence of the LMS algorithm and its residual error floor when a given adaptation step size is used. Over the last three decades, several improvements have been proposed to speed up the convergence of the LMS algorithm. These include normalized-LMS (NLMS) [5], [6], variable-length LMS algorithm [7], transform domain algorithms [8], and recently constrained-stability LMS (CSLMS) algorithm [9] and modified robust variable step size LMS (MRVSS) algorithm [10]. The CSLMS algorithm has been proposed for use in processing speech signals [9]. Because of its improved performance over other published LMS algorithms, it is included in this paper for performance comparison with the proposed LLMS algorithm. Yet another approach of attempting to speed up the convergence of LMS algorithm without having to sacrifice too much of its error floor performance is through the use of a variable step size LMS (VSSLMS) algorithm. All the published VSSLMS algorithms [7], [11]–[14] make use of an initial large adaptation step size to speed up the convergence. Upon approaching the steady state, smaller step sizes are then introduced to decrease the level of adjustment, hence maintaining a lower error floor. More recently, the MRVSS algorithm, a modified version of the VSSLMS algorithm, has been proposed to improve tracking ability of the robust VSSLMS algorithm (RVSS) [12], [13]. All the above previously published algorithms require an accurate reference signal for their proper operation. In some cases, several operating parameters are also required to be specified. For example, in the case of MRVSS algorithm, the algorithm makes use of twelve predefined parameters. As a result, the performance of such an algorithm becomes highly dependent on the nature of the input signal [15]. Furthermore, the computational complex multicomplexity of MRVSS algorithm involves plications and complex additions [16], while the CSLMS algorithm requires a total of complex multiplications, one complex division and complex additions, where is the number of antenna array elements. This paper presents a very different approach to achieve fast convergence with an LMS based algorithm. The proposed least mean square-least mean square (LLMS) algorithm involves the use of two LMS algorithm sections, and , separated by an array image factor, , as shown in Fig. 1. Such an arrangement maintains the low complexity generally associated with LMS algorithm. It can be shown that an -element antenna

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Fig. 1. The proposed LLMS algorithm with an external reference signal.

array employing the LLMS algorithm involves complex multiplications and complex additions, i.e., slightly doubling the computational requirements of a conventional LMS algorithm scheme. With the proposed LLMS algorithm scheme, as shown in , yielded from the first Fig. 1, the intermediate output, LMS or algorithm section, is multiplied by the array image factor, , of the desired signal. The resultant “filtered” signal is further processed by the second LMS algorithm or algorithm section. For the adaptation process, the error signal of algorithm, , is fed back to combine with that of algorithm, to form the overall error signal, , for updating the tap weights of algorithm. As shown in Fig. 1, a common external reference signal is used for both the two LMS algorithm sections, i.e., and . Moreover, this external reference signal may be replaced by in place of , and for to produce a self-referenced version of the LLMS algorithm scheme, as discussed in Section II-B. For the case of a moving target, it is necessary that the array image factor, , is made adaptive in order to follow the angle of arrival (AOA) of the wanted signal. A simple yet effective method of estimating is also presented in this paper. This adaptive version will from here on be simply known as LLMS algorithm in order to differentiate it from the scheme that makes use of prescribed . The latter will be referred to as algorithm. The rest of the paper is organized as follows. In Section II, the convergence of LLMS algorithm is analyzed in the presence of an external reference signal. This is then followed by an analysis involving the use of the estimated outputs, and in place of the external reference. The latter is referred to as selfreferencing from here on. Estimation of the array image factor and step size boundaries of LLMS algorithm are discussed in Section III. Results obtained from computer simulations for an eight element array are presented in Sections IV and V. Finally, Section VI concludes the paper.

II. CONVERGENCE OF THE PROPOSED LLMS ALGORITHM The convergence of the proposed LLMS algorithm, which employs a prescribed array image factor, , is analyzed with the following assumptions. (i) The propagation environment is time invariant. should be (ii) The components of the signal vector independent identically distributed (iid). (iii) All signals are zero mean and statistically stationary at least to the second order. A. Analysis With an External Reference First, we consider the case when an external reference signal is used. From Fig. 1, the error signal for updating the LLMS algorithm at the iteration is given by (1) with where for algorithm and 2 for algorithm; and represent LMS algorithm the input signal and weight vectors of the denotes the Hermitian matrix of section respectively, and . algorithm is derived from the The input signal of algorithm, such that

where is the image of the array factor of the desired signal and is assumed fixed for this analysis. The weight vector for the LMS algorithm section is updated according to [17] (2) where is the step size, and is a positive number that is inversely proportional to the input signal power.

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Now, the convergence performance of the LLMS algorithm such can be analyzed in terms of the expected value of that

(3) denotes expectation; signifies modulus; stands where , and is for conjugate operator; the correlation matrix of the input signals given by . It follows from the Appendix that after substituting (A10) in (A9), the minimum mean square error (MSE) becomes

(4) corresponds to the input signal cross-correlation where . vector given by Based on (A10) and (4), (3) becomes

(5) . The error values of (5) are plotted where as the theoretical curve in Fig. 2(b). yields another form for Differentiating (5) with respect to the gradient, such that (6) Using eigenvalue decomposition (EVD) of

in (6) yields

(7) is the diagonal matrix of where is the diageigenvalues of for an N element array; onal of . For steepest descent, the weight vector is updated according to (8) is the convergence constant that controls the stability where is the and the rate of adaptation of the weight vector, and gradient at the iteration. We may rewrite (8) in the form of a linear homogeneous vector difference equation using (5), (6) and (7) to give (9) Alternatively, (9) can be written as

(10)

Fig. 2. The convergence of LLMS, LLMS , CSLMS, MRVSS and LMS algorithms with the parameters given in the 2nd column of Table I, for three different values of input SNR. (a) SNR = 5 dB; (b) SNR = 10 dB; (c) SNR = 15 dB.

with being a unity matrix. By substituting (10) in (5), the MSE iteration is given by at the (11) From (11), the asymptotic value of . With the term

becomes zero since converging,

as discussed in Section III, the mean square error will finally approach its minimum value, such that

(12)

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B. Analysis of the Self-Referencing Scheme Next, consider the case when the external reference is being replaced by internally generated signals, such that

Substituting (16) into (19) and multiplying both sides of (19) by , gives

(13) (20) As a result of these changes and note that the error signal , we can redefine in (3) as

, and . To find the condition for convergence, we take the expected value of (20). This leads to the second and third right hand side and are uncor(RHS) terms of (20) to vanish as both related. As a result, we obtain where

(14) Based on the definition of (14), we reanalyze the MSE as defined in (3) to yield

(21) (15) corresponds to the input signal cross-correlation where . The error values obvector given by tained from (15) are plotted as the theoretical curve in Fig. 4. By following the same analyzing steps in the Appendix in conjunction with (4)to (15), it can be shown that the proposed LLMS algorithm will converge under the condition of self-referencing. C. Mean Weight Vector Convergence To simplify the analysis, we use the concept that once the two individual LMS algorithm sections that make up the LLMS algorithm are converging, the LLMS algorithm is also converging. This enables the range of allowed step size values to be separately determined for and algorithms. Those values that overlapping these two ranges of step sizes are then considered valid for use in the LLMS algorithm to ensure its convergence. In (2), define the error as [18], [19] (16) is a zero mean measurement noise and is the weight where vector error. Then, let the time-varying weight vector be modeled by a random walk process [11], such that

For

algorithm, using (7), (21) can be rewritten as (22)

Thus, for the first stage of the LLMS algorithm, i.e., gorithm, convergence can be satisfied if gives

al. This

(23) where For

is the largest eigenvalue of . algorithm, (21) is rewritten as (24)

where coefficient can be expressed as

is a matrix and its general

(25)

where and Assume that each other, then

are the and

and elements of . are statistically independent from

(26)

(17) is a zero mean white sequence vector with diagonal where and being the weight variance. Also, correlation matrix let the weight error vector be

According to assumptions (ii) and (iii), we conclude that, the in (26) is nonzero only when . Therefore, product (26) can be rewritten as (27)

(18) where is the variance of the output of the stage. In matrix form, (27) can be written as

From (2), (17) and (18), we obtain (19)

algorithm

(28)

SRAR et al.: ADAPTIVE ARRAY BEAMFORMING USING A COMBINED LMS-LMS ALGORITHM

where is a complex matrix having a rank of one, and it can be analyzed according to (24) using EVD [20], such that

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algorithm converges, its output tends to apWhen with both the interference and noise proach being suppressed. Taking expectation of (34) yields (36)

(29) unitary matrix, and is the diagonal matrix of eigenvalues of the array factor matrix . Since this matrix is singular of rank 1, it has only one eigen, this eigenvalue is equal to so that value. With . From (29), the convergence of algorithm can be satisfied if . This gives where

is

an

-by-

As the input signal is stationary, we can approximate . Thus (36) can be rewritten as (37) algoBy assuming that both the input signal and the rithm weights are independent, the expectation of (35) can be written as (38)

(30) From (37) and (38), the array factor elements are estimated as Thus, to ensure convergence of the LLMS algorithm, the step size values of and must satisfy (23) and (30), respectively. III. LLMS WITH ADAPTIVE ARRAY IMAGE FACTOR A. Estimation of Array Image Factor For self-adaptive beamforming, it requires that the array be adjusted automatically to always tracking image factor the AOA of the desired signal. A simple method for estimating is now described. Consider the array inputs to the algorithm section as

(31) , and are the desired signal, interference where and are complex signal and noise, respectively, and array factors for the desired signal and the cochannel interference, respectively. By referencing with respect to the first antenna element, then and are given by

(39) where is a small constant introduced to prevent the denominator of (39) from becoming zero. Its value is chosen such that . For the computer simulations described in this paper, has been set to 0.0001. It follows that the instantaneous values of the elements of can be expressed as (40) Thus, (40) provides a mean of calculating the elements of the array image factor for use in the LLMS algorithm. B. Range of Step Size Values

for the LLMS Algorithm

(32)

For the adaptive LLMS algorithm scheme, the convergence of follows the same condition given by (23). However, the algorithm differ the allowable step size values, , for somewhat from those of (30) because the array image factor calculated using (39) is highly correlated to the error of algorithm. The correlation coefficient between in (34) algorithm, , in (19) is defined as and the error of

(33)

(41)

where according to the far-field plane wave model, and ; is the spacing between adjacent antenna elements, and is the carrier wavelength. Now, rearranging (31) in element form gives

and where signal and the and As mean, as

are the standard deviations of the input algorithm error, respectively. are highly correlated and almost zero , so that (41) is approximated

(34) (42) where is the element of The output of the individual are then given by

with . algorithm tap weights

(35)

Substituting (37) in (42) gives (43)

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Rewriting (43) to obtain

such that

TABLE I VALUES OF THE CONSTANTS USED IN SIMULATION

(44) Since be written as

, the element form of

in (24) can

Substituting (44) in the above equation gives

(45) Now, multiply both sides of (22) by maximum eigenvalue , yields

and introducing the

terference signals that arrive at each antenna element undergo independent Rayleigh fading. To facilitate comparison with the published algorithms; such as CSLMS algorithm in [9] and MRVSS algorithm in [10], a brief outline of the weight adaptation process of these algorithms is given here. According to [9], the weights of the CSLMS algorithm are adapted as follow:

(46) (49)

Substituting (46) in (45), gives (47) Finally, from (24), the condition for convergence of gorithm is given as

al-

where is a small constant and is adjusted to yield the best possible performance in the operating environment under consideration, and

This leads to (48) and correspond to the maximum power of the where algorithm, respectively. input signal and the error of IV. SIMULATION ENVIRONMENT The performance of the proposed LLMS algorithm, with either external reference or self reference, has been evaluated by means of simulations. For comparison purposes, results obtained with the conventional LMS, CSLMS and MRVSS algorithms are also presented. For the simulations, the following parameters are used. • A linear array of 8 isotropic antenna elements. • A desired binary phase shift keying (BPSK) arrives at an or 10 . angle of 0 , or if specified at either • An AWGN channel. • All weight vectors are initially set to zero. with the • A BPSK interference signal arrives at same amplitude as the desired signal. • For the Rayleigh fading channel, the Doppler frequency, . It is considered that both the desired and in-

As for the MRVSS algorithm [10], the step size, , is updated, such that if if

if if where , , , and is the time averaged error correlation over two consecutive values. The time averaged error square signal is with its upper and lower bounds as and , respectively. The upper and lower bounds of , and , respectively. are Table I tabulates the values of the various constants adopted for the simulations of the five different adaptive algorithms. The parameter values for the MRVSS algorithm operating in an AWGN channel are those given in [10], [12], [13]. All other values adopted here for the MRVSS and CSLMS algorithms have been chosen for obtaining the best performance out of these algorithms.

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Often, performance comparison between different adaptive beamforming schemes is made in terms of the convergence errors and resultant beam patterns. Moreover, for a digitally modulated signal, it is also convenient to make use of the error vector magnitude (EVM) as an accurate measure of any distortion introduced by the adaptive scheme on the received signal at a given signal-to-noise ratio (SNR). It is shown in [21] that EVM is more sensitive to variations in SNR than bit error rate (BER). EVM is defined as [22] (50) where is the number of symbols used, is the output is the transmit symbol. is of the beamformer, and the average power of all symbols for the given modulation. V. SIMULATION RESULTS A. Performance With an External Reference , CSLMS, First, the performances of the LLMS, MRVSS and LMS algorithm schemes have been evaluated in the presence of an external reference signal. The convergence performances of these schemes are compared based on the obtained from 100 indiensemble average squared error vidual simulation runs. These results have been obtained for and . different values of input SNR, and step sizes, Figs. 2(a)–(c) show the convergence behaviors of the five adaptive schemes for SNR1 values of 5, 10, and 15 dB, respecalgorithm scheme, the theorettively. For the proposed is ical convergence error calculated using (5) for also shown in Fig. 2(b). It is observed that under the given conditions, the two variants of the proposed LLMS algorithm converge much faster than the other three schemes. Furthermore, their error floors are less sensitive to variations in the input SNR, even for input SNR as small as 5 dB. Also, as shown in Fig. 2(b), there is a close agreement between the simulated and theoretical algorithms. As error curves for the proposed LLMS and for the CSLMS and MRVSS algorithms, they share the same performance for all the three SNR values considered. As expected, the conventional LMS algorithm converges the slowest among the five algorithms. Next, it can be verified that for ensuring convergence of both versions of the LLMS algorithm schemes, the values of the step size used have to be within the bounds given in (23) and (30) for algorithm, and (23) and (48) for LLMS algorithm. For an 8-element array operating with an input SNR of 10 dB, and for we require that algorithm, and and for LLMS algorithm. When the step sizes are chosen to be well within their limits, such as for the values used in Fig. 2, the two LLMS algorithm schemes are able to converge within a few iterations to a low error floor. However, algorithm shows sign of insta1Signal-to-noise ratio (SNR) is defined as the ratio of the average signal power

to AWGN power, determined over the signal bandwidth. The signal power is obtained from averaging over 16 M symbols.

Fig. 3. The convergence behaviors of LLMS and LLMS algorithms at SNR = 10 dB for step size values set at their upper limits.

bility when operating with step sizes close to their upper limits. algorithm Such instability in convergence behavior of is demonstrated as shown in Fig. 3 for two cases that make use and , and and . of As discussed in Section III-B, the limits of the step size for algorithm. LLMS algorithm are different from those for This is due to the fact that the calculation of the array image factor in LLMS algorithm is correlated with the output of its and 1st LMS algorithm stage. The allowable upper limits of for LLMS algorithm are demonstrated in Fig. 3, for the two and , and cases involving the use and , respectively. Since, the elements of the array image factor , of LLMS algorithm are being determined adaptively, this can influence its convergence behavior. As shown in Fig. 3, the resulting error floors of LLMS algorithm operating with one of the step sizes close to the upper limit tend to first diverge before finally converging, resulting in a longer convergence time. B. Performance With Self-Referencing As shown in Fig. 2 and Fig. 3, both LLMS and algorithms can converge within ten iterations. Once this occurs, the intermediate output, , tends to resemble the desired , and may then be used in place of the external refersignal ence, , for the current iteration of the algorithm section. algorithm section converges, its output, , As the becomes the estimated . As a result, may be used as the reference for the algorithm section. to replace This feedforward and feedback arrangement enables the provision of self-referencing in LLMS algorithm, and allows the external reference signal to be discontinued after an initial four iterations. The ability of the LLMS algorithms to maintain operation with the internally generated reference signals is demonstrated in Fig. 4. On the other hand, it clearly shows that the traditional LMS, CSLMS, MRVSS algorithms are unable to converge without the use of an external reference signal. For comparison, the theoretical convergence errors calculated from (15) are also plotted in Fig. 4. C. Performance With a Noisy Reference Signal The performances of LLMS, , CSLMS, MRVSS and LMS algorithms have also been investigated when their refer-

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Fig. 4. The convergence of LLMS and LLMS algorithms with self-referencing using the parameters given in Table I (column 2), for SNR = 10 dB. An external reference is used for the initial four iterations before switching to self-referencing.

Fig. 5. The influence of noise in the reference signal on the mean square error  using the parameters given in column 2 of Table I.

ence signals used are corrupted by AWGN. This is done by examining the resultant mean square error when the noise in the reference signal is varied. Fig. 5 shows the ensemble average of the mean square error, , obtained from 100 individual simulation runs, as a function of the ratio of the rms noise to the amplitude of the reference signal. It is interesting to note that the conventional LMS, CSLMS and MRVSS algorithms are quite sensitive to the presence of noise in the reference signal. On the other hand, both and LLMS algorithms are very tolerant to noisy reference signal. As shown in Fig. 5, the values of associated with LLMS and algorithms remain very small even when the rms noise becomes as large as the reference signal. The effect of noise is even less pronounced on LLMS algorithm as is continuously updated rather than its array image factor being fixed as a prescribed constant vector. Next, Fig. 6 shows the resulting beam patterns obtained with the five algorithms when the reference signal is corrupted by . It is shown AWGN. The desired signal arrives at in Fig. 3(a) and Fig. 3(b) that the correct beam patterns can still be achieved using LLMS and algorithms when the ratios of rms noise to the reference signal are 3 dB and 9 dB, respectively. In the absence of noise, all the five algorithms are able to achieve the same beam forming performance, as shown in Fig. 6(c). Moreover, as noise in the reference signal increases,

Fig. 6. The beams patterns achieved with the LLMS, LLMS , CSLMS, MRVSS and LMS algorithms when the reference signal is contaminated by AWGN. The desired signal arrives at  = 20 . (a) ( =S ) = 3 dB; (b) ( =S ) = 9 dB; (c) Without noise.

0

0

0

the beam pattern of LLMS algorithm tends to deviate slightly from its designated direction as a result of the use of the estimated values for the array image factor . D. Tracking Performance of LLMS Algorithm algorithms in tracking The ability of LLMS and sudden interruptions in the input signal is investigated by ex. For this study, amining the behavior of the error signal the input signal is assumed to be periodically interrupted for 25 out of 100 iterations. The resulting tracking performances algorithms are shown in Fig. 7, which of LLMS and shows their respective mean square errors increasing very rapidly when the input signal is switched on or off. This algorithms indicates the fast response of LLMS and and to sudden interruptions in the input signal. Both LLMS algorithms behave similarly. Unlike the responses for LMS, CSLMS and MRVSS algorithms, which are also included in Fig. 7 for comparison purpose, the mean square errors associated with both and LLMS algorithms remain low when the input signal is interrupted.

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Fig. 7. Tracking performance comparison of LLMS, LLMS , CSLMS, MRVSS and LMS algorithms with  =  =  = 0:05 and SNR = 10 dB.

Fig. 8. The EVM values obtained with LLMS, LLMS , CSLMS, MRVSS and LMS algorithms at different input SNR.

E. EVM and Scatter Plot The performances of the five algorithms, based on the rms EVM computed using (50), for values of input SNR ranging from 0–30 dB in steps of 5 dB are shown in Fig. 8. These EVM values have been calculated after each of the adaptive algorithms has converged. It is observed that both the proposed algorithms achieve the two lowest EVM LLMS and values. Such superior performance becomes even more pronounced at lower values of input SNR. This further confirms the observation made from Fig. 2 showing that the operations of and LLMS algorithms are very insensitive to changes in input SNR. Next, the scatter plots of the BPSK signal recovered using the adaptive beamformer, based on LMS, CSLMS, MRVSS, and LLMS algorithms are shown in Figs. 9(a)–(e), respectively. The scatter plots are obtained from 100 signal samples for an input and . Again, the and LLMS algorithms show the least scatter plots of spreading, indicating their ability to retain the signal fidelity. F. Operation in Flat Rayleigh Fading Channel The ability of an adaptive beamformer to operate in a fast changing signal environment is examined by subjecting the input signals to undergo flat Rayleigh fading. In this case, the

Fig. 9. The scatter plots of the recovered BPSK signal obtained with (a) LMS, (b) CSLMS, (c) MRVSS, (d) LLMS , and (e) LLMS algorithms for input SNR = 10 dB and SIR = 0 dB.

rms EVM is again used as the performance metric for comparison between the different adaptive beamforming algorithms. The following conditions are considered in the performance evaluation by computer simulation. • The signals arriving at each antenna element, for both the desired and interference, undergo independent flat Rayleigh fading. and 45 • Two interfering signals are emanating from with the same amplitude as the desired signal. • The parameters as tabulated in column 3 of Table I are adopted for the different algorithms. • Each simulation involves a run of 16 Mbits. , corresponding to a mo• Doppler frequency bility of 72 km/h at 900 MHz. Fig. 10(a) and Fig. 10(b) show the resultant EVM values as a function of the input SNR achieved for the case with and without interference, respectively. From these figures, the following observations are made: • With the exception of MRVSS algorithm, the other four algorithms are able to operate in the presence of Rayleigh fading.

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, and are the iteration, of the desired average output powers, at the signal, the interference signal and the AWGN, respectively. The input amplitudes of the desired and interfering BPSK signals are and , respectively, while is the rms noise iteration, associated with a voltage. The weights, at the . given algorithm are represented by Figs. 11(a)–(f) show how the angle of arrival of an interfering signal is affecting the of the desired signal under three different values of input SNR, i.e., 5, 10 and 15 dB. As expected, of the desired the interference has little effect on the signal when its AOA is outside the main beam of the array. This and . Among the cases conis true for both algorithms achieve higher values sidered, LLMS and , and show less fluctuations in when the interof ference is arriving outside the beamwidth. The latter is the result of the lower side lobes associated with LLMS and algorithms.

where

VI. CONCLUSIONS

Fig. 10. The EVM values obtained with the LLMS, LLMS , CSLMS, MRVSS and LMS algorithms for different input SNR in the presence of Rayleigh fading. (a) Fading with interference; (b) fading without interference.

• Irrespective of whether interfering signals are present or not, LLMS and algorithms perform superior among the algorithms considered. • LLMS algorithm is the least affected by interference. G. Influence of the AOA,

, of the Interference

The effect of the AOA of an interfering signal on the desired signal recovered at the output of an adaptive beamformer is also investigated. In this study, the desired signal is arriving at ei, boreside, or , end-fire, with three difther ferent input SNR values, i.e., 5 dB, 10 dB or 15 dB. The interference has the same power as that of the desired signal, i.e., , and is arriving over a range of AOA from to 90 . The performance measure is the output signal-to-noise plus interference ratio, , obtained with a given algorithm after convergence has been achieved. At each iteration, the output signal-to-interference plus for each algorithm is calculated acnoise ratio, cording to

(51)

This paper presents a new algorithm, called LLMS algorithm, which employs an array image factor, , sandwiched in between two LMS algorithm sections, for use in array beamforming. This algorithm adopts a different approach compared with earlier VSSLMS algorithm based techniques, such as CSLMS and MRVSS algorithms, which make use of step size adaptation to enhance their performance. For proper operation, these VSSLMS algorithms often require many input signal dependent parameters to be specified. As noted in [23], it is difficult in practice to obtain the exact values simultaneously for all these parameters. On the other hand, it is shown that the rapid convergence of the proposed LLMS algorithm depends only on the step size values, one for each of the two LMS algorithm sections. This makes LLMS algorithm attractive for practical applications. As discussed in Section II, the proposed algorithm makes to interface between the two use of the array image factor LMS algorithm sections. In this way, an accurate fixed beam can be obtained by prior setting of the elements of with the prescribed values for the required direction. Alternatively, may be made adaptive to automatically track the target signal. A simple and effective method has been proposed for calculating the element values of , based on the estimated output signal of the first LMS algorithm section and its tap weights. The convergence of the two versions of the LLMS algorithm has been analyzed assuming the use of an external reference signal. This is then extended to cover the case that makes use of self-referencing. The convergence behaviors of these two LLMS algorithm schemes with different step size combinations of and have been demonstrated by means of simulations under different input SNR conditions. Also, their step size limits have been derived analytically and verified by computer simulations. It is shown that the LLMS algorithm performs quite similarly and sometimes better than the algorithm. These two

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Fig. 11. Influence of AOA of the interference on SINR for three different values of input SNR. The desired signal arrives at either  = 0 or  = 90 . (a) SNR = 5 dB,  = 90 ; (b) SNR = 5 dB,  = 0 ; (c) SNR = 10 dB,  = 90 ; (d) SNR = 10 dB,  = 0 ; (e) SNR = 15 dB,  = 90 ; (f) SNR = 15 dB,  = 0 .

variants of the LLMS algorithm scheme are shown to have rapid convergence, typically within a few iterations. Also, the resulting steady state MSE is quite insensitive to input SNR. Furthermore, unlike the conventional LMS, CSLMS and MRVSS algorithms, the proposed LLMS and algorithms are able to operate with noisy reference signal. Once initial convergence is achieved, usually within a few iterations, both and LLMS algorithms can maintain their operation through self-referencing. Moreover, the resultant EVM values and scatter plots obtained for operation in the presence of Rayleigh fading further demonstrate the superior performance of LLMS and algorithms over the other

three LMS-based algorithm schemes in a fast changing signal environment. From the study of the influence of the AOA of the interference on the desired signal at the output of an adaptive beamformer, it is observed that the proposed LLMS algorithm schemes outperform the other three algorithms. The superior performance of the proposed LLMS algorithm has been achieved with a complexity slightly larger than twice the conventional LMS algorithm scheme. Moreover, its complexity is lower than the MRVSS algorithm as well as our previously published RLMS algorithm scheme [24], [25], and approximately the same as CSLMS algorithm.

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APPENDIX As given in (3), the expected values of

Since as

is

, the last RHS term of (A1) may be written

(A7) Again, applying the assumptions (ii) and (iii), we obtain (A1) Consider the first term on the right hand side (RHS) of (A1). It can be expressed as

(A8) As a result, the mean square error as specified by (A1) can be rewritten to include the results of (A6) and (A8) to become

(A2) where stands for conjugate operator. and being zero mean and uncorrelated With based on the assumptions (ii) and (iii) given in Section II, the last RHS term of (A2) is therefore equal to zero. This gives (A3)

(A9) Differentiating (A9) with respect to the weight vector , and by equating the results to zero, we obtain the optimal weight vector as

From (1), the last RHS term of (A3) becomes

(A10)

(A4) Assume

, and let , (A4) can be rewritten as

where

where is a full rank matrix. For this analysis, the input signal is considered well excited as it has a non-singular correis non-sparse and its compolation matrix. That is true as nents are contaminated by AWGN. Equation (A10) represents the Wiener-Hopf equation in matrix form. ACKNOWLEDGMENT The authors thank the reviewers for their helpful comments. REFERENCES

(A5) corresponds to the input signal where cross-correlation vector. Substituting (A5) in (A3), the first term on the RHS of (A1) becomes

(A6)

[1] N. A. Mohamed and J. G. Dunham, “Adaptive beamforming for DS-CDMA using conjugate gradient algorithm in a multipath fading channel,” in Proc. IEEE Emerging Technologies Symp. on Wireless Communications and Systems, Richardson, Apr. 1999, pp. 1–5. [2] S. Chandran, “Performance of adaptive antenna arrays in the presence of varying noise power in WiMAX applications,” in Proc. IET Int. Conf. on Wireless, Mobile and Multimedia Networks, Mumbai, India, Jan. 2008, pp. 3–5. [3] J. A. Stine, “Exploiting smart antennas in wireless mesh networks using contention access,” IEEE Trans. Wireless Commun., vol. 13, pp. 38–49, 2006. [4] B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE ASSP Mag., vol. 5, pp. 4–24, 1988. [5] V. H. Nascimento, “The normalized LMS algorithm with dependent noise,” in Proc. Anais do 19 Simpòsio Brasileiro de Telecomunicações, Fortaleza, Brazil, Mar. 2001.

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[6] D. T. M. Slock, “On the convergence behavior of the LMS and the normalized LMS algorithms,” IEEE Trans. Signal Processing, vol. 41, pp. 2811–2825, 1993. [7] V. H. Nascimento, “Improving the initial convergence of adaptive filters: Variable-length LMS algorithms,” in Proc. 14th Int. Conf. on Digital Signal Processing, Santorini, Greece, Jul. 2007, pp. 667–670. [8] E. M. Lobato, O. J. Tobias, and R. Seara, “Stochastic modeling of the transform-domain "LMS algorithm for correlated Gaussian data,” IEEE Trans. Signal Processing, vol. 56, pp. 1840–1852, 2008. [9] J. M. Górriz, J. Ramírez, S. Cruces-Alvarez, D. Erdogmus, C. G. Puntonet, and E. W. Lang, “Speech enhancement in discontinuous transmission systems using the constrained-stability least-mean-squares algorithm,” J. Acoust. Society Amer., vol. 124, no. 6, pp. 3669–3683, Dec. 2008. [10] K. Zou and X. Zhao, “A new modified robust variable step size LMS algorithm,” in Proc. 4th IEEE Conf. on Industrial Electronics and Applications, Xian, China, May 2009, pp. 2699–2703. [11] S. Zhao, Z. Man, and S. Khoo, “A fast variable step-size LMS algorithm with system identification,” in Proc. 2nd IEEE Conf. on Industrial Electronics and Applications, Harbin, China, May 2007, pp. 2340–2345. [12] R. H. Kwong and E. W. Johnston, “A variable step size LMS algorithm,” IEEE Trans. Signal Processing, vol. 40, pp. 1633–1642, 1992. [13] T. Aboulnasr and K. Mayyas, “A robust variable step-size LMS-type algorithm: Analysis and simulations,” IEEE Trans. Signal Processing, vol. 45, pp. 631–639, 1997. [14] A. Wee-Peng and B. Farhang-Boroujeny, “A new class of gradient adaptive step-size LMS algorithms,” IEEE Trans. Signal Processing, vol. 49, pp. 805–810, 2001. [15] V. J. Mathews and Z. Xie, “A stochastic gradient adaptive filter with gradient adaptive step size,” IEEE Trans. Signal Processing, vol. 41, pp. 2075–2087, 1993. [16] I. H. Tarek, “A simple variable step size LMS adaptive algorithm,” Int. J. Circuit Theory Applicat., vol. 32, pp. 523–536, 2004. [17] E. Eweda, “Comparison of RLS, LMS, and sign algorithms for tracking randomly time-varying channels,” IEEE Trans. Signal Processing, vol. 42, no. 11, pp. 2937–2944, 1994. [18] S. Zhao, Z. Man, S. Khoo, and H. R. Wu, “Variable step-size LMS algorithm with a quotient form,” IEEE Trans. Signal Processing, vol. 89, pp. 67–76, 2009. [19] S. B. Gelfand, W. Yongbin, and J. V. Krogmeier, “The stability of variable step-size LMS algorithms,” IEEE Trans. Signal Processing, vol. 47, pp. 3277–3288, 1999. [20] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: The Johns Hopkins Univ. Press, 1996. [21] Q. Zhang, Q. Xu, and W. Zhu, “A new EVM calculation method for broadband modulated signals and simulation,” in Proc. 8th Int. Conf. on Electronic Measurement and Instruments, Beijing, China, Jul. 2007, pp. 2-661–2-665. [22] H. Arslan and H. Mahmoud, “Error vector magnitude to SNR conversion for nondata-aided receivers,” IEEE Trans. Wireless Commun., vol. 8, pp. 2694–2704, 2009. [23] Y. Li and X. Wang, “A modified VS LMS algorithm,” in Proc. 9th Int. Conf. on Advanced Communication Technology, Seoul, Korea, Feb. 2007, pp. 615–618. [24] J. A. Srar and K.-S. Chung, “Adaptive array beam forming using a combined RLS-LMS algorithm,” presented at the 14th Asia-Pacific Conf. on Communications, Tokyo, Japan, Oct. 2008. [25] J. A. Srar and K.-S. Chung, “Performance of RLMS algorithm in adaptive array beam forming,” presented at the 11th IEEE Int. Conf. on Communication Systems, Guangzhou, China, Dec. 2008.

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Jalal Abdulsayed Srar was born in, Libya, in 1970. He received the B.Sc. degree in electronics engineering from Garunis University, Libya, in 1993 and the M.Sc. degree from the Higher Industrial Institute (HII), Libya, in 2001. From 2001 to 2006, he worked with the Adaptive Antenna Research Group, HII, Libya. He has been a Lecturer in the Electrical Engineering Department, Misurata University, Libya, since 2003. He joined the CTRG group, Curtin University, Australia, in 2008. His research interests including beamforming algorithms, adaptive antenna, and signal processing for communications.

Kah-Seng Chung (SM’87) received the Ph.D. degree in electrical engineering from Cambridge University, Cambridge, U.K., in 1977. He began his engineering career in 1973 by joining GEC Hirst Research Centre, U.K., working on high-speed digital line transmission. In 1977, he took up a teaching position with the Department of Electrical Engineering, National University of Singapore. During 1979 to 1987, he was with Philips Research Laboratories, Eindhoven, The Netherlands, leading the research on spectral efficient digital modulation techniques for mobile radio communications, and monolithic integration of radio transceivers. Since 1987, he has been with the Curtin University of Technology, where he is now a Professor of mobile telecommunications. His current research interests are on broadband wireless backhauls, self-configurable wireless networks, broadband powerline communications, adaptive antenna arrays and transceiver architectures for SoC. He hold 12 U.S. patents, and has published more than 90 technical papers. Prof. Chung is a Fellow of the Institution of Engineering and Technology (IET), London, U.K. He is a Chartered Engineer under the Council of Engineering Institutes, U.K.

Ali Mansour received the M.S. degree in electronic electric engineering from Lebanese University (Tripoli-Lebanon), in September 1992, the M.Sc. and Ph.D. degrees in signal, image and speech processing from the “Institut National Polytechnique de Grenoble-INPG (France), in July 1993 and January 1997, respectively, and the HDR degree (Habilitation a Diriger des Recherches) from UBO, Brest, France, in November 2006. From January 1997 to July 1997, he held a Postdoctoral position at LTIRF-(INPG Grenoble, France). From August 1997 to September 2001, he was a Researcher at the Bio-Mimetic Control Research Center (BMC), Institut of Physical and Chemical Research (RIKEN), Nagoya, Japan. From October 2001 to January 2008, he held a Teacher-Researcher position at ENSIETA, Brest, France. Since February 2008, he has been a Senior Lecturer in the Department of Electrical and Computer Engineering, Curtin University of Technology (ECE-Curtin Uni.), Perth, Australia. During January 2009, he held an Invited Professor position at the Universite du Littoral Cote d’Opale, Calais, France. His research interests are in the areas of blind separation of sources, high-order statistics, signal processing and robotics and telecommunication. He is the author and the coauthor of three books. He is the first author of many papers published in international journals, such as IEEE TRANSACTIONS ON SIGNAL PROCESSING, IEEE Signal Processing Letters, NeuroComputing, IEICE and Artificial Life and Robotics. He is also the first author of many papers published in the proceedings of various international conferences.

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Adaptive Correction to Array Coefficients Through Dithering and Near-Field Sensing Ramakrishna Janaswamy, Fellow, IEEE, Dev V. Gupta, and Daniel H. Schaubert, Fellow, IEEE

Abstract—Adaptive correction of the excitation coefficients of a phased array achieved through dithering the magnitudes and phases of the element coefficients and sensing the fields through a near-zone probe is demonstrated by considering a linear array. Knowledge of the reference signal generated by the desired array at the near-zone probe is assumed. Deviations in the coefficients of the actual array from the desired array are corrected adaptively and simultaneously by means of a gradient based algorithm. Requirements for the algorithm to converge, its performance with and without a receiver noise and the effect of the dither parameters are studied. The effect of element mutual coupling on the performance of the array is also demonstrated by considering an array of half-wave dipoles. Index Terms—Antenna Array, dithering, log-normal distribution, near-field sensing, phased array, phased array random errors, random signals, uniform distribution.

I. INTRODUCTION HASED arrays are deployed in a number of electronic systems where high beam directivity and/or electronic scanning of the beam is desired. Applications range from radar systems [1], [2] to smart antennas in wireless communications [3], [4]. It has been known for quite sometime that errors (random and/or correlated fluctuations) present in the excitation coefficients of a phased array can degrade its performance [5], ([6], Ch. 2), ([7], Ch. 13), [8], and ([9], Ch. 7). Undesirable effects resulting from errors in the magnitude and phase of the array coefficients can include decrease in directivity, increase in sidelobes, andsteeringthebeaminawrongdirection.Thedegradationcanbe particularly severe for high-performance arrays designed to produce low sidelobes or narrow beam-width. For example, in satellite communications, where high directivity and low sidelobes are often required, degradation of the radiation pattern will result in requiring higher transmit power or causing interference to neighboring satellites, both of which are undesirable. The sources of these errors can be many ranging from those induced by environmental changes, mechanical variations, assembly inaccuracies, mutual coupling effects, etc. to those caused by mistuned or failed amplifiers and phase shifters. In this paper, we discuss a novel idea of correcting for the errors in the excitation coefficients of an array

P

Manuscript received October 14, 2009; revised March 20, 2010; accepted April 05, 2010. Date of publication September 02, 2010; date of current version November 03, 2010. The authors are with the Department of Electrical & Computer Engineering, University of Massachusetts, Amherst, MA 01003 USA (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.2010.2071359

by dithering the magnitude and phase of the individual elements and observing the field at a near-zone probe. By dithering here is meant deliberately introducing pseudo-random fluctuations into the array coefficients and performing expectation of the observed signal. In digital picture [10] and audio technologies [11], a dither signal consisting of a high-frequency pseudo-random noise is added to the actual analog signal before it is quantized and then subtracted back after the quantizing operation. The result is that the non-linearities caused by quantization are spread out over all amplitudescales,resultinginamuchmorepleasingvideooraudio perception. Dithering and its effect on quantization noise has been studied in detail in [12] and [13]. The dithering process considered in [10], [11], [12] and [13] and the one considered here share a common feature in the sense that they all involve introducing pseudo-random noise to the signal (the array coefficients here) under consideration. However, the noise applied by us is neither additive nor subtractive and is utilized for the purpose of regularizing a matrix involved in the error minimization procedure. Furthermore, its use in the context of adaptively correcting the coefficients of an array deployed in field is novel and has not been considered before to the best of the authors’ knowledge. The departure of the field from that produced by the desired array (the reference field) at one or more near-zone sensors is observed and corrected using an error minimization scheme in a manner similar to an adaptive array [14]. If the random fluctuations introduced vary at a rate faster than the rate of fluctuations of the array coefficients, the array can be made to continuously remain in sync with the desired array. Single or multiple near-zone probes have been employed in the past by several researchers to calibrate a phased array [15]–[17], and [18]. In the context of the present paper, the techniques considered in these works can be employed to efficiently generate the reference field. A calibration scheme for digital beamforming arrays using multiple near-field sensors has recently been presented in [19]. The references therein describe some of the other work carried out using near-field sensors. The novelty of the present approach is to use a near-field sensors in conjunction with dithering. The main advantage of the methodology presented in this paper is that it facilitates an adaptive correction to the coefficients so that the array is made to track a given design in near-real time. Furthermore, the correction is done simultaneously for all of the elements instead of the successive approach employed in some of the previous works. There need not be any relation between the random fluctuations introduced by us during the dithering process and the errors that exist in the array excitation coefficients, keeping in mind that the sole purpose of dithering and averaging is to correct for errors in the excitation coefficients. Still it is worthwhile

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JANASWAMY et al.: ADAPTIVE CORRECTION TO ARRAY COEFFICIENTS THROUGH DITHERING AND NEAR-FIELD SENSING

to compare our choice for the statistics with those employed in the antenna tolerance theory. Earlier works [20] dealing with errors introduced by tolerances in reflector antenna surface assumed Gaussian distributions for the phase to facilitate analysis. In the arena of antenna arrays, Gaussian distribution for phase has been assumed in [21] and uniform distribution has previously been assumed in [5] and [22], once again to facilitate analysis. However, [22] has shown that the uniform distribution in phase error is consistent with measurements made directly on the feed network of the antenna array and that phase errors with a maximum deviation of up to 14 are not uncommon. A uniform distribution in phase has also been assumed in [23]. In the present study we assume a uniform distribution in phase over an . As for the magnitude distributions, Ruze [20] interval assumed Gaussian distribution on a linear scale, while [5] and is the magnitude of [23] assumed a uniform distribution–if the th element, the magnitude with random errors is assumed , where is assumed to be Gaussian of the form . Note distributed or lie uniformly in the range that, strictly speaking, the Gaussian distribution is incorrect because it results in negative magnitudes with a finite probability and hence is only physical when the variance is small. There appears to be no preference of one choice over the other, other than the ability of a particular model to produce analytical results. What appears to be more important in these works is to gain an understanding on the deterioration in the average sidelobe level caused by these random errors, available through analytical formulas. In the present work we will assume a log-normal distribution for the dithering magnitudes such that they are of , where is a normal random variable, and the form depends on the RMS (Root Mean Square) deviation. For small , our model reduces to that employed in [20]. We do not anticipate our technique to not work with other choices of the magnitude and phase fluctuations. The near-zone sensor is assumed to sample the magnetic field, although the theory developed is equally valid for an electric field sensor. Correction for the array coefficients is achieved by employing a gradient based error minimization scheme. Theory is developed in Section II for both the noise-free and additive white noise cases. Finally, numerical results for a sample array with randomly affected magnitude and phase are presented in Section III to demonstrate the robustness of the algorithm. It is to be noted that the error minimization scheme employed in the paper is based primarily on the quadratic nature of the error function with respect to the array coefficients. In this regard, the algorithm presented here is equally applicable to non-linear (in spacing and geometry), planar, 3D conformal arrays or arrays with mutual coupling. For convenience and simplicity of analysis, we demonstrate the main idea behind our approach by first considering a uniform linear array comprised of Hertzian dipoles and a single near-zone sensor. The effect of mutual coupling on the performance of the approach is later demonstrated by considering a linear array of broadside coupled half-wave dipoles. II. THEORY Consider a linear array comprised of Hertzian dipoles arranged along the -axis with an inter-element spacing of as shown in Fig. 1. The axes of the dipoles are assumed to lie along

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the -axis and the total number of elements is denoted by . The normalized complex current excitation coefficient of the -th , where and are the element is denoted by magnitude and phase, respectively. An time convention is assumed, where is the radian frequency of operation and is the time variable. Treating the array as an aperture in the -plane, the total magnetic field vector for is given by [24]

(1) where , with being the entire term inside the summation sign, but excluding the factor , and represents the vector magnetic field due to a unit amwith current plitude Hertzian dipole located at , a prime denotes transpose, denotes a constant current amplitude, is the free-space wavenumber at the wavelength corresponding to is the -coordinate of the -th element, is the total length of the array, is the distance of the observation point from the -th element, and and are unit vectors in the - and -directions respectively. Note that the nature of the element is contained only in the terms . The array coefficients, , are usually designed to meet certain specifications on the far-field properties of the antenna such as its beam angle, its directivity, in or its side-lobe level. For convenience, we choose the subsequent development. Due to a variety of reasons, the array coefficients can get altered over time and we denote these with the corresponding magnetic as . In the following, we label the array with coeffield ficients as the true array (or the desired array) and that with as the actual array. Our objective is to devise a means for automatically correcting for the coefficients . To this end, we deliberately introduce noise-like fluctuations into the magnitude and phase of the array coefficients. This is done for both the true array and the actual array. We assume log-normal distribution with a standard deviation of dB for the magnitude and a uniform distribution with a maximum deviation of for the phase. Accordingly, we set the fluctuating magnitudes and phases of the true array as (2) (3) where is a unit-variance, zero mean Gaussian random variis a uniform random variable over . The able, and log-normal distribution employed here will rightly yield positive values for the magnitudes, in contrast to the Gaussian-on-alinear scale model that is prevalent in the antenna tolerance theory literature [20], [25], ([26], Ch. 12), and ([9], Ch. 7). Note that the noise applied is non-linear and does not appear as a additive term in the magnetic field expression (1). It is assumed that the magnitude and phase fluctuations are independent of each other and further that the fluctuations are independent from element to element. We denote the expectation with respect to these

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where a superscript denotes complex conjugation. The error signal will be a quadratic function of the array coefficients as can be easily verified by evaluating the following quantities:

(9) (10) where

is a notation for the

th element of a matrix and

(11) Substituting these expressions, the error signal can be expressed as Fig. 1. A uniform linear array comprised of Hertzian dipoles with a near-field sensor.

(12) where

fluctuations by the symbol ations, , is equal to

. The variance in the angle fluctu(13) Letting as

, the

th element of

can be written

(4) (14) , meaning that the peak From this it is evident that times the standard deviation. It can also deviation in angle is be easily verified that [27] (5) (6) (7) We label the coefficients pertaining to the true array as the dithered coefficients. The actual array coefficients are also dithered similarly and relations similar to (2) and and . The dithered fields due to the true array (3) hold for and the actual array are assumed to be observed at a near field sensor as shown in Fig. 1. These dithered fields are identified with a subscript on and .

A. Noise-Free Case We first consider the ideal situation of a receiver with no noise. An error signal based on the dithered signals is defined as

(8)

and Thus the error matrix is strictly convex in the variables gradient based algorithms are naturally suited for reducing the error starting from an arbitrary initial point. The quantities and are both positive with . Note that bold letters are used to indicate both vectors and matricies and the dot product in (12) is assumed to apply over vector quantities. Evidently, the is Hermitian. Another convenient form for is to matrix write it as

(15) where

is an diagonal matrix with elements along its principal diagonal and the superscript represents Hermitian conjugate. With no dithering (i.e., ), the matrix is simply seen to be with . In the noise-free case, the fields if ; consequently the error signal as can be clearly seen from (15). Hence the error signal will have a minimum at the true coefficients and a gradient based algorithm can be devised to nullify unwanted deviations. We follow the spirit of the LMS (Least Mean Square) algorithm [14], which is based on minimizing the error signal. Such a minimization takes place when the coefficients are corrected in the direction of the gradients of the error signal with respect to the actual at coefficients. Accordingly, we suppose the coefficients

JANASWAMY et al.: ADAPTIVE CORRECTION TO ARRAY COEFFICIENTS THROUGH DITHERING AND NEAR-FIELD SENSING

iteration as

to be related to the coefficients

at iteration

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The elements of the Hermitian matrix are seen to depend only on the dithering statistics, the free-space fields of the various elements, and the coefficients of the true array. Further, it is evident from (22) that is also positive definite. Hence its eigenvalues are all real and positive. Equation (22) is yet another form suitable for practical implementation of the dithering algorithm. In a matrix form, (21) reads

(16) (23) for

, where

is a positive real number and

(17)

where is an identity matrix of order . In order for the system , we need , where in (23) to converge as is the largest eigenvalue of the matrix . This requirement then implies that

is a notation for a complete complex derivative. The following results are easy to derive:

(24)

(18)

When this criterion for is met, the actual coefficients converge exponentially to the true values as the iteration progresses. B. Receiver With White Gaussian Noise

Combining (12), (13) with (18), it is straightforward to see that

(19) where is Kronecker’s delta. The correction term in (16) is then proportional to

(20) where is the th component of with . Thus the algorithm needs all states of the total component of the dithered field of the actual array at the sensor as well as all the states of the individual element fields of the true array. The latter can be generated once apriori in a controlled environment and then stored. The parameter has to be chosen appropriately so that the iterations do not diverge. To investigate this further, it is more convenient to look at the correction vector . From (16) and (19), it is clear that

Presence of receiver noise can have an impact on the effectiveness of the algorithm. To investigate this, we consider additive white Gaussian noise corrupting the actual signal. For ease of analysis, we treat the noise as if it originates in the array and received at the noise-free near-field sensor through the array coefficients. The noise considered here is assumed to be (i) zero mean, (ii) independent of the dithering process, and (iii) independent from element to element of the array. Furthermore, the noise fluctuations are assumed to take place much more rapidly than the dithering process. Consequently, the averaging times involved in carrying out the expectations of the noise processes are much shorter than those involving the dithering process. We shall use a symbol to denote expectation with respect to the white noise. The actual received signal is now written as , where is a complex-Gaussian noise vector generated at the array. Like the Green’s function , it will have - and - components and each entry of the column vector of the components is assumed to have a variance . Likewise the true signal is assumed to be corrupted by noise to . Note the corrupting noise for the acresult in tual and true received signals is distinguished by the presence of hat on the former. However, they will have the identical statiswill have a tics. Further note that the difference signal . The error signal noise floor even when in this case is

(21)

(25)

(22)

where we have used the fact that the expectation operator operates only on the noise related quantities and that , and , where is the noise power generated at each antenna element. The factor of 2 arises in the noise power because both the - and -components

where

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of contribute to it. From (25) it is clear that the component of is the gradient with respect to

and the signal-noise ratio Hence where we denote by the noise-to-signal ratio. From (31)

,

(26) (32) Note that, in contrast to the noise-free case, the error signal and its gradient do not vanish when . The gradient will, instead, vanish at another point in the variable space that is determined by the amount of noise power. As with the noise-free case, we write the following iteration equations for the array coefficients and their corresponding correction terms

is the smallest eigenvalue of and the second inwhere norm . Thereequality follows from the definition of fore the limiting value of the fractional residual error is upper bounded by

(33) (27)

(28) In a matrix form, (28) can be written as

where is the condition number of the matrix . The two terms in (29) offer competing trends–the first term decreases, while the second term increases as increases. Hence for sufficiently large signal-to-noise ratios, we expect the fractional residual error to first decrease, but eventually increase as the iteration in (29) progresses. It is to be noted from (33) that the convergence of the algorithm is strongly dependent on the condition number of the matrix in addition to the signal to noise ratio. C. Effect of Mutual Coupling

(29) where

(30) In the limit as

, one gets

(31) , where, as if is chosen such that before, is the largest eigenvalue of the matrix . Thus the actual array coefficients do not converge to the true coefficients, . At these coefficients, the error but instead to signal will have zero gradients. In order to assess the effect of noise quantitatively and to estimate its influence on the rate of convergence of the coefficients on the iterative procedure (28), we first need to define the signal power and the related signal to noise ratio. Using the representation shown in (15) and the definition of the matrix elements in (22), it can be shown that the mean signal power of the actual . Furthermore it is clear from (25) that array is the noise power in the receiver when the actual signal is mea. Observing that both powers consured is equal to tain the common pre- and post multiplicative factors of the form , we define the signal power, , as , where of a square matrix denotes its Euclidean norm and is equal to its largest eigenvalue [28], and the noise power .

The inevitable presence of element mutual coupling in a non-ideal array will not change the essence of the algorithm, although it will alter the characteristics of the matrix . We demonstrate the effect of mutual coupling by replacing the Hertzian dipoles with identical, center-fed half-wave dipoles of physical length and wire radius . The axis of the dipoles is still assumed to be along the -axis. The presence of mutual coupling on a transmit array can be treated in a manner similar to that in [29]. If the source impedance at the th element is and the array impedance matrix is denoted denoted by , where is a diagonal matrix by being the input impedance of an isolated with entries half-wave dipole, the coupling matrix relating the port currents with mutual coupling with those without mutual couis determined as . pling is equal to , where is the source The matrix diagonal matrix with entries . The mutual impedance matrix , which depends on the array geometry and the number of elements , can be easily generated by formulating an integral equation for the finite array and solving it by the method of moments [30]. Alternatively, it can be generated approximately by using analytical formulas that are available [30] for the mutual impedance between two separated dipoles. The magnetic field with mutual coupling is still given by (1) with replaced by . This has the effect of transforming the matrix to (34) where of a matrix

and extracts the principal diagonal . Of course in the absence of mutual coupling , the coupling matrix becomes an identity matrix

JANASWAMY et al.: ADAPTIVE CORRECTION TO ARRAY COEFFICIENTS THROUGH DITHERING AND NEAR-FIELD SENSING

Fig. 2. True, actual and corrected array amplitudes.

and one recovers (22). The rest of the steps in the algorithm remain the same as in the no mutual coupling case. III. NUMERICAL RESULTS dB sidelobe, broadside Results are presented below for a Taylor array with 32 elements to demonstrate the idea. The inter. The total length of the element spacing is chosen to be array is and the minimum far-zone distance . The aperture distribution, , versus element number is shown in Fig. 2 as a solid line. For the purpose of illustration, we perturb the true coefficients randomly with the magnitude varied on a dB scale using Gaussian fluctuations with an RMS deviation of 2 dB and the phase varied uniformly with an RMS deviation of 10 . The real and imaginary parts of the actual coefficients are also shown in Fig. 2 as dashed lines. The far-zone magnetic field strength for the true and actual array is shown in Fig. 3 as a function of lateral displacement for and . Clearly, the sidelobes have increased substantially and the mainlobe slightly broadened as a result of the fluctuations introduced. The actual array has a sidelobe dB, whereas the true array has a value level in excess of dB. of A near-field sensor is assumed to be located in the plane at and . dB The true and actual coefficients are dithered using . The actual and true near fields with and without and dithering are shown in Fig. 4. One effect of dithering is to raise the field levels in both the actual and true arrays and decrease the dynamic range of the signal variation. In a sense, dithering induces some spatial correlation of field fluctuation. The above and choice of parameters results in and . The maximum value of as per (24) is calculated to be and a value of was used to run the algorithm (22). The algorithm was terminated when reached 0.2% of . In practice,

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Fig. 3. True and actual far-zone magnetic fields of a linear array. A broadside, 25 dB sidelobe Taylor array comprised of Hertzian dipoles is assumed.

0

the algorithm may be terminated by considering errors in successive iterations. The initial 2-norm of the residual error was . The algorithm converged in iterations and the converged solution is also shown in Fig. 2 as a dashed-dot line. The converged coefficients are virtually indistinguishable from the true coefficients. Fig. 5 shows the effect of signal-to-noise ratio (SNR) on the residual error. The residual error for the noise-free case decreases exponentially with the iteration number , while it saturates to a finite value for the noisy case. The 30 dB SNR exhibits the situation where the benefits of large iteration number are felt initially, only to be overwhelmed by increasing contributions due to the noise term for large . The residual error is dB. It is seen that for this case, there is no benaround efit of increasing the number of iteration beyond about 500. The corrected coefficients along with the true and the actual coefficients are shown in Fig. 6. It is seen that the phase has been recovered very well, but the magnitudes have not converged to the true solution, even though the huge excursions present in the actual coefficients have been significantly reduced as a result of the dithering algorithm. Not surprisingly, the agreement is better for those elements of the array that are closer to the sensor. This may suggest for a more symmetric placement of sensors than the one deployed here. The corresponding far-zone pattern for the corrected coefficients is compared in Fig. 7 with the true pattern. By comparing with Fig. 3, it is seen that the even though the array coefficients have not been fully corrected, the sidelobes in the actual array have been lowered significantly by the dithering algorithm. The corrected and actual arrays has a sidelobe level dB and dB respectively. of Results with mutual coupling (indicated by Mut Coup in the chosen to be figure legend) are also shown in Fig. 5 with . For the dipole antennas, we chose the complex conjugate of and . The inter-element spacing remained . The impedance matrix was calculated by using at analytical formulas available for the mutual impedance between

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Fig. 5. Residual error as a function of iteration number for various signal-noise ratios for the 25 dB broadside Taylor array. Other parameters chosen are y = 4:8; x = 1:1L; 2L = 15:5;  = 3 dB and  = 12 .

0

Fig. 4. Magnetic fields at a near-zone sensor with and without dithering. Location of the near-field sensor is indicated by the dashed vertical line. (a) Undithered. (b) With dithered magnitudes and phases.

two separated dipoles. It is seen from Fig. 5 that for the noisefree case, mutual coupling has the effect of slowing the roll-off of error with iteration number, although the error continues to decrease monotonically. For the noisy case with an SNR of 30 dB, mutual coupling has the effect of slightly raising the error saturation floor. The converged array coefficients in this case are slightly inferior to those shown with no mutual coupling in Fig. 6. Overall, mutual coupling has the effect of slowing the convergence of the algorithm in both cases. The convergence rate and the residual error of the algorithm with ideal dipoles depends on the dithering parameters and . In general, larger values of and result in faster convergence with lower residual error. Conversely, the algorithm did not con. The convergence verge at all for no dithering

Fig. 6. True, actual and corrected array amplitudes for  = dB,  = 12 , and y = 4:8 .

030 dB.  = 3

rate also depended on the choice of , with faster convergence achieved for larger . For the SNR of 30 dB example considered above, the residual error after 500 iterations is decreased to dB when dithering was performed with dB, , and , all other parameters remaining constant. These results suggest that the deterioration of convergence of the algorithm caused by mutual coupling can be taken care of by optimizing the dither parameters and location of the sensor. The estimate for the upper bound in the residual error provided by (33) dB. The corrected coefficients and the corresponding is far-zone patterns are shown in Figs. 8 and Fig. 9 respectively. It is seen that the dithering algorithm has performed much better

JANASWAMY et al.: ADAPTIVE CORRECTION TO ARRAY COEFFICIENTS THROUGH DITHERING AND NEAR-FIELD SENSING

Fig. 7. True and corrected far-zone patterns after k dB,  , and y : . dB, 

030

=3

= 12

=48

Fig. 8. True and corrected array amplitudes for  , and y : . 

= 15

= 96

= 500 iterations for  =

= 030 dB  = 4 dB,

when compared to the values considered in Fig. 6. The condition number of the matrix is reduced to 209 for the parameters chosen here as opposed to a value 708 for the parameters chosen in Fig. 6. Hence for the same SNR, the algorithm performs better here. More detailed studies need to be done, however, before the algorithm can be optimized with respect to these parameters. To gain a perspective into signal power involved and the order of the SNRs achievable, let us consider some practical numbers. Assume that the near field sensor has a field coupling factor of (the sensor couples the field ). For an antenna current of mA, the signal power received in the sensor is , where that was made equal we have included back the factor to unity in the analysis. Assuming thermal noise in the receiver

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= 500 iterations for  =

Fig. 9. True and corrected far-zone patterns after k , and y : . dB with  dB, 

030

=4

= 15

=96

and a receiver noise figure of , the available noise power in is , a receiver bandwidth of is the Boltzman’s constant. For some realistic values where K, MHz, of the signal and noise powers are W, dBm. Using and for the parameters considered in Fig. 2, we get an SNR of 50.7 dB for every mA of the drive current on the dipoles. The SNR of 30 dB assumed in Fig. 6 is very pessimistic in this sense. The error gradient used in all of the plots shown thus far was obtained analytically in terms of the matrix . In practical arrays, one may want to implement the ensemble mean in expression (20) by means of Monte Carlo averaging. Fig. 10 shows with respect to the number the behavior of the gradient of realizations used in the averaging process. Results are shown for the first and the last element of the array. It appears that reasonable results could be obtained using about one thousand realizations. In general, more realizations are need for stronger dithering (larger and/or larger ), which partially offsets the advantage offered by needing fewer number of iterations in the correction process. When the error minimization process was carried out with no dithering, the algorithm did not correct for the amplitudes at all. A spectral analysis of the matrix revealed that its largest remained roughly the same as with eigenvalue of dithering. However, the condition number of the matrix jumped from its dithered value of 708. Hence from a to purely numerical standpoint, dithering has the effect of clustering the eigenvalues, thereby making more degrees of freedom available to the minimization procedure, and making it more immune to noise fluctuations. A second version of the algorithm was attempted with an error function defined as , which would require the storage of fewer field quantities. However, the algorithm did not converge at all.

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the algorithm to converge and 1 000 realizations per iteration to carry out the expectation, we estimate that the current algorithm would be able to track changes in the coefficients that vary at most at a rate of 1 Hz if the time per iteration is taken as 1 ms and the time per realization during the expectation operation is taken as 1 s. The LMS algorithm also needs an estimate of once during the largest eigenvalue of the Hermitian matrix the correction procedure. Although the current paper has not addressed the computational aspects of finding the dominant eigenvalue of such matrices, efficient schemes are available in the literature for determining it [31]. Although the paper considered the example of a practical array, several other practical and implementation issues remain unanswered. Extension to planar arrays as well as exploring other correction schemes such as those based on RLS (Recursive Least Square) or Kalman filtering are worth pursuing and will be the subject of investigation of a future paper. Furthermore, development of schemes for the efficient generation of the reference signal are highly desirable and the work of a recent study [18] is relevant in that regard. Speed-up of the algorithm by the consideration of multiple near-field sensors is another topic of further study and will also be taken up in the future. REFERENCES

Fig. 10. Implementation of the error gradient (@)=(@ c^ ) in (20) for j = 1 (farthest from the sensor) and j = N (closest to the sensor) by numerical averaging. Exact values obtained using (16) and (19) are shown by the dashed lines. (a) j = 1. (b) j = N .

IV. CONCLUSION An algorithm for automatically tracking the desired performance of an antenna array by dithering its coefficients and observing its field in the near-zone has been proposed and demonstrated by considering a uniform linear array comprised of Hertzian dipoles. An LMS type algorithm has been presented for correcting for the coefficients both in a noise-free and noisy environments. The robustness of the algorithm has been demonstrated by considering a realistic low-sidelobe, broadside array whose array coefficients experienced 2 dB RMS magnitude fluctuations and 10 RMS phase fluctuations. Mutual coupling has been shown to not change the essential character of the algorithm even though it deteriorates its performance somewhat. Assuming that one needs 1 000 iterations for

[1] M. I. Skolnik, Introduction to Radar Systems. New York: McGraw Hill, 2001. [2] D. C. Schleher, Introduction to Electronic Warfare. Boston, MA: Artech House, 1986. [3] J. J. C. Liberti and T. S. Rappaport, Smart Antennas for Wireless Communications. Upper Saddle River, NJ: Prentice Hall, 1999. [4] R. Janaswamy, Radiowave Propagation and Smart Antennas for Wireless Communications. Norwell, MA: Kluwer, 2001. [5] L. A. Rondinelli, “Effect of random errors on the performance of antenna arrays of many elements,” IRE Nat. Conv. Rec., Pt. I, vol. 7, pp. 174–189, 1959. [6] H. Jasik, “Fundamentals of Antennas,” in Antenna Engineering Handbook. New York: McGraw-Hill, 1961. [7] B. D. Steinberg, Principles of Aperture and Array System Design. New York: Wiley, 1976. [8] W. H. Kummer, “Basic array theory,” Proc. IEEE, vol. 80, no. 127–140, Jan. 1992. [9] R. J. Mailloux, Phased Array Antenna Handbook, 2nd ed. Boston: Artech House, 2005. [10] L. G. Roberts, “Picture coding using pseudo-random noise,” IRE Trans. Info. Theory, vol. 8, pp. 145–154, Feb. 1962. [11] K. C. Pohlmann, Principles of Digital Audio. Indianopolis, IN: Howard W. Sams & Co., 1985. [12] L. Schuchman, “Dither signals and their effect on quantization noise,” IEEE Trans. Commun. Technology, pp. 162–165, Dec. 1964. [13] G. Zames and N. A. Shneydor, “Dither in nonlinear systems,” IEEE Trans. Automat. Control, vol. AC-21, pp. 660–667, Oct. 1976. [14] D. H. Johnson and D. E. Dudgeon, Array Signal Processing: Concepts and Techniques. Upper Saddle River, NJ: Prentice Hall, 1993. [15] D. K. Alexander and J. R. P. Gray, “Computer-aided fault determination for an advanced phased array antenna,” in Proc. Antenna Applications Symp., Sep. 26–28, 1979, pp. 1–13. [16] H. M. Aumann, A. J. Fenn, and F. G. Willwerth, “Phased array antenna calibration and pattern prediction using mutual coupling measurements,” IEEE Trans. Antennas Propag., vol. 37, no. 7, pp. 844–850, July 1989. [17] M. Sarcione, J. Mulcahey, D. Schmidt, K. Chang, M. Russell, R. Enzmann, P. Rawlinson, W. Guzak, R. Howard, and M. Mitchell, “The design, development and testing of the THAAD solid state phased array,” presented at the IEEE Int. Symp. Phased Array Systems and Technology, Oct. 15–18, 1996. [18] T. Takahasha, Y. Konoshi, S. Makino, H. Ohmine, and H. Nakaguro, “Fast measurement technique for phased array calibration,” IEEE Trans. Antennas Propag., vol. 56, no. 7, pp. 1888–1899, Jul. 2008.

JANASWAMY et al.: ADAPTIVE CORRECTION TO ARRAY COEFFICIENTS THROUGH DITHERING AND NEAR-FIELD SENSING

[19] H. Pawlak and A. F. Jacob, “An external calibration scheme for DBF antenna arrays,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 59–67, Jan. 2010. [20] J. Ruze, “Antenna tolerance theory—A review,” Proc. IEEE, vol. 54, no. 4, pp. 633–640, 1966. [21] F.-I. Tseng and D. K. Cheng, “Gain optimization for arbitrary antenna arrays subject to random fluctuations,” IEEE Trans. Antennas Propag., vol. AP-15, no. 356–366, May 1967. [22] B. Kulke, “Dependence of sidelobe level on random phase error in a linear array antenna,” IEEE Trans. Antennas Propag., vol. AP-21, no. 4, pp. 569–571, 1973. [23] D. J. Ramsdale and R. A. Howerton, “Effect of element failure and random errors in amplitude and phase on the sidelobe level attainable with a linear array,” J. Acoust. Soc. Am., vol. 68, no. 3, pp. 901–908, 1980. [24] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1965. [25] R. E. Collin and F. J. Zucker, “Nonuniform Arrays, M. I. Skolnik,” in Antenna Theory, Pt. 1. New York: McGraw-Hill, 1969. [26] R. C. Hansen, Phased Array Antennas. New York, NY: John Wiley & Sons, 1998. [27] W. Feller, An Introduction to Probability Theory and Its Applications. New York: Wiley, 1966, vol. II. [28] R. A. Horn and C. R. Johnson, Matrix Analysis. New York, NY: Cambridge University Press, 1985. [29] R. Janaswamy, “Effect of element mutual coupling on the capacity of fixed length linear arrays,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 157–160, 2002. [30] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [31] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: Johns Hopkins, 1983.

Ramakrishna Janaswamy (F’02) received the Ph.D. degree in electrical engineering from the University of Massachusetts, Amherst, in 1986, the Master’s degree in microwave and radar engineering from IIT-Kharagpur, India, in 1983, and the Bachelor’s degree in electronics and communications engineering from REC-Warangal, India in 1981. From August 1986 to May 1987, he was an Assistant Professor of electrical engineering at Wilkes University, Wilkes Barre, PA. From August 1987–August 2001, he was on the faculty of the Department of Electrical and Computer Engineering, Naval Postgraduate School, Monterey, CA. In September 2001 he joined the Department of Electrical & Computer Engineering, University of Massachusetts, Amherst, where he is a currently a Professor. His research interests include deterministic and stochastic radio wave propagation modeling, analytical and computational electromagnetics, antenna theory and design, and wireless communications. Prof. Janaswamy is a Fellow of IEEE and was the recipient of the R. W. P. King Prize Paper Award of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION in 1995. For his services to the IEEE Monterey Bay Subsection, he received the IEEE 3rd Millennium Medal from the Santa Clara Valley Section in 2000. He is an elected member of U.S. National Committee of International Union of Radio Science, Commissions B and F. He served as an Associate Editor of Radio Science from January 1999–January 2004 and Associate Editor of IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY from 2003–2006. He is the author of the book Radiowave Propagation and Smart Antennas for Wireless Communications (Kluwer Academic, 2000) and a contributing author in the Handbook of Antennas in Wireless Communications, (CRC Press, 2001) and the Encyclopedia of RF and Microwave Engineering (Wiley, 2005).

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Dev Vrat Gupta received the Ph.D. degree in electrical engineering from the University of Massachusetts, Amherst (UMass), in 1977. He then joined Bell Laboratories and worked for eight years on advanced development projects building digital transmission systems, first as a Member of Technical Staff in the N. Andover, Massachusetts facility, and later as Supervisory Staff. In 1985 he joined a startup called Integrated Network Corporation in NJ and invented many data communication and multimedia technologies. In 1995, he started a company named Dagaz Technologies which designed and manufactured telephone exchange equipment that is used to provide DSL services to subscribers. This equipment terminates and multiplexes Digital Subscriber Loop wires (DSL loops) and connects the multiplexed output to the internet. The company was acquired by Cisco Systems in 1997. In 1998, he started Maxcomm Technologies in Chelmford, MA. Maxcomm designed and manufactured voice and data service multiplexing equipment for DSL loops. Cisco Systems bought Maxcomm in 1999. In 2000, he started Narad Network in Westford, MA. Narad provides Gigabit Ethernet Networking equipment to Cable Television companies so that they can provide Broadband Data services to medium and small businesses which are superior to those that are provided by the telephone company. By 2002, Narad had grown to over 150 employees but during the technology downturn the staffing had to be reduced. Narad still employs about 40 people and its equipment is in use at Time Warner Cable, Comcast and some international cable television service providers. In 2003, he started Newlans, Inc. in Westford, MA. Newlans is designing Wireless Gigabit Ethernet Local Area Networking equipment for use in American and international business premises. With Newlans technology, networking can be provided ubiquitously in the enterprise, without wires at data rates up to 1 Gbps. Finally, both the Capital Expense (CAPEX) and Operating Expense (OPEX) associated with providing networking services in the business’ facility, are greatly reduced by the technology. Dr. Gupta was name a Tech. Pioneer by The World Economic Forum for the years 2001 and 2002. He endowed the Gupta Chair at the Dept. of Electrical and Computer Engineering at UMass, is a Trustee at UMass, and a board member of the UMass. Foundation.

Daniel H. Schaubert (S’68–M’74–SM’79–F’89) received the Ph.D. degree in electrical engineering from the University of Illinois. He worked at the US Army Research Laboratory and the US Food and Drug Administration prior to joining the University of Massachusetts in 1982. Currently, he is Professor Emeritus of Electrical Engineering and Director of the Center for Advanced Sensor and Communication Antennas, a joint venture of the University of Massachusetts and the US Air Force Research Laboratory. His contributions have been mainly in the areas of antenna design and analysis. He has patents for conformal and printed circuit antennas. Several of his antenna designs have been used in military and civilian systems for radars, radiometers and communications, and he has designed low-cost antennas for commercial cellular and local area network products. He directed the design, fabrication and testing of antennas for the Cloud Profiling Radar System, a polarimetric 33-GHz and 95-GHz mobile radar, and the High-Altitude Wind and Rain Profiler, a dual-beam 13-GHz and 35-GHz airborne radar. He led the design efforts for several multioctave scanning array antennas, including the first prototypes for the Thousand Element Array demonstrator of the Square Kilometer Array project. Dr. Schaubert was President of the IEEE Antennas and Propagation Society, an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, Secretary-Treasurer of the Society, Newsletter Editor and Membership Chairman. He organizes the annual Antenna Applications Symposium. He received the John Kraus Antenna Award in 2008, the H. A. Wheeler Prize Applications Paper Award in 1997 and the IEEE Third Millennium Medal. He is listed in Who’s Who in America, American Men and Women of Science, Who’s Who in Science and Engineering, and Men of Achievement. He was an advisor to the European Antenna Centre of Excellence and is a member of the executive team for the IET Antenna and Propagation Professional Network.

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Ultrawideband All-Metal Flared-Notch Array Radiator Rick W. Kindt and William R. Pickles

Abstract—Simulations and measurements are presented for an all-metal flared-notch array element in both single and dual-polarization configurations. The ultrawideband radiator exhibits an operational bandwidth of 12:1 for broadside scan and 8:1 bandwidth at a 45-degree scan in all planes, maintaining active . The feed consists of a direct coax-to-slot-line transition that mounts directly into the base of the radiator. The all-metal flared-notches are machined from common metal stock and fed via SMA coaxial connectors. No soldering is required for any part of the design—including the feed—and assembly is simple and modular. The array parts are machined using a high-precision wire-EDM cutting technology, ensuring that measurements (in the 700 MHz–9 GHz range) are repeatable and give close agreement with theory, even through multiple assembly cycles of the modular construction system. This paper presents results for a 32-element linear array of horizontal elements and also an 8 8 planar array of dual-polarized elements, comparing measurements with full-wave simulations of the complete finite array structures.

VSWR 2

Index Terms—Domain decomposition, finite element method, phased array, tapered-slot, thick flared-notch element, ultrawideband arrays, Vivaldi array.

I. INTRODUCTION

T

HE flared-notch (Vivaldi) is a well-known element in ultrawideband (UWB) phased arrays [1], [2]. This radiator has seen wide-spread use for over 30 years and remains popular in part due to its robustness in design and range of manufacturability. Part of its popularity stems from the ease with which inexpensive printed-circuit board (PCB) manufacturing can be used to create large arrays [3], [4]. Over the years, many alternatives to traditional flared-notch designs have been proposed. The balanced antipodal Vivaldi antenna (BAVA) is related to the traditional flared notch but features a somewhat different feed mechanism [5]. As an alternative to PCB designs, all-metal flarednotch radiators provide a different range of manufacturing options and applications [6]–[9]. Typically the flared-notch class of radiator is considered electrically long—normally a wavelength or more at the highest frequency of operation when a bandwidth of 3:1 or greater is desired. However, in recent years, a class of reduced-height radiators has been proposed for which the element length is closer to half a wavelength from the array

backplane at the highest frequency of operation. Some of these elements are vertically-integrated such as the reduced-height BAVA [10] and the bunny ear [11], [12]. UWB designs have also been demonstrated using planar printed surface (or layered substrate) technology, some examples including fragmented apertures [13], [14] and planar dipoles [15], [16]. In general, lowerprofile designs exhibit better polarization purity for scans in the D-plane, though VSWR levels tend to be relatively higher than classic flared-notch designs. In this paper, an all-metal flared-notch radiator is presented that has a number of unique features. For broadside radiation, the UWB radiator exhibits an operational bandwidth of 12:1 and has for a large portion of the with frequency range [17]. At a 45-degree scan in all planes, the defor a full 8:1 bandwidth and 10:1 sign maintains is permissible. Structurally, the elebandwidth if ment is much thicker than typical flared-notch implementations (approximately 1/3 thickness-to-width ratio). The advantages of thick flared-notch radiators have previously been reported in [18]. The feed design consists of a direct coax-to-slot-line transition that operates over a 12:1 frequency range or better. The elements are fed via standard SMA coax connectors. No soldering is required for any part of the design, and assembly is simple and modular. In the presented implementation, the parts are machined using a high-precision wire-EDM cutting technology that achieves better than 10-mil tolerances for all dimensions of the machined parts. This precision helps to ensure that measurements (in the 700 MHz–9 GHz range) are repeatable and give close agreement with theory, even through multiple assembly cycles of the modular construction system. The remainder of the paper covers the design, simulation, and measurement of the array element performance. In Section II, the details of the mechanical and electrical design are presented. In Section III, performance characteristics of the radiator are given, including active VSWR, port isolation, array mismatch efficiency, element gain, and polarization purity. Results are presented for a 32-element linear array and also a dual-polarized 8 8 planar array of elements. Full-wave simulations of the entire finite array structures show close agreement with measurements. II. MECHANICALAND ELECTRICAL DESIGN

Manuscript received November 03, 2009; revised March 23, 2010; accepted May 09, 2010. Date of publication September 02, 2010; date of current version November 03, 2010. This work was supported by the U.S. Office of Naval Research in 2008-2009. The authors are with the Naval Research Laboratory, Washington, DC 20375 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.2010.2071360

The element design stems from an application with a need for 8:1 or better bandwidth [19]–[22]. Fig. 1 shows the profile of the flared-notch radiator, as constructed, listing the critical design parameters. The element is roughly a half-wavewide, thick, and long at the high end length of the frequency range. The actual element lattice spacing is -spacing at 7.4 GHz. For this project, 0.8’’, corresponding to

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Fig. 1. Dimensions of the flared-notch radiator; the taper meanders near the base of the element, meeting the slot-line cavity perpendicular to the direction of radiation to assist feed port insertion.

Fig. 2. 32-element linear array of flared notches; the array is constructed from modular sub-arrays of 8 elements.

two prototype arrays were built—a single polarization 32-element linear array shown in Fig. 2, and a dual-polarized 8 8 planar array of elements shown in Fig. 3. For conductivity and weight considerations, the elements are constructed entirely of 6061 Aluminum. The design is modular and machined to precise tolerances to facilitate easy reconfiguration, assembly and disassembly. For the linear array, the circuit path of the antenna is cut from 1/4’’-thick Aluminum sheet stock as a continuous trace along a single axis. In the case of the 8 8 dual-polarized array, the structure is made as a three-axis cut from a single block of metal. First, the square cavities created by the intersecting elements are drilled and cut in the xy-plane. Next, the circuit paths of the horizontal elements are cut as a continuous trace in the zx-plane, followed by the vertical elements cut as a single trace in the yz-plane. Though the dual-polarized array was made from a single block of material, it has been demonstrated that dual-polarized modules can also be constructed from machined 1/4’’ Aluminum sheet stock assembled in each polarization [21]. The 1/4’’ thickness was chosen because it is commonly available and thick enough to drill a 0.162’’ diameter hole directly into the back of the elements, through which a bulkhead SMA connector can be inserted and either pressed or screwed in place without the need for soldering. As a result, the elements are structurally much thicker than common embodiments of flared-notch designs, with a thickness of approximately 1/3 of the element width. The added thickness pushes undesirable impedance anomalies associated with the cavities (formed by the element intersections) well-above the highest frequency of operation. PCB-based flared-notch designs typically feature a feed based on a microstrip-to-slot-line transition using quarter-wave stubs as demonstrated in [24]. More wideband options include the Marchand balun [25] and the double-Y balun, which have been demonstrated to work over bandwidths of 5:1 or better [4]. The main disadvantage is that these techniques require a

2

Fig. 3. 8 8 dual-polarized planar array of flared-notches; 128 total ports, 64 in each polarization. The array is modular and can mate in any direction with arrays of the same size to create larger arrays.

two-step transition—from coax to microstrip/stripline to radiating slot. The design presented here transfers energy directly from coax to the radiating slot-line, as shown in Fig. 1. Fig. 4 shows the construction of the coaxial section of the feed port. This feed assembly is press-fit in place such that the center pin contacts the far wall of the slot, creating a short 96 mil from the slot-line cavity, to avoid the 81 mil-radius port hole breaching the cavity wall. A small, cylindrical spring-like device called a “fuzz-button” acts as an extension of the SMA center pin to create electrical contact with the cavity wall. The knurl of the assembled connector holds the port in place. Fig. 5 shows the base of the 8 8 array before and after the connectors are embedded. From the figure, it is clear the press-fit components create a very clean interface. Though the tolerances are not critical in the linear array, in the dual-polarized design the ports are spaced just close enough that a 5/16 SMA hex nut can turn without interference from the adjacent connector of the other polarization. It was determined through testing that more than 60 inch-lbs of torque could be applied to the connector without the knurl failing—far beyond the torque required to properly seat an SMA cable. In order to feed the coax straight into the base of the element, it is necessary to bend the slot such that it turns 90 degrees perpendicular to the direction of radiation. In part, this is difficult to do because of the cramped space within the element cell. To compensate, the slot must be meandered as shown in Fig. 1. It has been observed that the meandering slot helps alleviate some

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Fig. 7. Simulated infinite array VSWR of the dual-polarized element at a 60-degree scan angle shows VSWR below 3 for most of the operational frequency range. Fig. 4. Solder-free feed assembly for direct coax-to-slot-line transition (see Fig. 1 for placement).

Fig. 8. Simulated S12 (isolation) between vertical/horizontal ports, designed to be roughly better than 20 dB across band.

0

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Fig. 5. Backside of the 8 8 dual-polarized array (a) before and (b) after the ports are embedded. The straight-press connectors give a clean design.

III. RESULTS A. Simulated Results

Fig. 6. Simulated infinite array VSWR performance of the dual-polarized element at broadside (solid line) and 45-degree scans in three planes.

of the port coupling that occurs because of field asymmetry introduced by turning the slot 90-degrees to the direction of radiation. To ensure EMI isolation of the electronics behind the array, a back plate is used. By design, the cavity of the slot-line open is flush with the back plate of the array. This is done because the cavity geometry can excite resonances that interfere with the operation of the array. These impedance anomalies are different from the ones reported previously for single-polarized Vivaldi arrays [26]. It has been observed that moving the cavity further above the back plate causes these impedance anomalies to shift lower in frequency, reducing bandwidth. The size of the slot-line cavity and the position have been optimized to enhance low-end bandwidth. Further, the length and shape of the slot-line region have been adjusted to smooth the reactance of the impedance match across the frequency band. It is also important that the slot-line connects to the cavity at approximately the center, creating symmetry in the cavity fields. When the slot-line insertion point is moved relative to the cavity center, two resonant peaks can be identified in the VSWR response. By positioning the feed point correctly, the resonances partially cancel each other, creating a single peak in VSWR that is narrower and shorter (see spike in broadside VSWR at 9 GHz, shown in Fig. 6).

This section presents the predicted theoretical performance of the all-metal element design based on an infinite cell (Floquet) analysis [6], [27]. The analysis was performed using an in-house Navy code (CEMNAV-INF) based on the finite element method (FEM) that has been verified to give very similar results to commercial software [28]. The electromagnetic CAD models for the element simulations are very accurate representations of the mechanical models used for manufacturing the array. The reference plane of the coaxial feed port used in the simulations correlates approximately with the reference plane for the network analyzer cables used to collect S-parameter measurements. Fig. 6 shows the expected VSWR of the dual-polarized element in an infinite planar array for broadside radiation and scans to 45 degrees in the E-plane, H-plane, and D-plane. The results shown are for horizontal polarization with the vertical port passively terminated. At broadside radiation, a 12:1 bandwidth is achieved (725 MHz to 8.9 GHz) with VSWR levels below 2 across the entire band. The element maintains VSWR well below 2 for scans out to 45 degrees in all scan planes over more than 8:1 bandwidth (850 MHz to 8.1 GHz). If VSWR levels of 3 are acceptable, this can be achieved from 800 MHz to 8.4 GHz (10:1 bandwidth). This operational frequency range is reported with the caveat that scanning may be somewhat restricted at the higher frequencies element spacing of 7.4 GHz. While VSWR of 2 is due to the the baseline, for most of the operational band the actual VSWR is well below 1.5. Fig. 7 shows the simulated VSWR at a 60-degree scan. The VSWR is still below 2 for much of the band, with the exception of the H-plane scan showing slightly-worse degradation. Fig. 8 shows the port isolation for broadside radiation and scan angles at 45 degrees in three planes. This is a measure of energy leaking to the passive vertical port when the horizontal port is active.

KINDT AND PICKLES: ULTRAWIDEBAND ALL-METAL FLARED-NOTCH ARRAY RADIATOR

Fig. 9. Comparing the VSWR at broadside radiation for the all-metal element in a linear array versus a two-dimensional planar array.

With the exception of D-plane scanning, isolation is roughly 20 dB or better across the band. VSWR levels are very similar for both two-dimensional arrays of horizontal elements (i.e., single polarization only) and two-dimensional arrays of dual-polarized elements (i.e., horizontal/vertical element pairs). Linear arrays (as in Fig. 2) function somewhat differently, due to the lack of H-plane coupling. Fig. 9 shows the difference in VSWR levels for these two cases. Where the two-dimensional planar array achieves down to 725 GHz on the low end, a linear array based on the same element design (and spacing) only achieves down to 1.4 GHz [22]. However, Fig. 9 shows that the linear array does not have an impedance anomaly to limit the high-end bandwidth, though in practice, grating lobes would be a problem at these frequencies, limiting scan applications. B. S-Parameter Synthesis, Linear Array This section presents the measured (active) VSWR for the 32-element linear array and compares the results to full-wave simulations. Measurements of active impedance (and by extension, VSWR) at the element level are performed using an S-parameter synthesis procedure. To understand this measurement procedure, consider any -element array. The of the array is active input impedance seen at element obtained by synthesizing the active S-parameter at this measurements, i.e., element through the superposition of , where is the S-parameter seen at element with element excited, all other ports matched. can be used (in post-processing) to adjust the The factor amplitude and phase of individual elements for beam scanning and weighting. Consider the active VSWR at element 16, near the center of the 32-element linear array. The active VSWR at element 16 is obtained by collecting 32 unique measurements, or specifically: ), from which the active VSWR is computed. Obtaining a full characterization of the scattering matrix for the 32-element array requires unique S-parameter measurements. Once collected, it is possible to synthesize all combinations of phase scanning and amplitude weighting [29] and predict the VSWR seen at each specific element of the array. It is important to emphasize that this measurement procedure requires that a two-port calibration for non-insertable devices be performed. In other words, assuming all array elements have female connectors, it is not sufficient to do a standard two-port calibration on the usual set of male/female cables and then add a male-male adapter to the female cable post-calibration. This induces a phase error such that and cannot be added correctly.

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Fig. 10. Active VSWR of element 16 in the 32-element linear array at broadside scan and uniform amplitude weighting; exact finite and infinite simulations agree very closely with measurement over a wide bandwidth.

Fig. 11. Active VSWR of element 16 in 32-element linear array, positive 45-degree scan in E-plane; exact finite and infinite simulations track very well with measured data under scan.

Where the previous section used a simulation tool based on infinite cell analysis, this section requires a different type of analysis tool (CEMNAV-DD) for the full-wave simulation of the complete finite array structures, based on a non-matching grid Domain Decomposition-Finite Element Method (DD-FEM) [19], [30]. This rigorous design tool allows an engineer to predict with great accuracy exactly how each element in the array will function—including all mutual coupling and edge effects—giving nearly a one-to-one correspondence between numerical simulations and measurements. Here, the full-wave analysis of complete UWB finite array structure is presented, showing that simulations track very well with measurements, even under high scan angles. This is important for design engineers because it verifies that very accurate numerical studies can be performed on finite arrays prior to manufacturing, greatly increasing the chance of critical design flaws being caught early on. Elements that are not too near the array edges are expected to perform asymptotically similar to the infinite array case. Fig. 10 shows the comparison of the measured active VSWR at element number 16 (of 32) compared to the infinite array prediction and also the exact finite array simulation (broadside radiation and uniform amplitude weighting). The agreement is very close across the entire measurement band (from 0.5 GHz to 10.5 GHz), extending above and below the intended operational range of the design (roughly 1–8 GHz). Both tools (finite and infinite) do reasonably well predicting the VSWR of this element. Fig. 11 shows the comparison between simulations and measurements at a positive 45-degree scan. Here, VSWR is below 3 across the frequency band. Again, the results compare very favorably. Additional scan angles were reported in [17]. As a note, for elements near the center of the array, the negative scan is nearly identical. Fig. 12 shows the measured and simulated VSWR on elements (1), (4), and (8) of the 32-element linear array. VSWR levels in finite arrays are typically worse near the edge of the array and progressively converge to the infinite array result

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Fig. 13. Top view of the 8 8 dual-polarized flared-notch array. Circled is row number 6 (with numbering) upon which the S-parameter synthesis measurements were performed and reported here.

Fig. 12. Measured and simulated active VSWR of elements (1), (4), and (8) of the 32-element linear array, broadside radiation and positive 45-degree scan. Simulations exactly model edge effects in the finite array, such that results agree very closely with measurements, even under scan.

closer to the array center. What is important to show is that the simulation tools can very accurately predict the VSWR performance of the finite array, including the edge effects. Here, it is shown that simulations give very good agreement for broadside radiation and also the positive 45-degree scan case (scan to the right in Fig. 2). While only one scan case is reported here, several cases were considered and verified to give agreement to measurements. This gives confidence that these simulation tools can be used to validate larger array structures prior to incurring the cost of building them. C. S-Parameter Synthesis, 8

8 Dual-Polarized Array

A similar S-parameter measurement procedure was performed on the dual-polarized 8 8 planar array. For this array, in order to synthesize the active VSWR seen at any one element (assuming all other elements of a given polarization radiating), it is necessary to collect 64 S-parameter measurements. Complete characterization of the scattering matrix for this array would require over 2,000 S-parameter measurements for each polarization. For brevity, only row 6 of the 8 8 array has been characterized using the measurement procedure. Specifically, measurements are reported for elements (1), (4), and (6) of row 6 of the array, as depicted in Fig. 13. VSWR results are given in Fig. 14 for broadside radiation and a positive 45-degree (to the right in Fig. 13) E-plane scan, comparing measurements to a simulation of the entire 8 8 array using full-wave DD-FEM analysis. As expected, the VSWR is notably worse than the ideal results given in Fig. 6 for an infinitely-large array. While the VSWR is mainly below 2 across the frequency band at broadside radiation—even for this small array—it changes considerably from element to element. At the higher frequencies the VSWR levels are relatively stable and somewhat comparable to the results in Fig. 6. However, at lower frequencies for example, the local maximum in VSWR around 1.3 GHz moves around with element position and crests above 2 in some

Fig. 14. Simulated and measured VSWR for elements (1), (4), and (6) of row 6 in the 8 8 array, broadside radiation and E-plane scan to positive 45-degrees (to the right in Fig. 13).

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cases (for broadside). Also, note that the edge elements (e.g., element (1)) see a dip in VSWR at 675 MHz, which is below the low-frequency limit of the ideal element. VSWR levels are not symmetric for positive and negative scans, though this cannot be inferred from the limited plots of Fig. 14. This is mainly due to truncation effects and also the asymmetry in geometry created by element pairs—for example, the left side of the array is closed off by vertical elements, whereas the right side is not (see Fig. 13). What is perhaps most important is the fact that the full-wave simulation can very accurately predict the measured performance of the array, meaning simulation tools can be used to quantify finite array effects prior to building hardware, thereby minimizing risk. D. Array Mismatch Efficiency For finite array analysis, it is also important to consider array mismatch efficiency [19]. For antennas, mismatch efficiency is the component of the overall efficiency that relates specifically to impedance mismatch between the antenna port and the feed cable. In finite arrays, this will be dependent on the element position within the array. Array or aperture mismatch efficiency gives the average of this value over the entire array as a single-valued figure-of-merit to gauge how well an array performs at the system level, i.e., as if the entire array were fed via

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Fig. 15. Array mismatch efficiency for broadside radiation. The infinite cell (solid line) shows better than 90% efficiency from 725 MHz to 8.9 GHz with efficiency above 97% for most of the band, with measured data from the relatively small finite 8 8 array somewhat worse, mostly at the lower frequencies.

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Fig. 17. Radiation patterns for the 32-element linear array at 2 GHz. Measurements and simulations show close agreement.

Fig. 16. Array mismatch efficiency for various scan angles in the H-plane; results based on measured data from row 6 of the array.

a single port. Fig. 15 shows the computed array mismatch efficiency for the 8 8 array, computed using measured and simulated data. Measurement results are based on data from row 6 only, since data for the entire array was not collected (see Section III-C). Efficiency results for the full array must be simulated (dashed line). The solid (blue) line is the ideal efficiency that can be achieved for very large arrays. In general, the results for the small 8 8 array have converged at the higher frequency range (beyond 5 GHz), but are typically several percentage points worse than the ideal case at the low end. While this is the general rule, there is a range from 1 GHz–1.5 GHz where the small finite array has better mismatch efficiency than the infinite case. The important information to glean from the array mismatch efficiency figure-of-merit is that even for the small 8 8 array, the efficiency is better than 90% for almost the entire bandwidth of interest, and is better than 97% for the majority of the frequency band, meaning that on average the system is well matched, though larger arrays would have numbers closer to that of the infinite array case. To evaluate performance under scan, Fig. 16 gives the mismatch efficiency at various scan angles in the H-plane using measured data from row 6 of the array. As expected, some efficiency is lost when scanning, though even out to 60-degrees, efficiency is generally better than 90% across the band. E. Radiation Patterns Fig. 17 shows the results for the co- and cross-polarization radiation patterns of the 32-element array, measured and simulated at 2 GHz. The numerical results are generated using a full-wave simulation of the entire 32-element array including the backplane, using the in-house Navy DD-FEM code. The pattern measurements are collected in a 5’ 5’ near-field scanner. Several variations of measurement data were collected, including synthesis of composite array patterns from the coherent addition of individual element patterns and also array patterns formed from a single measurement using static power combiners. In general, the patterns are in good

2

Fig. 18. Measured co-polarization and cross-polarization patterns of the 8 8 array at 8 GHz. Patterns are reasonably symmetric and cross-polarization levels are below 20 dB.

0

Fig. 19. Simulated and measured gain of the dual-polarized array element.

agreement (measured vs. simulated). The lobe structures of the beam match closely, though the simulated cross-polarization levels are somewhat better than the measured results. This is typically the case when comparing numerical solutions to measured data, as it is much more difficult to achieve perfect alignment/positioning with physical hardware. Similarly, the radiation patterns of the 8 8 array are presented at 8 GHz. For this case, there were not enough passive beam-forming components on hand to perform a single measurement of the full array in receive mode, as was done for the linear array. Therefore, the measurements of the array were done in receive mode, superimposing stationary element patterns of individual element patterns for each of the 64 array elements. The patterns are fairly typical, and simulations agree reasonably-well with measurements. F. Element Gain In this section, the element gain vs. frequency is examined. Fig. 19 shows the gain from 0.5 GHz to 9.0 GHz. The simulated data is computed from the peak gain in the horizontal field

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TABLE I MEASURED CROSS-POL LEVELS VS. SCAN ANGLE AT 2 GHZ, 4 GHZ, AND 8 GHZ FOR THE 8

levels at broadside, using the unit cell of an infinite array simulation. The measured data is taken from the element in position of the 8 8 array (see Fig. 13). This element was chosen because it was determined to have the best VSWR behavior in the small 8 8 array. The measured data clearly follows the theoretical trend predicted by the simulation, though some variance in the data was encountered due to finite array effects. For the small array, it was found that the element pattern was not smooth and did not consistently give peak gain at broadside, leading to variation in the results. However, the usual gain trend is clear—approximately 5 dB of gain at half-wavelength spacing (7.4 GHz) falling off predictably towards lower frequencies. IV. CROSS-POLARIZATION VERSUS SCAN ANGLE Cross-polarization is defined as the difference in peak levels of vertically-polarized fields compared to the horizontally-polarized fields in the direction of scan, for horizontal polarization of the active array elements. The beams for the 8 8 array are synthesized from individual element pattern measurements taken in the near-field scanner. Element patterns for both polarizations are taken for each of the 64 elements and then superimposed coherently with a phase factor to synthesize a scanning beam. In order for this technique to work correctly, each of the element patterns must be collected under the same set of conditions. This assumes the data sets are collected from the same calibrated collection grid in space and the measurement setup remains stationary (both physically and electrically) during the entire measurement procedure. While there is a lot of room for user and system error with this procedure, in practice it gives remarkably-repeatable results. Measurements are collected at three frequencies an octave apart—2 GHz, 4 GHz, and 8 GHz. Cross-polarization data is then synthesized over a range of scan angles in both theta and phi. Table I lists the measured results for the 8 8 array. The simulated results agree with the measurements to within a few dB, and for brevity are not reported. For the given frequencies, cross-polarization levels are the lowest in the E-plane and . The worst results show H-plane—all below roughly . At 8 GHz, the cross-polarization up in the D-plane levels are as much as 12 dB higher than the co-polarization patterns for higher scan angles. It should be noted that the element spacing in the D-plane is more than half a wavelength at this frequency. Further, it is well known that arrays of flared-notches

2 8 DUAL-POLARIZED ARRAY

have good polarization purity in the principal planes with typically worse purity in the D-plane. In practice, for arrays with orthogonal element pairs such as the 8 8 array, polarization degradation can largely be corrected. It has been observed in both measurements and simulations that cross-polarization is better for shorter elements of the same general design, at the expense of bandwidth. As a note, the results are also somewhat affected by the fact that the small array does not scan well at lower frequencies. V. CONCLUSION This paper presents an all-metal UWB antenna element with 12:1 bandwidth at broadside scan, maintaining 8:1 bandwidth for 45-degree scans in all planes. The main limitation of the design is the relatively poor cross-polarization performance in the D-plane, which is similar to other electrically long radiators. However, independent control of the orthogonal feed ports can be used to control the cross-pol, and the element has many other desirable qualities, including large bandwidth, very low VSWR, modularity and simplicity of construction. The array shows reasonably good performance considering the small size and lack of edge treatment. The reason this design is presented in a small configuration is because it is part of a larger study on low-cost UWB arrays first presented here [21]. It is expected that larger array configurations and the inclusion of edge treatment such as parasitic elements would show improved overall performance. This paper clearly demonstrates that full-wave computational tools can be used to accurately predict the performance of finite arrays with complex element designs. While infinite cell analysis remains the most valuable tool for parametric design of antennas, finite array simulators can be used to improve the overall design of larger array systems, and will likely become an important tool for array design in the future. REFERENCES [1] P. J. Gibson, “The Vivaldi aerial,” in Proc. 9th Eur. Microwave Conf., 1979, pp. 101–105. [2] L. Lewis, M. Fassett, and J. Hunt, “A broadband stripline array element,” in Dig. IEEE Antennas Propagation Symp., Jun. 1974, vol. 12, pp. 335–337. [3] H. Holter, C. Tan-Huat, and D. H. Schaubert, “Experimental results of 144-element dual-polarized endfire tapered-slot phased arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 11, pp. 1707–1718, 2000. [4] M. Kragalott, W. R. Pickles, and M. S. Kluskens, “Design of a 5:1 bandwidth stripline notch array from FDTD analysis,” IEEE Trans. Antennas Propag., vol. 48, no. 11, pp. 1733–1741, Nov. 2000.

KINDT AND PICKLES: ULTRAWIDEBAND ALL-METAL FLARED-NOTCH ARRAY RADIATOR

[5] J. D. S. Langley, P. S. Hall, and P. Newham, “Balanced antipodal Vivaldi antenna for wide bandwidth phased arrays,” Proc. Inst. Elect. Eng. Microw., Antennas Propag., vol. 143, no. 2, pp. 97–102, 1996. [6] E. W. Lucas and T. P. Fontana, “A 3-D hybrid finite element/boundary element method for the unified radiation and scattering analysis of general infinite periodic arrays,” IEEE Trans. Antennas Propag., vol. 43, no. 2, pp. 145–153, Feb. 1995. [7] K. Trott, B. Cummings, R. Cavener, M. Deluca, J. Biondi, and T. Sikina, “Wideband phased array radiator,” in Proc. IEEE Int. Symp. on Phased Array Systems and Technology, 2003, pp. 383–386. [8] H. Holter, “Dual-polarized broadband array antenna with BOR-elements, mechanical design and measurements,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 305–312, Feb. 2007. [9] W. R. Pickles, J. B. L. Rao, D. P. Patel, and R. Mital, “Coincident phase center ultra wideband array of dual polarized flared notch elements,” in Dig. IEEE Antennas Propagation Symp., 2007, pp. 4421–4424. [10] M. W. Elsallal and D. H. Schaubert, “Reduced-Height array of BAVA with greater than octave bandwidth,” in Proc. Antenna Applications Symp., Sep. 21–23, 2005, pp. 226–242. [11] J. J. Lee and S. Livingston, “Wide band bunny-ear radiating element,” in Dig. IEEE Antennas Propagation Symp., 1993, vol. 3, pp. 1604–1607. [12] J. J. Lee, S. Livingston, and R. Koenig, “A low-profile wideband (5:1) dual-pol array,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 46–49, 2003. [13] P. Friedrich, L. Pringle, L. Fountain, P. Harms, G. Smith, J. Maloney, and M. Kesler, “A new class of broadband planar apertures,” in Proc. Antenna Application Symp., Allerton Park, IL, Nov. 2001, pp. 561–587. [14] B. Thors, H. Steyskal, and H. Holter, “Broad-band fragmented aperture phased array element design using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3280–3287, Oct. 2005. [15] B. Munk, R. Taylor, T. Durharn, W. Croswell, B. Pigon, R. Boozer, S. Brown, M. Jones, J. Pryor, S. Ortiz, J. Rawnick, K. Krebs, M. Vanstrum, G. Gothard, and D. Weibelt, “A low-profile broadband phased array antenna,” in Dig. IEEE Antennas Propagation Symp., Jun. 2003, vol. 41, pp. 448–451. [16] M. Jones and J. Rawnick, “A new approach to broadband array design using tightly coupled elements,” in Proc. IEEE Military Communications Conf., Oct. 29–31, 2007, DOI: 10.1109/MILCOM.2007. 4454764. [17] R. W. Kindt and R. Pickles, “12-to-1 bandwidth all-metal Vivaldi array element,” in Dig. IEEE Antennas Propagation Symp., Jun. 1–5, 2009, pp. 1–4. [18] B. T. McWhirter, S. K. Panaretos, J. Fraschilla, L. R. Walker, and J. L. Edie, “Thick flared notch radiator array,” U.S. patent 5 659 326, Aug. 19, 1997. [19] R. W. Kindt and M. N. Vouvakis, “Analysis of a wavelength-scaled array (WSA) architecture,” IEEE Trans. Antennas Propag., vol. 58, no. 9, pp. 2866–2874, Sep. 2010. [20] R. W. Kindt and M. N. Vouvakis, “Analysis of wavelength-scaled array architectures via domain decomposition techniques for finite arrays,” in Proc. Int. Conf. on Electromagnetics in Advanced Applications, Torino, Italy, Sep. 14–18, 2009, pp. 196–199. [21] R. Kindt and M. Kragalott, “A wavelength-scaled ultra-wide bandwidth array,” in Dig. IEEE Antennas Propagation Symp., Jun. 1–5, 2009, pp. 1–4.

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[22] R. Kindt, M. Kragalott, M. Parent, and G. Tavik, “Preliminary investigations of a low-cost ultrawideband array concept,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3791–3799, Dec. 2009. [23] S. Kasturi and D. H. Schaubert, “Effect of dielectric permittivity on infinite arrays of single-polarized Vivaldi antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 351–358, Feb. 2006. [24] J. Shin and D. H. Schaubert, “A parameter study of stripline-fed Vivaldi notch-antenna arrays,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 879–886, May 1999. [25] N. Marchand, “Transmission-line conversion transformers,” Electronics, vol. 17, pp. 142–146, Dec. 1944. [26] G. J. Wunsch and D. H. Schaubert, “Full and partial crosswalls between unit cells of endfire slotline arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 981–986, Jun. 2000. [27] T. F. Eibert, J. L. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3D doubly periodic structures using triangular Prismatic elements and a MFIE accelerated by the ewald transformation,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 843–850, May 1999. [28] Ansoft High Frequency Structure Simulation (HFSS), Version 11.1 Ansoft Corporation, 2008. [29] R. C. Hansen, Phased Array Antennas. New York: Wiley-Interscience, 1998. [30] M. N. Vouvakis, “A non-conformal domain decomposition method for solving large electromagnetic wave problems,” Ph.D. dissertation, Ohio State Univ., Columbus, OH, 2005.

Rick W. Kindt received the B.S.E. degree in electrical engineering and the M.S.E. and Ph.D. degrees from The University of Michigan, Ann Arbor, in 1998, 2000, and 2004, respectively He worked briefly as an Antenna Systems Engineer on airborne radar projects for Raytheon Systems Company, El Segundo, CA, before returning to the University of Michigan to complete his studies. As a graduate student, his research focused on hybrid finite element-boundary integral methods, with emphasis on domain decomposition techniques and fast methods for array-type problems. He worked as a Postdoctoral Researcher at The Ohio State University (2004–2005) where he did research on antenna design and hybrid numerical methods for array/platform analysis. Since 2005, he has been with the Electromagnetics Section, Radar Division, Naval Research Laboratory, Washington, DC. His current research interests include ultrawideband antenna array design as well as generalized computational methods for electromagnetic analysis with emphasis on domain decomposition techniques for very large structures.

William R. Pickles (S’87–M’87) received the B.S.E.E. degree (with honors) from the University of California at Davis, in 1980 and the M.S.E.E. degree from the University of Wisconsin at Madison, in 1987. From 1980 to 1985, he was a Staff Engineer with the McDonnell Douglas Electro-Optics Laboratory, Huntington Beach, CA. Since 1987, he has been with the Antenna Section, Radar Division, Naval Research Laboratory, Washington, DC. His main area of research there has been in antennas, specifically in ultra wideband flared notch antennas. In addition he has done work on the SPS-67 and SPS-49 radars and ultra-low sidelobe antennas. Mr. Pickles is a member of Tau Beta Pi.

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Analyzing Large-Scale Non-Periodic Arrays With Synthetic Basis Functions Bo Zhang, Gaobiao Xiao, Member, IEEE, Junfa Mao, Senior Member, IEEE, and Yan Wang

Abstract—A numerical method using synthetic basis functions (SBF) is introduced to analyze the scattering problem of the largescale non-periodic arrays. With the employment of the auxiliary sources, the scattering solution space of a specific scatterer can be determined, and its scattering characteristics will be effectively extracted from the solution space in the form of SBF, which is a high order basis function. The scattering field of an array can be calculated by using the scattering characteristics of every array unit. The number of the unknowns of the SBF linear system is reduced significantly compared with the conventional method of moments (MoM), leading to higher efficiency and lower cost of computation. The SBF approach is universal regardless of the geometrical differences. There are also specific mechanisms to control the precision and efficiency. The SBF method costs less computational time than the characteristic basis function (CBF) method. Moreover, by utilizing the SBF approach together with the generalized transition matrix (GTM) and the generalized surface integral equations (GSIE), the scattering analysis of dielectric bodies and chiral bodies and their arrays will be much more efficient with decreased unknowns and scale-reduced linear system. To deal with large-scale and complex scattering problems, the SBF method will be of great help to improve the performance. Index Terms—Electromagnetic scattering, generalized transition matrix, method of moments, non-periodic structure, synthetic basis functions.

I. INTRODUCTION

T

HIS paper focuses on large-scale scattering problems, which are comprised of multiple scatterers. One application is the periodic structure that is widely used in microwave engineering and antenna designs [1]–[3]. Since the periodic structure always has finite size in practice, it requires accurate numerical analysis, typically by using the method of moment (MoM). However, the conventional MoM is usually implemented with low order basis function such as RWG basis function [4]. The edge size of mesh is required to be of the order of 0.1 . When the scale of the structure becomes electrically large, the number of meshes as well as the unknowns will significantly increase, and the performance will be terrible because of the rapid growth of the computational complexity and the memory cost when solving matrix equations. Manuscript received February 06, 2009, revised October 29, 2009; accepted May 02, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. This work was supported in part by the Chinese National Natural Science Foundation (60971040), the National Basic Research Program of China (2009CB320202), and the Shanghai Aerospace Fund 2009. The authors are with the Department of Electronic Engineering, Shanghai Jiao Tong University, ShangHai 200240, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2071331

To deal with large-scale problems, the fast multi-pole method (FMM) [5] and the multilevel fast multi-pole algorithm (MLFMA) [6], [7] are proposed to speed up the solution of matrix equations. They are based on the sub-domain basis functions such as RWG. Recently, the approaches of aggregate basis function based on physical and geometrical features are developed, which try to use fewer high-order basis functions that are the combinations of lower order basis functions to represent scattering features of certain kinds of scatterer, and achieve efficient solution with enough accuracy. The characteristic basis function (CBF) was introduced in 2003 [8], followed by its various applications to antenna and scattering problems [9]–[12]. This approach divides a whole problem into several blocks. On each block, CBF is generated with two considerations: the self-interaction of the block under outside excitation and the mutual coupling effects from the rest of the blocks. By using this process, every block can be represented by only several characteristic basis functions, so the number of the unknowns of the matrix equation is greatly reduced. A similar method called sub-entire-domain (SED) basis function method has been proposed since 2004 [13], [14], which improved the CBF method by introducing dummy cells to represent several particular types of blocks in periodic structures. For each type of cell, the sub-entire-domain basis function (SEDBF) needs to be calculated only once. This reduces the complexity of generating CBFs. The idea of synthetic basis function (SBF) was put forward since the beginning of the 2000s [15], [16], but did not catch much attention until its systematical representation in 2007 [17]. The SBF method is also based on sub domain decomposition like CBF and SEDBF. The difference is that for each type of block, SBF method introduces the degree of freedom (DOF) of its solution. Based on this concept a number of SBFs are generated to approximately cover the solution space of each block. Therefore, the precision of numerical results can be controlled by choosing the number of SBFs on each block. Moreover, the SBF of a block can be re-used in a large system, hence greatly simplifying the work of SBF generation. In this paper, the applications of SBF method to the analysis of the scattering problems of perfectly electric conducting (PEC) bodies and inhomogeneous chiral dielectric bodies are represented. For the scattering of PEC structures, the linear system is based on surface integral equations (SIE) and basis functions on surface meshes such as RWGs. While for dielectric bodies, especially chiral bodies’ scattering problems, the conventional solution is based on volume basis functions such as Schaubert-Wilton-Glisson (SWG) basis functions [19]. The volume meshing of the chiral models usually causes a large

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ZHANG et al.: ANALYZING LARGE-SCALE NON-PERIODIC ARRAYS WITH SYNTHETIC BASIS FUNCTIONS

amount of unknowns for the MoM linear system, which is difficult to solve in a single computer. We utilize the generalized transition matrix (GTM) [20] and the generalized surface integral equations (GSIE) [21] to transfer the linear system of the volume integral equations (VIE) to that of SIE. Then the SBF approach is used to improve the performance further. With these two improvements, we are able to analyze the scattering problem of a large-scale chiral structure which is hard to deal with by the conventional MoM. The scattering solution space of a scatterer can be expanded by the responses from the excitations of a series of manually defined auxiliary sources. We can effectively extract the scattering characteristics from such solution space in the form of SBFs, thus leading to high accuracy with lower computational cost. The completeness of the solution space expanded by employing auxiliary sources is crucial to the accuracy of the numerical results. Therefore, we look into the mechanism of the auxiliary source definition. As mentioned in [17], the sources should be defined based on the equivalence theorem. They are placed on a surface around the scatterer under consideration to represent all the effects from outside scatterers. For each problem, we employ a closed surface surrounding the body and mesh it into triangle patches. Then the auxiliary sources are defined on every edge of the patches in the form of RWG. The set of SBFs obtained by these sources yields better approximation to the solution of the scatterer as compared to the basis set used in [17], where the auxiliary sources are equidistantly placed around the scatterer with each source being represented by a small RWG. For large-scale arrays, we can apply the source definition to each scatterer to generate the approximate solution space, and extract SBFs from the space for each scatterer. The choice of the threshold of the singular value ratios in the singular value decomposition (SVD) process will affect the generation of SBF and the overall computational performance. By setting up higher threshold, more SBFs are included and the accuracy tends to become higher. The application of the SBF method is presented in Section V, including numerical results of several large-scale non-periodic PEC arrays and chiral medium arrays. We will see that the SBF method is an effective approach to deal with such complex problems.

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is the th RWG basis function, and is the expansion coefficient to be solved. The linear system of the conventional MoM is created by the Galerkin test procedure, and can be represented in the form of matrix equation

where

(3) where the impedance matrix is an elements are calculated as follows,

matrix whose

(4) is the linear operator of EFIE in (1). The excitation vector and the vector are both column vectors. Since the maximum size of the RWG mesh should be of the order of , electrically large structures ususally require a terrific large number of RWGs, which may lead to unreasonably large-scale linear systems and make the computer inefficient or even unable to solve. The aggregate basis functions are introduced to reduce the number of unknowns, and relieve the heavy burden on both the memory and CPU. The approach using the aggregate basis functions can be divided into three steps: A. Domain Decomposition The whole structure is divided into blocks according to geometrical features. For example, if it is a periodic PEC array, each unit can be defined as one block. Here it is assumed that all blocks have the same structure for the sake of uniformity RWG basis functions on each of manipulation. If there are block, the total number of RWGs is . B. Define New Basis Functions In this process, the aggregate basis functions are defined on each block respectively. For the th block, we define as the th basis function

(5)

II. SBF APPROACHON PEC SCATTERING PROBLEMS The SBFs are linear combinations of the low-order basis functions, which are defined on each decomposed domain of the whole structure in order to reduce the dimension of the impedance matrix of conventional MoM. Consider the scattering problem of a three-dimensional PEC structure in free space. The current distribution is solved by the electrical field integral equation (EFIE) as follows: (1) where is the scalar Green’s function of 3D free space. Using RWG basis functions to discretize the EFIE, the current is represented as (2)

is the th RWG basis function on block- , is where the number of the aggregate basis functions of each block. There are basis functions for the whole problem. Because has already been determined, the aggregate basis functions are usually represented as vectors (6) The elements in (6) and the number of basis functions are determined by some specific mechanisms in different methods (CBF, SEDBF, and SBF, etc). Then the current distribution is a linear combination of the new basis functions

(7)

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This surface is around the block under consideration while excluding any part of other blocks. By solving responses from these sources, the solution space of such type of block can be approximately simulated. Assuming that we have defined auxiliary sources for the th block, in addition to the natural excitations on block- , therefore, excitation, there are is expanded by the current solution vecthe solution space tors corresponding to each excitation (as discussed in [17]) Fig. 1. Auxiliary sources displacement on a decomposed block in an array.

(11) are determined by solving the following The columns of equations, respectively:

C. Generate New Linear System The impedance matrix is regenerated with the basis functions defined in (8) is an matrix and is an where vector which can be represented in the form of partitioned matrix. The elements of the partitioned matrix generated from block- and block- are determined by the Galerkin test procedure (9)

(10) in which and are the global indices of the RWG basis functions and in the MoM matrix (3). and are the corresponding parts of the impedance matrix and excitation vector in (3). The number of unknowns of the new , much smaller than the conventional linear system is MoM linear system. Therefore, the solution will be accelerated. It can also be noticed that the new impedance matrix can be easily derived from the MoM impedance matrix. III. GENERATION OF SYNTHETIC BASIS FUNCTION The SBF method has specific mechanism to generate SBFs on blocks. It employs the concept of degree of freedom (DOF) of scattering fields [18]. It claims that for a certain structure with specified size and shape, the DOF of its scattering field is limited , where denotes the total between two bounds number of RWG basis functions and also the degrees of curdenotes the DOF extremum when rents on the body, and the observing point is in the far field. With the concept of DOF, we can notice that it is usually redundant to calculate scattering field with all the current basis functions defined on the body. This observation makes it possible to use fewer SBFs to approximate the solution space of the scattering fields. To achieve such goal of information compression, two important processes are introduced in generating SBF on the blocks: (1) adopting equivalent auxiliary sources and (2) singular value decomposition. Firstly, with domain decomposition, we establish a relatively isolated model by defining a number of auxiliary sources around the block under consideration and removing all the other blocks, as shown in Fig. 1. The auxiliary sources are set on a properly defined surface which can be called auxiliary source surface.

(12) (13) where denotes the corresponding part of block- in the is the natural excitation vector MoM impedance matrix. is the excitation vector of the th auxiliary while source. If the auxiliary sources are properly defined, will represent the solution space of block- approximately. Once the solution space is generated, the singular value decomposition (SVD) process is used to extract the independent basis of the solution space (14) stores the singular values, in which . and collect the singular vectors. The columns in the maare orthogonal to each other, and the singular value ratio trix is a measurement of the independency of the th column on the first column. If the ratio of the th column is very of small, we may claim that the columns after the th column are no longer independent. Therefore, we take the first columns of as the coefficients of SBFs of block(15) All SBFs of the whole structure are generated through the above processes. If all the blocks in the problem have the same structure, there will be SBFs in total and a matrix equation to solve. From the generation of SBF, we notice that there are several advantages of this method. Firstly, because of the employment of DOF theory we have the concept to approximate the accurate solution to the highest possible extent by carefully defining the auxiliary sources. For certain type of structures, we can approximately generate its solution space only once. When it is applied in different arrays, the SBFs on it can all be extracted from the solution space without additional calculations. Consider the CBF method, in which the CBFs on one block should be calculated from the mutual coupling problems between it and other solutions blocks. It will call for many sub-system solutions ( blocks) in the first step, and these solutions can hardly be for reused in a different problem. Contrast to CBF method, the SBF method is able to pack scattering characteristics of certain types of structures for reusing in different applications. Secondly, it is

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where denotes the th RWG basis function on the block. If we also define the excitation with the form of a RWG current , (16) will become function (17)

Fig. 2. Auxiliary source definition. (a) Equidistantly placed sources. Each source is in the form of small RWG. (b) Placed on a meshed source surface containing the scattering body.

flexible to control precision for complex problems when using SBF method. The number of SBFs in one block is not fixed. By , we setting different threshold of the singular value ratio can obtain different numbers of SBFs. If the threshold is small, , which are derived from SVD, more columns in the matrix will be included as new SBFs; and if the threshold is leveled up, the number of SBFs will decrease. This management is very important for the trade-off between efficiency and accuracy of the solution. A larger number of SBFs makes the solution more accurate, while less number of SBFs leads to greater reduction of unknowns and thus better efficiency of the solution. Compromise can be made by setting a well-assigned threshold to obtain good efficiency together with a satisfying precision. By introducing the auxiliary sources, the SBF generation is carried out without including information from other blocks (those blocks should be excluded outside the auxiliary source surface). All the coupling effects are represented by the solution with the auxiliary sources in the solution space. Therefore the SBF method is able to deal with various kinds of problems with the same mechanism. It is obvious that the arrangement of the auxiliary sources is crucial to the performance of SBF. For further discussion, we look into the auxiliary source definition. There are two major factors to consider: the form of sources and their placement. A simple idea is to employ point current sources and place them equidistantly on a regular shaped boundary around the block. For a PEC plate shown in Fig. 2(a), we may place the auxiliary sources on a cubical surface around it. To facilitate the computation of (13), we notice that the elements of the excitation vector is generated by the Galerkin test procedure

(16)

Note that (17) has the same form as (4). Hence, this procedure will be easily implemented in programming by just calling the same code for impedance matrix generation. For such consideration, the point sources in Fig. 2(a) will be replaced by small pairs of triangle, which have RWG current distribution on them. It seems reasonable that the auxiliary sources definition should include diverse polarizations. However, the polarization arrangement of the RWG form auxiliary sources used in [17] is not explicitly explained. Temporarily, there is no rigorous evidence to show that the sources with orthogonal polarizations will definitely lead to better result in our experiments. In fact, the RWG sources placed in single direction as in Fig. 2(a) also lead to satisfying numerical results in many cases that we have checked. It is probably because these sources are very small and the current on a RWG base is not strictly single-polarized, thus may produce relatively diverse polarizations to some extent. However, we still believe that these regularly displaced sources as mentioned above need further research on their effectiveness, and for now, we propose a modified sources definition as shown in Fig. 2(b). First we define the shape of surface where the sources are put. For the scatterer in Fig. 2(a), we employ the cubical surface. Then the surface is irregularly meshed into triangle patches, and the sources are defined as RWG current basis functions on each edge of the patches just like the definition of the RWG on the scattering body. As a result, these sources are irregularly placed with diverse polarizations. So far, both the modified sources and the original source arrangement have satisfactory numerical results in the experiments. It still needs further evidences or some theoretical proofs to obtain a universal conclusion about the relationship between the source definition and the excitation diversity. However, the modified auxiliary sources distribution on surface meshes are easier to generate with the mesh tools, and used in the following numerical examples. IV. SBF APPROACH ON DIELECTRIC BODY SCATTERING To further illustrate the effectiveness of the SBF method in extracting scattering characteristics, the SBF application to the analysis of the scattering problems of inhomogeneous dielectric body, especially inhomogeneous chiral body, is discussed in this section. The scattering of inhomogeneous structure is conventionally analyzed by the volume integral equations (VIE) as follows: (18) (19) where is the free space dyadic Green’s function. With the as well volume equivalence principle [19], the currents , can be represented by the equivalent as the total fields , displacement vector and the magnetic induction , which

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The detailed derivations of the GTM are not shown for the sake of simplicity. We notice that with the matrix, the scattering characteristics of the dielectric body are represented on the reference surface in the form of surface basis functions such as RWGs, whose number is much less than the SWGs. B. GSIE Solution

Fig. 3. A dielectric scatterer bounded by a reference surface.

are expanded by the so called SWG basis functions [19], [22] in tetrahedral meshes. After the discretization, the SWG-MoM linear system is represented as follows: (20) Such linear system usually has too many unknowns to solve. Therefore we introduce the generalized transition matrix (GTM) and the generalized surface integral equations (GSIE) [20], [21] to reduce the scale. Then the SBF method is applied to the simplified problem. A. GTM of Dielectric Body

In a scattering problem of multiple dielectric bodies, each body’s GTM is calculated by the process mentioned above. Assume that there are bodies. Each body forms a block, with its scattering characteristics described by an associated GTM and a specified reference surface. It is necessary to carefully evaluate the mutual effects of these blocks to get the whole scattering solution. We know that the total scattering field on one block (take block- for example) is the linear combination of the scattering fields from the natural excitation and those from excitations caused by other blocks. Therefore, the discretized form of surface integral equation on reference surface can be derived as (25) denotes the rotated tangential scattering field on block- . is the natural incident field is the GTM of block- . is the transmission on and matrix from the rotated tangential fields on block- to those on block- . The detailed derivation of GSIE solution can be found in [21]. If the dielectric bodies are identical, the GTMs of all blocks are the same. where

Consider an arbitrarily shaped dielectric body as shown in Fig. 3. We define a regular reference surface around it. For some scatterers with regular shapes, such as spheres or cubes, their natural boundaries can be used as the reference surface for simplicity. Since the scatterer is equivalent to free space region with volume equivalence currents, both the interior region and the exterior region of are considered as free space with , . of surface points outward. , The normal unit vector are incident fields illuminated on , and , are the , and , as the rotated scattering fields. Define tangential field components of the incident fields and scattering fields on respectively, (21) (22) We discretize the surface into triangle meshes and use RWG basis functions , to expand the fields on ,

C. SBF Implementation on GSIE As discussed in Section III, to generate SBFs on one block of the dielectric body, we first have to define a series of auxiliary auxiliary sources around the block. Assume that there are will be expanded sources around block- . The solution space by all the responses from the auxiliary sources as well as the natural excitation as in (18). Each response can be easily evaluated by the GTM of block(26)

(27) (23) where stands for or , and stands for the expanding coefficient of the corresponding field. By utilizing the SWG-MoM with Huygens’ equivalence principle as presented to describe the relations bein [20], we can generate GTM tween scattering fields’ and incident fields’ rotated tangential components on . With the above expansions, it can be represented as follows: (24)

where is the incident field from the th auxiliary source. With the SVD process as expressed in (14), we take the first th columns in as the SBF vectors like in (15). Each SBF is a linear combination of the RWG basis functions on reference surface with the expansion coefficients as in (5). Therefore, the discretized scattering fields on reference surfaces can be represented in the form of SBF

(28)

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where (29) Substituting (29) into (25) and testing it with SBFs (Galerkin’s scheme) result in a size-reduced linear system, and the number on all of unknowns equals to the total number of SBFs blocks. With the two-stage compression of scattering characteristics, GTM-GSIE and SBF, the solution scale for dielectric scattering problems will be significantly reduced. It is able to deal with those problems that may not be effectively solved by conventional methods. Recently, a similar scattering characteristic extraction approach is applied together with CBF method. As is discussed in [23], a series of dipole sources are placed on the scatterer’s boundary to represent the scattering characteristics of the scatterer itself. The CBFs of the scatterer can be extracted through SVD on the field space expanded by these sources. This method is in somewhat similar to GTM method, in the sense that a scatterer is replaced by equivalent sources in both methods. Although the auxiliary sources and SVD process are applied to both the SBF method and the method described in [23], their essential considerations are not the same: the auxiliary sources in [23] are used only to represent the scatterer’s natural scattering characteristics without considering the environment where the scatterer is placed, and the SVD process is used to extract the most significant modes corresponding to those most significant singular values; while in SBF method, auxiliary sources are used to create a special space to contain the key information of the scatterer in a specified environment, including the scattering characteristics of the scatterer itself and the affections of the environment such as the smallest distance to other scatterers and the excitation fields. The SVD process in SBF method is used to extract the key modes of this special space, hence enjoying the possibility of extracting scattering characteristics more efficiently. For example, when there are some other scatterers, the CBFs on the body under consideration does not change when the other scatterers get closer or farer. While in the SBF method, it is probably to use less SBFs when other scatterers are located farer away. Actually we have compared the method in [23] with GTM method, and think that the utilization of SVD in [23] is like applying SVD to the GTM directly in GTM method. V. NUMERICAL RESULTS In this section, we present numerical results of a series of scattering problems to illustrate the accuracy and efficiency of the SBF method. The results of the CBF method and SEDBF method are also given in PEC scattering problems for comparison. The detailed computation process of CBF is represented in [8], while the SEDBF method is proposed in [13] which are similar to SBF method. However, both of them have some distinctive features. For CBF, firstly, it includes both the self-interaction and the mutual coupling effects between every two blocks. Therefore the number of CBF on each block will be the same as the number of blocks (M in this paper), and the total number

2

2

Fig. 4. (a) 6 6 PEC array with square unit of 0:8  0:8 . The distance between the nearby two units is 0.1 , while the distance between two groups is 0.9 . (b) Non-periodic PEC array with radial structure placed on xoy plane. Each block is round plate with radius of r = 0:8 , in the radial direction, the distance between two nearby centers is 1.3 .

. Secondly, it uses geometrical extension for of unknowns is each block to better display the mutual effects with other blocks. Thirdly, the mutual coupling excitations from other blocks are derived from their primary basis functions, which reflect their self-interactions for natural excitation. SEDBF method is generally only used in periodic arrays, in which the scattering blocks can be classified into only a few types based on their locations. The SEDBFs of one block are calculated from the sub-problem only considering the coupling effect from the nearest blocks. For PEC scattering, two examples are presented. First, let’s see a PEC array as shown in Fig. 4(a). The unit of the array is a square PEC plate whose side length is 0.8 . The distance between the nearer four units is 0.1 while the larger distance between groups is 0.9 . There are 9 groups, and each group has 4 units. The array is placed on xoy plane, while the excitation is a -polarized plane wave whose incident direction is . By setting the maximum edge size of meshes as 0.1 , we discretize the whole structure into 4608 triangles and obtain 6336 RWGs from them. Each unit has 176 RWGs. By using SBF method, we define the sources as RWGs on the meshes of a cubical surface around one unit, the side length of the surface is 0.45 in order to keep the other blocks outside. The total number of extra sources is 378. The solution space is therefore a 176 378 matrix (the first column is derived from the natural excitation). After SVD process, we only take the first 13 columns as SBFs on one unit by setting a threshold of singular value ratio as 0.1. unknowns. Then we will face the problem with Using SEDBF method, there are 16 types of nearby coupling effects based on the theory in [13]. So there are 16 sub-problems to solve for the total 36 SEDBFs with one on each unit. While in . The the CBF method, the number of CBFs is numerical results of SBF, SEDBF, CBF and the conventional MoM results are shown in Fig. 5. We can notice that the SBF method leads to higher accuracy than SEDBF method, which is obvious in Fig. 5(b). This example reveals the deficiency of SEDBF in non-periodic array analysis. The CPU times are compared in Table I. In the non-periodic PEC structure shown in Fig. 4(b), there are 25 PEC units and a unit is a round plate with radius of 0.4 . The units are placed along the radial lines with the angle of , . The distance between the circle centers of adjacent layers is 1.3 . There are totally 3400 RWGs

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2

Fig. 5. RCS of 6 6 PEC array with dual period as shown in Fig. 4(a). Incident Each block is :  : , the smaller gap is angle of plane wave is  0.1 , the bigger gap is 0.9 . 13 SBFs are defined on each block.

Fig. 6. RCS of a non-periodic array with radial structure as in Fig. 4(b) Incident angle of plane wave is  Dashed line: 5 SBFs on each block. Dot line: 10 SBFs on each block.

TABLE I COMPARISONS OF THE COMPUTATION COST

example proves that the numerical result can be refined by employing more SBFs. This manageability of performance makes great sense when dealing with complex problems. Next, we present some examples of applying the SBF method to the analysis of dielectric arrays. Since the control group is the result from GSIE approach, we first use a small-scale problem to verify its accuracy. Consider a two-layered chiral sphere. The , and the radius outer radius of the sphere makes is half of . The outer layer has of the inner layer , , while the inner layer has , , . The number of volume meshes of the structure is 892, which makes 1995 SWGs for MoM. We define a spherical reference surface around it and generate 264 surface meshes on each surface. Then the total number of RWGs for GSIE approach is 396. In Fig. 7, the RCS results of the analytical solution [24], SWG-MoM, GSIE and SBF are plotted, showing that the result by the GSIE approach is in good agreement with the theoretic one. For large-scale dielectric arrays, it is usually difficult to use the conventional SWG-MoM because its linear system’s dimension is too large to solve. To illustrate the effectiveness of the SBF approach, we can use the GSIE solution for comparison. For a chiral array as shown in Fig. 8, whose unit is a two-lay( is the sphere radius), the ered chiral sphere with

=0

08 208

of the problem. In order to generate SBFs, we employ a spherical surface around one unit with the radius of 0.8 , then discretize the surface and obtain 201 RWGs as auxiliary sources. In SVD process, we set the threshold of SV ratio as 0.1, and 0.4, and take the first 10 and 5 columns as SBFs, respectively. Then the corresponding problems with 250 and 125 unknowns are solved. The RCS results and time consumption from conventional MoM, CBF and SBF are shown in Fig. 6 and Table I (The SEDBF is not applicable). The result from 10 SBFs reaches the same level of precision with CBF method, while consuming only 2/3 of the CBF computing time. It is shown that the numerical result with 10 SBFs on each block is consistent with MoM result. It is better than the result with 5 SBFs on each block. This

=0

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be twice of the number of basis functions. Therefore we get a 460 460 SBF linear system compared with the 7920 7920 linear system in the GSIE solution. Fig. 9 gives a comparison of the numerical results and it shows that the SBF method with 23 SBFs on each block reaches satisfying precision, whose number of unknowns is only 5.8% of the GSIE approach and 1.15% of the conventional SWG-MoM. This significant improvement makes the SBF method in conjunction with GSIE an effective approach in solving large-scale dielectric or chiral array scattering problems. VI. CONCLUSION

= =10

Fig. 7. Bistatic RCS of a two-layered chiral sphere with parameters of k r : ," : , , r =r ," : , : and   : . The incident wave is an x-polarized plane wave in z direction. 23 SBFs are used to generate the SBF solution.

15

=40

=0

=2

=20

=03 0

=

In this paper, the theoretical analysis of SBF method is discussed with its applications to PEC and dielectric scattering problems. SBF method has advantages in controlling the precision of complex problems. SBFs can extract enough features of the scattering fields of specific structures by introducing auxiliary sources to generate the solution space. The number of SBFs are controllable, enabling us to make a compromise between the precision and the efficiency. The advantages of SBF method in non-periodic multi-scatterer array problems are proved by several numerical examples. Since it can greatly reduce the number of unknowns, this SBF method would be very efficient in analyzing the scattering characteristics of large-scale arrays, especially dielectric and chiral cases. REFERENCES

Fig. 8. An array of 10 chiral spheres. Each sphere is two-layered with parameters of k r : ," : , , r =r ," : , : and  : .d r ,d r . The incident wave is an x-polarized  plane wave in z direction.

=

=15 =40 =0 = 10 = 4 =2 0

=2

=20

=03

Fig. 9. Bistatic RCS of the chiral array in Fig. 8 Dot line: 23 SBFs on each block. Dashed line: 11 SBFs on each block.

total number of volume meshes is 8920, which generate 19950 SWGs. After dividing the structure into blocks with reference surfaces, we get 3960 RWGs on the surfaces for the GSIE solution. By placing 320 auxiliary sources round one block and introducing the SBF generation processing, we get 23 SBFs on each block and 230 SBFs in total. Note that in the GSIE, both electrical field and magnetic field are needed to be calculated, the number of unknowns in the linear system should

[1] M. I. Aksun, A. Alparslan, and E. P. Karabulut, “Determining the effective constitutive parameters of finite periodic structures: Photonic crystals and metamaterials,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 6, pp. 1423–1434, Jun. 2008. [2] T. X. Wu and D. L. Jaggard, “Scattering of chiral periodic structure,” IEEE Trans. Antennas Propag., vol. 52, pp. 1859–1870, Jul. 2004. [3] G. Pelosi, A. Cocchi, and S. Selleri, “Electromagnetic scattering from infinite periodicstructures with a localized impurity,” IEEE Trans. Antennas Propag., vol. 49, pp. 697–702, May 2001. [4] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, pp. 409–418, May 1982. [5] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag., vol. 35, no. 3, pp. 7–12, Jun. 1993. [6] J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Tech. Lett., vol. 10, no. 1, pp. 14–19, Sep. 1995. [7] J. M. Song, C. C. Lu, and W. C. Chew, “MLFMA for electromagnetic scattering from large complex objects,” IEEE Trans. Antennas Propag., vol. 45, pp. 1488–1493, Oct. 1997. [8] V. V. S. Prakash and R. Mittra, “Characteristic basis function method: A new technique for efficent solution of method of moments matrix quations,” Microwave Opt. Technol. Lett., vol. 36, pp. 95–100, Jan. 2003. [9] J. Yeo, V. V. S. Prakash, and R. Mittra, “Efficient analysis of a class of microstrip antennas using the characteristic basis fucntion method (CBFM),” Microwave Opt. Technol. Lett., vol. 39, pp. 456–464, Dec. 2003. [10] J. Yeo and R. Mittra, “Numerically efficient analysis of microstrip antennas using the characteristic basis function method (CBFM),” in Proc. IEEE AP-S Int. Symp., Columbus, OH, Jun. 2003. [11] L. Xu, Z. Nie, and J. Hu, “Investigation on solving electromagnetic scattering by characteristic basis function method,” presented at the Int. Conf. Computational Electromagnetics, 2004. [12] C. Delgado, R. Mittra, and F. Cátedra, “Accurate representation of the edge behavior of current when using PO-derived characteristic basis functions,” IEEE Antenna Wireless Propag. Lett., vol. 7, pp. 43–45, 2008.

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[13] W. B. Lu, T. J. Cui, Z. G. Qian, X. X. Yin, and W. Hong, “Accurate analysis of large-scale periodic structures using an efficient sub-entiredomain basis function method,” IEEE Trans. Antennas Propag., vol. 52, pp. 3078–3085, Nov. 2004. [14] W. B. Lu, T. J. Cui, and H. Zhao, “Acceleration of fast multipole method for large-scale periodic structures with finite sizes using sub-entire-domain basis functions,” IEEE Trans. Antennas Propag., vol. 55, pp. 414–421, Feb. 2007. [15] L. Matekovits, G. Vecchi, G. Dassano, and M. Orefice, “Synthetic function analysis of large printed structures: The solution space sampling approach,” in Proc. IEEE AP-S Int. Symp., Boston, MA, Jul. 2001, pp. 568–571. [16] P. Focardi, A. Freni, S. Maci, and G. Vecchi, “Efficient analysis of arrays of rectangular corrugated horns: The synthetic aperture function approach,” IEEE Trans. Antennas Propag., vol. AP-53, no. 2, pp. 601–607, Feb. 2005. [17] L. Matekovits, V. A. Laza, and G. Vecchi, “Analysis of large complex structures with the synthetic-functions approach,” IEEE Trans. Antennas Propag., vol. 55, pp. 2509–2521, Sep. 2007. [18] O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag., vol. AP-37, no. 7, pp. 918–926, Jul. 1989. [19] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag., vol. AP-32, pp. 77–85, Jan. 1984. [20] G. Xiao, J. Mao, and B. Yuan, “Generalized transition matrix for arbitrarily shaped scatterers or scatterer groups,” IEEE Trans. Antennas Propag., vol. 56, pp. 3723–3732, Dec. 2008. [21] G. Xiao, J. 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. [22] M. Hasanovic, M. Chong, J. R. Mautz, and E. Arvas, “Scattering from 3-D inhomogeneous chiral bodies of arbitrary shape by the method of moments,” IEEE Trans. Antennas Propag., vol. 55, pp. 1817–1825, Jun. 2007. [23] O. Ozgun, R. Mittra, and M. Kuzuoglu, “Parallelized characteristic basis finite element method (CBFEM-MPI)—A non-iterative domain decomposition algorithm for electromagnetic scattering problems,” J. Comput. Phys., vol. 228, no. 6, pp. 2225–2238, Apr. 2009. [24] M. F. R. Cooray and I. R. Ciric, “Wave scattering by a chiral spheroid,” J. Opt. Society Amer. A: Optics, Image Sci. Vision, vol. 10, no. 6, pp. 1197–1203, Jun. 1993.

Gaobiao Xiao (M’02) was born in China, in 1965. He received the M.S. degree from Huazhong University of Science and Technology, Wuhan, China, in 1988, the B.S. degree from the National University of Defense Technology, Changsha, China, in 1991, and the Ph.D. degree from Chiba University, Chiba, Japan, in March 2002. He joined the Department of Electrical and Electronics Engineering, Hunan University, as a Research Associate from 1991 to 1997. Since April 2004, he has been an Associate Professor in the Department of Electronics, Shanghai Jiao Tong University, Shanghai, China. His research interests are in microwave filters, fiber-optic filters, inverse scattering problems, circuits analysis and numerical methods in electromagnetic fields.

Bo Zhang was born in China, in 1985. He received the B. S. degree from Shanghai Jiao Tong University, China, in 2007, where he is currently working toward the M.S. degree. His research interests are in scattering analysis and computational electromagnetics.

Yan Wang was born in 1983. She received the B.S. degree from Hangzhou Dianzi University, China, in 2007. She is currently working toward the M.S. degree at Shanghai Jiao Tong University, China. Her research interests are in microwave filters and numerical methods in electromagnetic fields.

Jun-Fa Mao (M’92–SM’98) was born in 1965. He received the B.S. degree in radiation physics from the University of Science and Technology of National Defense, China, in 1985, the M.S. degree in experimental nuclear physics form Shanghai Institute of Nuclear Research, Shanghai, China, in 1988, and the Ph.D. degree in electronic engineering from Shanghai Jiao Tong University, Shanghai, China, in 1992. Since 1992, he has been a Faculty Member in the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China, where he is currently a Chair Professor. He was a Visiting Scholar at the Chinese University of Hong Kong, Hong Kong, from 1994 to 1995, and a Postdoctoral Researcher at the University of California, Berkeley, from 1995 to 1996. His research interests include the interconnect problem of high-speed integrated circuits, microwave circuits and EMC. He has authored or coauthored more than 130 journal papers and 60 international conference papers. Dr. Mao received the National Natural Science Award of China in 2004, the National Technology Invention Award of China in 2008 and the Best Paper Award of 2008 APEMC in conjunction with 19th International Zurich EMC. He is a Cheung Kong Scholar of the Ministry of Education, China, an Associate Director of the Microwave Society of China Institute of Electronics, and the 2007–2008 Chair of the IEEE Shanghai Section. He was a Topic Expert of the High-Tech Program of China during 2001–2003, an Associate Dean of the School of Electronic, Information and Electrical Engineering, Shanghai Jiao Tong University, from 1999 to 2005.

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Combined Electromagnetic Energy and Momentum Conservation Equation Omid Zandi, Student Member, IEEE, Zahra Atlasbaf, Member, IEEE, and Mohammad Sadegh Abrishamian

Abstract—We have combined the two fundamental conservation theorems in electromagnetic theory—the energy and the momentum conservation theorems—to derive a vector wave equation for linear and homogeneous media with symmetrical constitutive matrix. We have demonstrated how the relation between the Poynting vector and the momentum density vector affects the electromagnetic quantities. Furthermore, a relation between the Poynting vector, the momentum density vector, and the phase velocity of the Poynting vector has been derived. In addition, what happens if the Poynting vector and the momentum density vector become mutually orthogonal or one of them vanishes has also been discussed. The angle between the above-mentioned vectors has been derived, and a relation between the momentum density vector and the wave vector for plane waves has been obtained. It has been shown that for plane waves, orthogonality of the phase velocity is not feasible. Index Terms—Momentum density vector, phase velocity, poynting vector, wave vector.

I. INTRODUCTION EDIA in modern macroscopic electromagnetism is probably the most important part of research. Special materials, such as phase reversal metamaterials [1], chiral, bi-isotropic and bi-anisotropic media [2], [3], and any unconventional material [4], offer special properties that include vast fields of interest. Novel boundary conditions also attract a huge amount of these researches to themselves, and some of the recent examples include PEMC boundary [5], [6], DB boundary [7], [8], impedance boundary [9], soft and hard surfaces [10], etc. The method used to analyze all the above-mentioned media and boundaries has been almost always based on analyzing the relations between the fields’ intensities and densities through their special constitutive relations. In this paper, we have developed a new method to analyze the wave propagation in different media. Instead of the fields’ intensities and densities, we have employed the Poynting vector (PV) and the momentum density vector (MDV). The energy conservation theorem (Poynting theorem) and the momentum conservation theorem have been combined to obtain a vector wave equation in terms of the PV, MDV, a reduced form of the Maxwell stress tensor, the Lorentz force, and the power supplied

M

Manuscript received May 27, 2009; revised April 26, 2010; accepted May 17, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. O. Zandi and Z. Atlasbaf are with Tarbiat Modares University, Tehran, Iran (e-mail: [email protected]; [email protected]). M. S. Abrishamian is with K. N. Toosi University of Technology, Tehran, Iran (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2071340

to the system. Based on this wave equation, some worthy conclusions have been obtained. Although by no means this method is claimed to be a complete substitute for the conventional method, it can be considered as a tool to obtain more insight into the problem. In particular, this method will be useful for media where the PV and MDV are in two different directions. For plane waves inside a linear medium, the group and the phase velocity vectors remain parallel to the PV and MDV, respectively. Hence, if the PV and MDV have different directions, the group and phase velocities will also have different directions. In this study, we have shown what happens when the PV and MDV become perpendicular to each other. In addition, the cases where the electromagnetic fields either lose their momentum or their energy or both have also been discussed. We developed a new approach to examine the electromagnetic boundary conditions, through the PV, MDV, Lorentz force, and Maxwell stress tensor. Some of the well-known boundary conditions were considered, and their effects on the normal and tangential components of the PV and MDV were demonstrated. We chose a rectangular coordinate system to extract three scalar equations from the vector wave equation, and then solved one of them. For time-harmonic fields, we determined a relation between the PV, MDV, and phase velocity of the PV for a special class of materials. Three different examples of isotropic, bi-isotropic, and anisotropic materials were considered to verify this result. The angle between the PV and MDV was derived. In macroscopic electromagnetism, these vectors can be perpendicular to each other; however, in this case, no electromagnetic wave can exist. For plane waves, the relation between the wave vector and MDV was derived, and it was shown that the orthogonality of the phase velocity is not feasible. In the following sections, bold letters have been used to denote vectors and italic letters denote the scalar quantities, components of a vector, or elements of a matrix. Furthermore, double-bar letters stand for tensors. II. WAVE EQUATION IN TERMS OF THE POYNTING AND THE MOMENTUM DENSITY VECTORS Poynting’s theorem, in conservation of the electromagnetic energy, can be written for linear bi-anisotropic media with symsystem as [11] metrical constitutive matrix in

0018-926X/$26.00 © 2010 IEEE

(1)

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where , , and stand for the total stored energy, the power density vector, and the power supplied by an external source, respectively. These are

(2a) (2b) (2c) On the other hand, for linear, homogeneous bi-anisotropic media with symmetrical constitutive matrix, the momentum conservation theorem states that [11]

(3) where , , and are the MDV, the Maxwell stress tensor, and the Lorentz force, respectively. These are

For a medium with no free charges, (8b) can be written in a rectangular coordinate system as

(8c) are considered to be the unit vectors in where , , and are their the rectangular coordinate system, and , , and and are the components of corresponding variables. and , respectively, in this coordinate system. Equation (7) is a general form of the electromagnetic wave equation in terms of the PV, MDV, etc. It governs the electromagnetic wave propagation in linear and homogeneous materials with symmetrical constitutive matrix. It also expresses a conservation theorem for both the energy and momentum simultaneously. From (1), the boundary condition for the normal component of the PV for two adjacent stationary media, shown by the indices of 1 and 2, can be given as

(4a) (4b)

(9)

where is a unit dyad. By calculating the gradient of (1) and also the time derivative of (3), we can get

where is a unit vector normal to the common surface from the first into the second media. From (3), another boundary condition for the Maxwell’s stress tensor in a stationary system can be derived as

(5a) (10) (5b) Now, if the stored energy, given in (2b), is a continuous and well-behaved function of time and position ([12], theorem 12.131), then we can say

The index of in (9) and (10) stands for the scalar or vector functions on the surface. The boundary conditions that govern the components of on the surface are more difficult to derive. Nevertheless, we determined the tangential component of it over the surface as

(6) (11a)

Subsequently, we can combine (5a) and (5b) to obtain and its normal component as (7)

(11b)

where

(8a) which can be termed as reduced Maxwell’s stress tensor. Furthermore, it can be shown that [13, pp.84-85]

(8b) 1That is, if F is a function of x and t in a region, we can say that if @F=@x and @F=@t exist and @ F=@x@t and @ F=@t@x are continuous in this region, @ F=@t@x. This theorem seems applicable to W in any then @ F=@x@t ordinary medium.

=

Clearly, the DB boundary condition [8] leads to the boundary , which filters the tangential comcondition of ponent of the MDV. Prefect electric conductor (PEC), prefect magnetic conductor (PMC), and prefect electromagnetic conductor (PEMC) boundaries [5], [6] filter the normal component of the PV and the tangential component of the MDV. However, in PEC, PMC, and PEMC media, none of the electromagnetic energy and momentum can propagate. On the soft and hard surfaces (SHS) [10], electrical and magnetic fields are parallel, and hence, it filters the normal component of the PV. A summary of some boundary conditions, in terms of the tangential and normal components of the PV and MDV, is shown in Table I.

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TABLE I SOME FAMOUS BOUNDARY CONDITIONS IN TERMS OF THE TANGENTIAL AND NORMAL COMPONENTS OF THE PV AND MDV

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From (1), we can see that

(14b) And from (3), we can see that

(14c)

M is a pseudoscalar. It should not be interpreted as the surface admittance, except for time-harmonic fields in the frequency domain.

Generally, the PV and MDV do not have the same orientations may not be parallel to in a homogeneous medium, because , and also may not be parallel to . We considered a plane where both the PV and MDV lie and built our rectangular coordinate system with two unit vectors in this plane and the third one perpendicular to it. We chose our coordinate system such that (12a) (12b) We assumed that the media is unbounded and there are no free charges. We substituted (8c) into (7) to rewrite the vector wave equation as a set of three scalar differential equations of

(13a)

From (14a), it is interesting to note that the PV and a component of the MDV (which is in the direction of the PV) have the potential to form a scalar wave equation. In the next section, we will discuss about this subject in more detail. III. SCALAR DIFFERENTIAL EQUATION FOR DIFFERENT MEDIA Equation (14a) is a partial differential equation in terms of two unknown functions. Hence, at this stage, another relation is necessary to express one of them in terms between and of the other. This additional equation makes (14a) as a partial differential equation in terms of one unknown function and it can be shown by (15a) (15b) to and is deterwhere (if exists) is an operator relating (again mined by the physical constitution of the media, and if exists) is its inverse. Inserting (15a) into (14a) leads to

(16) Indeed, the nature of determines whether (16) is a wave equation. We considered four arbitrary cases. At first, we assumed that and are nonzero and we have

(13b) (17) where is a constant or a function of and . In the second case, we assumed that is a null operator, indicating that (13c) (18) do not have any component It must be noted that and in the direction of ; hence, the divergence of the reduced Maxwell stress tensor in (8b) also does not have any component in the direction of . As a result, free charges, if any, do not change the form of (13a). In a medium with no macroscopic currents perpendicular to the direction of the PV and forces parallel to the direction of the PV, (13a) will be

Two situations lead to (18). First, the electromagnetic fields do not possess linear momentum, and second, the MDV is nonzero but is mutually orthogonal to the PV. When electromagnetic fields do not carry momentum, and are parallel or at least one of them is zero. is a null operator, In the third case, it is assumed that which means that

(14a)

(19)

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Equation (19) indicates that the electromagnetic fields cannot transfer energy. It takes place when and are parallel or at least one of them is zero. And finally, in the fourth case, we assumed that both and are zero. These four cases have been discussed individually as follows.

As another example, we considered an anisotropic medium with the following constitutive relations:

(26a) (26b)

A.

,

,

As mentioned earlier, we supposed that the ratio of over is a constant or a function of and . Hence, (14a) will be

(20) Then, the separated form of

It must be noted that in our rectangular coordinate, and are both zero. Furthermore, we also implicitly presumed that at and can be nonzero, because any time, only one of does not have any component in the direction of . We have discussed two cases that are known as the ordinary and extraordinary propagations [11]. and For ordinary wave propagation, we presumed that are zero, but and are nonzero. Thus the PV and MDV will be

(21) (27a) (27b)

helps us to solve the equation as (22)

which are parallel. We used (23) to obtain the phase velocity of the PV as

It is evident that (22) shows a propagating wave in the direction of . In addition, from (22), it can be observed that is the , and is the wave number. phase velocity of the PV or In media where the assumption of (17) holds, we can rewrite this in our rectangular coordinate system as

(27c) and For extraordinary propagation, we presumed that are zero, but and are nonzero. Thus, the PV and MDV will be

(23) If our assumption of (17) does not hold, (16) may become a complex partial differential equation, which governs the propagation in complex media. In this case, closed-form solutions in terms of elementary functions such as (22) may not be possible. However, these cases were excluded and we just considered the simplest case. A point that we can obtain from (20) and (23) in the considered media is that if casts a negative projection onto , there will be an evanescent wave. in (23) becomes for any One can verify that isotropic media, where and are its permittivity and permeability, respectively. Furthermore, for a bi-isotropic media with the following constitutive relations: (24a) (24b) the phase velocity of the PV in (23) will be

(25) which is exactly equal to what the traditional method gives [11, p. 78].

(28a) (28b) which are in different directions. We used (23) to obtain the phase velocity of the PV as

(28c) Equations (27c) and (28c) are exactly the same results that the traditional method gives [11, pp.68-69]. Let us consider the more general form of (13a), where the macroscopic currents exist. We only considered a theoretical case. We presumed that (29a) (29b) where and are constants (or functions of and ), and have the dimensions of velocity and one over the distance, respectively. This very special case makes (13a) a linear partial differential equation with constant coefficients, which is

(30)

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We used the separated form of (21) to divide (30) into two ordinary second-order differential equations of

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provided there are no macroscopic currents perpendicular to the direction of the PV and forces parallel to this direction. Equations (35a) and (35b) have solutions of

(31a) (36a) (36b)

(31b) where is a real constant or a real function of and . Both (31a) and (31b) can be solved straightforwardly. Here, we just considered the positive sign in the last terms of (31a) and (31b) to obtain (32a)

(32b) where and and . of The condition of

,

,

, and

are generally functions (37a)

(33) should be satisfied to have propagating waves. The effect of source and dissipation currents on the behavior of electromagnetic waves is known well. Depending on the positive or negative sign of , macroscopic currents can attenuate or amplify the electromagnetic waves, respectively. In this theoretical case, the existence of currents also changes the phase velocity. B.

where and are constants. Equation (36b) indicates that the stored energy is not a function of . Equation (36a) shows this important fact that in the considered media, when the PV and MDV are perpendicular to each other, the PV does not have a form of a propagating wave in its direction; rather it linearly depends on the variable . Of course, this kind of solution is not allowed in an unbounded media. When macroscopic currents exist, (13a) will be

where for the same special theoretical case of (29a) and (29b), it becomes

(37b) which is a parabolic equation. In fact, in the considered media, (37b) governs the electromagnetic power flow when it cannot propagate, because it does not possess momentum.

, Media with the constitutive relation of

C.

, Media with the constitutive relation of

(34a) is a pseudoscalar), does not allow electromagnetic (where fields to possess momentum; however, it generally exerts no constraint on the energy transport. For homogeneous and timeinvariant media, if we consider the two Maxwell’s curl equations of

(38a) is a pseudoscalar), does not allow electromagnetic (where fields to transfer energy; however, it generally exerts no constraint on the momentum transport. For homogeneous and timeinvariant media, if we consider the two Maxwell’s curl equations, we can easily show that (38a) leads to

we can easily show that (34a) leads to

(38b) (34b)

For such a medium, (13a) will be

For such a medium and also for media where the PV and MDV are mutually orthogonal, (14a) and (14c) get reduced to

(39a)

(35a) (35b)

Equation (14b) shows that the stored energy will not be a function of time.

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,

PEC, PMC, and PEMC as media do not let electromagnetic fields transfer energy and momentum. For such a medium, (13a) will be

(39b) Both (39a) and (39b) show a kind of balance among the exerted force (if any), electromagnetic momentum (again if any) and medium reaction. Because of this balance, no energy will propagate even though there are time-varying currents [14]. If the concept of “boundary” is defined as “a surface which does not let electromagnetic fields pass through it” [10], it seems reasonable to modify the definition of a prefect boundary to include the additional condition of

(40) which shows that such a surface does not let the MDV have a normal component on it. This modification is necessary, especially when an electromagnetic wave is incident on this surface and are not parallel to from an anisotropic media where and , respectively. Otherwise, a part of the electromagnetic fields will pass through it.

IV. ANGLE BETWEEN THE POYNTING AND THE MOMENTUM DENSITY VECTORS In this section, we will determine the cosine of the angle between the PV and MDV. First we will find the unit vectors that are in the directions of the above-mentioned vectors. We have (41a) (41b) where is a unit vector in the direction of the PV and is a unit vector in the direction of the MDV. Let us dot-multiply (41a) by (41b) to determine the cosine of . the angle between the PV and the MDV,

(42) If we use the vector identity

Fig. 1. (a) 

vs. " ="

(b) u

j

G=S j j

j

vs. 

.

From (44), it can be seen that the equation

(45a) is the condition in which the PV and MDV become mutually orthogonal, provided

(45b) for the extraordinary propagation in the Let us derive anisotropic media considered in Section III. If we insert (28a) and (28b) into (42), we will have

(46) (43)

Although depends on , the phase velocity in (28c) is independent of it. From (23), we have

we can rewrite (42) as (44) where

, for example, stands for the angle between

and .

(47) vs. and the ratio of In Fig. 1(a) and 1(b), vs. are depicted, respectively.

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TABLE II CONSTITUTIVE RELATIONS OF FOUR MEDIA WHERE ELECTROMAGNETIC FIELDS CANNOT PROPAGATE

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be mutually orthogonal and we will run into case B presented in Section III. However, for plane waves inside a source-free and lossless medium, orthogonality of the wave vector (or the phase velocity) leads to zero MDV. To prove it, we considered the Maxwell’s two curl equations for a monochromatic plane wave (with the , where and are the wave angular form of frequency and the position vector, respectively) in a source-free and lossless medium. These are [11]

In Table II, a summary of the media considered in this paper, where electromagnetic fields cannot propagate, is presented.

(53a) (53b)

where is the wave vector. Equations (48a) and (48b) indicate that the MDV is parallel to the wave vector. From (48a) and (48b), and can be written as

If is perpendicular to , then it has to lie on the plane of and . On the other hand, (53a) and (53b) imply that both and should be perpendicular to this plane, so that they will ,a be parallel and thus the MDV will vanish. When nontrivial solution for (51a) and (51b) is . When a material is lossy but without free charges, the dissipation current density appears on the right-hand side of the (53b). In this case, generally will not be perpendicular to the plane of and . Hence, and will no longer be parallel and the MDV will not be zero, but will be perpendicular to the PV. Let us take a step further by dot-multiplying both the sides of (51a) and (51b) by the PV

(49a) (49b)

(54a) (54b)

and are the well-known magnetic and electrical where vector potentials, respectively. The minus sign in (49b) is arbitrary. If we substitute (49a) into (4a), we can obtain

If condition (17) holds, one of its consequences is that the MDV must map a positive value onto the PV to have propagating waves. In this case and by considering (54a) and (54b), we find that

V. RELATION BETWEEN THE WAVE VECTOR AND THE MOMENTUM DENSITY VECTOR FOR PLANE WAVES For a monochromatic plane wave, the Maxwell’s two divergence equations in a source-free medium will be [11] (48a) (48b)

(50) By inserting (48b) into (50), we will have

(55) Therefore, if (51a)

Similarly, by inserting (49b) into (4a), we can get

(56a)

(51b)

which is regarded as the positive phase velocity [15], we must also have

Equations (51a) and (51b) show that (56b) (52) Equations (51a) and (51b) indicate that though and are parallel, they may have different orientations. Hence, the absolute value of the cosine of the angle derived in (44) equals the absolute value of the cosine of the angle between the phase and group velocities for plane waves. Equations (51a) and (51b) also state that for plane waves inside a source-free and lossless medium, and cannot be mutually orthogonal, because in this case, the MDV and PV will

Similarly, if

(57a) which is regarded as the negative phase velocity [15], we must also have

(57b)

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VI. CONCLUSION In this paper, we combined the two fundamental conservation equations in electromagnetism to obtain a vector wave equation in linear and homogeneous media with symmetrical constitutive matrix. Subsequently, we decomposed it into three scalar equations where one of them has a significant importance in analyzing the propagation of electromagnetic waves. We presumed a special relation between the PV and a component of the MDV which is in its direction. Based on this assumption, we determined the phase velocity of the PV and showed that evanescent waves come into being when the MDV casts a negative projection onto the PV. We also examined the situations where: 1) the MDV is zero or perpendicular to the PV; 2) the PV is zero; and 3) both the PV and MDV are zero. However, we found that in none of these situations, electromagnetic fields could propagate. Furthermore, the angle between the PV and MDV was derived and the relation between the wave vector and MDV for plane waves was demonstrated. REFERENCES [1] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, Engineering Approach. New York: Wiley, Nov. 2005. [2] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media. Norwood, MA: Artech House, 1994. [3] A. Serdyunkov, I. Semchenko, S. Tretyakov, and A. Sivola, Electromagnetics of Bi-Anisotropic Materials Theory and Applications. New York: Gordon and Breach Science, 2001. [4] O. N. Singh and A. Lakhtakia, Electromagnetic Fields in Unconventional Materials and Structures. New York: Wiley, 2000. [5] I. V. Lindell and A. H. Sihvola, “Transformation method for problems involving perfect electromagnetic conductor (PEMC) boundary,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 3005–3011, Sep. 2005. [6] I. V. Lindell and A. H. Sihvola, “Realization of the PEMC boundary,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 3012–3018, Sep. 2005. [7] I. V. Lindell and A. H. Sihvola, “DB boundary as isotropic soft surface,” presented at the Asian Pacific Microwave Conf., Hong Kong, Dec. 2008. [8] I. V. Lindell and A. H. Sihvola, “Zero-axial-parameter (ZAP) medium sheet,” Progress In Electromagn. Res., vol. PIER 89, pp. 213–224, 2009. [9] I. V. Lindell and A. H. Sihvola, “Realization of impedance boundary,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3669–3676, Dec. 2006.

[10] I. V. Lindell, “Ideal boundary and generalised soft and hard conditions,” Proc. IEE Microw. Antennas and Propagat., vol. 147, no. 6, pp. 495–499, Dec. 2000. [11] J. A. Kong, Electromagnetic Wave Theory. New York: Wiley, 1986. [12] T. M. Apostol, Mathematical Analysis, 2nd ed. New York: AddisonWesley Publishing Company, 1974. [13] C. T. Tai, Generalized Vector and Dyadic Analysis; Applied Mathematics in Field Theory, 2nd ed. Piscataway, NJ: IEEE, 1997. [14] J. Nachamkin, “The effect of material stresses on electromagnetic boundary conditions,” IEEE Antennas Propag. Mag., vol. 46, no. 5, pp. 11–17, Oct. 2004. [15] T. G. Mackay and A. Lakhtakia, “Positive-, negative-, and orthogonalphase-velocity propagation of electromagnetic plane waves in a simply moving medium: Reformulation and reappraisal,” Optik, vol. 120, pp. 45–48, 2009. Omid Zandi (S’08) was born in Tehran, Iran, on April 25, 1982. He received the B.S. degree from K. N. Toosi University of Technology, Tehran, in 2005 and the M.S. degree from Tarbiat Modares (T.M.) University, Tehran, in 2007, both in electrical engineering. He is currently working toward the Ph.D. degree at T.M. University. His main fields of research are propagation in complex media, antenna theory, and design of passive microwave components. He is also interested in some courses of the theoretical physics such as the special theory of relativity and the quantum electrodynamics.

Zahra Atlasbaf (M’08) received the B.S. degree in electrical engineering from the University of Tehran, Iran, and the M.S. and Ph.D. degrees in electrical engineering from the University of Tarbiat Modares, Tehran. She is currently an Assistant Professor on the faculty of Electrical and Computer Engineering, Tarbiat Modares University. Her research interests include numerical methods in electromagnetics, theory and applications of meta materials, and microwave and antenna design.

Mohammad Sadegh Abrishamian received the B.S. degree from the High Institute of Telecommunication, Iran, the M.S. from the Northrop University, Inglewood, CA, and the Ph.D. degree from Bradford University, Bradford, U.K., all in electrical engineering. He has been a faculty member at K.N. Toosi University of Technology, Tehran, for the past 28 years. His research interests include penetration and scattering of EM waves, photonic crystals, plasmonics and computational electromagnetics.

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A Novel Approach for Evaluating Hypersingular and Strongly Singular Surface Integrals in Electromagnetics Mei Song Tong, Senior Member, IEEE, and Weng Cho Chew, Fellow, IEEE

Abstract—Solving electromagnetic (EM) problems by integral equation methods requires an accurate and efficient treatment for the singular integral kernels related to the Green’s function. and operFor surface integral equations (SIEs), there are ators which include hypersingular integrals (HSIs) and strongly singular integrals (SSIs), respectively. The HSIs are generated from the double gradient of the Green’s function while the SSIs come from the single gradient of the Green’s function. Although the HSIs could be reduced to weakly singular integrals (WSIs) in the method of moments (MoM) implementation with divergence conforming basis function such as the Rao-Wilton-Glisson (RWG) basis function, they do appear in Nyström method (NM) or boundary element method (BEM) and one has to tackle them. The SSIs always exist in the operator and could also exist in the operator when the testing function is not the RWG-like basis function. The treatment for the HSIs and SSIs is essential because they have a significant influence on the numerical solutions. There have been many publications dealing with the singular integrals, but they mainly focus on the WISs or SSIs, and the HSIs were seldom addressed. In this work, we develop a novel approach for evaluating those HSIs and SSIs based on the Stokes’ theorem. The derived formulas are much simpler and more friendly in implementation since no polar coordinates or extra coordinate transformation are involved. Numerical experiments are presented to demonstrate the effectiveness of the approach. Index Terms—Electromagnetic scattering, integral equations, singular integrals, Stokes’ theorem.

I. INTRODUCTION

I

NTEGRAL equation methods (IEMs) are efficient numerical methods for solving electromagnetic (EM) problems and may be preferred in many applications [1], [2]. This is because the IEMs usually require a smaller number of unknowns and have a better scaling property for computational costs if compared with the differential equation methods (DEMs). The IEMs seek the solution of source distribution on boundaries or within objects first, instead of the field distribution at spatial

Manuscript received January 28, 2010; revised April 15, 2010; accepted April 29, 2010. Date of publication September 02, 2010; date of current version November 03, 2010. This work was supported in part by Y. T. Lo Endowed Chair Professorship. M. S. Tong is with the Center for Computational Electromagnetics and Electromagnetics Laboratory (CCEML), Department of Electrical and Computer Engineering (ECE), University of Illinois at Urbana-Champaign (UIUC), Urbana, IL 61801 USA (e-mail: [email protected]). W. C. Chew is with the University of Hong Kong, Pokfulam, Hong Kong. Digital Object Identifier 10.1109/TAP.2010.2071370

points directly, and the fields are calculated from the source distribution once available. Therefore, the solution domain is much smaller in the IEMs than in the DEMs, and also there is no need to implement absorbing boundary condition (ABC) for open field domains, leading to ease of implementation. However, the system matrix of IEMs is inherently dense and efficient accelerators like multilevel fast multipole algorithm (MLFMA) [2] are needed for solving large problems. Moreover, the IEMs are more complicated to implement in general due to the need of the Green’s function. The Green’s function in EM is singular and much effort is required generally to evaluate matrix elements with singular kernels. EM integral equations include surface integral equations (SIEs) and volume integral equations (VIEs). The SIEs are preferred whenever available because they require less unknowns in discretization. The SIEs can be the electric field integral equation (EFIE), magnetic field integral equation (MFIE), combined field integral equation (CFIE), Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) equation [3] or Müller equation [4]. All these equations include both and operators if the objects are penetrable or one of them if impenetrable. The kernel of the operator is the dyadic Green’s function which includes a double gradient operation on hypersingular the scalar Green’s function and results in is the integrals (HSIs) in evaluating matrix entries, where distance between a source point and an observation point or weakly field point. However, the HSIs could be reduced to singular integrals (WSIs) in the method of moments (MoM) solution if divergence conforming basis function such as the Rao-Wilton-Glisson (RWG) basis function [5] is used as an expansion and testing function. Without the help of RWG-like basis function, we must carefully handle the HSIs and this happens in the implementation of Nyström method (NM) or boundary element method (BEM) because they do not use any basis or testing function. In the operator, the kernel is a single gradient operator on the scalar Green’s function, yielding strongly singular integrals (SSIs) in the matrix elements. Note that we have adopted the convention of mechanical engineering [6] and mathematics [7] to categorize the degrees of singularity in the above. This convention could be different from the EM singular integrals are called HSIs convention in which the sometimes [8], and also different from the definition in [9] where the HSIs are named Cauchy singular integrals and SSIs are called WSIs. The SSIs always exist in the operator even in the MoM implementation. It could also exist in the operator in the MoM when the RWG-like basis function represents

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both the electric and magnetic current in penetrable structures has to be a testing function for producing and the a well-conditioned impedance matrix [10]. The accurate and efficient evaluation for the HSIs and SSIs is essential in solving SIEs because they have a significant impact on the numerical solutions and we must treat them wisely. There have been many publications dealing with the singular integrals for EM integral equations [11]–[31], but they have mainly focused on the WSIs and SSIs. For HSIs, only our previous work presented a systematic solution [32]. Although the formulas in those publications are effective, there is certain inconvenience in their implementations due to the introduction of polar coordinate system and extra coordinate transformation. In this work, we develop a novel approach for evaluating those HSIs and SSIs. The new derivation is based on the Stokes’ theorem [34] which changes a surface integral over a surface patch into a line integral along its boundary. We assume that the surface patches are the flat triangular patches which are most widely used in geometrical discretization. To derive the HSIs and SSIs, we also assume that the observation point is initially off the source patch, so the corresponding integral is regular. We then derive analytical solutions for the line integrals after applying the Stokes’ theorem and take a limit of the derived formulas by letting the observation point approach the source area to obtain the solutions of the HSIs and SSIs. This is a Cauchy-principal-value-like (CPV-like) approach for handling the HSIs and SSIs. Compared with other approaches, the new derivation is simpler and the formulas are more convenient in implementation since no polar coordinates and extra coordinate transformation are involved. Numerical experiments are presented to demonstrate the validity of the derived formulas by comparing with the solutions of our previous formulas. We also show several numerical examples to illustrate the applications of the new formulas to solve EM scattering problems.

II. SIES AND SINGULARITY SUBTRACTION There are several types of SIE in EM, depending on the property of objects with which EM wave is interacting. For EM wave scattering by an impenetrable object like perfectly electric conductor (PEC), the SIE can be an EFIE, i.e.

surrounding medium in the above. The SIE can also be an MFIE for the same problem, i.e.

(3) where is the incident magnetic field. If we add the MFIE to the EFIE with an appropriate coefficient in the above, then we can obtain a combined field integral equation (CFIE) in which there is no internal resonance problem. The above EFIE and MFIE can also be written as

(4) where

(5) are operator and operator, respectively. If the object is penetrable, such as a dielectric object, the corresponding EFIE and MFIE can be written as

(6) and (7) respectively. In the above, and are the equivalent electric and magnetic current on the object surface, respectively, the subscripts 1 and 2 are the domain indices for the object and its surrounding medium, respectively, and the superscripts “ ” and “ ” on imply the interior and exterior side of the surface , respectively. The above electric field and magnetic field can be by recalling the vector related to the source currents and and scalar potentials, i.e.

(1)

is the induced surface current density on the obwhere ject surface, is the unit normal vector on the object surface, is the incident electric field, and (2) is the dyadic Green’s function in which is the scalar Green’s function and is the distance between an observation point and a source point . Also, is the wavenumber and is the intrinsic impedance related to the

(8)

TONG AND CHEW: A NOVEL APPROACH FOR EVALUATING HYPERSINGULAR AND STRONGLY SINGULAR SURFACE INTEGRALS

where and are magnetic and electric vector potential, and and are electric and magnetic scalar potential, respectively. Also, and are the permittivity and permeability of the related medium, respectively, and is the angular frequency of wave in free space. As for the PEC object, if we add the MFIE to the EFIE for the same regions with appropriate coefficients, then we can get the CFIE. However, in addition to these three equations, we also have the PMCHWT equation and Müller equation for penetrable objects. They are produced by adding the internal field equations to the external field equations in the EFIE and MFIE with a weighted-sum method [1]. It can be seen that all equations in the above include both operator and operator or one of them. To solve these SIEs, we can use the MoM, NM or BEM to discretize the equations and transform them into algebraic matrix equations. The solutions are then relying on the accurate evaluation for matrix elements. When an observation point is far away from a source patch, the corresponding matrix element is regular and numerical quadrature rules can be applied directly to evaluate the matrix element. However, when the observation point is inside or very close to a source patch, the corresponding matrix element is singular or badly behaved; a special treatment is needed since numerical quadrature rules cannot be applied directly. Although the singularity cancelation method is popular for WSIs and has also been applied to SSIs [8], the special treatment for HSIs could only be the singularity extraction or subtraction method in which the singular core is extracted out from the integrand and added back to keep the integrand unchanged. The integrand after extracting the singular core is bounded now and numerical quadrature rules can be used to evaluate the integral but a Duffy-like procedure [35] should be employed in order to obtain a higher-order accuracy. The work reduces to evaluating the integral with the singular core as an integrand. For example, the hypersingular integrand in the operator involves the double gradient of the scalar it Green’s function and in a global coordinate system can be expressed as

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(11) where is a

source

is an observation point, point within a source patch

,

is the distance between them, and . Note that the subtracted terms in the above are those singular . The first terms in the Taylor expansion of the integrand integral in the above is bounded and can be evaluated using numerical quadrature rules. However, we need to divide the source patch into subtriangles by connecting the vertices with the observation point, so that the integrand in the first integral is piecewise smooth in each subtriangle. Applying higher-order quadrature rules to each subtriangle for the integral can lead to a higher-order accuracy in numerical integrations. The second or more weakly singular core integral only includes and can be evaluated using the Duffy’s method [35] or those techniques in [11]–[31]. The third integral (12) is a HSI and deriving the solution for it and for other HSIs or SSIs using a novel approach is the purpose of this work. From other components in (9), we can find other HSIs as follows

(9) where (10) and only six components are independent due to symmetry. The singularity subtraction for the above integrand can be illustrated with the first component as follows

(13) For the kernel

in the

operator

(14) we can find the SSIs similarly, i.e.

(15)

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(17)

From the above equation, we can prove that [34]

Fig. 1. A local coordinate system (u; v; w ) whose uv plane coincides with the flat triangular patch and w axis is parallel to the unit normal vector of the patch.

To derive the solutions for those HSIs and SSIs, we need to inwhose plane cotroduce a local coordinate system incides with the flat triangular patch and axis is parallel to the unit normal vector of the patch, which points to the side of observation point. The coordinate system is sketched in Fig. 1. In such a local coordinate system, the observation point is located , the source point is located at within the at triangular patch, and . The expressions of the above HSIs and SSIs should also be changed with , with , with by replacing , and with . Once the solutions in the local coordinate system are found, the solutions in the global coordinate system can be found conveniently by a coordinate transformation.

(18)

plane and , then If we choose the patch plane as the , , and , and the above equations we have can be simplified as

(19)

The first two equations in the above can be used to evaluate HSIs and SSIs.

III. STOKES’ THEOREM The existent techniques of evaluating singular integrals are mostly based on the multiple integrals for surface integrations [33]. The novel approach to deriving the HSIs and SSIs in this work is the Stokes’ theorem which changes a surface integral over a patch into a line integral along its boundary. This change can dramatically simplify the derivation of analytical solutions for the singular integrals. The Stokes’ theorem can be written as [34]

IV. DERIVATION OF NEW FORMULAS FOR HSIS AND SSIS Before deriving the solutions for HSIs and SSIs, we have to derive the formulas for the integrals when the observation point is not inside the source triangular patch (see Fig. 1). The integrals in the local coordinate system take the following forms

(16) is a vector defined over a surface where patch , is the unit normal vector on , is the boundary , and is the unit vector representing the direction of of (see Fig. 1). For a flat patch, is a constant , so we have vector and

(20)

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from which the integrals in (12), (13), and (15) can be obtained by combination. We can derive the integrals in (20) by using the first two equations in (19) with the help of integral tables [36], namely,

(21) where

(22) and it is assumed that . Note that we can derive the solutions for more integrals, such as for the WSIs , , , etc. by using the approach. The derived formulas are effective no matter how close the observation point is to the source triangular patch and they can be used to accurately evaluate the near-interaction matrix elements. When the observation point is inside the source patch, i.e. for the HSIs and SSIs, we only need to take a limit of those to obtain the solutions for the inteformulas by letting grals in (12), (13), and (15). By doing so, we have

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TABLE I SOLUTION COMPARISON OF NEW FORMULA WITH OLD FORMULA AND NUMERICAL INTEGRATION FOR

0I

V. NUMERICAL TESTS To examine the effectiveness of the above formulas, we compare the solutions of the new formulas with the solutions of the numerical integrations and the solutions of our previous formulas [32]. The source triangle is defined by ran, domly choosing three vertices and , and the observation point is located at , where can be changed and the projection of observation point is inside the source triangle. The numerical integrations are performed with a 24-point product Gaussian quadrature rule. Tables I, II and III show the results for the , , and , respectively, when the observation integrals point is approaching the source triangle (the minus sign is added in and so that their values are mostly positive in the tables and the width of tables can be reduced). It can be seen that the solutions of the new formulas are almost the same (first 14 digits) as the ones of our previous formulas no matter how close the observation point is to the source patch. However, the solutions by numerical integrations are only the same as the analytical solutions when the observation points are far away from the source patch as expected and there is a big difference between them when the observation point is close to the source patch, indicating that the results of numerical integrations are incorrect and they cannot be used. VI. NUMERICAL EXAMPLES

(23) where

(24) and the superscript

denotes the limit.

We use the newly derived formulas to evaluate the HSIs or SSIs in solving the EFIE via the Nyström method for EM scattering by PEC, dielectric, or composite objects. Both and operators appear in the EFIE for the scattering by dielectric or composite objects, so all those formulas will be used in the solutions. The geometry of objects is chosen to be spherical so that the available analytical solutions or Mie series solutions can be used for comparison. It is assumed that the incident wave has a and is propagating along direction frequency in free space in all examples. We then calculate the bistatic radar cross section (RCS) observed along the principal cut ( and to 180 ) for the scatterers with both vertical polarization (VP) and horizontal polarization (HP). The solutions are compared with the analytical solutions which can be derived from [37]. The first example is the scattering by a PEC sphere and we discretize its surface into 2,146 with a radius triangular patches. The bistatic RCS solutions in VP and HP are shown in Fig. 2 and they are close to the analytical solutions. The second example is the scattering by a dielectric sphere with a radius and a relative permittivity (the relative permeability for dielectric materials). We

TONG AND CHEW: A NOVEL APPROACH FOR EVALUATING HYPERSINGULAR AND STRONGLY SINGULAR SURFACE INTEGRALS

TABLE II SOLUTION COMPARISON OF NEW FORMULA WITH OLD FORMULA AND NUMERICAL INTEGRATION FOR

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0I

TABLE III SOLUTION COMPARISON OF NEW FORMULA WITH OLD FORMULA AND NUMERICAL INTEGRATION FOR I

Fig. 2. Bistatic RCS solutions for a PEC sphere with a radius a

= 0:25.

Fig. 4. Bistatic RCS solutions for a PEC sphere with one-layer full dielectric coating. The inner radius is a : , the outer radius is a : , and the : . relative permittivity of coating material is "

=02

=40

= 0 25

Fig. 3. It can be seen that the solutions are also close to the corresponding analytical solutions. The third example illustrates the scattering by a composite sphere, i.e. a PEC sphere fully coated with one-layer dielectric material. The radii of the inner and , respecand outer interfaces are . The tively, and the relative permittivity of coating is discretization for the outer interface is the same as for the PEC sphere and the inner interface is meshed into 1,782 triangular patches. The bistatic RCS solutions in VP and HP are sketched in Fig. 4 and again the solutions agree well with the corresponding analytical counterparts. Fig. 3. Bistatic RCS solutions for a dielectric sphere with a radius a : . and "

= 40

= 0:25

use the same geometrical discretization as for the PEC sphere and the bistatic RCS solutions in VP and HP are plotted in

VII. CONCLUSION The IEMs for solving EM problems include singular kernels because of the Green’s function. Depending on the type of integral equations and the numerical approaches, there exist HSIs,

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SSIs, and WSIs according to the definition for the degree of singularities in mechanical engineering. The accurate and efficient treatment for HSIs and SSIs is very essential in evaluating matrix entries because they have a major impact on the numerical solutions. Although many robust techniques have been developed for SSIs and WSIs and we have also developed a systematic solution for HSIs in our previous work, the better solutions are still desirable due to the importance of those integrals. In this work, we develop a novel approach for evaluating the HSIs and SSIs based on the Stokes’ theorem. The Stokes’ theorem allows us to change a surface integral over a surface patch into a line integral along its boundary and the line integral can be evaluated in a closed form more easily. We assume that the observation point is off the source patch initially and the resultant integrals are regular before applying the Stokes’ theorem. The HSIs and SSIs are then derived under a CPV-like sense, i.e. taking a limit by letting the observation point approach the source patch. The approach appears simpler and more friendly in implementation because there is no need to introduce a polar coordinate system and further coordinate transformation. Numerical experiments are performed to validate the new formulas for the HSIs and SSIs, and numerical examples for EM scattering problems are shown to illustrate their applications.

REFERENCES [1] W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves. San Rafael, CA: Morgan & Claypool, 2008. [2] W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics. Boston: Artech House, 2001. [3] A. J. Poggio and E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, Ed. Oxford: Pergamon Press, 1973, ch. 4. [4] C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin: Springer-Verlag, 1969. [5] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, no. 3, pp. 409–418, 1982. [6] L. J. Gray, A. Salvadori, A.-V. Phan, and A. Mantic, “Direct evaluation of hypersingular Galerkin surface integrals, II,” Electron. J. Boundary Elements, vol. 4, no. 3, pp. 105–130, 2006. [7] A. R. Krommer and C. W. Ueberhuber, Computational Integration. Philadelphia: SIAM, 1998. [8] P. W. Fink, D. R. Wilton, and M. A. Khayat, “Simple and efficient numerical evaluation of near-hypersingular integrals,” IEEE Antennas Wireless Propag., vol. 7, pp. 469–472, 2008. [9] G. C. Hsiao, R. E. Kleinman, and D.-Q. Wang, “Applications of boundary integral equation methods in 3D electromagnetic scattering,” J. Comput. Appl. Math., vol. 104, no. 2, pp. 89–110, 1999. [10] X. Q. Sheng, J. M. Jin, J. M. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1718–1726, 1998. [11] D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag., vol. 32, pp. 276–281, 1984. [12] R. D. Graglia, “Static and dynamic potential integrals for linearly varying source distributions in two- and three-dimensional problems,” IEEE Trans. Antennas Propag., vol. 35, pp. 662–669, 1987. [13] C. Schwab and W. L. Wendland, “On numerical cubatures of singular surface integrals in boundary element methods,” Numer. Math., vol. 62, pp. 343–369, 1992.

[14] R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag., vol. 41, pp. 1448–1455, 1993. [15] S. Caorsi, D. Moreno, and F. Sidoti, “Theoretical and numerical treatment of surface integrals involving the free-space Green’s function,” IEEE Trans. Antennas Propag., vol. 41, no. 9, pp. 1296–1301, Sept. 1993. [16] T. F. Eibert and V. Hansen, “On the calculation of potential integrals for linear source distributions on triangular domains,” IEEE Trans. Antennas Propag., vol. 43, pp. 1499–1502, Dec. 1995. [17] R. Klees, “Numerical calculation of weakly singular surface integrals,” J. Geodesy, vol. 70, no. 11, pp. 781–797, 1996. [18] P. Arcioni, M. Bressan, and L. Perregrini, “On the evaluation of the double surface integrals arising in the application of the boundary integral method to 3-D problems,” IEEE Trans. Microw. Theory Tech., vol. 45, pp. 436–439, Mar. 1997. [19] M. J. Bluck, M. D. Pocock, and S. P. Walker, “An accurate method for the calculation of singular integrals arising in time-domain integral equation analysis of electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 45, no. 12, pp. 1793–1798, Dec. 1997. [20] L. Rossi and P. J. Cullen, “On the fully numerical evaluation of the linear-shape function times the 3-D Green’s function on a plane triangle,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 398–402, Apr. 1999. [21] A. Herschlein, J. V. Hagen, and W. Wiesbeck, “Methods for the evaluation of regular, weakly singular and strongly singular surface reaction integrals arising in method of moments,” ACES J., vol. 17, no. 1, pp. 63–73, Mar. 2002. [22] W. Cai, Y. Yu, and X. C. Yuan, “Singularity treatment and high-order RWG basis functions for integral equations of electromagnetic scattering,” Int. J. Numer. Methods Eng., vol. 53, pp. 31–47, 2002. [23] S. Järvenpää, M. Taskinen, and P. Ylä-Oijala, “Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra,” Int. J. Numer. Methods. Eng., vol. 58, pp. 1149–1165, 2003. [24] B. M. Johnston and P. R. Johnston, “A comparison of transformation methods for evaluation two-dimensional weakly singular integrals,” Int. J. Numer. Methods. Eng., vol. 56, pp. 589–607, 2003. [25] D. J. Taylor, “Accurate and efficient numerical integration of weakly singular integrals in Galerkin EFIE solutions,” IEEE Trans. Antennas Propag., vol. 51, pp. 1630–1637, Jul. 2003. [26] L. J. Gray, J. M. Glaeser, and T. Kaplan, “Direct evaluation of hypersingular Galerkin surface integrals,” SIAM J. Sci. Comput., vol. 25, no. 5, pp. 1534–1556, 2004. [27] M. A. Khayat and D. R. Wilton, “Numerical evaluation of singular and near-singular potential integrals,” IEEE Trans. Antennas Propag., vol. 53, pp. 3180–3190, 2005. [28] S. Järvenpää, M. Taskinen, and P. Ylä-Oijala, “Singularity subtraction technique for high-order polynomial vector basis functions on planar triangles,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 42–49, Jan. 2006. [29] W.-H. Tang and S. D. Gedney, “An efficient evaluation of near singular surface integrals via the Khayat-Wilton transformation,” Microw. Opt. Tech. Lett., vol. 48, no. 8, pp. 1583–1586, 2006. [30] R. D. Graglia and G. Lombardi, “Machine precision evaluation of singular and nearly singular potential integrals by use of Gauss quadrature formulas for rational functions,” IEEE Trans. Antennas Propag., vol. 56, pp. 981–998, 2008. [31] A. G. Polimeridis and T. V. Yioultsis, “On the direct evaluation of weakly singular integrals in Galerkin mixed potential integral equation formulations,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 3011–3019, Sept. 2008. [32] M. S. Tong and W. C. Chew, “Super-hyper singularity treatment for solving 3D electric field integral equations,” Microw. Opt. Technol. Lett., vol. 49, pp. 1383–1388, 2007. [33] A. H. Stroud, Approximate Calculation of Multiple Integrals. Englewood Cliffs: Prentice-Hall, 1971. [34] O. D. Kellogg, Foundations of Potential Theory. New York: Frederick Ungar, 1944. [35] M. G. Duffy, “Quadrature over a pyramid or cube of integrands with a singularity at a vertex,” SIAM J. Numer. Anal., vol. 19, pp. 1260–1262, 1982. [36] H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. New York: Macmillan, 1961. [37] G. T. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook. New York: Plenum Press, 1970.

TONG AND CHEW: A NOVEL APPROACH FOR EVALUATING HYPERSINGULAR AND STRONGLY SINGULAR SURFACE INTEGRALS

Mei Song Tong (S’01–M’04–SM’07) received the Ph.D. degree in electrical engineering from Arizona State University (ASU), Tempe, in 2004. He is currently a Visiting Research Scientist at the Center for Computational Electromagnetics and Electromagnetics Laboratory (CCEML), Department of Electrical and Computer Engineering (ECE), University of Illinois at Urbana-Champaign (UIUC), Urbana. His research interests include numerical techniques in electromagnetics, acoustics and elastodynamics, simulation and design for RF/microwave circuits and systems, fast and efficient solutions for antenna analysis, interconnect and packaging analysis, and inverse electromagnetic scattering for imaging. He has authored or coauthored more than 50 papers in refereed journals and conference proceedings, and coauthored a book. He has served as a technical reviewer for many international journals and conferences, served as an associate editor and editorial board member for Progress in Electromagnetics Research (PIER) and Journal of Electromagnetic Waves and Applications (JEMWA), and served as an associate editor and guest editor for a special issue for Waves in Random and Complex Media (WRCM). In addition, he has also served as a session organizer, session chair and technical program committee (TPC) member for Progress in Electromagnetics Research Symposium (PIERS) and IEEE International Symposium on Antenna and Propagation (IEEE APS) and USNC/URSI National Radio Science Meeting. Dr. Tong is a fellow of Electromagnetics Academy (EMA), a full member (Commission B) of the U.S. National Committee for the International Union of Radio Science, and a member of the Applied Computational Electromagnetics Society (ACES) and the Sigma Xi Honor Society.

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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, from the Massachusetts Institute of Technology, Cambridge, MA, all in electrical engineering. 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. 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 of 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, authored and coauthored over 300 journal publications, over 400 conference publications and over ten book chapters. Prof. Chew is an IEEE Fellow, an OSA Fellow, an IOP Fellow, and was an NSF Presidential Young Investigator (USA). He received the Schelkunoff Best Paper Award for AP Transaction, the IEEE Graduate Teaching Award, UIUC Campus Wide Teaching Award, IBM Faculty Awards. He was a Founder Professor of the College of Engineering, and currently, 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 on the IEEE Adcom for Antennas and Propagation Society as well as Geoscience and Remote Sensing Society. He has been active with various journals and societies. 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.

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Source Decomposition as a Diakoptic Boundary Condition in FDTD With Reflecting External Regions Sharon Malevsky, Ehud Heyman, Fellow, IEEE, and Raphael Kastner, Fellow, IEEE

Abstract—The diakoptic approach is utilized for the creation of boundary conditions for the truncation of the FDTD computational grid in the presence of reflecting external media. The interfaces between the regions can be of arbitrary shape and are in close proximity to the scatterers in both regions. Surface Green’s functions over these interfaces serve to characterize the external domain from a viewpoint of the computational domain. Previously used field definitions of such Green’s functions, that parameters in microwave netcan be viewed as analogs of works, have involved effective hard boundaries at the interfaces. These boundaries, however, can cause artificial interactions with physical reflecting bodies in the external domain. To eliminate this phenomenon, the Green’s functions are now defined in terms of outgoing and incoming wave constituents, rather than fields. This distinction between the aforementioned constituents is made by viewing them as originating from sources within and outside the computational domain. The Green’s function—based boundary conditions are demonstrated to operate successfully in the two-dimensional case, resolving the aforementioned resonance issues efficiently and accurately. Index Terms—Electromagnetic analysis, electromagnetic scattering, finite difference time domain (FDTD) methods.

I. INTRODUCTION HE finite difference time domain (FDTD) method [1], [2] for solving electromagnetic scattering problems is traditionally limited to closed region, or unbounded region in free space. For the latter case, a host of absorbing boundary conditions (ABCs) are available in the literature, including local ABCs that are based either on asymptotic representations of the one way wave equation [3], [4], or on perfectly matched layers (PMLs) [5]. These formulations usually require the boundary to be a convex surface, typically a box or sphere, which may not conform very well to the shape of the scattering obstacle. A homogeneous “white space” thus created between the scatterer and the ABC, whose size may compromise the inherent efficiency of the local formulations. More importantly, local ABCs are restricted to non-reflecting external domains such as free space or an infinite microstrip substrate. In contrast to local ABCs, global boundary conditions (GBCs) involve integration over the entire boundary of the

T

Manuscript received November 29, 2009; revised March 16, 2010; accepted March 20, 2010. Date of publication June 14, 2010; date of current version November 03, 2010. This work was supported in part by the Israel Science Foundation under grants 674/07 and 1237/06. S. Malevsky is with Delft University of Technology, 2629 HS Delft, The Netherlands. E. Heyman and R. Kastner are with the School of Electrical Engineering, Tel Aviv University, Tel Aviv 69987, Israel (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.2010.2052577

computational domain. Early formulations [6] and many subsequent versions of GBCs were based on Kirchhoff-like integration of the boundary field using the free-space Green’s function. GBCs provide the advantage of treating non-convex boundaries [7]–[11]; however, they are applicable to non reflecting external domains only. GBCs also suffer from efficiency issues due to the need to integrate over the entire boundary. Theses issues are addressed in [11] by a multilevel space-time integration scheme that manages to reduce the numerical complexity substantially, again in the context of non-reflecting external domains only. The objective of this paper is to suggest boundary conditions that can handle reflective, as opposed to absorptive external domains. Reflecting external domains can be handled with the diakoptic approach. Diakoptics (Greek for “tearing apart”) is a domain decomposition approach whereby the physical space is divided into several distinct regions whose interfaces are referred to as “seams.” In our case, these regions would comprise the computational and external domains. This method was initially introduced into time domain problems in the form of the Johns matrix for the transmission-line modeling (TLM) formulation in [12]. Electromagnetic version has been introduced in a general multi-dimensional formulation of the Green’s function method (GFM) [13], [14]. In the GFM, adjacent regions are linked via impedance or admittance type Green’s functions defined over the seams, that can be of arbitrary shape and in close proximity to scatterers. A pre-processing stage is devoted to the construction of the Green’s functions transforming the magnetic (electric) field into the electric (magnetic) field over the seam. The Green’s functions serve to relate fields over the seam in the presence of a reflecting external domain as viewed from the computational domain. They are thus used as boundary conditions in the course of the main FDTD computational phase. The main advantage of this formulation is its capability of treating seams of general shapes, concave cases included, and thus economize of the size of the “white space.” The GFM also accommodates external regions that are reflective rather than absorptive. It is shown, however, (Section II) that the Green’s function used in the GFM introduces spurious late time multiple reflections between the external scatterers and the seam that are canceled in the overall summation yet may be a source of late time errors1. This phenomenon is attributed to the field-type character of the GFM, reminiscent of and matrix representation in microwave circuit theory. Another possible formulation is wave, or spectral domain decomposition, analogous to cascaded scattering matrices, where the field is resolved into outgoing and incoming harmonics 1A

similar problem might occur with concave geometries.

0018-926X/$26.00 © 2010 IEEE

MALEVSKY et al.: SOURCE DECOMPOSITION AS A DIAKOPTIC BOUNDARY CONDITION IN FDTD

Fig. 1. 1-D problem configuration. The area in the vicinity of the seam z is homogeneous.

=z

across the seam. This approach can be considered as analogous the -matrix representation in microwave circuit theory, as opposed to the field approach of the GFM. Definitions of the outgoing and incoming wave operators thus involve matched rather that short circuited “ports” at the seam. This eliminates the aforementioned spurious mechanism of reflections between the artificial seam and the external reflector. The main issue in this formulation is the restriction to a convex and separable seam that can compromise the efficiency of the procedure, having an excessively large white space. In order to resolve these issues, an alternative source decomposition method (SDM) is presented in this work. It is a spatial domain method, however it is formulated under terminated conditions, similarly to a wave formulation, and yet accommodates seams of general shapes. The SDM is based on the distinction between the fields originating from sources within and outside the computational domain, respectively, as opposed to outgoing and incoming spectral constituents. In the SDM, in contrast to the conventional wave-type formulation, the computational domain boundary needs not be convex. To implement this approach, a two-step pre-processing stage is executed. The SDM is applied successfully in the one- and two-dimensional cases, as shown in described in Sections III and IV, respectively. Numerical examples are given in Section V, demonstrating that the aforementioned resonance issues of the GFM are resolved efficiently and accurately without resorting to the wave-type representation. Conclusions to that effect are drawn in Section VI. II. THE FIELD-TYPE GREEN’S FUNCTION METHOD (GFM): ISSUES WITH REFLECTING EXTERNAL DOMAINS For a brief review the GFM [13], [14], consider first the 1-D time dependent Maxwell’s equations within the “computational (see Fig. 1) domain”

main. Note that the external domain can be of arbitrary composition. The immediate vicinity of the seam on both sides, however, , . is considered homogeneous with The discrete equivalent of (2), to be used in the context of the Yee grid, requires a pre-processing phase to evaluate the input admittance in a manner that is compatible with an FDTD computation of the entire space. Assuming a seam at the plane , the discrete magnetic curl equation in its vicinity is (3) where is defined in the homogeneous medium , with . The about the seam and rightmost term in (3), which lies within the external domain field, and therefore is not available, is replaced by the GFM BC. Similarly to its continuous domain counterpart (2), the BC is expressed as convolution of boundary field historical values with that reprethe discrete admittance-type Green’s function sents the external domain, as defined, in accordance with [13], by (4) where denotes the discrete-time convolution. The definition in and (4) includes the effect of the shift between the samples of . Note that in practice, the time series is truncated to its first terms, so that the convolution in (4) involves a history behind the wavefront. A dual impedance type of terms of Green’s function can also be defined by starting with the electric field curl equation instead of (3). Usage of the BC (4) in an FDTD code is contingent upon . Therefore, is evaluated in a pre-prothe availability of cessing stage that precedes the main FDTD computation. This stage is performed over the external domain, whereby an impulat the boundary sive (Kronecker delta) electric field is used as the input. This type of input, with zero electric field implies an impulsive magat for all times except at netic source backed by a perfect electric conductor (PEC) at that boundary. Under these conditions, the solution of the external medium in the preprocessing stage produces the input admittance Green’s function via

(1) These equations apply to a wave of the form , arising from the source terms , and with , that are, in general functions of . This domain is sewn together with the “external domain” using the following BC at the seam :

(5) (see also the left hand term in (4) with ). When the external medium is homogeneous, we denote , where is the “intrinsic” admittance that is the discrete counter part of the continuous free space admittance (see also explanation before (8)). For the dispersion-free case

(2) where the asterisk denotes a temporal convolution and is the aforementioned “input admittance” into the external do-

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,

a general values of ,

, as one may expect, whereas for is a rapidly decaying series. Results

, obtained via direct FDTD computations, -transform for techniques and combinatorics are in the form of a converging that can be truncated after say terms (see also series in similar phenomena in [15], [16]).

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Fig. 2. GFM model within the external domain: A PEC-backed magnetic I serves to excite the external domain for the purpose source at the seam i of generating the impedance/admittance type Green’s functions at the seam. In the case of an external domain with inhomogeneities like the step discontinuity shown, spurious multiple reflections will occur.

=

The main advantage in the usage of the GFM (3), (4) as boundary conditions is their capability of treating seams of general shapes, concave cases included, and thus economize of the size of the “white space”. These boundary conditions also accommodate reflective external domains. In reflective cases, the conceptual PEC boundary, implied in (5), reflects the scattered waves generated in the external domain back includes a series of into the same domain. Consequently, spurious multiple reflections between the external domain and this conceptual PEC boundary, as seen in Fig. 2.2 In principle, the GFM has a built-in mechanism to offset the transmissions of the spurious reflections into the computational domain, at least over a limited time scale. Each of these reflections, arriving at a given time, comprises sub-reflections that have been excited earlier, and then propagated by different components of the admittance vector. The aggregated effect of these sub-reflections produces a zero field, as shown in Fig. 3. To facilitate the annulment over unlimited time scale, however, the in (4) should be represented by an infinite admittance vector causes time spurious signals series. Truncation of the vector to be transmitted into the computational domain due to the absence of the high order terms needed to offset these signals. The need for pairwise cancellation of these spurious reflection may be a source of error. The series of spurious reflection converges 2In analogy, consider the continuous-domain input admittance to an external medium with reflections. The frequency domain input admittance can then be defined in terms of the outward-looking reflection coefficient 0 (z ; ! ) via (recall that the medium homogeneity in the immediate vicinity of the seam)

0!

0! y (z ; ! ) = 

! 100 0 (z ! 1+0 0 (z

! 0 (z =  [1 0 0

;! ;!

) )

;!

)]

Fig. 3. Interpretation of individual terms in (4) For the external domain with the step discontinuity of Fig. 2. The true reflection is seen at n = n +(2S= ). At later times, n = n +(4S= ) on, additional terms in the admittance function generate spurious reflections that are cancelled out by other terms.

at the rate of , that may be slow, hence long filters may be required. The main issue in this formulation is the restriction to a convex and separable seam that can compromise the efficiency of the procedure, having an excessively large white space. The length of the filter is also compatible with the distance of the farthest discontinuity from the seam. III. SOURCE DECOMPOSITION IN ONE DIMENSION In the one dimensional case, the SDM coincides with the wave-type approach discussed in the Introduction, since the seam is trivially planar. To implement this approach, one first resolves the electric field into the outgoing and incoming wave components, that are also the contributions of the sources in the computational and external domains, respectively, in this case. constituents via (recall that the immediate vicinity of the seam is homogeneous with , ): (7a) (7b) and . As noted above, the with analogous expressions for outgoing and incoming constituents originate from internal and external sources, respectively. The electric and magnetic fields of each constituent are then related by the intrinsic impedance that plays a role analogous to the characteristic impedance in transmission line representation. The discrete counterpart of this impedance is , with being the corresponding intrinsic admittance (see discussion after (5)). These operators are used in the context of the Yee grid as follows: (8a)

! [00 0 (z

;!

)]

(8b) :

(6)

Define also an immitance-type propagator as In the time domain, the second equality in (6) describes an infinite series of reflections between the external medium and the PEC at z .

(9)

MALEVSKY et al.: SOURCE DECOMPOSITION AS A DIAKOPTIC BOUNDARY CONDITION IN FDTD

whose physical meaning is revealed, for example, in the following reduction of (8): (10)

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that matched load in transmission line theory. The field is due to constrained test sources in the internal domain is then calculated. The 0 order designation is used for this first stage. Specifically, we identify the

term

as the desired

, . Note that for Next we assume, that, as in (3), the seam coincides with a sampling point of the E-field, hereafter defined as an E-type boundary. Since the medium in the vicinity of the seam is con, , one can apply the sidered homogeneous with decomposition (8) about the seam as follows:

near the seam. In the second stage, the outgoing field test problem includes an external domain that contains the true inhomogeneities whereas the internal domain is replaced with a homogeneous medium. Using the same internal sources, the

(11)

(15)

Note that the two magnetic field constituents in (11) are defined and outside the seam, respectively. just inside We now note that the external sources, that have given rise

from which the reflection coefficient can be extracted by de-convolving (12), now written in the form

to

is calculated. In this stage, the 1 order destotal field ignation is used. We can now identify the backscattered field as

(16)

, are in fact dependent sources, excited in

response to the outgoing wave impinging on external scatterers, when present. Therefore, the incoming and outgoing constituents are interrelated via a spatial domain reflection that characterizes the external domain such that operator

In order to perform the de-convolution operation, the convolution (16) is re-cast in matrix representation, as follows: (17) where

(12) (18a)

Substituting (12) into (11), we have a BC in the following format that contains only outgoing constituents of the magnetic field:

(18b) (13) Equation (13) serves as the BC for the internal domain at the seam, applied concurrently with the FDTD procedure. The quantities and used there are also evaluated recursively in the course of the FDTD computation, using the following combination of (10) and (12):

(18c) and is the causal-circulant matrix shown in (19) at can the bottom of the page. Then, the elements of be extracted recursively, starting from the first row and adding one equation at a time for each element.

(14) that characterizes the The reflection coefficient external domain, is independent of the excitation and needs to be available beforehand. This mandates a pre-processing stage . In line with the comment for the calculation of is truncated after (4), note that in practice the temporal series to its first terms, such that the BC in (13) involves an -term history behind the wavefront. The pre-processing stage for finding proceeds in two stages as follows. In the first stage, the entire space is replaced with a homogeneous medium, creating an analog of a

.. .

IV. SOURCE DECOMPOSITION IN TWO DIMENSIONS The one dimensional formulation (12) of Section III can branch out into either a wave or source decomposition two dimensional representations. The wave representation is less useful in two or three dimensions, where it is related only to the spectral constituents. For this wave representation to hold in multidimensional problems, the enclosing surface has to be convex. The SDM is still valid, though, for concave as well as convex surfaces. In this representation, we redefine the outgoing and incoming field constituents as the outcomes of sources in the computational and external domain, respectively.

.. .

.. .

.. .

(19)

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by the “external return operator” follows:

, defined as (22)

the two dimensional generalization of (14) is

Fig. 4. 2-D example: Parallel plate waveguide with @=@x external domain.

= 0 and a reflective (23)

Then, the generalization to more dimensions can be carried out in the spatial (field) rather than the spectral (wave) domain in the form of the source decomposition method detailed below. In the one-dimensional case treated above, the two definitions coincide. In order to develop the source decomposition method (SDM) for the two dimensional case, consider a typical geometry with , that can be defined in the form of a parallel plate waveguide of width in the direction (see Fig. 4), supporting a propagating mode. The discrete index notations in this case are where is any field component, and the corresponding FDTD equations are (note that and are now functions of ) (20a) (20b)

(20c) and . In the sequel we where and . The field at use the shorthand notation in now a summation each point over the seam at of contributions from all nearby points along the seam. This summation is expressed, for the case of an -type boundary, in the following generalization of (11):

(21) where , and the medium is assumed to be homogein the immediate surrounding of the neous seam.3 The generalization of the wave reflection coefficient of (12) would apply to a wave representation, however it does not apply the SDM. Instead, it is replaced 3In general, the spatial summation is not a convolution because the spatial matrices are not circulant.

where is defined in analogy with (10). As with (14), usage of the boundary condition (23) is contingent upon the avail, that has been computed in a preability of processing stage. The procedure for this stage is again analogous to the one dimensional case. One sets the medium within the internal domain to be homogeneous. Then, the outgoing fields on , and the total field both sides of the seam, are computed for the same trial sources in the presence of a homogeneous and the actual external domains, respectively. These quantities are used to construct the two dimensional equivalent is then extracted. of (16), from which Following the comment after (4), note again that in practice the temporal series are truncated to their first terms, so that the BC in (13) involves an -terms history behind the wavefront. V. NUMERICAL EXAMPLES The performance of the SDM is evaluated in comparison with a FDTD reference simulation in a pseudo-infinite computational domain that extends over both the original computational and external domains. Scatterers within the original external domain can be included. The figure of merit that quantifies the BC performance is the normalized RMS error over the entire original computational domain, evaluated at the maximal temporal point:

(24)

where and are the results of the SDM and the reference problem, respectively, and the summations are done over the computational domain. The SDM is first tested for a 1-D case as an ABC with a homogeneous external domain for several values of . The excitation is defined by the electric field at the aperture plane (see Fig. 1) that can be replaced by its equivalent magnetic current backed by a perfect electric conductor at the same plane. The exciting pulse for this case is a Blackman-Harris window, shown in (25) at the bottom of the following page, with and , , , . The temporal size of the vector in (12) and –(13) is referred to henceforth as “filter length.” The multiple reflection issue is not present in this case, and indeed the results track those found for the impedance/admittance approach

MALEVSKY et al.: SOURCE DECOMPOSITION AS A DIAKOPTIC BOUNDARY CONDITION IN FDTD

Fig. 5. RMS normalized error vs. the BC filter length (defined as the temporal size the vector R in (12) –(13)) for a 1-D problem as an ABC for homogeneous external domain (no multiple reflections issue). Size of computational domain: I 100. Simulation was carried for N = 1000 times steps. Results track those in [13].

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Fig. 7. The normalized RMS error for the case of Fig. 6.

=

Fig. 8. Twice differentiated Gaussian pulse in x = i1x direction with the TE profile in the y = |1y direction, for the parallel plate waveguide structure in Fig. 4. Fig. 6. The reflection filter R response for a 1-D problem, where the external domain has a step discontinuity  = 9. The response is presented for several values of over the normalized discrete time axis n .

2

in [13], see Fig. 5. Ideally, for an infinite filter length, the error , the error is would be negligible. For filter lengths of . Next, an external domain that contains a still under at step discontinuity in the electrical permittivity from to at . Results for the filter funcnow account for the reflections in the external tion domain, as can be seen in Fig. 6 for several values of . The continuous value of has more dispersion errors as is decreased. The 2D SDM of Section IV is tested in the parallel plate waveguide configuration, with the excitation again defined by

(see Fig. 4). The the electric field at the aperture plane doubly differentiated Gaussian excitation is designed as a pulse whose spectrum fits below the FDTD upper frequency limit. Also, to show a propagating wave in the context of this example, the spectrum fits above the waveguide cutoff frequency, a condition necessary in general (see Fig. 8). The total normalized RMS error compared with the FDTD solution is shown in the error is Fig. 7. The error for finite filter lengths of . typically under Results are shown in Fig. 9 for both homogeneous and step discontinuity external domains, the latter being similar to the 1-D case of Figs. 6 and 7. In order to achieve a normalized or lower for both cases, a filter length RMS error of is needed. This order of accuracy is the same as the of one obtained in the GFM, however the SDM requires a shorter

otherwise

(25)

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Fig. 9. The normalized RMS error of the field for the 2-D case of Fig. 4 vs. the filter length for both a homogeneous medium and step discontinuity external domains. The discontinuity is the same as in Figs. 6 and 7. Size of computational 100. Simulation was carried for N = 1000 times steps. domain: I

=

Fig. 11. simulation of the total electric field near the boundary as a function of time for a step discontinuity in the dielectric constant at 20 samples beyond the boundary. the sd (solid line) method reconstructs an incident pulse plus one physical reflected pulse, while the GFM (dashed line) generates spurious reflections. = 0:7.

Fig. 10. The SDM run times, in comparison, remain about stationary. Consequently, although the GFM is more efficient for a nearby scatterer, the SDM becomes much more efficient in more general cases. If one keeps the filter length fixed, then the GFM computation is bound to show higher errors at later times. This phenomenon is seen in Fig. 11 that shows the electric field near the boundary as a function of time for both methods. A discontinuity in the dielectric constant is introduced at 20 samples beyond the boundary. For this case, the SDM reconstructs an incident pulse plus one physical reflected pulse, while the GFM generates spurious reflections. VI. CONCLUSIONS Fig. 10. Run time of the SD compared with the GFM for a normalized RMS error of 30 dB vs. the scatterer distance. The ensure this level of error, the computational domain for the GFM case is chosen to include the external discontinuity, while the GFM filter length remains constant, describing the free space boundary conditions only. For the SD case, both size of the computational domain and the filter length remain stationary.

0

computation time (not including the pre-processing stage) as seen in Fig. 10. This calculation is performed once for all possible configurations within the computational domain and all excitations, therefore it cannot be uniquely quantized. Regarding memory consumption, it is roughly proportional to the total number of samples, similarly to the computational time behavior. In this simulation, a comparison is made between the two methods. The normalized simulation time is measured for a fixed normalized while varying the distance of the external RMS error of scatterer from boundary. In the GFM, the length of the filer has to be increased in accordance with the distance of the external scatterer from the boundary, in order to include the scatterer within the computational domain and thus avoid the multiple reflection phenomenon described in Section II. Therefore, run times increase roughly in proportion to this distance as seen in

The time domain diakoptic/domain decomposition method approach is capable of stitching together regions of arbitrary shape and composition, that may contain reflective objects. The source decomposition method (SDM), presented above, has been shown to be a viable variant of this approach that eliminates spurious reflections between physical and artificial boundaries. Since the entire computation is done within the framework of the second order accurate, explicit Yee scheme, no issues of instability are encountered as long as the CFL condition is adhered to. Although implementation of this method requires a pre-processing stage, this added burden is small relative to the main computational stage. The method is particularly suitable for multiple scatterers scenarios, where the scatterers are distinct. It is most useful in a design process, where the computational domain undergoes several modifications with many excitations while the external boundary remains unchanged. Another computational advantage is seen in the reduction of the size of the white space by letting the computational domain boundary track the boundary of a scatterer. This advantage is especially important for a multi-body problem with many excitations, where an outright FDTD computation would involve an expanded computational domain that includes all scatterers, internal and external, with a substantial white space between

MALEVSKY et al.: SOURCE DECOMPOSITION AS A DIAKOPTIC BOUNDARY CONDITION IN FDTD

them. Clearly, for tightly packed scatterers with a small number of excitations, the conventional FDTD method suffices. The formulation in Section IV, although written in two-dimensional terms for the sake of relative simplicity, is the same for three dimensions, except for extra variables necessary to model the three dimensional dyad involved. Although outside the scope of the present work, three dimensional problems are routinely implemented in FDTD simulations. Once combined with the SDM, problems with highly complex structures (e.g., the human body) could be torn into smaller regions, that can be dealt with efficiently. This is the topic of future work. Although the pre-processing stage of calculation of is done only once per a given configurations for all excitations, acceleration of this stage may be in order. For the homogeneous case, methods for by-passing the direct computation of FDTD-compatible Green’s functions can be found in [15], [16]. More complex cases would probably involve hybridization with integral equations techniques. REFERENCES [1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302–307, May 1966. [2] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [3] B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comp., vol. 31, pp. 629–651, 1977. [4] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic,” IEEE Trans. Electromagn. Compat., vol. 23, pp. 377–382, 1981. [5] J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 114, no. 1, pp. 185–200, 1994. [6] McDonald and Wexler, “Finite element solution of unbounded field problems,” IEEE Trans. Microwave Theory Tech., vol. MTT-20, pp. 841–847, Dec. 1972. [7] J. Liu and J. Jin, A novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation problems 2000, Research Rep.. [8] E. M. D. Jiao and M. Lu, “A fast time-domain finite element—Boundary integral method for electromagnetic analysis,” IEEE Trans. Antennas Propag., vol. 49, pp. 1453–1461, 2001. [9] A. Boag, U. Shemer, and R. Kastner, “Hybrid absorbing boundary conditions based on fast non-uniform grid integration for non-convex scatterers,” Microw. Opt. Technol. Lett., vol. 43, no. 2, pp. 102–106, Oct. 2004. [10] A. Boag, U. Shemer, and R. Kastner, “Non-uniform grid accelerated global boundary condition for acoustic scattering,” Comput. Methods Appl. Mech. Engrg., vol. 195, pp. 3608–3621, Jun. 2006. [11] M. Lu, A. A. Ergin, B. Shanker, and E. Michielssen, “Multilevel plane wave time domain-based global boundary kernels for two-dimensional finite difference time domain simulations,” Radio Sci., vol. 39, no. RS4007, 2004. [12] P. B. Johns and A. Akhtarzad, “The use of time domain diakoptics in time discreet models of fields,” Int. J. Numer. Methods Eng., vol. 17, pp. 1–14, 1981. [13] R. Holtzman and R. Kastner, “On the time-domain discrete Green’s function method (GFM) characterizing the FDTD grid boundary,” IEEE Trans. Antennas Propag., vol. 49, pp. 1079–1093, 2001. [14] R. Holtzman, R. Kastner, E. Heyman, and R. Ziolkowski, “Stability analysis of the Green function method (GFM) used as an ABC for arbitrary boundaries,” IEEE Trans. Antennas Propag., vol. 50. [15] R. Kastner, “A multi-dimensional Z -transform evaluation of the discrete finite difference time domain Green’s function,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1215–1222, Apr. 2006.

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[16] N. Rospsha and R. Kastner, “Closed form FDTD-compatible Greens function based on combinatorics,” J. Comput. Phys. (2007), pp. 1017–1029, 2002.

Sharon Malevsky was born in Holon, Israel, in 1971. He received the B.Sc. degree in electrical engineering from the Technion, Israel Institute of Technology, Haifa, in 1997 and the M.Sc. degree in electrical engineering from Tel-Aviv University, Israel, in 2005. He is currently working toward the Ph.D. degree at Delft University of Technology, The Netherlands. From 1997 to 2001, he was employed in Intel Cellular Communication Division as an RF Engineer, where he researched advanced power amplification schemes for 3rd generation cellular applications. From 2002 to 2005, he was with DSP Group Limited, Herzelia, Israel, as an RFIC designer, where he was part of a Software Defined Radio team, designing IC’s for cordless phones.

Ehud Heyman (S’80–M’82–SM’88–F’01) was born in Tel Aviv, Israel, in 1952. He received the B.Sc. degree (summa cum laude) in electrical engineering from Tel Aviv University, the M.Sc. degree in electrical engineering (with distinction) from the Technion, Israel Institute of Technology, Haifa, and the Ph.D. degree in electro-physics from the Polytechnic Institute of New York, (now Polytechnic Institute of NYU), Brooklyn, in 1977, 1979, and 1982, respectively. While at the Polytechnic he was a Research Fellow and later a Postdoctoral Fellow, as well as a Rothschild, a Fullbright, and a Hebrew Technical Institute Fellow. In 1983, he joined the Department of Physical Electronics at Tel Aviv University, where he is now a Professor of electromagnetic theory. He served as the Department Head, the Head of the School of Electrical Engineering, and since 2006 he is serving as Dean of Engineering. From 1991 to 1992, he was on sabbatical at Northeastern University, Boston, the Massachusetts Institute of Technology, Cambridge, and A. J. Devaney Association, Boston. He spent several summers as a Visiting Professor at various universities. He has published over 100 journal articles and has been an invited speaker at many international conferences. His research interests involve analytic methods in wave theory, including asymptotic and time-domain techniques for propagation and scattering, beam and pulsed beam fields, short-pulse antennas, inverse scattering, target identification, imaging and synthetic aperture radar, propagation in random medium. Prof. Heyman is a Member of Sigma Xi and the Chairman of the Israeli National Committee for Radio Sciences (URSI). He is an Associate Editor of the IEEE Press Series on Electromagnetic Waves and was an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Raphael Kastner (F’08) was born in Haifa, Israel, in 1948. He received the B.Sc. (summa cum laude) and the M.Sc. degrees in electrical engineering, engineering from the Technion, Israel Institute of Technology, Haifa, in 1973 and 1976, respectively, and the Ph.D. degree from the University of Illinois, Urbana, in 1982. From 1976 to 1988, he was with RAFAEL, Israel Armament Development Authority, where from 1982 to 1986 he headed the antenna section. He was a Visiting Assistant Professor at Syracuse University, from 1986 to 1987, and a Visiting Scholar at the University of Illinois in 1987 and 1989. Since 1988, he has been with School of Electrical Engineering, Tel Aviv University, where is now a Professor. In 2000, he co-founded XellAnt Inc. and acted as its CEO until 2004. His research interests are in computational electromagnetics and antennas. Prof. Kastner is a recipient of the IEEE Third Millenium medal and several excellence in teaching awards. He is a member of Tau Beta Pi and Eta Kappa Nu.

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MAS Pole Location and Effective Spatial Bandwidth of the Scattered Field James E. Richie, Senior Member, IEEE

Abstract—The concept of effective spatial bandwidth (EBW) is introduced for periodic domains. The EBW is applied to the incident and scattered fields along the boundary of an infinite circular cylinder. The scattered field is formulated using the method of auxiliary sources (MAS). In MAS, monopoles on an auxiliary surface (AS) are used to model the scattered field. It is shown that the EBW of the incident field can provide some insight regarding the placement of poles for the MAS scattered field model. Example simulations are provided to demonstrate the usefulness of EBW with respect to monopole placement rules in MAS. Index Terms—Boundary value problems, electromagnetic scattering. Fig. 1. Geometry of the two-dimensional scattering problem.

I. INTRODUCTION HE generalized multipole technique (GMT) [1] and its variations can be used to compute the scattering from objects in a variety of scenarios. GMT and related methods compute the scattering from perfectly conducting objects by placing canonical sources within the object to model the scattered field. Often, discrete multipoles are used for this purpose. One variation within the family of GMT methods is the method of auxiliary surfaces (MAS) [2]. In MAS, an auxiliary surface (AS) is defined within the scatterer. The canonical sources are placed on the AS. Typically, monopoles are used for two-dimensional scattering problems and Hertzian dipoles are used for three-dimensional scatterers. Recently, a three-dimensional quasistatic MAS formulation has been reported in [3]. The major questions that arise when implementing GMT methods pertain to the location and number of poles necessary to obtain a sufficiently accurate solution. One approach to determining the location and number of poles is to develop rules based on qualitative information and experience. In [4] an empirical scheme is proposed to determine the location and number of monopole origins for two-dimensional scattering problems. In [5], a rule-based algorithm is used to determine appropriate multipole origins for GMT. Pole location, number and placement issues are also discussed in [6], and [7]. Monopole location in MAS is governed by the AS. Desirable characteristics for the AS are reviewed in [2]. In particular, the auxiliary surface must enclose the singularities of the scattered field. Studies of this requirement appear in [1, Ch. 5] and [8].

T

Manuscript received July 15, 2009; revised April 13, 2010; accepted April 14, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The author is with the Department of Electrical and Computer Engineering, Marquette University, Milwaukee, WI 53233 USA (e-mail: james.richie@mu. edu). Digital Object Identifier 10.1109/TAP.2010.2071346

A second approach to determining the location and number of poles is to study the convergence and accuracy of the numerical method. It is possible to infer useful guidelines from the results of such studies. Investigations concerning the convergence and accuracy of MAS for the perfectly conducting circular cylindrical scatterer (as shown in Fig. 1) have been reported in [9] and [10]. In [9], a monopole line source in the vicinity of the cylinder is investigated. For a cylinder of radius and a monopole line (point L in Fig. 1), the singularity in the scattered source at field is at a radius of . Therefore, the AS radius must be and . In [9], AS radius choices both chosen between inside and outside this requirement are investigated. In [10], the numerical accuracy and analytical accuracy of MAS are described in detail. It is shown that the numerical accuracy dominates the error when the AS radius is very small. To understand the significance of a small AS radius, consider the geometry of Fig. 1. The addition theorem is used to write the field at due to a unit strength monopole at as an expansion ) [11] of multipoles at the origin (with (1) where k is the wavenumber , is the Bessel function of the first kind of order , and is the Hankel function of the second kind of order representing outward traveling waves. Eqn. (1) demonstrates that a monopole at is equivalent to a multipole expansion at the origin. When the AS radius is zero (an extreme case), only the term of the expansion is non-zero. When is small, the multipole expansion has . Thus, there is only a small small coefficients except near amount of variation in the monopole field along the boundary if the AS radius is small.

0018-926X/$26.00 © 2010 IEEE

RICHIE: MAS POLE LOCATION AND EFFECTIVE SPATIAL BANDWIDTH OF THE SCATTERED FIELD

However, as , the left hand side of (1) indicates that . In other words, the the field becomes nearly singular at becomes very large amount of variation in the field along as the AS radius approaches the cylinder radius. In this paper, the following question is investigated. For the geometry of Fig. 1, how much variation in the field along the boundary due to the MAS monopoles is necessary to obtain a suitable solution? Certainly a small AS radius admits small variation and an AS radius near admits large variation of the fields along the boundary. It is the incident field variation along the boundary that determines how much variation is needed from the MAS monopole field. It is believed that a fundamental understanding of the relationship between the amount of incident field variation and monopole placement can be used in a wide variety of situations and lead to additional guidelines for monopole placement in general problems. The intent of this work is to use a well-known problem to obtain some physical insight into the effect of monopole placement in the MAS method. The work presented here is not intended to introduce a new, more efficient implementation of MAS; rather, the results obtained by this investigation provide valuable physical insight to the more general problem. The analysis presented here includes a procedure to quantify the amount of variation of fields along the boundary. The result will be denoted as the effective spatial bandwidth (or EBW) of the field. Next, the EBW for the incident field along the boundary will be computed both analytically and numerically. The scattered field EBW will also be presented, both for the analytic solution to the circular scatterer, and for the MAS monopole. Example simulations will then be described and discussed. The examples shall demonstrate the effectiveness of EBW as an engineering tool to aid in the placement of monopoles in the MAS technique. II. BOUNDARY FIELD BANDWIDTH The concept of spatial bandwidth of fields for non-periodic domains is discussed in [12]. Bandwidth can be thought of as a measure of the frequency content or amount of variation of a signal or function. In this paper, the terms “bandwidth” and “frequency” refer to the degree of spatial variation of field quantities. In many cases, the absolute bandwidth is infinite because there is non-zero energy over the entire spectrum. The energy asymptotically approaches zero as the frequency becomes large enough. This asymptotic behavior is typical for scattered fields [12]. Thus, it is convenient to define the bandwidth as the range of frequencies that contain a (usually large) percentage of the total energy of the function. can be computed either in The bandwidth of a function the spectral domain or in the original domain using convolution. In the spectral domain, the spatial frequency content is computed and the bandwidth can be estimated from the spectrum. Consider estimating the bandwidth via convolution. One approach is as follows. First, the function is bandlimited. Consider with domain . The convolution (2)

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with maximum freresults in the band-limited function quency . For frequencies below , the spectral content of and are identical. To estimate the bandwidth, the funcis computed for increasing until the difference in tion and becomes smaller than some energy between pre-defined threshold. In the present problem, the function is a field quantity on the boundary of a scatterer. The field quantity has a spatial or effective bandwidth (EBW) that quantifies the amount of variation (or spatial frequency content) of the field quantity. Consider the scattering of an incident field by a conducting circular cylinder as shown in Fig. 1. The cylinder is uniform and of infinite extent in . The incident fields considered will be . The uniformity in allows the scattering problem to be plane. solved in the A field quantity (e.g., an incident or scattered field component) will be a periodic function of around the surface of the scatterer. Because the period of is , the fundamental spatial frequency is one and the harmonics are integers. The bandlimlimited to a maximum spatial frequency of ited function (an integer) can be determined using (3) where is the circumference of the scatterer, is the distance from the origin to the point on the scatterer at angle , and (4) term represents the average value of . Note that the The functions , or equivalently and with integer , represent the spatial harmonic functions for periodic domain considered in this work. The effective spatial bandwidth of the field shall be denoted , define such as EBW. For a periodic function where that is the smallest integer with (5) and (6) where

is the bandlimited form of , as given by (3) and (4).

A. Incident Field Effective Bandwidth In this section, the effective bandwidth of the incident field is computed. Both the case of a plane wave incident field and a monopole line source are considered. In each case, the incident field is normalized to unit strength, i.e., . 1) Plane Wave Incident Field: Consider a plane wave incidirection, as shown in dent on the cylinder, traveling in the Fig. 1. The electric field is given by where (7)

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c

for case a = 0:5; EBW for incident field is N = 5.

Fig. 2. EBW for a plane wave incident on a cylinder of radius a().

Fig. 3.

where the superscript p indicates that the incident field is a plane wave. on the surface of a cylinder of The effective bandwidth of radius can be investigated using (3). After some manipulation

Again, the expression for indicates that the absolute bandwidth is infinite. The effective bandwidth depends on the behavior of the term in brackets. Calculation of EBW indicates and approaches the plane that the EBW is large for small wave bandwidth as becomes large, as expected.

(8)

is the bandlimited form (with maximum frequency where ) of . Consider (8). Since all values of are allowed in the sum, the is infinite. However, the coefficient of absolute bandwidth of each term is proportional to the quantity in brackets in (8). The . quantity in brackets converges to zero as For the case of a circular cross section, (8) also demonstrates that the wave transformation in (7) results in the spectral content of the field around the circular boundary. Note that this is only true for the circular case. The effective bandwidth for the plane wave can also be computed by numerically integrating (3) and (6). A graph showing the EBW vs. cylinder radius (in wavelengths) is shown in Fig. 2. As the radius increases, the EBW for a plane wave around the circumference of the cylinder increases, as expected. 2) Monopole Line Source Incident Field: Consider a , labeled L in Fig. 1. The monopole line source at incident field is . The field along the cylinder boundary can be written using the addition theorem [11] (9) where the superscript o indicates a monopole line source, and is 1 if and 2 otherwise. can be computed The bandlimited form of

(10)

j

j

B. Scattered Field Bandwidth In this section the effective bandwith of the scattered field along the boundary is considered. The analytic solution model and the MAS model for the scattered field are discussed. 1) Analytic Solution Case: First, consider the analytic solution to the circular cylindrical problem shown in Fig. 1 (11) where the superscript a indicates the analytic solution. For a plane wave incident field (8), the coefficients are given by (12) Equation (11) represents the scattered field as a multipole expansion where all poles are at the origin. The bandlimited function corresponding to can be found using (3)

(13)

Once again, the absolute bandwidth is infinite and the effective bandwidth depends on the coefficients in (13). In this case, however, as increases. As seen in (12), approaches 0 at a faster rate. Each term of the sum in (13) can also be considered to have an EBW. Since (11) is a spectral expansion of the scattered field term is . along the boundary, the EBW for the It is illuminating to compare the coefficients with the effective bandwidth of the incident field for a particular problem. vs. for the case of a radius cylinder. Fig. 3 shows . Note that the incident field, a plane wave, has an EBW of The fifth term in the series has an EBW that matches

RICHIE: MAS POLE LOCATION AND EFFECTIVE SPATIAL BANDWIDTH OF THE SCATTERED FIELD

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for chosen with a radius equal to or larger than . Note that the AS radius depends a cylinder of radius on the incident field as well as the geometry of the scatterer. III. RESULTS AND DISCUSSION In this section, example simulations based on the results of the previous section are discussed. A measure is introduced that quantifies whether the solution is stable (or well-behaved). The examples shall demonstrate the significance of the preceding development and the usefulness of EBW. In each case, the incident . field is normalized to unit strength, i.e., The MAS formulation for the scattered field is a set of monopoles at origins Fig. 4. EBW for a monopole of various radius ( ) for cylinder with a = 0:5; EBW for incident field is N = 5.

the incident field EBW. Clearly, the coefficients converge to 0 . quickly once 2) MAS Monopole Case: Consider a single monopole inside labeled S in Fig. 1. The the cylinder at some location . The field strength is normalized to unit strength, i.e., electric field due to the monopole can be written (14) where the superscript M indicates MAS monopole. Applying an addition theorem and computing the bandlimited form of around a cylinder of radius , we obtain

(15)

where an infinite absolute bandwidth is noted, as before. Consider the location of the monopole. A monopole located at the origin has a constant field along the circular boundary. , which is The EBW for a monopole at the origin is easily verified using (3). However, as the monopole moves away from the origin, the EBW of the resultant field along the cylinder boundary increases. Fig. 4 shows the numerically computed EBW for a monopole as it is moved from the origin toward the boundary of a cylinder . Fig. 4 verifies the variation of the field due with to a MAS monopole as the AS radius varies, as described in Section I. The variation (or EBW) of the incident field can now be compared to the EBW of the MAS monopole. Assuming the incident field is a plane wave, the EBW for the . The monopole EBW matches the plane incident field is . Certainly, choosing wave EBW at should result in solutions that are well-behaved in some sense, may result in while choosing significantly less than poorly behaved solutions. In other words, to avoid poor numerical accuracy, as described in [10], the AS radius should be

(16) where is the vector from the origin to monopole m, and the is the complex amplitude (or strength) of the coefficient monopole. An example monopole location is shown in Fig. 1, labeled S. Generally, the monopoles are placed on an auxiliary surface (AS); for the cylinder, the AS is a circle with radius . The monopoles are equally spaced in around the AS, begin. ning with The MAS method as implemented here computes the coeffiusing the system of linear equations cients (17) where is a vector from the origin to the cylinder boundary at and . The are angle computed using an LU decomposition on the MAS matrix. , the average boundary condition error, is used Typically, to quantify the accuracy of the MAS solution. The average boundary condition error is computed using 360 points equally spaced along the boundary: (18) Before the simulation results are presented, the bandlimited form for the MAS scattered field (16) is computed. To compute the bandlimited form of the MAS series, (16) is substituted into (3), an addition theorem is applied and terms are rearranged to obtain

(19)

Comparing (19) to (13), it is apparent that the terms in brackets for each equation should be nearly equivalent for an accurate is small, then the terms may solution. Thus, if in (13). The MAS solution will result be much smaller than

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in

that are very large. Next, consider (16) with very large . To satisfy the boundary conditions, the coefficient phases will differ by nearly 180 to keep the magnitude of the scattered field on the order of the incident field. Coefficients with large magnitude and oscillating phase are typically not stable and therefore not desirable. Formally, the stability of numerical solutions can be inferred from the condition number of the matrix. The condition number can be estimated from the eigenvalues of the system as done for MAS in [10]. An alternative measure of stability is proposed here based on values of a particular solution. As described above, it is the expected that poor stability will be characterized by large with phases differing by nearly 180 . To quantify this possible behavior of the coefficients, define a measure V as (20)

where the numerator is the magnitude of the coefficient with the maximum magnitude; the denominator consists of an average of the coefficients and not the average of their magnitudes. Thus, will have a large numerator and a relatively large oscillating small denominator in (20).

Fig. 5. Plot of EBW vs. MAS monopole radius for a 1:25 radius cylinder. Solid line: the MAS monopole EBW (radius =  in figure); dashed line: EBW (radius = a = in figure). for a line source at  corresponding to 

TABLE I MAS RESULTS (a = 0:5, PLANE WAVE INCIDENT FIELD)

A. Monopole Line Source Incident Field The performance of MAS for the circular cylinder in the presence of a monopole line source has been discussed in [8] and [9]. The monopole radius must enclose the singularities of the scattered field. For the circular cross section, the singularities are when the line source known to be at a radius of is at a radius , as shown in Fig. 1. Calculation of the EBW for the monopole line source has been performed. It has been found that the MAS monopole radius required for the EBW of the scattered field closely matches . the radius given by and , EBW for the For example, with line source incident field is 6. The minimum MAS monopole compared to . radius using EBW is , placing the line source at , Again for , compared EBW calculations indicate an AS radius of to . . We shall Consider a larger cylinder with radius now compare the EBW for a MAS monopole at AS radius and the EBW for a monopole line source at the corresponding , i.e., the line source is at . Fig. 5 shows the comparison of EBW for the MAS pole and line source at the singular positions. The line source EBW besince the line source has moved comes flat at far enough away to approximate a plane wave source (approxior ). mately when , the EBW of the incident field matches closely For q the EBW of the MAS monopole. This verifies the result found . previously; the poles should be on an AS of radius at least , the line source has . This result For . matches the plane wave EBW for a cylinder with in this case. Therefore, the AS radius must be larger than Certainly, there must be a minimum AS radius defined by the

plane wave EBW. In the next section, we discuss this minimum AS radius. B. Plane Wave Incident Field A plane wave incident field can be considered as a special case of the monopole line source incident field where . For this limiting case, the singularity approaches the origin. In principle, the AS radius can be made very small. In [10], it is shown that a small AS radius can result in large numerical errors. In this section, it is shown that a minimum AS radius can be estimated using EBW results. radius cylinder with a plane wave incident Consider a field as shown in Fig. 1. The EBW for the incident field is 5. The and 20 monopoles equally spaced MAS results for in are summarized in Table I. The monopole radius is varied and . between , the EBW for the monopoles is 2 (recall, For incident field EBW is 5). Table I shows the boundary condition error is less than 3% for 10 monopoles, and improves greatly for 20 monopoles. However, is an excessively large number. The maximum coefficient magnitude (the numerator of , ) is over 150 for 10 monopoles and is over 1,000 for 20 monopoles.

RICHIE: MAS POLE LOCATION AND EFFECTIVE SPATIAL BANDWIDTH OF THE SCATTERED FIELD

Choosing , the monopole EBW is 3, a value closer is small for to the incident field EBW of 5. Again, and improves greatly for . The measure for 10 monopoles is much smaller than the case and nearly . Values of for , 20 are doubles for would be 1.91, 1.43, respectively. The solution for solution. considered more stable than the For , monopole EBW matches the incident field inEBW of 5. The boundary condition error improves as creases. The measure is less than 4 and increases somewhat increases. Values of for and 20 are as 1.60 and 0.977, respectively. demonstrate that numerical accuResults for racy has been achieved; the analytical accuracy now dominates the boundary condition error. To improve the solution accuracy, more monopoles are necessary. The stability of the solution, as increases. indicated by smaller , is expected to remain as Simulations have been performed to investigate the effect of on and for the case of with increasing . As increases up to 50, remains at approximately 1.5 while the average boundary condition error decreases. In summary, it has been demonstrated that EBW can be used to obtain well-behaved solutions. In particular, the EBW of the MAS monopoles along the boundary must be equal to or greater than the EBW for the incident field to avoid numerical inaccuracies. Optimization of a simulation by minimizing the boundary condition error may not guarantee suitable solutions. In Table I it can be seen that increasing MAS monopole radius for a fixed increases . However, the decreasing as increases can be interpreted as obtaining solutions that are increasingly well-behaved. Certainly, when the scatterer is a more complex object, well-behaved solutions will be desired. IV. CONCLUSIONS In this work, the effective spatial bandwidth (EBW) of fields along the scatterer boundary has been introduced. The EBW for a variety of incident fields and scattered field models have been investigated. The EBW concept clearly indicates a lower limit for the radius of the MAS monopoles in both the plane wave incident field and the line source incident field cases. For the line source incident field, minimum AS radius as determined from EBW closely matches the well-known singularity radius. In general, the EBW of the MAS monopoles along the boundary must be equal to or greater than the EBW for the incident field to avoid numerical inaccuracies. A measure, , has been proposed and used to quantify the suitability of an MAS solution. It has been demonstrated that in some cases, the monopole coefficients have very large magis designed to nitudes and oscillating phases. The measure

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extract the severity of this effect, and thus indicate whether a solution is well-behaved or not well-behaved. The results reported here are for scatterers with a circular cross section; the concepts and tools developed can also be applied to scatterers with non-circular cross sections. However, for non-circular cross sections, application of the addition theorem will not result in the spatial harmonic content of the fields along the boundary; therefore, numerical methods to determine the EBW will be required. REFERENCES [1] Generalized Multipole Techniques for Electromagnetic and Light Scattering, ser. Computational Methods in Mechanics, T. Wriedt, Ed.. New York: Elsevier Science, 1999, vol. 4. [2] D. I. Kaklamani and H. T. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag., vol. 44, no. 3, pp. 48–64, Jun. 2002. [3] F. Shubitidze, K. O’Neill, S. A. Haider, K. Sun, and K. D. Paulsen, “Application of the method of auxiliary sources to the wide-band electromagnetic induction problem,” IEEE Trans. Geosci. Remote Sens., vol. 40, no. 4, pp. 928–942, Apr. 2002. [4] K. Beshir and J. Richie, “On the location and number of expansion centers for the generalized multipole technique,” IEEE Trans. Electromagn. Compat., vol. 38, no. 2, pp. 177–180, May 1996. [5] E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Amer. A, vol. 19, no. 1, pp. 101–111, Jan. 2002. [6] C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics. Boston, MA: Artech House, 1990. [7] P. B. Leuchtmann, “Automatic computation of optimum origins of the poles in the multiple multipole method (MMP method),” IEEE Trans. Magn., vol. M-19, no. 6, pp. 2371–2374, Nov. 1983. [8] Y. Leviatan, “Analytic continuation considerations when using generalized formulations for scattering problems,” IEEE Trans. Antennas Propag., vol. 38, no. 8, pp. 1259–1263, Aug. 1990. [9] G. Fikioris, “On two types of convergence in the method of auxiliary sources,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2022–2033, Jul. 2006. [10] H. T. Anastassiu, D. G. Lymperopoulos, and D. I. Kaklamani, “Accuracy analysis and optimization of the method of auxiliary sources (MAS) for scattering by a circular cylinder,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1541–1547, Jun. 2004. [11] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1965. [12] O. M. Bucci and G. Franceschetti, “On the spatial bandwidth of scattered fields,” IEEE Trans. Antennas Propag., vol. 35, no. 12, pp. 1445–1455, Dec. 1987.

James E. Richie (M’80–SM’96) received the B.S. degree in electrical engineering from Lafayette College, Easton, PA, in 1983, and the M.S. (electrical engineering) and Ph.D. degrees from the University of Pennsylvania, Philadelphia, in 1985 and 1988, respectively. He has been on the faculty of the Electrical and Computer Engineering Department, Marquette University, Milwaukee, WI, since 1988, where he is currently an Associate Professor and Coordinator of the Electrical Engineering Program. Dr. Richie is a member of Tau Beta Pi, Eta Kappa Nu, and Sigma Xi.

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High Efficiency Wideband Aperture-Coupled Stacked Patch Antennas Assembled Using Millimeter Thick Micromachined Polymer Structures Sumanth Kumar Pavuluri, Changhai Wang, and Alan J. Sangster

Abstract—Micromachined stacked patch antenna devices with high efficiency and wideband characteristics are reported. Polymer based fabrication and assembly processes have been developed in order to produce the stacked suspended antenna devices. Millimeter thick micromachined SU8 based polymer rings are used to create air gaps between the patches and the microwave substrate for optimized high efficiency operation. Thin film liquid crystal polymer (LCP) and polyimide substrates are used to support the radiating and parasitic patch elements. The polymer rings also form cavities to protect the patches and substrate from moisture and dust. The antenna structures are fabricated in layers and then assembled to obtain 3D devices. The antenna devices have been designed using an electromagnetic simulation package. The aperture coupled devices are impedance matched for wideband operation. RF measurements show wideband operation of the devices and the results are in good agreement with that of simulation. Typical gain and bandwidths are 7.8 dBi and 39% for a microstrip fed antenna device while they are 7.6 dBi and 44% for a CPW fed device. The predicted efficiency from the results of simulation is above 97% for the antenna devices. Index Terms—Aperture coupling, microassembly, microelectromechanical devices, micromachining, microstrip antennas, stacked microstrip antennas.

I. INTRODUCTION HE need for high data transmission rate coupled with ever increasing demand for mobile devices has generated a great interest in low cost, compact microwave and millimeter wave antennas exhibiting high gain and wide bandwidth. Conventional patch antenna devices suffer from low gain and narrow bandwidth. Some of the problems with the large self-reactances associated with probe excited patch antennas can be eliminated using aperture based coupling techniques. Aperture coupling also simplifies the construction process by eliminating soldered connections that would otherwise lead to additional parasitic effects [1]. Impedance-matching in these antennas can be realized by adjusting the dimensions of the excitation slot and by adding a small tuning stub. It is well known that aperture coupled antennas operating at different resonances can lead to a signifi-

T

Manuscript received April 03, 2009; revised March 10, 2010; accepted May 11, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The authors are with the School of Engineering and Physical Science, HeriotWatt University, Edinburgh EH14 4AS, U.K. (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.2010.2071334

cant increase in bandwidth, a desirable feature for many applications in the field of telecommunications. A stacked patch antenna structure has been considered as an efficient radiating device that offers compactness and wide bandwidth operation [2]–[5]. Multilayered PCB and PTFE techniques have already been used to create stacked antenna devices [5]–[8]. An aperture-coupled stacked patch antenna fed by a coplanar waveguide has been described by Mestdagh et al. [5] operating at GHz. However, the efficiency of the arrangement was limited to around 75% due to the dielectric and conductor losses within the PCB substrates, employed in preference to PTFE substrates which are expensive. This problem becomes worse as the operating frequency of the antenna moves into the millimeter wave region. Consequently, there is a growing demand for new technologies capable of producing microwave and millimeter wave antennas with improved bandwidth and efficiency relative to the currently available devices. Microelectromechanical (MEMS) systems techniques and materials that have become available in recent years can be used to eliminate some of the fundamental limitations inherent to the present generation of printed antennas. High temperature fabrication processes such as silicon and LTCC (Low temperature co-fired ceramic) micromachining have been used previously to fabricate suspended patch antennas [9], [10]. However, the LTCC process involves expensive drilling while silicon micromachining involves inefficient wet/dry etching techniques as well as the use of relatively expensive silicon substrate material. Recently it has been shown that the gain and efficiency of antennas can be improved considerably by using partial and complete substrate independent antenna designs [11]. This is achieved by creating an air gap or cavity between the substrate surface and the radiating patch using micromachined polymer pillars or a polymer rim [11]–[13]. However, the bandwidth of these antenna devices is % using a single patch. In this paper we present limited to the design, fabrication and assembly of an aperture coupled antenna device with two stacked patches and a CPW fed device with three stacked patches. SU8 polymer is utilized to create the suspended radiating patch elements. Excellent packaging properties of the SU8 polymer particularly the low temperature processing characteristics, enable integration of antennas and filters with other technology platforms such as LTCC, silicon micromachining and MMIC technologies. II. ANTENNA DESIGN AND SIMULATION Aperture coupled stacked patch antenna devices have been designed for high efficiency wideband operation using micro-

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PAVULURI et al.: HIGH EFFICIENCY WIDEBAND APERTURE-COUPLED STACKED PATCH ANTENNAS

machined polymer rims. Fig. 1(a) and (c) show the cross sectional view of a microstrip fed stacked antenna and the top view of the coupling aperture and the microstrip on the substrate surfaces. The suspended patches were used to improve the bandwidth of the resultant antenna device. The device consists of a double cladded microwave PCB substrate and two suspended patches to form a stacked antenna device. The microstrip feed line on the bottom surface of the microwave substrate excites the device through a rectangular coupling slot in the ground plane on the top surface of the substrate. The patches on thin film substrates are supported by micromachined polymer spacers to reduce the loss and hence to improve the efficiency of the device. The cavities formed by the bonded polymer rings also provide protection of the antenna elements from moisture and particles. A microwave PCB material (AD300A, Arlon MED) [14] was used as the base substrate and polyimide thin films (Du Pont) were used as the supporting substrates for the two suspended patches. Fig. 1(b) and (d) show the schematic of a CPW fed stacked patch antenna and the layout of the aperture and the feeding line on the top surface of the base substrate. In this device, three stacked patches were used to increase the bandwidth. A single cladded PTFE substrate (Taconic TLY-3-0200-CH/CH) [15] was used to produce the CPW line and the coupling aperture. The stacked patches are suspended symmetrically above the aperture using micromachined SU8 polymer rims. The micromachined stacked antenna devices were designed using an approach similar to those described previously for the development of multiplayer stacked wideband antenna devices [7], [16], [17]. The devices were designed for operation around 10 GHz of frequency for ease of characterization using in-house facilities. The design and modeling work was carried out using a commercial electromagnetic (EM) simulation package (Ansoft HFSS). For wideband operations impedance matching techniques based on stacked parasitic patches have been employed to improve the bandwidth of the devices with optimum antenna gains. The air gaps between the stacked patches and the substrate (Fig. 1) were chosen to increase the antenna bandwidth and at the same time to maintain a low profile for the overall antenna structures. For the microstrip fed device, the top patch and the aperture were designed to be in close resonance while the microstrip line and the lower patch were used as the impedance tuning elements for wideband operation. In the design process the approximate dimensions of the patches were determined for optimum performance at the frequency of operation. The aperture length was selected to be resonant at a frequency close to that of the patches and the width was about one tenth of the length. The heights of the air cavities were chosen based on a trade-off between the bandwidth and the challenges for fabrication. The length of the feedline was then varied to obtain sufficient bandwidth. Fine impedance tuning was achieved by adjusting the dimensions of the lower patch. As the SU8 polymer is a lossy microwave material, the shape and dimensions of the SU8 spacer rings have to be adjusted carefully to ensure good antenna performance. The dimensions of the SU8 rims were determined to obtain a small footprint and high efficiency for the resultant antenna devices. It was necessary to carry out sev-

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Fig. 1. Schematic cross-sectional views of the stacked high gain patch antennas using micromachined polymer spacers, (a) microstrip fed antenna and (b) CPW fed device. The top views of the apertures and feed lines on the substrate surfaces corresponding to (a) and (b) are shown in (c) and (d) respectively.

eral iterations of the above steps to obtain an optimized antenna structure for high gain wideband operation. The CPW fed device was designed using a similar method. In this case three patches were used to produce a bandwidth of % as the air gap between the base substrate and the bottom patch is smaller than in the microstrip fed device. The top two patches have the same dimensions and the electromagnetic coupling between them increases the bandwidth of the antenna device. The aperture and the top two patches were designed to be stub were adin close resonance. The bottom patch and the justed to obtain impedance matching operation with wide bandwidth. Table I shows a summary of the physical dimensions of the structure layers for both antenna devices. The thickness, dielectric constant and loss tangent of the substrates are given in Table II. The dielectric constant and loss tangent of the SU8 polymer are 4.2 [18] and 0.042 respectively [19]. III. ANTENNA FABRICATION AND ASSEMBLY Fig. 2 shows the schematic of the process flow for construction of the antenna devices. The microstrip fed antenna device

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TABLE I SUMMARY OF THE DESIGN PARAMETERS FOR THE SUSPENDED STACKED PATCH ANTENNA DEVICES

TABLE II SUMMARY OF THE THICKNESS AND MICROWAVE PROPERTIES OF THE SUBSTRATES

is used to illustrate the fabrication and assembly process. A MEMS based fabrication process has been implemented to obtain precision metal patterns on the substrates with the aim of reducing metal losses at higher frequencies. Fig. 2(a) shows the fabrication of the microstrip and the aperture on the microwave substrate followed by the fabrication of the SU8 polymer rim. The details of the fabrication method have been described previously [16], [17]. The microstrip and aperture structures are produced on the Arlon microwave substrate shown in Table II. The thickness of the SU8 rim fabricated on the substrate is 1.2 mm, half of the total height of the air gap between the Arlon substrate and the polyimide thin film substrate of the lower patch. The corresponding spacer rim for bonding was fabricated on the polyimide substrate after fabrication of the patch in order to produce a total air gap of 2.4 mm after assembly. The height of the SU8 rim on the polyimide substrate with the upper patch is 0.67 mm as necessary to produce impedance matched performance. The assembly process is illustrated in Fig. 2(c) and (d). The polyimide film supporting the lower patch was attached to the substrate by bonding of the SU8 rim on the polyimide film to the corresponding one on the base substrate. Then the polyimide film with the top patch was attached to the polyimide substrate of the lower patch by bonding the SU8 rim to the polyimide film. Precision alignment marks on the polyimide films and the base substrate were used to facilitate the accurate alignment of different layers. A viscous SU8 solution (SU8-100) was used to bond the structures together. The bonding process was carried out at 55 C for 10 minutes in order to bond the two polymer rims together. Fig. 3 shows the images of an assembled antenna device with two stacked patch elements.

Fig. 2. Schematic of the fabrication flow on (a) microwave substrate (b) polyimide film (c) assembly of the first patch (d) assembly of the second patch.

Fig. 3. (a) Optical picture of the front side of a fabricated antenna. (b) Optical picture of the reverse side of the antenna.

The fabrication process for the CPW fed antenna device is similar to that of the microstrip fed antenna device. A copper cladded LCP (Rogers Corp. ultralam 3000 series) substrate of 100 m thickness is used to support the radiating patches while a PTFE substrate supplied by Taconic is used as the base substrate. The use of different substrate materials from that of the microstrip fed device was due to the availability of the materials. Fig. 4 shows optical pictures of the CPW fed antenna device with three stacked patch elements. Fig. 4(a) shows the CPW line and the coupling aperture fabricated on PTFE substrate prior to the assembly of the patches. Fig. 4(b) shows a picture of the bottom patch and a SU8 polymer rim on a LCP substrate. The opening of 15 mm in length in the SU8 rim is used to facilitate the attachment of an SMA connector to the ground and the

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Fig. 4. (a) Photograph of a fabricated CPW line and aperture on the PTFE substrate, (b) optical picture of a patch and a SU8 polymer rim on a LCP substrate, (c) optical picture of an assembled CPW fed antenna device.

CPW line on the base substrate for RF characterization. Fig. 4(c) shows the assembled device after stacking three patch layers on the PTFE base substrate. The metal patch elements are fabricated on the lower surface of the LCP substrate for better protection. In contrast the top patch of the microstrip fed device was fabricated on the upper surface of the polyimide thin film substrate in order to show the top patch after assembly. IV. RESULTS AND DISCUSSION As described in Section II, the design and simulation of the wideband antenna devices was carried out using the Ansoft HFSS package. All of the design features as shown in Fig. 1 and the associated parameters were included in full-wave electromagnetic simulation. RF characterization of the aperture coupled patch antennas was carried out using an HP 8510 network analyzer. The gain measurements were conducted using the gain comparison method [22]. A 15 dBi standard gain horn was used as the transmitting antenna and a 20 dBi standard gain horn as the reference antenna. The measurements were carried out using a separation distance of one meter between the transmitting and the receiving antennas. The radiation characteristics were obtained from far field measurements using the 20 dBi gain horn antenna. All of the measurements were carried out in an anechoic chamber. Fig. 5 shows the measured and simulated results of reflecfor the micromachined antenna devices. The tion coefficient dB bandwidth for the microstrip fed stacked patch antenna device was determined to be 3.7 GHz or 39% from the measured results. The corresponding result for the CPW fed stacked patch antenna was 3.6 GHz or 44%. It can be seen from Fig. 5 that the measured results agree well with that of the simulation apart from the downward frequency shift. The shift, particularly in the vicinity of the main resonance, can be attributed to the effect of the SMA connector overlay which was not included in the simulation. Figs. 6 and 7 show the measurement and simulation results of the normalized radiation patterns at the mid-band frequency for the antenna devices. The corresponding gain values are 7.6 dBi and 6 dBi for the microstrip fed antenna and the

Fig. 5. Simulation and measurement results of reflection coefficient, S for the high gain wideband stacked antennas, (a) microstrip fed device and (b) CPW design.

CPW fed device respectively. The predicted radiation efficiencies for the devices are 98% and 97% for microstrip and CPW fed devices respectively based on the results of simulation using HFSS. The simulation results also show that the thin film polyimide and LCP substrates have negligible effect on the performance of the devices. Fig. 8 shows the results of gain variation as a function of frequency for the antenna devices. The measured results agree well with that of simulation. The gain varies from 5 dBi at GHz to around 7.8 dBi at GHz for the microstrip fed device. For the CPW fed device the gain varies from 5 dBi at GHz to around 7.6 dBi at GHz. Therefore wideband operation of the devices has been realized. This is achieved by using the suspended patches and multiple tuning mechanisms. The performance of the antenna devices is comparable to that of a foam based stacked device reported previously [6]. This is not surprising as the foam is a low loss material and has a dielectric constant of close to unity. However the foam material is prone to moisture absorption and is difficult to use for device construction. V. CONCLUSION High efficiency, wideband stacked patch antenna devices have been successfully fabricated on microwave substrates using micromachining and micro-assembly methods. By creating air cavities between the radiating patches and the substrates using polymer rims, the gain of the devices has been improved significantly over conventional microstrip patch antennas. This polymer based stacked antenna process has a dual function of supporting the air suspended radiating patches and for antenna module packaging. Antenna gains as high as 7.8 dBi and 7.6 dBi have been obtained at the frequency of

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Fig. 6. Simulated and measured E-plane (a) and H-plane (b) radiation patterns for the microstrip fed antenna device at 9.5 GHz.

Fig. 8. Frequency dependent gain for (a) microstrip fed and (b) CPW fed stacked patch antenna with respect to frequency.

stacked single antenna devices can also be used to form antenna arrays for high efficiency and wideband applications. ACKNOWLEDGMENT The authors would like to thank the European Division of Taconic for supplying the copper foil to this work. Fig. 7. Simulated and measured E-plane (a) and H-plane (b) radiation patterns for the CPW fed antenna device at 8.25 GHz.

10.5 GHz for the microstrip fed stacked antenna device and at 10 GHz for the CPW fed device. The measured results are generally in good agreement with that of the simulation. The potential radiation efficiency of the device is as high as 98% based on the simulation results. The wideband operation was achieved by optimally tuning resonances associated with the antenna device through the use of stacked patches as impedance transformers and by optimizing the dimensions of the lower patches, the tuning stub and the aperture. The bandwidths of the antenna devices were determined to be 39% and 44% respectively, about a factor of 4 larger than the figure of 10.5% obtained for the probe fed device [11]. The associated fabrication process provides an alternative low cost packaging process as compared to the conventional LTCC and PCB technologies. The suspended antennas are lightweight and can be easily fabricated with micromachining of the SU8 photopolymer. The method of integration has potential applications in the emerging field of UWB networks, WLAN networks at 60 GHz, automotive collision radar and terahertz antenna devices. The

REFERENCES [1] D. M. Pozar, “Microstrip antennas,” Proc. IEEE, vol. 80, pp. 79–91, Jan. 1992. [2] J.-F. Zürcher, “The SSFIP: A global concept for high performance broadband planar antennas,” Electron. Lett., vol. 24, pp. 1433–1435, Nov. 1988. [3] E. Nishiyama, M. Aikawa, and S. Egashira, “Three-element stacked microstrip antenna with wide-band and high-gain performances,” in Int. IEEE AP-S Symp. Dig., Jun. 2003, vol. 2, pp. 900–903. [4] R. Q. Lee and K. F. Lee, “Experimental study of the two-layer electromagnetically coupled rectangular patch antenna,” IEEE Trans. Antennas Propag., vol. 38, pp. 1298–1302, Aug. 1990. [5] S. Mestdagh, R. Walter De, and G. A. E. Vandenbosch, “CPW-fed stacked microstrip antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 74–83, Jan. 2004. [6] S. D. Targonski, R. B. Waterhouse, and D. M. Pozar, “Design of wide-band aperture-stacked patch microstrip antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 9, pp. 1245–1251, Sep. 1998. [7] F. Croq and D. M. Pozar, “Millimeter-wave design of wide-band aperture-coupled stacked microstrip antennas,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1770–1776, Dec. 1991. [8] G. S. Kirov, A. Abdel-Rahman, and A. S. Omar, “Wideband aperture coupled microstrip antenna,” in Proc. IEEE APS Symp., Jun. 22–27, 2003, vol. 2, pp. 888–891. [9] I. K. Kim, N. Kidera, S. Pinel, J. Papapolymerou, J. Laskar, J. G. Yook, and M. M. Tentzeris, “Linear tapered cavity-backed slot antenna for millimeter-wave LTCC modules,” IEEE Antennas Wireless Propag. Lett., pp. 175–178, 2006.

PAVULURI et al.: HIGH EFFICIENCY WIDEBAND APERTURE-COUPLED STACKED PATCH ANTENNAS

[10] G. P. Gauthier, J. P. Raskin, L. P. B. Katehi, and G. M. Rebeiz, “A 94-GHz aperture-coupled micromachined microstrip antenna,” IEEE Trans. Antennas Propag., vol. 47, no. 12, pp. 1761–1766, Dec. 1999. [11] B. Pan, Y. K. Yoon, G. E. Ponchak, M. G. Allen, J. Papapolymerou, and M. M. Tentzeris, “Analysis and characterization of a high-performance Ka-band surface micromachined elevated patch antenna,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 511–514, Dec. 2006. [12] K. Jeong-Geun, L. Hyung Suk, L. Ho-Seon, Y. Jun-Bo, and S. Hong, “60-GHz CPW-fed post-supported patch antenna using micromachining technology,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 635–637, Oct. 2005. [13] Y. H. Cho, K. Sung-Tek, C. Wonkyu, H. Man-Lyun, C. Pyo, and K. Young-Se, “A frequency agile floating-patch MEMS antenna for 42 GHz applications,” in Proc. IEEE APS Symp., Jul. 3–8, 2005, vol. 1A, pp. 512–515. [14] Data Sheet for AD 300A PTFE Arlon Materials for Electronics (MED). Orlando, FL, (Woven Fiberglass/Micro-Ceramic Filled Laminate for RF & Microwave Printed Circuit Boards), [Online]. Available: http://www.ctsind.com.sg/arlon.html [15] Data Sheet for TLY RF & Microwave Laminate, Taconic, specializing in coating of glass fabrics with PTFE, presented at Woven fiberglass fabric coated with PTFE interleaved with thin sheets of pure PTFE, [Online]. Available: http://www.taconic-add.com/pdf/tly.pdf [16] S. K. Pavuluri, C. H. Wang, and A. J. Sangster, “A high-performance aperture-coupled patch antenna supported by a micromachined polymer ring,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 283–286, Sep. 2008. [17] C. H. Wang and S. K. Pavuluri, “Fabrication and assembly of high gain MEMS antennas for wireless communications,” Proc IEEE 58th ECTC, pp. 1941–1945, May 27–30, 2008. [18] J. R. Thorpe, D. P. Steenson, and R. E. Miles, “High frequency transmission line using micromachined polymer dielectric,” Electron. Lett., vol. 34, pp. 1237–1238, Jun. 1998. [19] S. Lucyszyn, “Comment on terahertz time-domain spectroscopy of films fabricated from SU-8,” Electron. Lett., vol. 37, p. 1267, Sept=. 27, 2001. [20] Dupont Inc., Summary of Properties for Kapton Polyimide [Online]. Available: http://www2.dupont.com/Kapton/en_US/index.html [21] D. C. Thompson, O. Tantot, H. Jallageas, G. E. Ponchak, M. M. Tentzeris, and J. Papapolymerou, “Characterization of liquid crystal polymer (LCP) material and transmission lines on LCP substrates from 30 to 110 GHz,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1343–1352, Apr. 2004. [22] C. A. Balanis, Antenna Theory Analysis and Design. New York: Wiley, 1997, ch. ch. 16, pp. 858–874.

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Sumanth Kumar Pavuluri received the B.Tech. (Bachelor of Technology) degree in electronics and communications engineering from Jawaharlal Nehru Technological University, Hyderabad, India, in 2004 and the M.Sc. degree in microsystem engineering from Heriot-Watt University, Edinburgh, U.K., in 2005, where he is working toward the Ph.D. degree. He is also a Research Associate working on the Frequency Agile Microwave Oven Bonding System (FAMOBS) project at Heriot-Watt University. His current research interests include modelling, fabrication and testing of RF-MEMS devices, micromachined antennas, waveguide cavity ovens and IC packaging.

Changhai Wang received the B.Sc. degree in semiconductor physics and devices from Jilin University, Changchun, China, in 1985, and the M.Sc. degree in optoeletronic and laser devices and the Ph.D. degree in low power all-optical switching devices from Heriot Watt University, Edinburgh, U.K., in 1988 and 1991 respectively. He is currently a Lecturer in electrical and electronic engineering in the School of Engineering and Physical Sciences at Heriot-Watt University. His current research interests include fabrication and assembly of microstructures, MEMS devices and sensors, MEMS packaging, laser assisted processes for MEMS and electronics manufacture and biochip technology. He held a Royal Society of Edinburgh Enterprise Fellowship in Optoelectronics from 1997 to 1998. He has published over 80 journal and conference papers.

Alan J. Sangster received the B.Sc. degree in electrical and electronic engineering in 1963, the M.Sc. degree in 1964, and the Ph.D. degree in 1967, all from the University of Aberdeen in Scotland. He spent four years with Ferranti plc, Edinburgh, Scotland, doing research into wideband travelling wave tubes, and three years with Plessey Radar Ltd., Cowes, U.K., investigating and developing microwave devices and antennas for microwave landing systems and frequency scanned radar systems. Since 1972, he has been with the Heriot-Watt University, Edinburgh, where he became Professor of Electromagnetic Engineering in 1989. His current research interests lie in the areas of microwave antennas, mm-wave sensing, electrostatically driven micromotors and microactuators, electromagnetic levitation, medical applications of microwaves, and the numerical solution of electromagnetic radiation and scattering problems. He is the author of over 200 papers and presentations on these topics. Prof. Sangster is a Fellow of the Institution of Engineering and Technology (IET), London, U.K., and has been a Member of the Electromagnetics Academy of New York since 1993.

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Novel Reconfigurable Defected Ground Structure Resonator on Coplanar Waveguide Heba B. El-Shaarawy, Fabio Coccetti, Member, IEEE, Robert Plana, Senior Member, IEEE, Mostafa El-Said, and Essam A. Hashish, Member, IEEE

Abstract—A novel reconfigurable defected ground structure (DGS) resonator fabricated on coplanar waveguide (CPW) technology is presented. The resonator is endowed with an original design which enables the generation of multiple transmission zeros at arbitrary frequencies. The chosen design is indeed based on a slot defect created on the lateral ground planes of the CPW with the double advantage to allow a simple reconfiguration, by means of surface mounted (or fabricated) components, and a very compact solution, by exploiting the transversal dimension of the coplanar wave transmission line (CPW-TL). Four different states of the diodes configuration are investigated, where in each state multiple transmission zeros are produced in the frequency range from 1 GHz to 11 GHz. The equivalent circuit of each state is obtained using a conventional circuit parameter extraction method. Moreover, the slotline design equations are used to identify the transmission zeros and validated using the magnetic field distribution inside the slot. In this work, the reconfigurability is first proven by means of short bridges mounted in specific locations. These bridges are then replaced by PIN diodes. Simulated and measured results are in good agreement. Index Terms—Circuit parameter extraction, coplanar waveguide (CPW), defected ground structure (DGS), transmission zero.

I. INTRODUCTION INCE the late 1980s, electromagnetic bandgap (EBG) structures [1]–[7] and defected ground structures (DGS) [7]–[14] have attracted the interest of many researchers, due to their interesting properties in terms of size miniaturization, suppression of surface waves and arbitrary stopbands. Since then, they have been used in many applications like lowpass filters [8], bandpass filters [6], [9], [10], antennas [7], waveguides [3] and others. Recently, a great trend towards the implementation of a reconfigurable DGS where the location of the transmission zeros can be controlled and tuned may be seen from a number of recent publications [11]–[13]. However, still not a large amount

S

Manuscript received May 04, 2009; revised September 17, 2009; accepted May 17, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. H. B. El-Shaarawy, F. Coccetti, and R. Plana are with Laboratoire d’Analyse et d’Architectures des Systèmes, Centre National de la Recherche Scientifique (LAAS-CNRS), F-31077 Toulouse, France and also with the UPS, INSA, INP, ISAE, Université de Toulouse, F-31077 Toulouse, France (e-mail: [email protected]; [email protected]; [email protected]) M. El-Said and E. A. Hashish are with the Faculty of Engineering, Electronics and Communications Department, Cairo University, Cairo, Egypt (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.2010.2071336

of work has been published in this domain. In addition, most of the work that has been published so far has considered the structures by means of parallel RLC equivalent circuit to represent the transmission zeros in the response [8], [11], providing no insight into the underlying electromagnetic wave propagation principle and hence providing no design rules. Further attempts to investigate these structures have been made by using more complex methods such as the Block Floquet’s Theorem [12], the Linpar method [10], or neural networks [14]. The disadvantages of these methods are their mathematical complexity and so their limits in providing initial design rules. In this paper, we present a novel reconfigurable DGS unit cell on coplanar waveguide technology. This unit cell may be considered as a development to that presented in [13], where we have an additional square resonator added to the structure. In addition, the cell, hereby presented, contains a number of PIN diodes on each side of the coplanar waveguide ground planes to give complete control of the number of transmission zeros obtained and their resonant frequencies. Four different states corresponding to four different diode configurations are presented here. The equivalent circuit of each state is obtained using a conventional circuit parameter extraction method. In addition to the parallel RLC equivalent circuit, electromagnetic explanation of the structure, using the slotline waveguide design equations of Janaswamy and Schaubert [15], is hereby presented providing simple and efficient design rules. These rules have been applied to the design of the structures centered at different arbitrary frequencies and validated by the simulated magnetic field distribution inside the etched slots using an FEM (Ansoft HFSS) simulation tool. Simulated and measured responses are in good agreement with each other. The paper is organized as follows: Section II presents briefly the description of the novel reconfigurable DGS. Section III addresses the design, fabrication, characterization and modeling of the reconfigurable DGS with the different configurations. Section IV presents the final measurements of the reconfigurable DGS resonator using PIN diodes as switching components. Finally, conclusions are outlined in Section V of this paper. II. STRUCTURE DESCRIPTION OF THE NOVEL RECONFIGURABLE DGS The structure consists of a DGS resonator based on coplanar waveguide. The lattice shaped unit DGS is shown in Fig. 1. The cell is made of two square resonators of different sizes, and , whose patterns are etched in the ground plane of the CPW with a slot width . The smaller square is connected to the slot of the coplanar waveguide by means of a transverse gap of length

0018-926X/$26.00 © 2010 IEEE

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Fig. 1. Schematic diagram of the proposed reconfigurable DGS resonator (half symmetrical view).

, while another is used to connect the larger square to the . The structure contains three diodes smaller one of length is used in each side of the ground plane. The first diode to disconnect the larger square from the rest of the structure. is used to divide the smaller resonator The second diode (smaller square) into two resonators of different lengths. Adding , is used to divide another degree of freedom, a third diode the bigger ring resonator into two. This structure is designed and fabricated on Teflon substrate , dielectric thickness of dielectric constant mm, and loss tangent 0.0035. The dimensions of the fabricated mm, mm, mm, structure are: mm and mm, where the CPW parameters are mm in order to attain 50 of input line impedance. In the fabrication of these structures, as a first iteration, we modeled the diodes as ideal ones, i.e., the diodes are modeled either as short or open circuits, and then PIN diodes have been used for the sake of reconfigurability in the final implementation of the circuits.

Fig. 2. (a) Layout of the DGS for configuration one (S is ON, S and S are OFF). (b) Magnetic current distribution in the slot. (c) Electromagnetic (EM), and circuit (Ct) simulated and measured S-Parameters. (d) Equivalent circuit model.

III. STUDY OF THE RECONFIGURABLE DGS In the previous section, we explained the structure of the DGS resonator. As mentioned, there are three diodes in each side of the ground plane, consequently different responses according to the different diode configurations will be considered. In each configuration, we will give the EM simulation, measured response and the equivalent circuit model of the DGS when the diodes are modeled as open and short circuits. Then using the slotline equations and the magnetic field distribution inside the slot, the resonant frequency of the transmission zeros will be proven. A. Configuration One:

: ON,

: OFF, and

: OFF

In this case, the upper larger square is isolated from the rest of the structure as shown in Fig. 2(a). The structure has a transmission zero at 6 GHz as shown from the simulated and measured responses in Fig. 2(c). Since the structure gives a single transmission zeros then it can be represented by one parallel resonator as shown in Fig. 2(d). The extraction method of the resistance , inductance and capacitance has been explained in [11]. To investigate the EM wave propagation in the CPW structure, we considered it as two slotline structures in parallel to

each other, then considering the upper part of the structure as shown in Fig. 2(a), it is found that the path difference between path (1), denoted by the continuous blue line, and path (2), denoted by the dotted red line, is 14 mm. This electrical path at 6 GHz according to the slotline difference is equal to equations used; therefore a transmission zero is produced due to 180 phase difference obtained at point “P.” Fig. 2(b) shows the magnetic field distribution in the slot, using the HFSS software, which validates our theory that this path difference corresponds to /2 at 6 GHz causing the existence of a transmission zero. The equations of the slotline used to explain the structure are the closed form expressions of Janaswamy and Schaubert [15] for low substrates obtained by curve-fitting the numerical results of Galerkin’s method in the Fourier transform domain (Appendix). B. Configuration Two:

: ON,

: ON, and

: OFF

In this case, the upper square resonator is isolated as in the previous configuration, and the smaller resonator is divided into two unequal resonators as shown in Fig. 3(a). Simulated and measured responses are shown in Fig. 3(d). The structure forms

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Fig. 3. (a) Layout of the DGS for configuration two (S and S are ON, and S is OFF). (b) Magnetic current distribution in the slot at 4.8 GHz, (c) at 10.8 GHz. (d) Electromagnetic (EM), and circuit (Ct) simulated and measured S-Parameters. (e) Equivalent circuit model.

Fig. 4. (a) Layout of the DGS for configuration three (S ; S , and S are OFF). (b) Magnetic current distribution at 2.28 GHz, (c) at 7.3 GHz, (d) at 9.7 GHz. (e) Electromagnetic (EM), and circuit (Ct) simulated and measured S-Parameters. (f) Equivalent circuit model.

two transmission zeros at 4.8 and 10.8 GHz. This is represented resonators in Fig. 3(e). by the two parallel Using the same technique, this response is verified using the resslotline design equations, where each portion forms a onator that is terminated by a short circuit. Therefore, each resonator forms a transmission zero at the corresponding frequency as illustrated in the responses in Fig. 3(d). This explanation is confirmed by the magnetic field distribution at 4.8 and 10.8 GHz in Fig. 3(b) and 3(c) respectively.

Comparing that to the slot line curve, it is found that when this electrical path difference equals to odd multiples of half of the guided wavelength, a transmission zero is produced due to the 180 phase difference at “P.” In Fig. 4(b), (c), and (d), the magnetic field distribution in the slot is illustrated. Fig. 4(b) shows the magnetic field distribution in the slot at 2.28 GHz. We observe only one maximum for the field and no field in the CPW slot after the point “P,” which validates our theory. Fig. 4(c) shows the H-field at 7.44 GHz. In this case, we have ), causing three maxima along the path corresponding to 3 the second transmission zero. Finally, in Fig. 4(d), the magnetic field exhibits five maxima, which means that the path difference , causing the third transmission zero at corresponds to 9.85 GHz.

C. Configuration Three:

: OFF,

: OFF, and

: OFF

Here, all diodes will act as open circuits leading to the structure in Fig. 4(a), where both squares are connected to the CPW slots. Simulated and measured responses are shown in Fig. 4(e), where we have three transmission zeros at 2.28, 7.44 and 9.85 GHz, represented by three parallel RLC resonators shown in Fig. 4(f). Using the slot line equations, it is found that the path difference between path (1), and path (2) in Fig. 4(a) is 41.5 mm.

D. Configuration Four:

: Off,

: OFF, and

: ON

When is ON, the structure is as shown in Fig. 5(a). The structure shows three resonance frequencies at 2.04 GHz, 4.26 GHz, and 8.39 GHz, Fig. 5(e), presented by the three parallel

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Fig. 6. Photograph of the fabricated DGS resonator and detailed schematic view of the diode setup for S .

Fig. 5. (a) Layout of the DGS for configuration four (S and S are OFF, and S is ON). (b) Magnetic current distribution at 2.04 GHz, (c) at 4.26 GHz, (d) at 8.4 GHz. (e) EM, and circuit simulated and measured S-Parameters. (f) Equivalent circuit model.

circuits shown in Fig. 5(f). In this case the larger square has a short bridge along its path causing the whole structure to divide into two basic resonators. The first resonator, denoted by resonator ending by a short circuit, as the red path, is a in the previous case, producing a 180 phase shift at point “P” which causes the first resonant frequency at GHz. The same goes for the second resonator, denoted by the green path, giving the second transmission zero at frequency GHz, Fig. 5(b) and (c). The third transmission zero at GHz occurs when the branch, starting from point “M” to and 3 forming a the short circuits, corresponds to transmission zero at point “M,” and consequently a zero in the whole structure response as illustrated in Fig. 5(d). IV. MEASUREMENT RESULTS USING PIN DIODES In this section we present the measurement results when these short circuits are replaced by PIN diodes. The PIN diodes used here are the Philips silicon PIN diodes BAP 63-02. They have

been chosen due to their low diode capacitance pF), and very low series induclow forward resistance nH), in addition to their small size (1.1 0.7 tance mm) which makes them compatible with our design. In this structure we have three diodes in each ground plane. Two diodes ( and ) are connected between the inner patch of the square resonator and the ground plane. Fixing of these diodes is as shown in Fig. 1 where the voltage for biasing the diode is applied on the square patch. For the diodes across the transverse slot between the square , the biasing is more complex. According to the resonators initial structure in Fig. 1, the two terminals of the diodes are connected to the ground plane therefore the biasing in this case is impossible. In this case, a small patch is etched in the ground plane where two back to back diodes are connected. When the voltage is applied to this patch, both diodes will be ON, short circuiting the upper resonator, and when no biasing is applied, the path is open for the signal to pass. It should be noted that the etching is done as small as possible in order not to cause large perturbation in the ground plane for the calculated signal, at the same time to provide high impedance, and so to prevent the signal from passing in the other paths instead of the main designed path. A photograph of the fabricated structure is shown in Fig. 6. From the measurement results shown in Fig. 7, in configuration one, we have one transmission zero at 4.35 GHz of dB, and a 10-dB bandwidth of 0.88 GHz. In configuration two, we have two transmission zeros at 3.75 and 8.47 dB, and dB, and 10-dB bandwidths GHz of of 0.36 and 1.29 GHz, respectively. In configuration three, we have three transmission zeros at 1.86, 6.75 and 8.34 GHz, of , and dB, and 10-dB bandwidths of 0.36, 0.52, and 0.09 GHz, respectively. Finally in configuration four, we have three transmission zeros at 1.68, 3.48, and 7.17 GHz, , and , and 10-dB bandwidths of 0.27, of 0.18 and 0.84 GHz respectively. A shift in the resonance frequencies of the transmission zeros, between the measurement results with PIN diodes and those previously mentioned, where diodes have been modeled as open and short circuits, can be observed. The effect of the diode has not been previously considered, in order to facilitate obtaining clear design rules for the transmission zeros using the slotline design equations. As expected, adding the diodes yields a shift of the resonant frequencies due to the intrinsic capacitance and inductance introduced by these components. This shift is not the same in all cases as it depends on the diode working condition, whether they are biased or not, in addition to their location in

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Fig. 7. Measurement results with PIN diodes compared to simulation results when diodes are modeled as open and short circuits, (a) configuration one, (b) configuration two, (c) configuration three, and (d) configuration four.

Fig. 8. Slotline waveguide wavelength and characteristic impedance for the substrate used versus frequency.

the circuit. Also the quality factor of the transmission zeros has decreased as a consequence of the diode resistance introduced. By observing the first zero of the four configurations, it is evident that there is a shift of less that 4% in all cases be-

tween measured results with open/short circuits, theory, EM and circuit simulations; and a much larger shift (15–25%) between measured results with diodes and those with open/short circuits. This means that there are two mechanisms taking part to this drift which sum to each other. The first is the unavoidable over-etching of the metal (affecting both cases; open/short and diodes), the second is the parasite capacitance associated with the diodes package (SMD metal pads), beside those aforementioned. This drift is frequency dependent and seems to validate the above hypothesis. All these factors may be neglected in a first design iteration, and considered afterwards once the location of the transmission zeros have been found by means of the aforeintroduced slotline theory. Eventually the effect of the diodes can be considered for a final adjustment of the structure. It should be clear that these reported results are obtained with a single iteration and no redesign has been done. V. CONCLUSION In this paper, we presented a novel reconfigurable DGS resonator capable to yield arbitrary transmission zeros over the band 1–11 GHz. The structure was studied by means of slot-

EL-SHAARAWY et al.: NOVEL RECONFIGURABLE DEFECTED GROUND STRUCTURE RESONATOR ON COPLANAR WAVEGUIDE

line design equations which allow a direct insight of the EM propagation taking place within the structure, and provide in addition, straight forward design rules for the resonator. This later have been presented and discussed here for the first time. A good agreement has been observed between simulation and measurement results. The reconfigurability of the structure presented here is achieved by PIN diodes acting as switches. In case of slight tuning requirements, varactor diodes are to be used instead. Better performance (closer to the ideal case) are expected by using RF MEMS switches. Finally, this reconfigurable DGS resonator is a good candidate for a number of applications where suppression of passbands and reconfigurability are required. For example, this DGS cell may be combined with a multiband antenna to allow or suppress one of the bands according to the need, or after a bandpass filter to suppress higher order harmonics. APPENDIX Janaswamy and Schaubert [15] for low substrates obtained by curve-fitting the numerical results of Galerkin’s method in the Fourier transform domain. and (the range For where our design exists)

(1) %.

(2) %. REFERENCES [1] E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Amer. B, vol. 10, pp. 283–295, Feb. 1993. [2] F. Yang, K. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic bandgap (UC-EBG) structure and its applications for microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1509–1514, Aug. 1999. [3] F. R. Yang, K. P. Ma, Y. Qian, and T. Itoh, “A novel TEM waveguide using uniplanar compact photonic-bandgap (UC-PBG) structure,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2092–2098, Nov. 1999. [4] T.-Y. Yun and K. Chang, “Uniplanar one-dimensional photonic-bandgap structures and resonators,” IEEE Trans. Microw. Theory Tech., vol. MTT-49, pp. 549–553, Mar. 2001.

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[5] C. Caloz and T. Itoh, “Multilayer and anisotropic planar compact PBG structures for microstrip applications,” IEEE Trans. Microw. Theory Tech., vol. MTT-50, pp. 2206–2208, Sep. 2002. [6] S.-G. Mao and Y. Zhi, “Coplanar waveguide BPF with compact size and wide spurious free stopband using electromagnetic bandgap resonators,” IEEE Microw. Wireless Comp. Lett., vol. 7, no. 3, pp. 181–183, Mar. 2007. [7] H. Liu, Z. Li, X. Sun, and J. Mao, “Harmonic suppression with photonic band gap and defected ground structure for a microstrip patch antenna,” IEEE Microw. Wireless Comp. Lett., vol. 15, pp. 55–56, Feb. 2005. [8] J. Lim, C. Kim, D. Ahn, Y. Jeong, and S. Nam, “Design of low-pass filters using defected ground structure,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 8, pp. 2539–2545, Jan. 2005. [9] H. B. El-Shaarawy, F. Coccetti, R. Plana, M. El-Said, and E. A. Hashish, “Compact bandpass ring resonator filter with enhanced wide-band rejection characteristics using defected ground structures,” IEEE Microw. Wireless Comp. Lett., vol. 18, no. 8, pp. 500–503, Aug. 2008. [10] M. Abdelaziz, A. M. E. Safwat, F. Podevin, and A. Vilcot, “Narrow bandpass filter based on the modified DGS,” in Proc. 37th Eur. Microw. Conf., Oct. 2007, pp. 75–78. [11] E. K. I. Hamad, A. M. E. Safwat, and A. S. Omar, “A MEMS reconfigurable DGS resonator for K-band applications,” IEEE J. Microelectromech. Syst., vol. 15, no. 4, pp. 756–762, Aug. 2006. [12] M. F. Karim, A. Q. Liu, A. Alphones, and A. B. Yu, “A reconfigurable micromachined switching filter using periodic structures,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 6, pp. 1154–1162, June 2007, Part 1. [13] A. M. E. Safwat, F. Podevin, P. Ferrari, and A. Vilcot, “Tunable bandstop defected ground structure resonator using reconfigurable dumbbell-shaped coplanar waveguide,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 9, pp. 3559–3564, Sept. 2006. [14] L. Yuan, L. Jiao, and Y. Chunhui, “The back propagation neural network model of non-periodic defected ground structure,” in Proc. Global Symp. Millimeter Waves (GSMM), Apr. 2008, pp. 29–32. [15] R. Janaswamy and D. H. Shaubert, “Characteristic impedance of a wide slot line on low permittivity substrates,” IEEE Trans. Microw. Theory Tech., vol. 34, no. 6, pp. 900–902, Jun. 1986, Part 1. Heba B. El-Shaarawy was born in Cairo, Egypt, in 1981. She received the Bachelor’s degree (with honors) and the Master’s degree in the miniaturization of microstrip filters from Cairo University, Cairo, Egypt, in 2003 and 2005, respectively, and the Ph.D. degree from the Laboratoire d’Analyse et d’Architectures des Systèmes, Centre National de la Recherche Scientifique (LAAS-CNRS) and the University of Toulouse, Touslouse, France in 2009. Previously, she worked as a teaching assistant on the Faculty of Engineering, Electronics and Communication Department, Cairo University. Currently, she is a Postdoctoral Researcher in the LAAS-CNRS. Her fields of interest are microstrip components and antennas, electromagnetic bandgap and defected ground structures, reconfigurable structures and ferroelectric materials.

Fabio Coccetti (M’10) as born in Italy, in 1972. He received the Laurea degree from the University of Perugia, Perugia, Italy and the Ph.D. degree from the Technische Universität München, Germany, in 1999 and 2004, respectively. From February to July 2000, he was a Visiting Scientist at the Radiation Lab, University of Michigan. During 2001 and 2002, he worked jointly on a design and fabrication project with German companies in the area of microelectromechanical systems (RFMEMS). Since September 2004, he has been working as a Research Scientist at the Laboratoire d’Analyse et d’Architectures des Systèmes, Centre National de la Recherche Scientifique (LAAS-CNRS), Toulouse, France. During this period, he has been working on numerous research projects including the theoretical and experimental investigation of power handling in RF-MEMS and the electromagnetic design and modeling of RF microsystems. He has held the position of Coordinator for the European Network of Excellence on “RF-MEMS and RF Microsystems” (AMICOM NoE) and is a member of the ARRRO Consortium, a specific support action aiming to define the RF-MEMS

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and RF-NEMS technology roadmap. Both projects were funded by the European Commission under the 6th Framework Program. His research interests include design, modeling and fabrication and of RF MEMS based circuits, numerical electromagnetic and multiphysics modeling of distributed microwave and millimeter wave networks.

Robert Plana (SM’08) was born in Toulouse, France, in March 1964. He received the Ph.D. degree from the Laboratoire d’Analyse et d’Architectures des Systèmes, Centre National de la Recherche Scientifique (LAAS-CNRS) and Paul Sabatier University, Toulouse, France, in 1993. His dissertation was on the “Noise modeling and characterization of advanced microwave devices (HEMT, PHEMT and HBT) that includes the reliability.” In 1993, he was an Associate Professor at LAASCNRS, and started a new research area concerning the investigation of millimeterwave capabilities of Silicon based technologies. More precisely, he has focused on the microwave and millimeterwave properties of SiGe devices and their capabilities for low noise circuits. In 1995, he started a new project concerning the improvement of the passives on silicon through the use of MEMS technologies. In 1999, he was involved with SiGe Semi-conductor in Ottawa, Canada, where he was working on the low power and low noise integrated circuits for RF applications. In the same year, he received a special award from CNRS for his work on Silicon based technologies for millimeterwave communications. Since 2000, he has been a Professor at Paul Sabatier University and Institut Universitaire de France, he started a research team at LAAS-CNRS in the field of micro and nanosystem for RF and millimeterwave communications. His main interests are on the technology, design, modelling, test, characterization and reliability of RF MEMS for low noise and high power millimeterwave applications and the development of the MEMS IC concept for smart microsystem. He built a network of excellence in Europe, “AMICOM,” regrouping 25 research groups. He has authored and coauthored more than 300 international journals and conferences. In 2004, he was appointed Deputy Director of the Information and Communication Department at CNRS

Headquarters. From January 2005 to January 2006, he was Director of the Information and Communication Department at CNRS. Since 2006, he has headed a research group at LAAS-CNRS in the field of micro and nanosystem for wireless communications. From November 2007 to November 2009, he was with the “French Research Agency” where he was the Project Officer of the National Nanotechnology Initiative. In November 2009, he was appointed head of the Department of Mathematics, Nanosciences and Nanotechnology, Information and Communication Technology at the Ministry of Research where he is in charge of defining the French strategy for research and innovation.

Mostafa El-Said was born in Cairo, Egypt, in May 1942. He graduated from Cairo University, in 1963, and received the Dipl.Ing. and Dr.Ing. degrees from Karlsruha University, West Germany, in 1970 and 1974, respectively. From 1975 to 1981, he worked as an Assistant Professor at Cairo University, and from 1981 to 1986, he was an expert at Riyad University, Saudi Arabia. Since 1992, he has been a Professor at Cairo University. His fields of interest are microwave engineering, radiation and circuits.

Essam A. Hashish (M’96) received the B.Sc., M.Sc., and the Ph.D. degrees from Cairo University, Cairo, Egypt, in 1973, 1977, and 1985, respectively. He is now an acting Professor in the Antennas and Propagation Group, Department of Electronics and Communications, Faculty of Engineering, Cairo University. His main interest is electromagnetic remote sensing, wave propagation and microwave antennas.

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Artificial Impedance Surfaces for Reduced Dispersion in Antenna Feeding Systems George Goussetis, Member, IEEE, José Luis Gómez-Tornero, Member, IEEE, Alexandros P. Feresidis, Senior Member, IEEE, and Nikolaos K. Uzunoglu, Fellow, IEEE

Abstract—An impedance surface is presented that reduces the dispersion experienced upon propagation of broadband pulses within rectangular waveguides. The surface impedance is selected so that, within a frequency range, the transverse resonance condition is satisfied for longitudinal wavenumber that varies linearly with frequency. A synthesis procedure for practical surface topologies consisting of periodic dipole arrays is described. An example involving a finite structure is employed to illustrate the reduced dispersion. Numerical simulation results obtained from in-house mode-matching method as well as HFSS are presented. A prototype is fabricated and tested experimentally validating the theoretical predictions. Index Terms—Artificial surfaces, dispersion compensation, frequency selective surfaces, pulse transmission, surface impedance, waveguide dispersion.

I. INTRODUCTION HE dispersive nature of hollow waveguides is often an unwanted propagation characteristic. Transmitting pulses of a short duration through a finite length of a waveguide, results in broadening of the pulse profile in the time domain due to dispersion, as illustrated in the schematic of Fig. 1. This poses problems such as degradation of signal quality in wideband and UWB systems and reduction of the spatial resolution in radar systems. Yet hollow waveguides are often preferred in the feeding network of many antenna systems due to their excellent signal confinement, low losses and power handling capability. The dispersion in hollow waveguides arises as a result of the cutoff frequency associated with each waveguide mode in conjunction with the requirement for single mode operation. The latter imposes operation close to the modal cutoff, where the waveguide is strongly dispersive.

T

Manuscript received October 13, 2009; revised March 02, 2010; accepted May 03, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The work of G. Goussetis was supported by the Royal Academy of Engineering under a five-year research fellowship. G. Goussetis is with the Institute of Electronics Communications and Information Technology, Queen’s University Belfast, BT3 9DT, U.K. (e-mail: [email protected]). J. L. Gómez-Tornero is with the Department of Communication and Information Technologies, Technical University of Cartagena, Cartagena 30202 Spain (e-mail: [email protected]). A. P. Feresidis is with the Wireless Communications Research Group, Department of Electronic and Electrical Engineering, Loughborough University, Leicestershire LE11 3TU, U.K. (e-mail: [email protected]). N. K. Uzunoglu is with School of Electrical and Computer Engineering, National Technical University of Athens, Zografou 15773, 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.2010.2071358

Fig. 1. Schematic of a horn antenna fed by a waveguide.

Engineered surfaces, whose interaction with electromagnetic waves can be tailored and described by equivalent homogenized parameters, such as generalized sheet transition conditions [1], averaged boundary conditions [2] or equivalent impedance/admittance [3] are recently attracting increased attention. Their practical realization typically involves 2D periodic arrangements of scatterers that often resembles frequency selective surfaces [4], although the term is more general to account for various geometrical configurations and functionalities. For example, a periodic array grounded with vertical vias has been proposed to produce an equivalent high surface impedance for both surface and normally incident plane waves [5]. Similar functions were also demonstrated for a via-free structure [6] as well as a ground plane with densely populated vias in the absence of the planar array [7]. Theoretical investigations of artificial materials (metamaterials) in metallic waveguide housing have predicted several unusual characteristics [8] that allow for increased design flexibility. Several practical realizations of waveguides loaded with engineered surfaces have also been reported in the literature to produce unusual characteristics for tangible applications. In [9], a uniplanar photonic band gap structure was employed to produce a quasi-TEM rectangular waveguide. The authors of [10] predicted that a waveguide below cutoff will support left-handed propagation when loaded with split rings along the E-plane, which was later demonstrated in [11]–[13], leading to miniaturized waveguides. In [14], [15], a surface consisting of a doubly periodic dipole array was introduced between two parallel metallic plates to produce compact cavity resonators, while in [16] one of the two parallel plates was substituted by a doubly periodic dipole array to produce a directive leaky wave antenna. Combination of these concepts has led to subwavelength high-gain antennas [17], [18]. Electromagnetic and photonic band gap structures have also been demonstrated to reduce unwanted dispersion in waveguides for microwave [19], [20] and optical [21] applications.

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and are the susceptances seen to the left and right where plane, given by from the (2) Numerical solution of the transverse resonance eigenvalue (1) together with the separation of variables regarding the propagation in the - and -directions (no variation along the -axis for the lowest order mode) (3) yields a numerical solution for the well known dispersion relamode in rectangular waveguides. tion of the along the Next, assume an equivalent shunt susceptance symmetry plane of the waveguide cross-section, as shown in can vary with both the frequency Fig. 2(c). In general, and the transverse (with respect to the surface) wavenumber , . In this case the transverse resso that we can write onance condition (1) can be rewritten as Fig. 2. Rectangular waveguide with an impedance surface along the E-plane: schematic representation (a). Transmission line and lumped representation of the equivalent transverse resonance circuit of unloaded waveguide (b) and waveguide loaded with an impedance surface (c).

In this paper, we extend on previous work [19], [20] suggesting the use of a practical impedance surface in order to reduce the unwanted dispersion of hollow rectangular waveguides commonly utilized as antenna feeds. We propose a synthesis procedure for a practical impedance (admittance) surface that when introduced transversely in the waveguide reduces the dispersion. In particular, we employ the transverse resonance method in order to extract the required admittance characteristics that would result in dispersion-free propagation if introduced along the E-plane of a rectangular waveguide (Fig. 2(a)). We then proceed to outline a synthesis procedure for practical surfaces that within a frequency range reproduce the required equivalent sheet admittance. Numerical results from in-house mode-matching code corroborated by commercial finite element method software as well as experimental measurements are presented for validation and demonstration of the reduced dispersion. II. SURFACE IMPEDANCE FOR DISPERSION-FREE WAVEGUIDE A. Transverse Resonance

(4) In principle, can be realized by an impedance surface located along the E-plane of the waveguide, as shown schematically in Fig. 2(a). By suitably choosing the equivalent admittance of this surface, we can synthesize the dispersion properties of the loaded waveguide. In particular we are interested in a propagation constant that varies linearly with frequency, which can be described by (5) where is the free-space wavenumber. The group velocity in the loaded waveguide is equal to , where is the speed of and in (5) are light in vacuum. The conditions necessary to ensure compatibility with causality and the positive cutoff frequency in a waveguide respectively. In order to satisfy the transverse resonance condition (4) for the linear dispersion relation given in (5), the required equivalent susceptance of the , should be surface in Fig. 2(a), (6) where the wavenumber normal to the surface

satisfies (7)

Note that fast wave propagation occurs along the loaded guide when

Propagation in a hollow rectangular waveguide can be conveniently analyzed employing a transverse resonance circuit such as the one shown in Fig. 2(b). Lowest order mode propagation for the conventional waveguide suggests that the waveguide cross-section can be viewed as a TE transmission line short-circuited at both ends (Fig. 2(b)). For the lossless case, propagation at a given frequency occurs for transverse wavenumbers that satisfy the transverse resonance condition [22]:

According to the bouncing plane wave model of waveguide propagation [17], in this regime the impedance surface experiences TE incident waves at an angle that satisfies (see Fig. 2(a))

(1)

(9)

(8)

GOUSSETIS et al.: ARTIFICIAL IMPEDANCE SURFACES FOR REDUCED DISPERSION IN ANTENNA FEEDING SYSTEMS

Fig. 3. Light line (dashed line) and dispersion relation of the conventional x-band waveguide (dotted line). Dispersion of an ideal dispersion free wave: and B = (dash-dotted line). Disguide satisfying (5) with A persion of the waveguide loaded with a practical impedance surface consisting : of a periodic dipole array described in Section III (Fig. 6) with h and d as obtained with transverse resonance (solid line) and HFSS (crosses).

=17

= 4 mm

= 145rad m

= 7 16 mm

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Fig. 4. Left y-axis: incidence angle (real part) on the impedance surface as theoretically calculated by the transverse resonance equation (4). Right y-axis: reflection coefficient (TE) from the theoretically required impedance surface (R , dashed line). The reflection coefficient from a practical surface described in Section III (shown in Fig. 6) with h : and d as calculated from the pole and zero function (0 , dotted line) and the full wave MoM results , solid line, shown only within the real angle of incidence range) are also (0 superimposed on the right y-axis.

= 7 16mm

= 4mm

At the frequency where the left hand side of (8) is equal to zero, is equal to 90 and the waveguide propagation constant, , is equal to the free-space propagation constant, . At higher frequencies, is complex and the waveguide supports slow waves, associated with the surface modes of the impedance surface. B. Dispersion-Free Waveguiding and Surface Admittance In this section, by means of an example, we derive the properties of a fictitious impedance surface required to produce a non-dispersive waveguide when loaded in the configuration of Fig. 2. Following [19], [20], we concentrate in the range between 7 GHz and 11 GHz and commence by selecting a standard X-band waveguide housing ( , ). are selected taking into account the The parameters and group delay requirements, thermal dissipation and realizability. For a practical impedance surface, Foster’s reactance theorem imposes that the equivalent susceptance can only increase with frequency [22]. In addition, higher values of correspond to reduced group velocity and in turn increased time average energy stored in the periodic structure [22], suggesting higher dissipation. The above considerations should be taken into account . Next, when selecting and . In this example we set considering realization constraints we select the cutoff param, so that an x-band waveeter, , here set to guide housing is suitable for single mode operation. The linear dispersion relation obtained by (5) for these values of and is shown in Fig. 3, where the dispersion of the X-band waveguide and the light line are also shown for comparison. The linear dispersion relation crosses the light line at 9.89 GHz. The angle of incidence, , (real part) of TE incident plane waves experienced by the impedance surface as calculated by (9) is plotted against frequency in Fig. 4 and represents a physical angle only up to 9.89 GHz. Assuming a loaded waveguide characterized by this linear plane to the dispersion, the susceptances seen from the

=0

Fig. 5. The susceptances seen to the left and right hand side of the x plane Y , dash-dotted line) and the required admittance of of the waveguide (Y the surface (Y dashed line) for a loaded X-band waveguide (Fig. 1) yielding the linear dispersion of Fig. 2. The susceptance of a practical impedance surface described in Section III (Fig. 6) with h : and d (Y , solid line). The transverse resonance summations (4) are shown with blue dashed and black dotted line respectively.

=

= 7 16mm

= 4 mm

right and left hand side ( and respectively) are shown in Fig. 5 (dash-dotted line). Fig. 5 also depicts the shunt sus, , to identically satisfy the ceptance required at transverse resonance condition (4) for the chosen linear dispersion. The transverse resonance summation (4) is also marked (dash-dotted line) in Fig. 5, and as expected is identically zero. In reading Fig. 5, attention is drawn to the fact that each frequency point implies a defined wavenumber , which is given by (5). The reflection coefficient experienced by TE waves incident on the fictitious impedance surface can be readily derived by

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the required susceptance, , (Fig. 5) and the longitudinal wavenumber, , (Fig. 2) according to (10) where is the TE susceptance of free space [18]. The magfor the example under consideration is shown nitude of in Fig. 4 with dashed line. Note that the reflection coefficient shown in Fig. 4 has a physical meaning only for frequencies that correspond to a purely real angle of incidence (fast waves). For this example, this corresponds to frequencies below 9.89 GHz. III. IMPEDANCE SURFACE REALIZATION Practical impedance surfaces are often realized as 2D periodic metallodielectric planar arrays, such as those employed to produce frequency selective surfaces [4]. These structures are typically characterized by equivalent sheet susceptances that vary monotonically with frequency. Since the required suscep, exhibits a minimum (Fig. 5), a surface whose equivtance, , matches at all frequencies is not alent susceptance, feasible. However, it is possible to design a complex surface matches within a finite frequency range. The so that waveguide will then produce a quasi-linear dispersion within that frequency range. In accordance to the example of the previous section, here we specify a frequency range between 7 GHz and 11 GHz, where the required susceptance of the complex surface increases monotonically with frequency. In general, it is possible to select the dimensions of the waveguide as well as parameters of the dispersion relation so that the the and varies monotonically with frequency required susceptance within the range of interest. A. Physical Layout or the Impedance Surface To maintain the simplicity of the structure, we propose to realize the impedance surface by a free-standing dipole array. The initial structure therefore consists of a rectangular waveguide that is periodically loaded in the E-plane with symmetrically located metallic dipole elements (Fig. 6). The geometrical symmetry of the structure for a TE10 waveguide mode suggests an electric symmetry along the H-plane and a magnetic symmetry along the E-plane of the waveguide (Fig. 6). The former ensures that the structure is electromagnetically equivalent with an E-plane waveguide periodically loaded with ridges [23], as shown in Fig. 6. By virtue of this transformation, the need to support the otherwise free-standing dipoles is removed and the structure is compatible with the well-established all-metal insert split-block housing E-plane technology [23]. Furthermore, image theory suggests that the structure is also equivalent to an infinite parallel plate waveguide loaded with a planar periodic array of metal strips, as in [14]. Therefore the linear array of dipoles can be modeled by a doubly periodic array of strips, such as those considered in [3]. The degrees of freedom associated with the design of a planar periodic array of metal strips are three, namely the length, , and width, , of the dipole as well as the longitudinal periodicity, , along the -axis (Fig. 6). The transverse periodicity (along the y-axis) is fixed by the waveguide height, , and equal to the height, , of the waveguide, since for TE modes the metallic

Fig. 6. Layout of the waveguide with a dipole impedance surface and an equivalent transformation using the electric symmetry into a more practical configuration. ES and MS stand for the electric and magnetic symmetry planes assuming TE10 incidence.

walls at (Fig. 2) act as symmetry planes. Provided that it is significantly smaller than the length, the width, , of the dipoles only weakly affects the electromagnetic response of the periodic array. We therefore fix it to a practical value, in this case equal to 2 mm. The synthesis problem therefore reduces to plane. a two-dimensional search in the B. Synthesis of the Impedance Surface Next, is the synthesis of the physical dimensions of a planar , periodic array of dipoles, whose equivalent susceptance, , shown in Fig. 5 within the frematches the required one, quency range of interest. As discussed above, each frequency defined by the point in Fig. 4 corresponds to a wavenumber dispersion relation (5), which is shown in Fig. 3 (dash-dotted is to reproduce the line). Therefore the design target for values of the required susceptance along the specific path plane. Although the scattering of of interest within the electromagnetic waves from doubly periodic planar metallodielectric structures has been treated extensively in the literature (e.g. [2]–[4], [24]), a synthesis procedure is not available. Here we propose a two-step synthesis process. The first step of the proposed synthesis technique is based on the fast wave regime, corresponding to real angles of incidence on the impedance surface. The aim is to narrow down the values to those that reproduce the required repossible for real angles of incidence along the flection coefficient path of interest. To this end, we employ a fast and efficient solver based on the method of moments (MoM) to calculate the reflection coefficient of TE waves by free-space arrays for frequencies in the with varying length and periodicity range 7 GHz to 9.89 GHz and angles of incidence determined by the dash-dotted curve in Fig. 4. In order to compare the re, sponse from the array with dipole length and periodicity , with the required reflection coefficient , we use the cost function (11) path shown in Fig. 4 and for i.e. the summation over the frequencies between 7 GHz and 9.89 GHz of the absolute value of the reflection coefficient obtained for each pair of ,

GOUSSETIS et al.: ARTIFICIAL IMPEDANCE SURFACES FOR REDUCED DISPERSION IN ANTENNA FEEDING SYSTEMS

O( )

Fig. 7. Value of the objective function h; d given by equation (10) for impedance surfaces consisting of periodic arrays of metallic strips with varying strip length, h, and periodicity, d (as defined in Fig. 6).

, reduced at each point by the theoretically required . Minima of the cost function (10) suggest value, possible optimum solutions. A plot of versus variables is shown in Fig. 7. The line of minima corresponds to the array geometries whose equivalent susceptance converges to the , in the fast wave regime. required, Fig. 7 provides no information for the admittance of the complex surface above 9.89 GHz, i.e. in the slow wave (complex angle of incidence) regime. At the second step of the synthesis process, the optimum dimensions among those determined initially (line of minima in Fig. 7) are selected. To this end the surface admittance in the slow wave regime is considered. A semianalytical method termed as “pole-zero method”, proposed in [3], is employed to extract the surface admittance. The “polezero method” is based on approximating the surface admittance as a ratio of polynomials. This reduces the task of estimating the equivalent admittance, into estimating a set of poles and zeros. In [3], this is achieved by extracting the poles and zeros for few real angles of incidence, interpolating those using a polynomial function and subsequently use this polynomial to extrapolate the poles and zeros in the slow wave regime. The method is well described in [3], and it is employed here directly. Employing the pole and zero technique, we proceed to derive the admittance for arrays with dimensions along the line of minima in Fig. 7 for the remaining frequencies (between 9.89 GHz up to 11 GHz) and wavenumbers determined by the given dispersion relation (5). MoM was used for modeling the reflection characteristics of the arrays, as in [3]. The optimum , surface design is the one whose equivalent susceptance, agrees best with the required one, , in the frequency range of interest (here 7 GHz to 11 GHz). Although the computational is in general greater in effort for each pair of dimensions the second step of the design procedure, significantly the search plane. Further now is limited to one dimension in the simplification during this optimization procedure arises upon consideration of the physical significance of susceptance poles. These correspond to resonances of the arrays, which in turn are associated with their surface wave electromagnetic band gap [6].

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Emergence of the band gap at frequencies close to the range of interest is unwanted, since these are associated with highly non-linear dispersion. Instead, susceptance poles should appear at frequencies higher than those of interest. Recognizing that shorter dipoles produce resonances at higher frequencies, the search during this step is preferably commencing from dipoles with reduced length, . The extracted susceptance for a periodic array with dipole and periodicity is shown in length ). As shown, beyond 7 GHz it conFig. 5 (black solid line, . The transverse resonance summaverges to the required, tion is shown with black dotted line and is approximately zero above 7 GHz. The reflection coefficient as obtained and with MoM is shown in Fig. 4 with black solid line is restricted to the fast wave regime, up to 9.89 GHz. The reflection coefficient as obtained from the pole and zero admittance across the real and complex angle of incidence domain is also and is in agreement with the theoretically shown in Fig. 4 required value beyond 9.89 GHz. The dispersion of the loaded waveguide as obtained from transverse resonance employing the synthesized impedance surface is shown in Fig. 3 (black continuous line). Full wave results obtained with HFSS, [25], are also shown in Fig. 3 with crosses to validate this dispersion. As it can be seen in Fig. 3, very good agreement is observed between HFSS and transverse resonance, validating that the waveguide loaded with this surface is nearly dispersion-free within the frequencies 7 GHz to 11 GHz. IV. NUMERICAL AND EXPERIMENTAL RESULTS In this section by means of a practical example we demonstrate the reduced dispersion in a waveguide of finite length loaded with the impedance surface designed above. Initially, a short section of a waveguide loaded with the optimised impedance surface is matched to a rectangular waveguide. An in-house code based on mode-matching method [23] was employed for this purpose. Matching is achieved by gradually reducing the length of the dipoles toward the input and output ports, in order to achieve acceptable reflection levels. To validate the performance of the tapered matching, Fig. 8 shows the response for a loaded waveguide with total length 7.8 cm as obtained with the in-house code as well as HFSS. In this example 6 unit cells with linearly reduced length have been employed for the matching section on either end of the loaded waveguide. A prototype was fabricated (shown as inset in Fig. 8) and tested. The experimental results are also shown in Fig. 8. The good agreement between the simulations and the measurements validates the accuracy of the modeling tools. In particular, Fig. 8(a) shows the magnitude of the reflection and transmission coefficient, being the former largely below the level of 20 dB within the frequency range of interest. Fig. 8(b) shows the unfolded phase of the transmission coefficient for this loaded waveguide. In order to evaluate the effect of the transition section, the unfolded phase for a loaded waveguide of the same length but without the matching section, as calculated assuming the propagation constant of the infinite case (dispersion relation of Fig. 3), is shown for comparison. The unfolded phase for an equal length of an unloaded X-band waveguide is also shown for comparison. As shown, the matching section slightly

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Fig. 8. Fabricated prototype (shown as inset) loaded with the optimised impedance surface (h : , ) of total length 78 mm. (a) Measured and simulated (in-house mode matching as well as FEM [22]) magnitude of and . (b) Measured and simulated unfolded transmission (unfolded phase for transmission along 78 mm of unloaded phase of x-band waveguide shown for comparison).

S

S

= 7 16 mm d = 4 mm S

reduces the linearity for this example, which is still considerably improved compared to the unloaded waveguide. For longer waveguides sections, as discussed below, the perturbation from the matching section will be further reduced. Next, a loaded waveguide of total length 84.6 cm is considered. The finite structure consists of 192 unit cells of the dipole surface and a tapered matching section of 10 cells with linearly reducing length at input and output respectively. For the modeling of the complete structure the mode matching based tool has been employed, whose accuracy has been validated above. The simulated magnitudes of the transmission and reflection coefficient for this prototype are depicted in Fig. 9(a). As shown, is below the level of 20 dB above 7 GHz. In order to quantify the dispersion performance, Fig. 9(b) shows the group velocity for this finite section of loaded waveguide (including the transition). For comparison, the group velocity of an unloaded X-band waveguide is also shown. The reduced dispersion is evidenced by the significantly reduced variation in the group velocity, from 70% in the unloaded case to 20% in the loaded case. The reduced waveguide dispersion realized by virtue of the impedance surface loading is demonstrated for a pulse with a carrier at 9 GHz and modulated by a Gaussian profile with vari. The spectrum of the pulse is depicted in ance Fig. 10(a), which shows that more than 90% of the total power content of this pulse is above 8 GHz. The time domain representation of the pulse during propagation along 84.6 cm of X-band

Fig. 9. Matched waveguide loaded with the optimised impedance surface (h = 7:16 mm, d = 4 mm) and total length 846 mm (a) simulated (in-house mode matching) reflection and transmission coefficient (d) group velocity (averaged). For comparison the group velocity of an unloaded x-band waveguide is also shown. TABLE I PULSE CHARACTERISTICS UPON PROPAGATION ALONG 846 mm

waveguide as well as the waveguide of Fig. 9 is demonstrated in Fig. 10(b) and 10(c). The pulse at input is also shown for comparison. Table I summarizes the main characteristics of the pulse following propagation along 84.6 cm for the two cases. For comparison, Table I also includes the cases of free-space mode propagation in a custom rectpropagation as well as angular waveguide with cutoff at 5.6 GHz (equal to the cutoff of the loaded waveguide). As shown, the proposed impedance surface loaded waveguide of Fig. 9 maintains nearly 80% of the power initially within a 400 ps window compared to 42% for the X-band and 67% for the custom rectangular waveguide. This is also reflected in the peak magnitude of the pulse after propagation. On the other hand, there is an increase of about 14% in the group delay in the loaded waveguide compared to the hollow case, due to the reduced group velocity.

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Fig. 10. Spectrum of pulse under consideration (a) and time domain representation of the pulse at the input (dashed line in b, c). The pulse at the output of 84.6 cm length of hollow x-band waveguide (b) and the loaded waveguide of Fig. 9(c).

V. CONCLUSION A technique to reduce the dispersion of rectangular waveguides by means of loading them with an artificial impedance surface has been presented. Based on the transverse resonance technique, the equivalent admittance of the surface required to produce linear dispersion has been rigorously obtained. An efficient and generic synthesis technique has been proposed for the design of practical impedance surfaces consisting of arrays of periodic strips producing the required admittance. Numerical and experimental results have been presented that demonstrate the reduction of waveguide dispersion. REFERENCES [1] E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2641–2651, Oct. 2003. [2] S. Tretyakov, Analytical Modeling in Applied Electromagnetics. Norwood, MA: Artech House, 2003. [3] S. Maci, M. Caiazzo, A. Cucini, and M. Casaletti, “A pole-zero matching method for EBG surfaces composed of a dipole FSS printed on a grounded dielectric slab,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 70–81, Jan. 2005. [4] B. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [5] D. Sievenpiper, Z. Lijun, R. F. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [6] G. Goussetis, A. P. Feresidis, and J. C. Vardaxoglou, “Tailoring the AMC and EBG characteristics of periodic metallic arrays printed on grounded on grounded dielectric substrate,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 82–89, Jan. 2006. [7] M. G. Silveirinha, C. A. Fernandes, and J. R. Costa, “Electromagnetic characterization of textured surfaces formed by metallic pins,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 405–415, Feb. 2008. [8] A. Alu and N. Engheta, “Guided modes in a waveguide filled with a pair of single negative (SNG), double negative (DNG) and/or double positive (DPS) layers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 199–210, Jan. 2004. [9] F.-R. Yang, K.-P. Ma, Y. Qian, and T. Itoh, “A novel TEM waveguide using Uniplanar Compact Photonic-Bandgap (UC-PBG) structure,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2092–2098, Nov. 1999. [10] R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in sub-wavelength split-ring resonator loaded metallic waveguides,” Phys. Rev. Lett., vol. 89, no. 18, Oct. 28, 2002. [11] S. Hrabar, J. Bartolic, and Z. Sipus, “Waveguide miniaturization using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 110–119, Jan. 2005.

[12] J. Carbonell, L. J. Rogla, V. E. Boria, and D. Lippens, “Design and experimental verification of backward-wave propagation in periodic waveguide structures,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1527–1533, Jun. 2006. [13] E. Rajo-Iglesias, O. Quevedo-Teruel, and M. N. M. Kehn, “Multiband SRR loaded rectangular waveguide,” IEEE Trans Antennas Propag., vol. 57, no. 5, pp. 1571–1975, May 2009. [14] M. Caiazzo, S. Maci, and N. Engheta, “A metamaterial surface for compact cavity resonators,” IEEE Antennas Wireless Propag. Lett, vol. 3, pp. 261–264, 2004. [15] G. Goussetis, A. P. Feresidis, and R. Cheung, “Quality factor assessment of subwavelength cavities at FIR frequencies,” J. Optics A, vol. 9, pp. s355–s360, Aug. 2007. [16] P. Kosmas, A. P. Feresidis, and G. Goussetis, “Periodic FDTD analysis of a 2-D leaky-wave planar antenna based on dipole frequency selective surfaces,” IEEE Trans. Antenna Propag, vol. 55, no. 7, pp. 2006–2012, Jul. 2007. [17] A. Feresidis, G. Goussetis, S. Wang, and J. C. Vardaxoglou, “Artificial magnetic conductor surfaces and their application to low profile highgain planar antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 209–215, Jan. 2005. [18] S. Wang, A. P. Feresidis, G. Goussetis, and J. C. Vardaxoglou, “Highgain subwavelength resonant cavity antennas based on metamaterial ground planes,” Proc. Inst. Elect. Eng. Antennas Propag., vol. 153, no. 1, pp. 1–6, Feb. 2006. [19] G. Goussetis, N. Uzunoglou, J.-L. Gomez-Tornero, B. Gimeno, and V. E. Boria, “An E-plane EBG waveguide for dispersion compensated transmission of short pulses,” presented at the IEEE Antenna and Propagation Symp., Honolulu, Jun. 9–15, 2007. [20] G. Goussetis, C. Mateo-Segura, J.-L. Gomez-Tornero, and N. Uzunoglu, “Waveguide loaded with metamaterial surface for dispersion linearization: Numerical and experimental results,” presented at the IEEE Antennas and Propagation Symp., San Diego, Jul. 7–12, 2008. [21] Y. S. Chen, Y. Zhao, A. Hosseini, D. Kwong, W. Jiang, S. R. Bank, E. Tutuc, and R. T. Chen, “Delay-time-enhanced flat-band photonic crystal waveguides with capsule-shaped holes on silicon nanomembrane,” IEEE J. Sel. Topics Quant. Electron., vol. 15, pp. 1510–1514, 2009. [22] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: Wiley-IEEE press, 2000. [23] D. Pozar, Microwave Engineering. Boston, MA: Addison-Wesley, 1993. [24] G. Goussetis, A. P. Feresidis, and P. Kosmas, “Efficient analysis, design and filter applications of EBG waveguide with periodic resonant loads,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 11, pp. 3885–3892, Nov. 2006. [25] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Raisanen, and S. A. 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, June 2008. [26] “High Frequency Structure Simulator,” ver. 11, Ansoft Corporation.

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George Goussetis (S’99–M’02) received the Electrical and Computer Engineering degree from the National Technical University of Athens, Athens, Greece, in 1998, the Ph.D. degree from the University of Westminster, Westminster, U.K. and the B.Sc. degree in physics (first class) from University College London (UCL), London, U.K., in 2002. In 1998, he joined the Space Engineering, Rome, Italy, as Junior RF Engineer and in 1999 the Wireless Communications Research Group, University of Westminster, as a Research Assistant. Between 2002 and 2006 he was a Senior Research Fellow at Loughborough University, UK. Between 2006 and 2009 he was a Lecturer (Assistant Professor) with the School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, U.K. He joined the Institute of Electronics Communications and Information Technology at Queen’s University Belfast, U.K, in September 2009 as a Reader (Associate Professor). He has authored or coauthored over 100 peer-reviewed papers three book chapters and two patents. His research interests include the modeling and design of microwave filters, frequency-selective surfaces and EBG structures, microwave heating as well numerical techniques for electromagnetics. Dr. Goussetis received the Onassis foundation scholarship in 2001. In October 2006, he was awarded a five-year research fellowship by the Royal Academy of Engineering, U.K.

José Luis Gómez-Tornero (M’06) was born in Murcia, Spain, in 1977. He received the Telecommunications Engineer degree from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 2001, and the “laurea cum laude” Ph.D. degree in telecommunication engineering from the Technical University of Cartagena (UPCT), Cartagena, Spain, in 2005. In 1999, he joined the Radio Communications Department, UPV, as a research student, where he was involved in the development of analytical and numerical tools for the automated design of microwave filters in waveguide technology for space applications. In 2000, he joined the Radio Frequency Division, Industry Alcatel Espacio, Madrid, Spain, where he was involved with the development of microwave active circuits for telemetry, tracking and control (TTC) transponders for space applications. In 2001, he joined the Technical University of Cartagena (UPCT), Spain, as an Assistant Professor. From October 2005 to February 2009, he held the position of Vice Dean for Students and Lectures affairs in the Telecommunication Engineering Faculty at the UPCT. Since 2008, he has been an Associate Professor at the Department of Communication and Information Technologies, UPCT. His current research interests include analysis and design of leaky-wave antennas and the development of numerical methods for the analysis of novel passive radiating structures in planar and waveguide technologies. Dr. Gómez-Tornero received the national award from the EPSON-Ibérica Foundation for the Best Ph.D. Project in the field of Technology of Information and Communications in July 2004. In June 2006, he received the Vodafone foundation-COIT/AEIT (Colegio Oficial de Ingenieros de Telecomunicación) award for the Best Spanish Ph.D. thesis in the area of Advanced Mobile Communications Technologies. This thesis was also awarded in December 2006 as the Best Thesis in the area of Electrical Engineering, by the Technical University of Cartagena. In February 2010, he was appointed CSIRO Distinguished Visiting Scientist by the CSIRO ICT Centre, Sydney.

Alexandros P. Feresidis (S’98–M’01–SM’08) was born in Thessaloniki, Greece, in 1975. He received the Physics degree from Aristotle University of Thessaloniki, Greece, in 1997, the M.Sc.(Eng.) degree in radio communications and high frequency engineering from the University of Leeds, U.K., in 1998, and the Ph.D. degree in electronic and electrical engineering from Loughborough University, U.K., in 2002. During the first half of 2002, he was a Research Associate and in the same year he was appointed Lecturer in Wireless Communications in the Department of Electronic and Electrical Engineering, Loughborough University. In 2006, he was promoted to a Senior Lecturer in the same department. He has published more than 100 papers in peer reviewed international journals and conference proceedings and has coauthored three book chapters. His research interests include analysis and design of artificial periodic metamaterials, electromagnetic band gap (EBG) structures and frequency selective surfaces (FSS), high-gain and base station antennas, small/compact antennas, computational electromagnetics and passive microwave/mm-wave circuits.

Nikolaos K. Uzunoglu (M’82–SM’97–F’06) was born in Constantinople, Turkey, in 1951. He received the B.Sc. degree in electronics from the Technical University of Istanbul, Istanbul, Turkey, in 1973, and the Ph.D. degree from the University of Essex, Essex, U.K., in 1976. Since 1987, he has been a Professor with the School of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece. He has authored or coauthored over 300 papers in refereed international journals and three books. His research interests include electromagnetic scattering, propagation of electromagnetic waves, fiber-optics telecommunications, and biomedical engineering. Prof. Uzunoglu was the recipient of many honorary awards including the 1981 International G. Marconi Award in telecommunications.

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Multilayered Wideband Absorbers for Oblique Angle of Incidence Alireza Kazemzadeh and Anders Karlsson

Abstract—Design procedures of Jaumann and circuit analog absorbers are mostly formulated for normal angle of incidence. Only a few design methods considering oblique angle of incidence are published. The published methods are restricted to single resistive layer circuit analog absorbers or multilayered Jaumann absorbers with low permittivity spacers. General design procedures are developed in this paper for multilayered Jaumann and capacitive circuit absorbers. By expanding the scan and frequency compensation techniques to multilayered structures, Jaumann absorbers with outstanding performances are designed. A capacitive circuit absorber is presented with a stable frequency response up to 45 for both polarizations, having an ultrawide bandwidth of 26 GHz. Index Terms—Absorbing media, electromagnetic scattering by absorbing media, frequency selective surfaces.

I. INTRODUCTION BSORBERS are mounted on the surface of objects to reduce their radar cross sections. Dielectric absorbers are suitable candidates for reduction of specular reflections of large metallic objects. The simplest type of absorber called Salisbury screen [6] is formed by a single homogenous resistive sheet (at absorption frequency) in front of the perfectly placed conducting ground plane. By increasing the number of the resistive sheets, apart from each other (Jaumann absorber) larger bandwidths are achieved [4], [5]. Further improvement of the bandwidth is possible by replacing homogeneous resistive sheets with lossy band-stop frequency selective surface (FSS) sheets. This results in a complex sheet admittance that by proper design increases the bandwidth. Such absorbers are referred to as circuit analog absorbers [8]–[10]. In a recent publication we proposed the capacitive circuit absorber, a modification to the circuit analog absorber method that not only simplifies the design procedure but also leads to new applications [7]. One of the new applications is the design of ultrawideband absorber for oblique angle of incidence. This particular feature is explained and demonstrated in this paper. Jaumann and circuit analog absorbers are mostly investigated for normal angle of incidence. This simplifies the design procedure since the variation of the frequency response at oblique angles of incidence is not considered. Moreover, if symmetric array elements are used (e.g., crossed dipoles [10])

A

Manuscript received January 15, 2010; revised April 08, 2010; accepted May 09, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The authors are with the Department of Electrical and Information Technology, Lund University, Lund SE-221 00, 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.2010.2071366

the frequency response becomes polarization insensitive at normal angle of incidence. The situation is totally different when the absorber is designed for oblique angles of incidence. The frequency response shifts up in frequency and becomes polarization dependent [9], [10]. These unwanted features should be compensated as much as possible if the absorber is going to operate under large scan angles (usually up to 45 ) and different polarizations. Therefore, the designer faces constraints and requirements that are not present in the design of conventional absorbers. This complicates the design problem such that previous investigations described in literatures, preferred to consider only special cases that are simpler to solve. For the Jaumann absorber Chambers and Tennant have proposed a method of design based on genetic algorithm optimization [3]. It is a powerful approach that results in large bandwidths in comparison to the other published designs [9], [10]. Since the designs are based on genetic algorithm optimization and many degrees of freedom exist in a multilayered absorber, they limit the number of optimization variables. To speed up the optimization, the permittivity of the dielectric layers are considered to be very low (foam or honeycomb material) and fixed in their designs [3]. For normal angle of incidence foam or honeycomb materials are used extensively in wideband absorbers but they are not optimal choices for designs considering large scan angles. Munk et al. have explained and demonstrated that high permittivity dielectric layers are essential for scan and frequency compensations and increase of bandwidth [9], [10]. This paper modifies the Chambers and Tennant’s approach by suggesting a very accurate model for the equivalent impedance of a resistive sheet embedded in dielectric cover. It is shown that when proper dielectric layers are used, a two layered Jaumann absorber can offer almost the same bandwidth as the three layered optimized design of Chambers and Tennant [3]. The design approach of Munk et al. has magnificent compensation features but is limited to single resistive layer absorbers with moderate bandwidths (at most ) [9], [10]. If larger bandwidths are desired, the number of frequency selective surfaces must increase. Accurate modeling of the frequency selective surfaces (FSS) over a large bandwidth for different angles of incidence and polarizations, is a challenging problem. The variation of the resonant frequency, the bandwidth, the harmonic and anti-resonance frequencies with respect to polarization and angles of incidence must be considered in the model [7], [8], [10]. In addition, the process of finding proper FSS elements that are able to match the ground plane to the free space over a large bandwidth for different polarizations and incident angles is complicated. This paper provides a design tool for multilayered ultrawideband capacitive circuit absorbers [7]. These absorbers are designed by low-pass FSS elements instead

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Fig. 1. The schematic of a single resistive layer absorber. The resistive layer is a homogeneous sheet in the Jaumann absorber and a periodic array (FSS) in case of circuit analog absorber.

of band-stop resonating arrays and consequently do not suffer from the harmonics and anti-resonance problems that occur in circuit analog absorbers [9], [10]. An absorber is presented possessing the largest bandwidth among the published designs [1], [2], [8]–[10]. It is also shown that in contrast to Munk’s approach [9], [10] it is possible to have a wider range of selections for the permittivity of the dielectric layers in multilayered absorbers. This is essential in some applications where the mechanical and thermal properties of the absorbers are important. II. THE DESIGN METHOD OF MUNK FOR SINGLE RESISTIVE LAYER ABSORBERS The design of single resistive layer Jaumann and circuit analog absorbers are investigated by Munk et al. in [9], [10]. Since the number of resistive layers has been limited to one, it is expected that their approach cannot be applied to the general case of multilayered absorbers without modification. The significance of their investigation is the illustration of the important role that dielectric layers play in the performance of the absorber, when oblique angle of incidence is considered. They have introduced brilliant compensation techniques that are extended to multilayered absorbers in the following sections. The schematic of a single resistive sheet/FSS layer absorber is shown in Fig. 1. According to the design method of Munk et al., each dielectric layer and the resistive element have specific roles in the performance of the absorber. The first dielectric layer ( in Fig. 1) is responsible for frequency compensation. Upward shift in frequency is a general characteristic of the absorber response at oblique angles of incidence [9], [10]. Proper selection in Munk’s deof the permittivity of the first layer ( sign ) can stabilize the frequency response. The second dielectric layer ( in Fig. 1) is used for increasing the bandwidth and scan compensation. A proper selection of its permittivity is done according to the following relation [8]–[10]:

(1)

Fig. 2. The schematic of a two resistive layer Jaumann absorber. The resistive sheets are embedded in thin dielectric covers not shown in the figure. The ex, ). ternal skin is used in design No.2 (

= 3 Thickness = 30 mil

is the (maximum) angle of incidence in air. A 45 where angle of incidence corresponds to . The thicknesses of both layers ( , , see Fig. 1) are selected slightly larger than a quarter of a wavelength. The only remaining parameters to be determined are the resistivity of the sheet, and in case of the circuit analog absorber, the shape and dimensions of the FSS unit cell element. This is done by synthesizing the required sheet admittance that matches the ground plane to the free space for the given values of the dielectric permittivities and thicknesses. The details of the broadband matching technique are explained in [8]–[10]. Since the absorbers of Munk et al. are single resistive layer with few degrees of freedom, the role of each dielectric layer is predetermined. The permittivity of the layers cannot be selected arbitrarily, otherwise the performance degrades. The situation is completely different in multilayered absorbers. As our design examples demonstrate, there are some freedom in selection of the dielectric layers. This is essential in practical designs where the mechanical or thermal properties of the absorber are also important. The absorbers of Munk et al. have at most relative band) of 2–2.7 depending on polarization [9], [10]. To widths ( broaden the bandwidth ratio and stabilize it for different polarizations, more resistive layers are required. Systematic methods are presented in the following sections to design wideband multilayered absorbers for oblique angle of incidence. III. MULTILAYERED JAUMANN ABSORBER FOR OBLIQUE ANGLE OF INCIDENCE The Jaumann absorber consists of several homogenous resistive sheets separated by dielectric layers. An example is shown in Fig. 2. By proper selection of the resistivity of the sheets and the thicknesses and permittivities of the dielectric layers, power can be absorbed from the incident wave over a frequency range [4], [5], [8]. The design of the absorber is usually done for normal angle of incidence. This section presents a general method of design for multilayered Jaumann absorbers with

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wideband frequency responses for oblique angle of incidence. First, the pioneering work of Chambers and Tennant is studied and the shortcomings of their approach is explained. Then, their method is modified by providing accurate model for the impedance of a resistive sheet embedded in a dielectric cover. Different design examples are provided and important results are demonstrated by them. A. Generalization of the Chambers and Tennat’s Design Approach Chambers and Tennant have proposed a method of design for oblique angle of incidence based on optimization [3]. They have published design examples that are the most wideband absorbers among the published papers [9], [10]. These large bandwidths have been achieved in the cost of using many resistive layers. This results in absorbers with large total thicknesses. It is proved in this section that more efficient designs are possible. Since their approach is based on genetic algorithm optimization and there are many degrees of freedom in a multilayered absorber, they try to minimize the number of unknown variables. Based on experience from wideband designs done for normal angle of incidence, Chambers and Tennat have considered the permittivities of the dielectric layers fixed, with values ). very close to unity (foams or honeycomb materials, It was shown in the previous section that for oblique angles of incidence proper selection of the permittivity of the dielectric layers are vital for optimal performance of the absorber. The scan and frequency compensations require dielectric layers with higher permittivities than foams or honeycomb materials. It is illustrated that when proper dielectric layers are used, absorbers with fewer resistive sheets can offer the same bandwidths as the Chambers and Tennant’s designs. This clearly demonstrates the important role the permittivity of the dielectric layers play in the performance of the absorber. Since the models proposed by Chambers and Tennant are not applicable to multilayered dielectric absorbers with arbitrary values of permittivity, a more general model is required. Consider the schematic of a two resistive layer Jaumann absorber as shown in Fig. 2. It is well-known that the dielectric layers in a homogenous stratified medium can be modeled by equivalent transmission lines [11]. The length and the intrinsic impedance of the equivalent transmission lines are functions of polarization and angle of incidence. The free space surrounding the structure is also modeled by a port with proper impedance [11]. For the dielectric layer ( , ) the equivalent transmission line length ( ) and intrinsic impedance ( ) are obtained as follows: (2) (3) (4) (5) In the above formulas and are the free space intrinsic is the impedance and admittance, respectively. The angle

Fig. 3. Equivalent circuits models of a single homogeneous resistive sheet surrounded by free space. Only circuit (a) can be used in a general multilayered absorber. The circuit (b) has been used by Chambers and Tennant and is accurate with only low permittivity spacers.

angle of incidence and is the direction cosine in layer , calculated from the Snell’s law [11]. For the equivalent . port impedance one can use the above formulas with This results in for TE polarization and for TM. To complete the absorber analysis, an accurate model for the resistive sheets is required. Chambers and Tennant suggest the following relations between the sheet ( ) and the equivalent resistance ( ) for resistivity different polarizations [3]: (6) (7) These relations are valid if and only if the reference impedance (for calculating the reflection/transmission coefficient) is selected to be the free space intrinsic impedance. In other words, the equivalent port impedance is assumed to in their approach. This is in conflict be fixed and equal to with (4) and (5) for which the port impedance is a function of incident angle and polarization. To resolve the conflict two important questions must be answered. Why (6) and (7) do not result in perceptible errors in Chambers and Tennant’s designs? and How can these equations be generalized to multilayered structure with arbitrary values of permittivity? Consider the equivalent circuits of Fig. 3. Simple calculations show that the two circuits result in the same reflection/transmission response. If a single layer resistive sheet is considered in free space, each of the equivalent circuits of Fig. 3 can be used to model its frequency response. The parameter ’ ’ in Fig. 3 is equal to for TM polarization and for TE polarization. The relations of Chambers and Tennant ((6), (7)) correspond to the equivalent circuit model (b) in Fig. 3 with fixed port impedance. In multilayered structures with only low permittivity layers this model does not result in a perceptible error, but it can not be used in a general design with arbitrary permittivities. Therefore, the model with variable port impedance is used (part (a) in Fig. 3) in this paper. Consequently, (6) and (7) are no longer valid for the equivalent impedance of the resistive sheets and new accurate relations must be proposed. From now on, the homogenous resistive sheets, and later the periodic square arrays of the capacitive circuit absorbers, are all embedded in dielectric covers, see Figs. 4 and 10. This is usually a requirement from the fabrication points of view, but it also

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TABLE I THE THICKNESSES OF THE DIELECTRIC LAYERS FOR THE JAUMANN ABSORBERS

TABLE II THE PERMITTIVITIES OF THE DIELECTRIC LAYERS FOR THE JAUMANN ABSORBERS

Fig. 4. The schematic of the resistive sheet embedded in a thin dielectric cover and its accurate equivalent circuit model. l is the equivalent length (function of angle of incidence) and Z is the equivalent impedance (function of both polarization and angle of incidence).

simplifies the modeling of the resistive sheets (both homogenous and periodic patterns) significantly. Without the dielectric cover the equivalent impedance model has a complicated dependence on polarization and angle of incidence for periodic arrays, which makes modeling almost impossible. For homogenous resistive sheets embedded in a dielectric cover, the sheets can be modeled by the equivalent circuit in Fig. 4. The transmission line length ( ) and the intrinsic impedance ( ) of the equivalent circuit model, are calculated by (2)–(5). The resistance ( ) in this case has a simple relation to the sheet resistivity ( ). They are identical for different angles of incidence and polarizations. It is simply the generalization of the circuit model of part (a) in Fig. 3 for a single homogenous resistive sheet in free space, to the multilayered dielectric structures.

TABLE III THE RESISTIVITY OF THE SHEETS FOR THE JAUMANN ABSORBERS

Fig. 5. The frequency response of the Jaumann absorber (design No.1) for normal and oblique angles of incidence 45 .(Full-wave simulation).

B. Design Examples, Explanations and Comparisons The whole absorber of Fig. 2 can now be modeled by equivalent circuits for both polarizations and different angles of incidence. There exists no restriction on the permittivities of the layers. This enables us to look for optimal permittivities of the layers for scan and frequency compensations and bandwidth increase. Two different designs are presented. The aim is to demonstrate that there might be more than one solution, when multilayered absorbers are considered. This valuable result permits the designer to take other physical properties of the absorber into consideration besides the electrical properties. Also it shows that the design approach suggested by Munk et.al [9], [10] is not general and it is applicable only to single resistive layer absorber. Both of the designs are two resistive layer Jaumann absorbers. The schematic of the absorbers are the same (see Fig. 2) except , ) is used that the external skin ( in the second design. The thicknesses and permittivities of the spacers and sheet resistivity values of the designs are tabulated in Tables I–III. All the resistive sheets are embedded in dielectric and permittivity covers with thickness (see Fig. 4). The frequency response of the absorbers are shown in Figs. 5 and 6. Despite the employment of different dielectric layers in the designs, the absorbers have almost equal bandwidths. The pos) to sibility of applying a high permittivity external skin (

Fig. 6. The frequency response of the Jaumann absorber for the design No.2 , ).(Full-wave simulation). with external skin (

= 3 Thickness = 30 mil

the second design without degrading the performance is remarkable. According to the Munk’s formulation it is unattainable. This is expected since Munk’s approach is optimal if and only if single resistive layer absorbers are considered. In multilayered absorbers the whole structure takes care of the frequency and scan compensations. If a layer must have a certain value of permittivity or thickness (for example the external skin in the second design), the values of other layers and the resistivity of the sheets can be adjusted to compensate for the deteriorations introduced by the selection. The rule of thumb of the Munk’s

KAZEMZADEH AND KARLSSON: MULTILAYERED WIDEBAND ABSORBERS FOR OBLIQUE ANGLE OF INCIDENCE

Fig. 7. The frequency response of the Jaumann absorber circuit model (design No.1) at normal angle of incidence. The circled numbers correspond to the reflection coefficient seen at the locations marked in Fig. 2.

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Fig. 8. The frequency response of the Jaumann absorber circuit model (design No.1) for TE polarization, oblique angles of incidence 45 . The circled numbers correspond to the reflection coefficient seen at the locations marked in Fig. 2.

approach is optimal for single layer absorbers, but the ideas can be extended and used in multilayered absorbers. For example it can explain why our two resistive sheet absorbers have the same bandwidth as the three resistive sheet, genetic algorithm optimized absorber of Chambers and Tennant [3]. Their absorber is ) while in our made of low permittivity spacers only ( designs the dielectric layers are optimized to increase the bandwidth and to perform the scan and frequency compensations with fewer number of resistive sheets. The freedom in selecting the dielectric layers is also important from the application point of view. One application Chambers and Tennant are aiming for, is to suppress the waves entering aircraft engine ducts [3]. Due to high thermal and pressure shocks in the engine ducts, the mechanical and thermal properties of the absorber are important. This can not be fulfilled only by foams or honeycomb material used in Chambers and Tennant’s designs. C. A Detailed Illustration of the Wideband Matching Technique In multilayered absorbers the broadband matching of the ground-plane to free space is a complicated process. It must be done not only for the normal angle of incidence but also for different polarizations at oblique angle of incidence, simultaneously. In addition in multilayered absorbers, there are more steps of impedance transformation and addition of sheet admittances to perform, compared to the single resistive layer absorbers [9], [10]. It is instructive to have a look at the matching steps of a multilayered absorber at normal and oblique angles of incidence. For this purpose the Jaumann absorber of the first design (the one without the external skin) is selected. The matching phases that the absorber goes through are illustrated step by step in Smith charts. Three cases, the normal angle of incidence and the frequency response for the

Fig. 9. The frequency response of the Jaumann absorber circuit model (design No.1) for TM polarization, oblique angles of incidence 45 . The circled numbers correspond to the reflection coefficient seen at the locations marked in Fig. 2.

TE/TM polarizations at 45 (angle of incidence), are considered. The Smith charts of Figs. 7–9 represent the reflection coefficient at each step of the matching process. The circled numbers in Figs. 7–9 correspond to the reflection coefficient seen at the locations marked in Fig. 2. IV. CAPACITIVE CIRCUIT ABSORBER FOR OBLIQUE ANGLE OF INCIDENCE Homogenous resistive sheets are employed in Jaumann absorbers. If these sheets are replaced by proper lossy frequency

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selective surfaces, the complex admittance of the sheets can increase the bandwidth. Circuit analog absorbers are one class of FSS based absorbers, designed by band-stop resonating arrays [8]–[10]. Another important subgroup of FSS based absorbers are capacitive circuit absorbers [7]. In this class of absorbers the resonating elements are replaced by low-pass FSS arrays. This has several advantages and leads to new applications as explained in a recent publication [7]. Another new application of the proposed method is the ultrawideband absorber for oblique angle of incidence. This particular feature of the capacitive circuit absorber is explored in this section. It is explained why circuit analog absorbers are not proper candidates for ultrawideband designs under oblique angles of incidence and how to overcome the difficulties by capacitive circuit approach. A. Resonating FSS Elements, Difficulties and the Alternative Solution The main problem with the resonating structures is the harmonics of the fundamental resonance and the anti-resonance effects that limit the bandwidth of the absorber [9], [10]. By replacing the resonating elements by low-pass arrays this unwanted feature disappears in the whole frequency range of interest [7]. Moreover, the design of multilayered absorber capable of handling normal and oblique angles of incidence for both polarization is a complicated problem. It can not be done unless accurate models for the behavior of the equivalent impedance of the FSS elements are available. The equivalent impedance of the periodic array is a function of polarization and angle of incidence. For the resonating structures the resonating frequency and the bandwidth vary with angle of incidence and polarization. Also the harmonic and the anti-resonance frequencies are slightly shifted by polarization and incident angle [7]. Thus it is very complicated to provide an accurate circuit model of resonating FSS elements for a large bandwidth and scan angle. Fortunately, the square patch element used for synthesizing the low-pass RC elements of the capacitive circuit absorbers [7] does not suffer from the mentioned problems and can be modeled accurately for a wide range of angles of incidence at both polarizations. This permits us to simulate accurately the behavior of the absorber by circuit models, for different incident angles and polarizations. An ultrawideband absorber is designed by the method. B. Accurate, Wideband Model of Square Patch Periodic Array Consider a periodic square patch array embedded in a dielectric cover as shown in Fig. 10(a). The array can be modeled accurately by the equivalent circuit model shown in Fig. 10(b). The periodic array is embedded in a dielectric cover to stabilize the and values of the equivalent circuit. Like before the equivalent length ( ) and the intrinsic impedance ( ) of the transmission lines (see part(b) in Fig. 10) are calculated from (2)–(5). Fortunately, the and values of the equivalent circuit (when the square patch array is embedded in a proper cover) do not vary with angle of incidence. They fluctuate insignificantly around their values at normal angle of incidence for both polarizations (see the values in Tables IV–V). Therefore, with a very high accuracy, they can be considered constant in the absorber model. If necessary, final adjustments can be done in a

Fig. 10. Square patch array and its circuit model, (a) the front and the side view of the FSS structure (b) The equivalent circuit model and its parameters.

TABLE IV THE RESISTANCE (R) AND THE CAPACITANCE (C) VALUES OF THE CIRCUIT MODEL FOR THE SQUARE PATCH FSS AT DIFFERENT ANGLES OF INCIDENCE AND POLARIZATIONS. ( = 2:3, T = 0:2 mm, a = 4:3 mm, w = 4:1 mm, R = 100 =Sq .)

TABLE V THE RESISTANCE (R) AND THE CAPACITANCE (C) VALUES OF THE CIRCUIT MODEL FOR THE SQUARE PATCH FSS AT DIFFERENT ANGLES OF INCIDENCE AND POLARIZATIONS. ( = 2:3, T = 0:2 mm, a = 4:3 mm, w = 3:8 mm, R = 100 =Sq .)

full-wave simulation but experience has shown that the circuit models are sufficiently accurate (for example see Figs. 13–15). The stability of the and values of the equivalent circuit to changes of the incident angle is demonstrated in Tables IV–V.

KAZEMZADEH AND KARLSSON: MULTILAYERED WIDEBAND ABSORBERS FOR OBLIQUE ANGLE OF INCIDENCE

Fig. 11. The schematic of the capacitive circuit absorber. The first resistive layer is a homogeneous resistive sheet but others are square patches with different resistivity and dimensions.

Fig. 12. The frequency response of the capacitive circuit absorber at normal and oblique angle of incidence.(Full-wave simulation).

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Fig. 14. Comparison of the frequency response of the equivalent circuit model and the full-wave simulation (CST Microwave Studio) for TE polarization at 45 angle of incidence.

Fig. 15. Comparison of the frequency response of the equivalent circuit model and the full-wave simulation (CST Microwave Studio) for TM polarization at 45 angle of incidence.

imported from the full-wave simulation. The accuracy of the code is demonstrated clearly by the examples of the next subsection, specially by Figs. 13–15 that show very close agreement between the full-wave frequency responses of a multilayered absorber and its equivalent circuit models frequency responses. This cannot happen unless the square patches have been modeled very accurately. C. An Ultrawideband Design

Fig. 13. Comparison of the frequency response of the equivalent circuit model and the full-wave simulation (CST Microwave Studio) at normal angle of incidence.

Two different widths are considered for a typical square patch array embedded in a dielectric cover. The and values of the Tables IV–V are determined such that the circuit model of Fig. 10(b) results in a frequency response (reflection coefficient) as close as possible to the fullwave simulation (CST Microwave Studio) frequency response of the corresponding square patch. To facilitate the parameters estimation process a Matlab code is written that extracts the and the values very accurately for a given frequency response

It was explained that the low-pass square patches, when embedded in proper dielectric covers, exhibit valuable properties. They can be modeled accurately over large bandwidths for different polarizations and angles of incidence. This section demonstrates how these significant properties can be utilized to design ultrawideband capacitive circuit absorbers for oblique angle of incidence. The first step in the absorber design process is to find an equivalent circuit model solution that fulfills the absorption requirements at normal and oblique angles of incidence for both TE and TM polarizations, concurrently. Then, the circuit model is synthesized by proper square patches to form the actual absorber. To proceed, three different circuit models must be considered simultaneously: 1) The equivalent circuit model for normal angle of incidence. 2) The equivalent circuit model for 45 angle of incidence and TE polarization.

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3) The equivalent circuit model for 45 angle of incidence and TM polarization. Because symmetric elements (square patches) will be used later to synthesize the absorber, the frequency response is polarization insensitive at normal angle of incidence. Consequently, one circuit model is sufficient for this illumination angle. It is important to note that these three circuit models are not independent from each other. If the circuit model for the normal angle of incidence is selected as the independent system, then all the parameters of the other two equivalent circuits are known. As explained, the lengths of the equivalent transmission lines and their intrinsic impedances at oblique angle are calculated by (2)–(5) for the given polarization. Moreover, from the previous subsection, it is known that the values of the series RC elements, modeling the resistive layers (see Fig. 10), are the same for all the three circuit models when the square patches are embedded in proper dielectric covers. Henceforth, the parameters of the equivalent circuit model for normal angle of incidence are selected as the independent variables, since this circuit model is the one which is eventually synthesized to obtain the actual absorber. Since the equivalent circuit model of a multilayered absorber involves many parameters to determine and the design process is a multi-objective problem that requires absorption for three dependent circuits simultaneously, the Smith chart matching technique of Munk [8]–[10] becomes impractical to follow. On the other hand, the chance of finding a solution out of a blind optimization is low and time consuming. Good choice of parameters, for example based on experience, may speed up the design process significantly. Also it should be remembered that a useful circuit model solution is the one that can be synthesized eventually to construct the real absorber. It happens that a circuit model satisfies all the absorption requirements but is not realizable. For example, sometimes the values of the capacitances may not be practical for being synthesized by periodic square patches. Consequently, the circuit models should be guided by a well-experienced designer to result in a practical solution. Therefore, a combination of educated guesses, optimizations, realizability tests and reconsideration of the design parameters, is the best strategy to achieve a practical solution. A three layered capacitive capacitive absorber is designed by the above method. The schematic of the absorber is shown in Fig. 11. The first resistive sheet used in the absorber is a homogeneous resistive sheet and the rest are lossy periodic square patches. All the resistive sheets are embedded in similar dielecand tric covers, with the parameters (see Figs. 4 and 10). The parameters of the dielectric layers used in the design (see Fig. 11) are tabulated in Table VI. The dimensions of the square patches and the resistivity of the sheets ) are given in Tables VII–VIII. These square patches are ( used to synthesize the RC circuits of the equivalent circuit model with the parameter values listed in Table IX. It should be noted that the periodicity of the last square patch ( ) is half of the ). fundamental spatial period of the absorber ( This is done to make the synthesis of the required capacitances possible. The absorber has an ultrawideband frequency response, at least 26 GHz. The frequency response of the absorber at normal

TABLE VI THE PARAMETERS OF THE DIELECTRIC LAYERS USED IN THE CAPACITIVE CIRCUIT ABSORBER

TABLE VII THE RESISTIVITY OF THE SHEETS USED IN THE CAPACITIVE CIRCUIT ABSORBER

TABLE VIII THE DIMENSIONS (PERIODICITY (a), WIDTH (w ) ) OF THE PERIODIC SQUARE PATCHES. THE FUNDAMENTAL SPATIAL PERIOD OF THE ABSORBER IS 3.6 mm

TABLE IX THE VALUES OF THE PARAMETERS OF THE RC CIRCUITS USED IN THE EQUIVALENT CIRCUIT MODEL OF THE UWB ABSORBER

and oblique angle of incidence is shown in Fig. 12. The fullwave simulation is carried out 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. The ability of the absorber to operate for such a wide range of angles of incidence (up to 45 ) for both polarizations over its huge bandwidth is remarkable. Comparisons demonstrate that our design possesses the largest bandwidth among the published designs, keeping in mind that some of them are designed only for the simple case of normal incidence [1], [2], [8]–[10]. It is worth to mention that the total thickness of the absorber is only 16.9 mm. Since the design of the absorber is based on equivalent circuit models, the accuracy of such equivalent models and their agreement with the actual full-wave simulations must be verified. For this reason the frequency responses of the absorber obtained from full-wave simulations are compared to the frequency responses of the equivalent circuit models for different angles of incidence and polarizations. Comparison of the frequency responses in Figs. 13–15 illustrates the accuracy of our equivalent circuit models. In particular, it verifies the possibility of accurate modeling of the square patches with low-pass RC circuits for a large scan angles and different polarizations. D. Sensitivity Analysis of the UWB Design An important aspect of a design method is its practicality. It should be shown that our proposed design approach is suitable for realistic implementation of the absorber. In fabrication of the absorber unwanted deviations from the values calculated in the theory might occur for some parameters. Although it is expected that achieving an outstanding performance, like the ultrawide-

KAZEMZADEH AND KARLSSON: MULTILAYERED WIDEBAND ABSORBERS FOR OBLIQUE ANGLE OF INCIDENCE

Fig. 16. Sensitivity test for the UWB design when the sheet resistivity of the second resistive layer (R ) is varied. Full-wave simulation (CST Microwave Studio) for TE polarization at 45 angle of incidence.

Fig. 17. Sensitivity test for the UWB design when the sheet resistivity of the second resistive layer (R ) is varied. Full-wave simulation (CST Microwave Studio) for TM polarization at 45 angle of incidence.

band absorber of this section, requires care and accuracy in the fabrication process but there are always parameters that are difficult to control during the manufacturing. A good design method should be capable of tolerating small variations of its parameters. In the following it is shown by examples that our ultrawide band absorber has this property and the design approach is a practical method. Since our multilayered wideband design involves many parameters it is impractical to show the effect of small variation of all the parameters on the performance of the absorber within a subsection. Consequently, two parameters that might have the highest chance of being altered in the fabrication process are selected and it is verified that our design has a very good stability against these unwanted changes. Also, the oblique angle of incidence (45 ) is selected during the investigations because the frequency response for this case is more sensitive to changes than the frequency response for the normal angle of incidence. The first parameter to be considered is the resistivity of the resistive sheet that a periodic square patch array is made of. For this ) reason, the sheet resistivity of the second resistive layer ( is deviated from its optimal value and the effect of the variation on the absorber frequency response is studied. The results of the sensitivity test for this parameter are illustrated in Figs. 16 and 17. As seen from the figures the absorber has very good stability against small changes in the value of the sheet resistivity. Another important change of parameter that might influence the performance of the absorber is the perturbation of the permittivity of a dielectric layer. The change in the permittivity of a

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Fig. 18. Sensitivity test for the UWB design when the permittivity of all three dielectric covers ( ) are varied simultaneously. Full-wave simulation (CST Microwave Studio) for TE polarization at 45 angle of incidence.

Fig. 19. Sensitivity test for the UWB design when the permittivity of all three dielectric covers ( ) are varied simultaneously. Full-wave simulation (CST Microwave Studio) for TM polarization at 45 angle of incidence.

layer might occur due to utilization of glue layers, which are essential in any realistic design, or formation of possible air-gaps between layers in addition to the usual changes in the electrical properties of the materials used in the fabrication. Because the size of air gaps and the thickness of the different parts of the glue layers (depending on the type of the technology used in the design, the glue layer might consist of several very thin layers) are much smaller than the wavelength, their effects can be modeled by effective permittivity values through homogenization techniques with high accuracy. For example, a dielectric cover in our model may include the effect of glue layers in its effective permittivity and thickness. In this way, the total effect of these layers and the statistical variations of their parameters can be studied by sensitivity test for the permittivity of the dielectric covers. If the absorber shows a stable frequency response to this sensitivity test, it has a great chance of being fabricated without a perceptible change in its frequency response. It is shown that our proposed ultrawideband absorber has this property and is capable of tolerating small changes of the dielectric cover permittivity. For this purpose the worst scenario is selected, i.e., the permittivity of all the dielectric covers (three in our example) are altered simultaneously and it is shown that the changes have negligible effect on the absorber performance. The sensitivity test for different polarizations are illustrated in Figs. 18–19. It should be noted that effect of glue layers or the air-gaps can also be modeled by introducing more dielectric layers to both the equivalent circuit models or the full-wave simulations,

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if more accurate models are desired than the homogenized dielectric layers. These extra layers do not influence our design methodology since the formulation of the problem is developed for a general number of dielectric and resistive layers, but the increased complexity of the models usually do not results in much more information than the simpler models obtained by homogenization. It is the application of the absorber that determines to what extent the details should be considered in the analysis before a prototype is fabricated and tested. The purpose of this subsection is to verify that our design method is practical and to emphasize that it can even model small details if required. V. CONCLUSION Design of an ultrawideband absorber operating for a large range of incident angles and different polarizations is a challenging problem. Usually the design of Jaumann and circuit analog absorbers are restricted to normal angle of incidence. Therefore, the variation of the frequency response with respect to incident angle and polarization are not taken into consideration in the design. No general methods have so far been described in the literatures although some specific cases have been investigated [3], [9], [10]. In case of multilayered Jaumann absorbers, only low permittivity spacers are used in the designs [3]. It is shown that according to scan and frequency compensation techniques, this is not a reasonable choice. The circuit analog absorbers of Munk et al. are single FSS layer designs with moderate bandwidths [9], [10], that are not adequate for invisibility against modern radars operating at different frequency intervals. Therefore, a systematic method for achieving larger bandwidths with multilayered absorbers is proposed in the paper. A general model of Jaumann absorber is provided. The model is applicable to arbitrary permittivities of the dielectric layers, leading to two vital advantages. First, from the scan and frequency compensation techniques [10], it is known that using only low permittivity spacers is not optimal for oblique angles of incidence. Second, there are applications where mechanical and thermal properties of the absorbers are important for the designer. This can not be fulfilled by foams and honeycomb materials alone, as used in the Chambers and Tennant’s designs [3]. Different designs of two resistive layer Jaumann absorbers are presented, possessing almost the same bandwidth as the three layered genetic algorithm optimized design of Chambers and Tennant [3]. A three layered capacitive absorber is presented with an ultrawide bandwidth of 26 GHz. The absorber can operate at normal and oblique angles of incidence for both polarizations. It is explained that because of harmonics and anti-resonance effects

[10] of the resonating elements such a large bandwidth is not achievable by circuit analog absorbers. By replacing the bandstop resonating FSS elements with low-pass periodic square patches, the problems associated with harmonics and anti-resonances are avoided. Also it is shown that the periodic square patch array has a special property that enables us to model it very accurately for a large scan angle. All these fascinating properties of square patches lead us to designs that are outstanding in comparison to the earlier published designs [1], [2], [8]–[10]. ACKNOWLEDGMENT The authors would like to thank the Swedish Research Council for their support of this project. REFERENCES [1] S. Chakravarty, R. Mittra, and N. R. Williams, “On the application of the microgenetic algorithm to the design of broadband microwave absorbers comprising frequency-selective surfaces embedded in multilayered dielectric media,” IEEE Trans. Microwave Theory Tech., vol. 49, pp. 1050–1059, 2001. [2] S. Chakravarty, R. Mittra, and N. R. Williams, “Application of the microgenetic algorithm (MGA) to the design of broad-band microwave absorbers using multiple frequency selective surface screens buried in dielectrics,” IEEE Trans. Antennas Propag., vol. 50, pp. 284–296, 2002. [3] B. Chambers and A. Tennant, “Design of wideband Jaumann radar absorbers with optimum oblique incidence performance,” Electron. Lett., vol. 30, no. 18, pp. 1530–1532, Sep. 1994. [4] L. J. Du Toit, “The design of Jauman absorbers,” IEEE Antennas Propag. Mag., vol. 36, no. 6, pp. 17–25, 1994. [5] L. J. Du Toit and J. H. Cloete, “Electric screen Jauman absorber design algorithms,” IEEE Trans. Microwave Theory Tech., vol. 44, pt. 1, pp. 2238–2245, 1996. [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, pp. 1443–1454, 1988. [7] A. Kazemzadeh and A. Karlsson, “Capacitive circuit method for fast and efficient design of wideband radar absorbers,” IEEE Trans. Antennas Propag., vol. 57, pp. 2307–2314, Aug. 2009. [8] B. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [9] B. Munk, Metamaterials: Critique and Alternatives. New York: Wiley, 2009. [10] B. A. Munk, P. Munk, and J. Pryor, “On designing Jaumann and circuit analog absorbers (CA absorbers) for oblique angle of incidence,” IEEE Trans. Antennas Propag., vol. 55, pp. 186–193, Jan. 2007. [11] J. R. Wait, Electromagnetic Wave Theory. New York: Wiley, 1987.

Alireza Kazemzadeh, photograph and biography not available at the time of publication.

Anders Karlsson, photograph and biography not available at the time of publication.

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A Split-Step FDTD Method for 3-D Maxwell’s Equations in General Anisotropic Media Gurpreet Singh, Eng Leong Tan, Senior Member, IEEE, and Zhi Ning Chen, Fellow, IEEE

Abstract—A split-step finite-difference time-domain (FDTD) method is presented for 3-D Maxwell’s equations in general anisotropic media. The general anisotropic media can be characterized by full permittivity and permeability tensors. Stability analysis of the proposed split-step FDTD method using the Fourier method is presented. The eigenvalues of the Fourier amplification matrix are numerically shown to have unity magnitude even for time steps greater than the Courant limit time step, thereby illustrating the stable and non-dissipative nature of the split-step FDTD method in general anisotropic media. Numerical results are presented to further validate the accuracy and stability of the proposed split-step FDTD method in general anisotropic media. Index Terms—Alternating direction implicit FDTD, anisotropic media, finite-difference time-domain (FDTD) method, locally 1-D FDTD, split-step FDTD, unconditionally stable FDTD methods.

I. INTRODUCTION

T

HE finite-difference time-domain (FDTD) method has been widely used to obtain solutions of Maxwell’s equations for a wide range of electromagnetic problems [1]. For the conventional explicit FDTD method [2], the computational efficiency is constrained by the Courant-Friedrich-Levy (CFL) stability condition which imposes a maximum time step size depending on the spatial mesh sizes. To overcome the CFL condition, several unconditionally stable FDTD methods have been developed based on techniques such as alternating direction implicit (ADI) [3], [4], split-step [5], [6], locally 1-D (LOD) [7]–[10] and Crank-Nicolson [11], [12]. Unconditionally stable FDTD methods have been extended to treat complex materials that are frequency dispersive [13], [14] and lossy [15], [16]. Anisotropic media represent another class of complex materials that have been successfully treated by several variants of the conventional explicit FDTD method. For non-dispersive anisotropic media, several explicit FDTD methods have been presented in [17]–[23]. The conventional FDTD method was initially extended in [17] for treating diagonal anisotropic media. In [18], an explicit FDTD method in terms of electric and magnetic fields was presented to treat anisotropic media with full permittivity and conductivity tensors. Several extensions of this method were presented in [19] by considering Manuscript received May 31, 2009; revised March 28, 2010; accepted May 11, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. G. Singh and E. L. Tan are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]; [email protected]). Z. N. Chen is with the Institute for Infocomm Research, Singapore 138632, Singapore. Digital Object Identifier 10.1109/TAP.2010.2071342

higher-order approximations and alternate grid structures. In [20], an alternate explicit FDTD method for 3-D anisotropic dielectrics was proposed using electric and magnetic fields as well as electric flux. An extension of this method that includes magnetic flux for treating magnetic anisotropic materials has been presented in [21]. In [22], an explicit FDTD method in 3-D cylindrical grids was presented to model full anisotropic conductive media. Several efforts have also been made to treat materials that are both anisotropic and frequency dispersive with the explicit FDTD method [21], [24]–[29]. Recently, researchers have extended the treatment of anisotropic media using the unconditionally stable FDTD methods. The ADI-FDTD method has been employed in [30] for the analysis of anisotropic substrate with diagonal permittivity tensor. In [31], the complex-envelope (CE) ADI-FDTD method was extended to treat anisotropic dielectrics, with 2-D permittivity tensors (anisotropy in the plane), for modeling degenerate band edge photonic crystals. The proposed CE-ADI-FDTD method was further extended in [32] to model lossy anisotropic dielectrics with 2-D permittivity and conductivity tensors. Several efforts have also been made to develop anisotropic based perfectly matched layer (PML) absorbing boundary conditions for unconditionally stable FDTD methods [33], [34]. These anisotropic based PML absorbers are generally characterized by diagonal material tensors. Despite the abovementioned efforts, one of the existing challenges lies in the development of unconditionally stable FDTD methods for the treatment of anisotropic media characterized by full 3-D constitutive tensors. In this paper, we present a split-step FDTD method for 3-D Maxwell’s equations in general anisotropic media. The general anisotropic media can be characterized by full permittivity and permeability tensors. Formulation of the proposed split-step FDTD method in general anisotropic media is first presented in Section II. Stability analysis of the proposed split-step FDTD method using the Fourier method is next presented in Section III. The eigenvalues of the Fourier amplification matrix are numerically shown to have unity magnitude even for time steps greater than the Courant limit time step, thereby illustrating the stable and non-dissipative nature of the proposed split-step FDTD method in general anisotropic media. Numerical results from four different experiments are then presented in Section IV to further validate the accuracy and stability of the proposed split-step FDTD method in general anisotropic media. II. FORMULATION In this section, we present the formulation of the proposed split-step FDTD method for 3-D Maxwell’s equations in gen-

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eral anisotropic media. The general anisotropic media can be and permeability tencharacterized by full permittivity sors. The proposed formulation is achieved by first splitting the and magnetic flux components into update of electric 6 sub-steps. These 6 sub-steps are written as Sub-step 1

Sub-step 6

(6a) (6b) (6c) (6d)

(1a) (1b)

Then, by applying the inverse constitutive relations

(1c) (7) (8)

(1d) Sub-step 2

(2a) (2b)

to each sub-step and after some manipulations, the split-step FDTD method for 3-D Maxwell’s equations, in general anisotropic media, is written in terms of only - field components as Sub-step 1

(2c)

(9a)

(2d)

(9b)

Sub-step 3

(9c) (9d) (3a) (3b)

(9e) (9f)

(3c) (3d)

Sub-step 2

Sub-step 4 (10a) (4a) (4b) (10b)

(4c)

(10c)

(4d)

(10d)

Sub-step 5

(10e) (10f) (5a) (5b)

Sub-step 3

(5c) (5d)

(11a)

SINGH et al.: A SPLIT-STEP FDTD METHOD FOR 3-D MAXWELL’S EQUATIONS IN GENERAL ANISOTROPIC MEDIA

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(14c) (11b) (11c) (11d)

(14d) (14e) (14f)

(11e) (11f) Sub-step 4

Next, we discuss the actual numerical update of the electric and magnetic field components. For spatial discretization, we assume the use of Yee’s grid. Consider Sub-step 1 as an example. By substituting the right-hand side (RHS) of (9b) into (9a) and after some manipulations, the implicit update of is performed as

(12a)

(12b) (12c) (12d) (12e) (12f) Sub-step 5 (15)

(13a)

With (15), a system of linear equations with tridiagonal coefis effificient matrix is formed and the update of ciently carried out. Subsequently the update of is determined as

(13b) (13c) (13d) (13e) (13f) Sub-step 6 (16)

(14a)

(14b)

and , the updates Upon updating of remaining field components are performed using the 4 explicit update (9c)–(9f). Note that at every field point on the Yee’s grid, only one field component is directly available. The remaining field components that are not directly available are attained by interpolation from four values in the neighbouring grid points. Such interpolations are performed in (9c)–(9f). As

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an example, in the update of is attained by

in (9c), the value of

(17) Similar type of interpolations were performed in [18]. The updates of electric and magnetic field components in Sub-step 2 to Sub-step 6 are performed in a similar manner discussed above. Note that when and are isotropic or diagonal, the formulation Sub-step 1 to Sub-step 6 reduces to the conventional splitstep FDTD method (SS1 or LOD1) [10]. Furthermore, apart from the 4 explicit equations in each sub-step, such as (9c)–(9f) in Sub-step 1, the entire update from n to n+1 requires the solution of 6 pairs of implicit 1-D scalar wave equations. This requirement is the same as the conventional split-step FDTD method (SS1 or LOD1) in [10], where the solutions of 6 pairs of implicit 1-D scalar wave equations are achieved over 2 substeps.

Fig. 1. Eigenvalues (dots) of amplification matrix case of medium 1 (electric anisotropic).

G around unit semicircle for

magnetic) anisotropic media and denote them as anisotropic and ), medium 2 ( medium 1 ( and ) and medium 3 ( and ) respectively with

III. STABILITY ANALYSIS In this section, we analyze the stability of the proposed split-step FDTD method for 3-D Maxwell’s equations in general anisotropic media. Due to the complexity, there is not much literature for stability analysis in general anisotropic media. Hence, we resort to the Fourier method and numerically investigate the eigenvalues of the Fourier amplification matrix. Let the 6 6 amplification matrix of each sub-step be denoted as , where is the sub-step number ranging from 1 to 6, and the amplification matrix of the entire update from n to n+1 be is determined from denoted as . The amplification matrix the proposed split-step FDTD formulation in terms of - field components. The computation of amplification matrix and are presented in the Appendix. Note that some elements of amplification matrix contain certain cosine terms. These cosine terms arise upon Fourier transformation of field components as determined by the interpolation operation described earlier, cf. (17). The relationship between amplification matrix and the field vector in Fourier domain is written as

(18) where Fourier domain

and

is the 6

1 field vector in

(19) To analyze the stability of the proposed split-step FDTD method, we numerically scan the eigenvalues of the amplification matrix over a range of Fourier wavenumbers. In this numerical scan, we consider three arbitrary cases of anisotropic media, namely, electric, magnetic and general (both electric and

is the 3 3 identity matrix. and are the free space permittivity and permeability respectively. We assume uniform cell of 6.255 mm, 1.033 mm and 1.615 size mm for the case of medium 1, 2 and 3 respectively. We denote where is the the Courant limit time step as speed of light in free space. The eigenvalues of the amplification matrix for medium 1, 2 and 3 are scanned over four independent parameters, namely where CFLN is the the time step size Courant number and the angles , and , which are related to the Fourier wavenumbers , and (see Appendix). The CFLN number is set to 1, 5, 10, 20, 60 and 100. The angles , and are each set to values across a range of 0 to radians and interval of 0.15 radians. For each medium case and CFLN value, we determine amplification matrices and their corresponding eigenvalues. These eigenvalues are plotted on the unit semicircle and shown in Figs. 1–3. , the eigenFrom the figures, we observe that, for values corresponding to anisotropic medium 1, 2 and 3 still lie on the unit semicircle and have unity magnitude, thereby illustrating the stable and non-dissipative nature of the proposed split-step FDTD method for 3-D Maxwell’s equation in general anisotropic media.

SINGH et al.: A SPLIT-STEP FDTD METHOD FOR 3-D MAXWELL’S EQUATIONS IN GENERAL ANISOTROPIC MEDIA

Fig. 2. Eigenvalues (dots) of amplification matrix case of medium 2 (magnetic anisotropic).

Fig. 3. Eigenvalues (dots) of amplification matrix case of medium 3 (general anisotropic).

G around unit semicircle for

G around unit semicircle for

IV. NUMERICAL RESULTS In this section, we present numerical results from four experiments to validate the accuracy and stability of the proposed split-step FDTD method for 3-D Maxwell’s equations in general anisotropic media. A. Experiment 1 In experiment 1, we simulate a 30 30 30 uniform grid rectangular cavity filled with homogeneous anisotropic medium characterized by medium 1, 2 and 3 in Section III and with perfect electric conducting (PEC) boundary conditions. We of 0.447 mm, 1.033 mm and assume uniform cell size 1.615 mm for anisotropic medium 1, 2 and 3 respectively. These cell sizes correspond to cell per minimum wavelength (CPW) of 40, 38 and 23 respectively. The minimum wavelengths in the anisotropic media are approximated using the , where and relation are the maximum absolute elements from and respectively and is the free space wavelength corresponding

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H

Fig. 4. Time-domain computed using Split-Step FDTD (CFLN = 10) and Explicit FDTD for case of medium 1 (electric anisotropic).

to the upper frequency of the source function. The source function is modeled by a bandpass Gaussian function: where , and , with , 1000 and 1000 and with , 225 and 322 for case of medium 1, 2 and 3 respecfield at source tively. The current source is applied to the point (16,25,16). field is At observation point (16,7,7) the time domain first recorded over 10000, 5000 and 7500 iterations for experiment with anisotropic medium 1, 2 and 3 respectively. For each medium case, the experiment is repeated using the proposed split-step FDTD method with CFLN values, ranging from 1 to 10. The experimental results are compared with the explicit . FDTD method for anisotropic media [18]–[21] at Plots of the normalized field using the proposed split-step and the explicit FDTD method FDTD method at are shown in Fig. 4–6. The figures show stable time domain fields using the proposed split-step FDTD method for all media cases. Good agreement is also shown between the proposed split-step FDTD method and the explicit FDTD method. The resonant frequencies are next extracted from the time fields, recorded over longer iterations of 100000 domain time steps. The required CPU time of the proposed split-step FDTD method at various CFLNs and the explicit FDTD method is compared in Table I for the case of medium 3. The computing platform is C++ environment with Intel Dual Core 3.00 GHz processor. From the table, we observe that the proposed split-step FDTD method achieves a lower CPU time and is computationally efficient compared to the explicit However, the price to pay for FDTD method for this efficiency is a reduction in accuracy that can be observed from the resonant frequency versus CFLN plots in Fig. 7. The top and bottom plots in Fig. 7 correspond to the resonant frequencies 1.257 GHz and 1.187 GHz respectively. From the figures, we observe that the relative frequency error between the proposed split-step FDTD method and the explicit FDTD method increases as CFLN increases. Specifically, the relative are 0.32% and 0.28% for the resonant errors at

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H

Fig. 5. Time-domain computed using Split-Step FDTD (CFLN = 10) and Explicit FDTD for case of medium 2 (magnetic anisotropic).

Fig. 7. Resonant frequencies extracted using Split-Step FDTD and Explicit FDTD for case of medium 3 (general anisotropic). Top and bottom plots correspond to resonance at 1.257 GHz and 1.187 GHz respectively.

TABLE I COMPARISONOF CPU TIME BETWEEN PROPOSED SPLIT-STEP FDTD AND EXPLICIT FDTD FOR EXPERIMENT 1

H

Fig. 6. Time-domain computed using Split-Step FDTD (CFLN = 10) and Explicit FDTD for case of medium 3 (general anisotropic).

frequencies 1.257 GHz and 1.187 GHz respectively. The resonant frequencies extracted using the proposed split-step FDTD method is still in good agreement with the explicit FDTD method. Similarly, good agreement is observed for cases of medium 1 and 2. Next, we validate the late time stability of the proposed splitfield step FDTD method. A point source is initialized at the at source point (16,25,16). At observation point (16,7,7) the field is recorded over 80000 iterations with time domain (which corresponds to 8 million time steps at ) for case of medium 3. In Fig. 8, we plot the norfield using the proposed split-step FDTD method malized . From the figure, we observe stable time dowith field even after large number of iterations. Late time main stable results were similarly observed for smaller CFLN and for cases of medium 1 and 2. B. Experiment 2 In experiment 2, we simulate a 30 30 30 uniform grid rectangular cavity with a centered 24 24 22 anisotropic dielectric resonator surrounded by air and with PEC boundary

H

Fig. 8. Time-domain computed using Split-Step FDTD (CFLN = 100) over 80000 iterations for case of medium 3 (general anisotropic).

condition, as illustrated in Fig. 9 inset ( plane view). Anisotropic dielectric resonator in cavities have previously been investigated in [35] using the finite element method to extract resonant modes. Here, we investigate the resonant modes extracted using the proposed split-step FDTD method. The anisotropic dielectric resonator is characterized by medium 1 of Section III. We assume uniform cell size of 6.255 mm that . The source function is modeled corresponds to by a bandpass Gaussian function:

SINGH et al.: A SPLIT-STEP FDTD METHOD FOR 3-D MAXWELL’S EQUATIONS IN GENERAL ANISOTROPIC MEDIA

Fig. 9. Magnitude of E (!) versus frequency computed using Split-Step FDTD (CFLN = 5) and Explicit FDTD for case of medium 1 (electric anisotropic) dielectric resonator.

TABLE II COMPARISON OF CPU TIME BETWEEN PROPOSED SPLIT-STEP FDTD AND EXPLICIT FDTD FOR EXPERIMENT 2

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Fig. 10. Resonant frequencies extracted using Split-Step FDTD and Explicit FDTD for case of anisotropic dielectric resonator. Top and bottom plots correspond to resonance at 6.48 10 Hz and 7.78 10 Hz respectively.

2

2

imum tolerable error levels of 1%, 2% and 3% can be met by operating the proposed split-step FDTD method with CFLN of 5, 8 and 10 respectively. C. Experiment 3

with

, and . The current source is applied to the field at source point (10, 10, 15). field is At observation point (22, 22, 22), the time domain recorded over 8000 iterations using the explicit FDTD method and the proposed split-step FDTD method with CFLN values ranging from 1 to 10. The required CPU time of the proposed split-step FDTD method at various CFLNs and the explicit FDTD method is compared in Table II. From the table, we observe that the proposed split-step FDTD method achieves a lower CPU time and is computationally efficient compared to . the explicit FDTD method for The accuracy of the proposed split-step FDTD method is next field in freanalyzed. We first compare the magnitude of the quency domain using the proposed split-step FDTD method at and the explicit FDTD method. This comparison is shown in Fig. 9. From the figure, we observe reasonably good agreement between both methods. Next, we compare the resonant frequencies extracted using the explicit FDTD method and the proposed split-step FDTD method with CFLN ranging from 1 to 10. This comparison is highlighted in Fig. 10, where the top and bottom plots correspond to the resonance at 6.48 Hz and 7.78 Hz respectively. From the figure, we observe that the relative errors at are 1.63% and 2.56% for Hz and 7.78 Hz rethe resonant frequencies 6.48 spectively. The resonant frequencies extracted by both methods are still in good agreement. Fig. 10 also illustrates that the max-

In experiment 3, we simulate 3-D scattering from a 40 40 40 uniform grid anisotropic dielectric cube centered in a 100 100 100 computational domain. The anisotropic dielectric cube and background media are characterized by medium 1 of Section III and free space respectively. We assume . uniform cell size of 2 mm that corresponds to The total-field/scattered-field (TF/SF) formulation is used to introduce into the total field region a -polarized incident plane wave, propagating towards the positive direction [36]. The incident plane wave is modeled using a bandpass Gaussian pulse: with , and . The Mur absorbing boundary condition (ABC) [37], [38] is used to terminate the boundaries of the computational domain. The cross sectional view of the computational domain is shown in Fig. 11 inset. At the observation point (90, 90, 51), in the scattered field field is recorded over 5000 iterations region, the scattered using the explicit FDTD method and proposed split-step FDTD method with CFLN of 1, 5 and 10. The normalized scattered field extracted by both methods are plotted and compared in Fig. 11. From the figure, we observe that as the CFLN into creases from 1 to 10 (time step size increases from and number of iterations reduces from 5000 to 500), the matching of time-domain field plots with the explicit FDTD method slightly degrades but is still acceptable. This signifies the trade-off between efficiency and accuracy of the split-step FDTD method. D. Experiment 4 In experiment 4, we verify 2-D simulation with ( , and ) scattering from an infinitely long anisotropic dielectric cylinder of radius 40 grids, centered in a 200 200 uni-

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Fig. 11. Time-domain scattered E computed using Split-Step FDTD (CFLN = 1, 5 and 10) and Explicit FDTD.

Fig. 12. Radar cross section (dB) computed using Split-Step FDTD (CFLN = 1 and 5) and Explicit FDTD.

form computational domain. The anisotropic dielectric cylinder is characterized by a relative permittivity tensor

Four numerical experiments have been presented to further validate the stability and accuracy of the proposed split-step FDTD method in general anisotropic media. Closed region problems involving PEC rectangular cavities filled with homogeneous and inhomogeneous anisotropic media have been considered in experiments 1 and 2 respectively. Open region problems involving scattering off anisotropic cube and infinite cylinder have been considered in experiments 3 and 4 respectively. Numerical results using the proposed split-step FDTD method have shown stable fields until late time even for time steps greater than the Courant limit time step. Furthermore, it has been shown that the proposed split-step FDTD method is more efficient compared to the explicit FDTD method when The proposed split-step FDTD method has also been shown to attain results that are in good agreement with the explicit FDTD method.

and the background media is free space. We assume uniform cell . The TF/SF forsize of 2.5 mm that corresponds to mulation is used to introduce into the total field region a -polarized incident plane wave, propagating towards the positive direction. The incident plane wave is modeled using a bandpass Gaussian pulse: with , and . is the Courant limit time step size in a 2-D free space grid. The Mur ABC is used to terminate the boundaries of the computational domain. After 10000 iterations, the bistatic radar cross section (RCS) is determined from the scattered fields in the scattered field region. Fig. 12 compares the RCS extracted using the proposed split-step FDTD method with CFLN of 1 and 5 and the explicit FDTD method. From the figure, we observe good agreement between both methods. V. CONCLUSION This paper has presented a split-step FDTD method for 3-D Maxwell’s equations in general anisotropic media. The general anisotropic media can be characterized by full permittivity and permeability tensors. The proposed split-step FDTD method splits the field update from n to n+1 into 6 sub-steps. In each sub-step, two field components are first solved from a system of linear equations with tridiagonal coefficient matrix. These two field components are then used in explicit update equations to determine the remaining four field components. Stability analysis of the proposed split-step FDTD method using the Fourier method was presented. The eigenvalues of the Fourier amplification matrix were numerically shown to have unity magnitude even for time steps greater than the Courant limit time step, thereby illustrating the stable and non-dissipative nature of the proposed split-step FDTD method in general anisotropic media.

APPENDIX COMPUTATION OF AMPLIFICATION MATRIX

AND

Let and denote an element in row r and column c of 6 6 matrices and respectively. Let with , and . Except the following elements of matrices and , the remaining diagonal elements are unity and offdiagonal elements are zero:

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The amplification matrix and using

and

can then be determined .

REFERENCES [1] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 2005. [2] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, pp. 302–307, May 1966. [3] F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 1550–1558, Sep. 2000.

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[4] T. Namiki, “3-D ADI-FDTD method—Unconditionally stable timedomain algorithm for solving full vector Maxwell’s equations,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 1743–1748, Oct. 2000. [5] J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math., vol. 158, pp. 485–505, 2003. [6] W. Fu and E. L. Tan, “Development of split-step FDTD method with higher-order spatial accuracy,” Electron. Lett., vol. 40, no. 20, pp. 1252–1253, Sep. 2004. [7] J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett., vol. 41, no. 19, pp. 1046–1047, Sep. 2005. [8] E. L. Tan, “Unconditionally stable LOD-FDTD method for 3-D Maxwell’s equations,” IEEE Microw. Wireless Comp. Lett., vol. 17, no. 2, pp. 85–87, Feb. 2007. [9] I. Ahmed, E. K. Chua, E. P. Li, and Z. Chen, “Development of the three-dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antennas Propag., vol. 56, pp. 3596–3600, Nov. 2008. [10] E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antennas Propag., vol. 56, pp. 170–177, Jan. 2008. [11] G. Sun and C. W. Trueman, “Efficient implementations of the CrankNicolson scheme for the finite-difference time-domain method,” IEEE Trans. Microwave Theory Tech., vol. 54, pp. 2275–2284, May 2006. [12] E. L. Tan, “Efficient algorithms for Crank-Nicolson-based finite-difference time-domain methods,” IEEE Trans. Microwave Theory Tech., vol. 56, pp. 408–413, Feb. 2008. [13] X. T. Dong, N. V. Venkatarayalu, B. Guo, W. Y. Yin, and Y. B. Gan, “General formulation of unconditionally stable ADI-FDTD method in linear dispersive media,” IEEE Trans. Microwave Theory Tech., vol. 52, pp. 170–174, Jan. 2004. [14] J. Shibayama, R. Takahashi, J. Yamauchi, and H. Nakano, “Frequencydependent LOD-FDTD implementations for dispersive media,” Electron. Lett., vol. 42, no. 19, pp. 1084–1085, Sep. 2006. [15] C. Yuan and Z. Chen, “On the modeling of conducting media with the unconditionally stable ADI-FDTD method,” IEEE Trans. Microwave Theory Tech., vol. 51, pp. 1929–1938, Aug. 2003. [16] W. Fu and E. L. Tan, “Stability and dispersion analysis for ADI-FDTD method in lossy media,” IEEE Trans. Antennas Propag., vol. 55, pp. 1095–1102, Apr. 2007. [17] D. H. Choi and W. J. R. Hoefer, “The finite-difference time-domain method and its application to eigenvalue problems,” IEEE Trans. Microwave Theory Tech., vol. 34, pp. 1464–1470, Dec. 1986. [18] J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antennas Propag., vol. 41, pp. 994–999, Jul. 1993. [19] S. G. Garcia, T. M. Hung-Bao, R. G. Martin, and B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech., vol. 44, pp. 2195–2206, Dec. 1996. [20] A. P. Zhao, J. Juntunen, and A. V. Raisanen, “An efficient FDTD algorithm for the analysis of microstrip patch antennas printed on a general anisotropic dielectric substrate,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1142–1146, Jul. 1999. [21] L. Duo and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microwave Opt. Technol. Lett., vol. 48, no. 10, pp. 2083–2090, Oct. 2006. [22] H. O. Lee and F. L. Teixeira, “Cylindrical FDTD analysis of LWD tools through anisotropic dipping-layered earth media,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 2, pp. 383–388, Feb. 2007. [23] G. R. Werner and J. R. Cary, “A stable FDTD algorithm for non-diagonal anisotropic dielectrics,” J. Comput. Phys., vol. 226, pp. 1085–1101, May 2007. [24] F. Hunsberger, R. Luebbers, and K. Kunz, “Finite-difference time-domain analysis of gyrotropic media-I: Magnetized plasma,” IEEE Trans. Antennas Propag., vol. 40, pp. 1489–1495, Dec. 1992. [25] J. A. Pereda, L. A. Vielva, A. Vegas, and A. Prieto, “A treatment of magnetized ferrites using the FDTD method,” IEEE Microw. and Guided Wave Lett., vol. 3, no. 5, pp. 136–138, May 1993. [26] M. Celuch, A. Moryc, and W. K. Gwarek, “Numerical stability of FDTD algorithms in gyrotropic media analysed through their dispersion relations,” in Proc. IEEE MTT-S Int. Microw. Symp., 2007, pp. 725–728.

[27] H. Mosallaei, “FDTD-PLRC technique for modeling of anisotropicdispersive media and metamaterial devices,” IEEE Trans. Electromagn. Compat., vol. 49, pp. 649–660, Aug. 2007. [28] F. L. Teixeira, “Time-domain finite-difference and finite-element methods for Maxwell equations in complex media,” IEEE Trans. Antennas Propag., vol. 56, pp. 2150–2166, Aug. 2008. [29] A. Grande, J. A. Pereda, O. Gonzalez, and A. Vegas, “Stability and accuracy analysis of an extension of the FDTD method to incorporate magnetized ferrites,” Int. J. Numer. Modeling: Electron. Netw., Devices Fields, vol. 22, no. 2, pp. 109–127, Mar. 2009. [30] C. Ma, Z. Chen, and A. P. Zhao, “Development of an unconditionally stable full-wave 2D ADI-FDTD method for analysis of arbitrary waveguiding structures,” in Proc. IEEE MTT-S Int. Microw. Symp., 2002, vol. 3, pp. 2049–2052. [31] K.-Y. Jung and F. L. Teixeira, “CE-ADI-FDTD analysis of photonic crystals with a degenerate band edge (DBE),” in Proc. IEEE Antennas and Propag. Soc. Int. Symp., Jun. 2007, pp. 4449–4452. [32] K.-Y. Jung, F. L. Teixeira, and R. Lee, “Complex envelope PML-ADIFDTD method for lossy anisotropic dielectrics,” IEEE Antennas. Wireless Propag. Lett., vol. 6, pp. 643–646, 2007. [33] A. P. Zhao, “Uniaxial perfectly matched layer media for an unconditionally stable 3-D ADI-FD-TD method,” IEEE Microw. Wireless Comp. Lett., vol. 12, no. 12, pp. 497–499, Dec. 2002. [34] Z. Y. Huang and G. W. Pan, “Universally applicable uniaxial perfect matched layer formulation for explicit and implicit finite difference time domain algorithms,” IET Microw. Antennas Propag., vol. 2, no. 7, pp. 668–676, 2008. [35] J. M. Gil, “CAD-oriented analysis of cylindrical and spherical dielectric resonators in cavities and MIC environments by means of finite elements,” IEEE Trans. Microwave Theory Tech., vol. 53, pp. 2866–2874, Sep. 2005. [36] G. Singh, E. L. Tan, and Z. N. Chen, “Implementation of total-field/ scattered-field technique in the 2-D LOD-FDTD method,” in Proc. Asia Pacific Microwave Conf., Dec. 2009, pp. 1505–1508. [37] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat., vol. 23, pp. 377–382, Nov. 1981. [38] T. Stefanski and T. D. Drysdale, “Improved implementation of the Mur first-order absorbing boundary condition in the ADI-FDTD method,” Microwave Opt. Technol. Lett., vol. 50, no. 7, pp. 1757–1761, Jul. 2008.

Gurpreet Singh received the B.Eng. (electrical) degree with first class honors from Nanyang Technological University, Singapore, in 2007, where he is currently working toward the Ph.D. degree. His research interests include computational electromagnetics, RF and microwave circuit design.

Eng Leong Tan (SM’09) received the B.Eng. (electrical) degree with first class honors from the University of Malaya, and the Ph.D. degree in electrical engineering from Nanyang Technological University, Singapore. From 1991 to 1992, he was a Research Assistant at the University of Malaya. From 1991 to 1994, he worked part-time at Commercial Network Corporations Sdn. Bhd., Malaysia. From 1999 to 2002, he was a Member of Technical Staff at the Institute for Infocomm Research, Singapore. Currently, he is an Associate Professor at the School of Electrical and Electronic Engineering, Nanyang Technological University. His research interests include electromagnetic and acoustic simulations, RF and microwave circuit design.

SINGH et al.: A SPLIT-STEP FDTD METHOD FOR 3-D MAXWELL’S EQUATIONS IN GENERAL ANISOTROPIC MEDIA

Zhi Ning Chen (M’99–SM’05–F’07) received the B.Eng., M.Eng., and Ph.D. degrees from the Institute of Communications Engineering, China, and the Do.E. degree from the University of Tsukuba, Japan, in 1985, 1988, 1993, and 2003, all in electrical engineering. From 1988 to 1997, he worked at the Institute of Communications Engineering, Southeast University, and City University of Hong Kong, China with teaching and research appointments. In 1997, he was awarded a JSPS Fellowship to join the University of Tsukuba, Japan. In 2001 and 2004, he visited the University of Tsukuba under a JSPS Fellowship Program (senior level). In 2004, he worked at IBM T. J. Watson Research Center, New York, as Academic Visitor. Since 1999, he has worked with the Institute for Infocomm Research where his current appointments are Principal Scientist and Department Head for RF & Optical. He is concurrently holding Adjunct/Guest Professors at Southeast University, Nanjing University, Shanghai Jiao Tong University, Tongji University and National University of Singapore. He has published 280 journal and con-

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ference papers as well as authored and edited the books Broadband Planar Antennas (New York: Wiley, (2005), UWB Communications (New York: Wiley, 2006), and Antennas for Portable Devices (New York: Wiley, 2007). He also contributed chapters to the books UWB Antennas and Propagation for Communications, Radar, and Imaging as well as the Antenna Engineering Handbook. He holds 28 granted and filed patents with 17 licensed deals with industry. His current research interest includes applied electromagnetic engineering, RF transmission over bio-channel, and antennas for wireless systems, in particular at mmW, submmW, and THz for medical and healthcare applications. Dr. Chen is a Fellow of the IEEE for his contribution to small and broadband antennas for wireless and IEEE AP-S Distinguished Lecturer (2008–2010). He is the recipient of the CST University Publication Award 2008, IEEE AP-S Honorable Mention Student Paper Contest 2008, IES Prestigious Engineering Achievement Award 2006, I2R Quarterly Best Paper Award 2004, and IEEE iWAT 2005 Best Poster Award. He has organized many international technical events as key organizer. He is the founder of International Workshop on Antenna Technology (iWAT).

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Locally-Conformal FDTD for Anisotropic Conductive Interfaces Hwa Ok Lee and Fernando L. Teixeira

Abstract—Locally-conformal discretizations have been developed for the finite-difference time-domain (FDTD) method to mitigate staircasing errors that arise in the modeling of arbitrary (e.g., curved, slanted) interfaces by regular orthogonal grids. In particular, locally-conformal algorithms based on the computation of effective material parameters in partially-filled FDTD cells are attractive because they are simple to implement and maintain the basic conditional stability of FDTD, with only minor changes on the Courant limit. In this paper, we propose a new locally-conformal implementation for FDTD tailored for anisotropic conductive interfaces. Instead of employing arithmetic or geometrical averages, we invoke the quasi-static relationship between the current density and the electric field to derive the effective conductivities of partially-filled FDTD cells. Index Terms—Anisotropic media, finite-difference time-domain (FDTD) methods, staircasing approximation.

I. INTRODUCTION

T

HE use of regular, orthogonal grids in the finite-difference time-domain (FDTD) method leads to a “matrix-free” algorithm [1], [2] that does not necessitate any linear algebra. This allows FDTD to solve very large problems beyond the reach of other numerical methods [2], [3], but presents a well-known drawback: the presence of staircasing error in arbitrary (slanted, curved) interfaces. Short of refining the FDTD grid or using subgridding, the standard way of tackling this problem in FDTD is to use locally deformed integration paths or local effective material parameters in partially-filled FDTD cells [4]–[12]. These solutions are often referred to as “contour-path” or “(locally-) conformal” discretizations. The use of effective parameters is typically preferred because of its ease of implementation and because it does not fundamentally modify the conditional stability of FDTD [9], [11]–[13], except for a change on the value of the Courant limit. Local, effective parameters are very useful to avoid excessive FDTD grid refinement in other contexts as well [14], [15]. FDTD simulations sometimes encounter the need to accurately model interfaces between anisotropic conductive Manuscript received November 04, 2009; revised May 05, 2010; accepted May 07, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. This work was supported in part by the NSF through grants EECS-0347502 and EECS-0925272, and in part by the OSC through grants PAS-0061 and PAS-0110. The authors are with ElectroScience Laboratory and Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210 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.2010.2071362

media, such as in Earth media problems [16]. To simulate problems involving isotropic conductive media interfaces, locally-conformal FDTD with either edge- or face-based [17] effective conductivities have been utilized. These approaches are less suited for anisotropic interfaces because of the more complicated spatial stencils that arise in the latter [16], [18] and ambiguities that ensue in the definition of the directional conductivities using simple averages. For anisotropic media, a stable FDTD algorithm for non-diagonal lossless anisotropic dielectrics with second order accuracy was recently developed in [19]. Late time-stability in this case was obtained by enforcing symmetry and positive-definiteness of the discretized tensor operators. It is of note that the need to accommodate anisotropic materials in FDTD discretizations also arises in some approaches aimed at modeling interfaces (including isotropic ones) while preserving the second-order accuracy of FDTD [20]. In this paper, we propose and develop a 3-D “locally-conformal” FDTD algorithm for anisotropic conductive interfaces where the conductivity tensor is, in general, a full 3 3 tensor. Instead of arithmetic or geometrical averages, we derive effective conductivities of partially-filled FDTD cells (with generally slanted interfaces) based on a quasi-static approximation. II. FORMULATION The derivation of the proposed locally-conformal approach is very simple and based on the well-known quasi-static conductance properties for a material block (representing a rectangular FDTD cell) with cross-sectional area , length and material and , as illustrated in Fig. 1. Recall that in conductivities the parallel (to the current) arrangement depicted in Fig. 1(a), the effective conductivity can be calculated from the equivalent conductance obtained by taking an weighted area average as

(1) where is the effective conductivity. For the series arrangement as depicted in Fig. 1(b), we have

(2) This quasi-static approximation is justified when the dimensions of the discretization cell are much smaller than both the wavelength of operation and (for problems involving finite conductivities) the skin-depth, which is typical in FDTD. Letting , the conductance determines the current across

0018-926X/$26.00 © 2010 IEEE

LEE AND TEIXEIRA: LOCALLY-CONFORMAL FDTD FOR ANISOTROPIC CONDUCTIVE INTERFACES

Fig. 1. (a) Parallel and (b) series arrangement of two conductive media, for a current density J~ along the direction indicated.

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Fig. 3. Geometry and current excitation for the computation of  fx; y; zg.

;u =

A. Effective Conductivities for Fig. 2(a) Case Let us consider first the geometry of Fig. 2(a). For the equivaand , the partially-filled FDTD lent conductances cell exhibit two different conductivities in series (i.e., along ) but with non-uniform length along as shown in Fig. 3. Hence, the equivalent conductance is obtained by evaluating the distributed parallel connection along of two series conductances per unit area that depend on , that is,

(4) where and

is the length of medium segment along is the length of medium segment determined by

Fig. 2. Top-view geometries of partially-filled FDTD cells crossed by slanted interfaces between media with conductivity tensors   and   .

the cell given a voltage drop (or vice-versa) by means . In anisotropic media, the derivaof Ohm’s law tion of the 3 3 effective conductances follow the same general steps, from the successive application of , for , where the direction of the current with respect to the material interface determines the series or parallel nature of the connection. Here, we use these basic properties to derive an effective , that relates and via Ohm’s law conductivity tensor , on partially-filled FDTD cells with two material , given by conductivity tensors

(5) and the parameters

and are given by

(6) Using (2) and (4), effective conductivities obtained as

and

are

(7) (3)

For a planar interface parallel to the axis and with arbitrary slanted intersection in the - plane, we illustrate four possible geometries of partially-filled FDTD cells in Fig. 2. We describe for these geometries. next the derivation of

and also for the partially-filled FDTD Next, for cell geometry in Fig. 2(a), we note that two different conductivities are connected in parallel with a cross-section that varies along , as illustrated in Fig. 4. Hence, the equivalent conductance is obtained by evaluating the distributed series connection

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Fig. 4. Geometry and current excitation for the computation of fx; y; zg.

 ;u

= Fig. 5. Geometry and current excitation for the computation of fx; y; z g.

along of two parallel conductances per unit length that depend on , that is

and where the and

and

=

are the cross-sectional areas corresponding to regions, respectively, given by

(8) and are the cross-sectional areas of where regions, respectively, determined by

 ;u

(13) Using (12), effective conductivities pressed as

and

are ex-

(14) (9) are cross-sectional areas over the left side and where right side (similarly for ), respectively, as indicated in Fig. 4, and the parameters and are given by

The derivation of effective conductivities for the remaining three geometries: (b), (c), and (d) in Fig. 2 follows the same simple lines as above and, for brevity, will not be shown in full detail here. Only the key expressions will be discussed here. B. Effective Conductivities for Fig. 2(b) Case

(10) Effective conductivities and (8) as

and

Effective conductivities and shown in Fig. 2(b) are expressed as

are obtained from (1)

for the geometry

(15) with

(11)

Last, to obtain and in Fig. 2(a), we note that the equivalent conductance corresponds to the parallel connection of two conductances obtained from material blocks of same length but different cross-sectional areas, as illustrated in Fig. 5. and are obtained The equivalent conductivities from (1) as

(and , resp.) are the cross-sectional where ( , resp.) region, areas over the front and back side of the respectively. and are given by Effective conductivities

(12)

(17)

(16)

LEE AND TEIXEIRA: LOCALLY-CONFORMAL FDTD FOR ANISOTROPIC CONDUCTIVE INTERFACES

with

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with

(24)

(18) and in Fig. 2(b) can be Effective conductivities again obtained using (14), as they correspond to two uniformly filled cells in parallel. Due to the symmetry of the problem, it can be shown that the effective conductivities just presented for the Fig. 2(b) case are related to those of Fig. 2(a) by a proper exchange of and indices and some of the geometric parameters. C. Effective Conductivities for Fig. 2(c) Case In the geometry in Fig. 2(c), the media interface does not cut through opposite faces, but instead through adjacent faces. Because of this, the effective conductivities are derived in two steps. The first step follows the basic procedure of handling a non-uniform filled region as discussed above, with effective conductivity given by

and in Fig. 2(c) can Effective conductivities be obtained by using (14), as they again correspond to two uniformly filled cells in parallel. D. Effective Conductivities for Fig. 2(d) Case Effective conductivities of the geometry as illustrated in Fig. 2(d) are derived analogously to the two-step procedures discussed for the geometry in Fig. 2(c). Effective conductivities and are given by

(25) where

is

(26)

(19) with

with

(27)

(20) The second step combines the conductance resulting from in the non-uniform filled region with the conductance of the region with uniform conductivity, in a parallel connection of conductances along , cf. Fig. 2(c) (note an alternative derivation considering a series connection along is equally possible). This gives

is the cross-section area of the back side over where again the region. and for the geometry Effective conductivities of Fig. 2 are expressed as

(28) (21) Effective conductivity fashion, giving

with

is calculated in a similar two-step

(22)

is obtained from the first step calculation for the where non-uniform filled part, and expressed as

(23)

(29) and in Fig. 2(d) Finally, effective conductivities are once more obtained by using (14). Due to the symmetry of the problem, it can be shown that the effective conductivities just derived for the Fig. 2(d) case are and related to those of Fig. 2(c) by a proper exchange of conductivities, and indices, and some of the geometric parameters.

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For simplicity, the above analysis have considered material interfaces that are planar and parallel to the -axis. In principle, the same general methodology can be applied to cases where the interface forms an arbitrary angle to the axis, or to nonplanar interfaces. The basic difference is the additional amount of bookkeeping necessary and the more involved integrals to be evaluated. For example, instead of (4), the generic expression in that case would read for

(30) where now the lengths and coordinates.

and

are functions of both

III. RESULTS We stress that the effective conductivity tensor elements need to be computed only once for a given FDTD simulation (prior to the time-marching loop) [16]. We first consider the results from locally-conformal FDTD simulations of an electric dipole source embedded in an anisotropic media with parameters taken from conductive Earth media. Earth formations often exhibit that is different a conductivity along the vertical direction, from the conductivity along the horizontal direction, . When the source is inclined by an oblique angle with respect to the vertical direction, the conductivity appears as a full 3 3 tensor in a coordinate system aligned with the source [16]. In the following, we assume an Earth formation with S/m and S/m, and consider throughout. The dipole source is located near a cylindrical rod filled with S/m that acts as a scatterer. an isotropic conductivity Locally-conformal FDTD results are compared against conventional FDTD results for different discretization scales. A cylindrical grid aligned to the dipole source is used; hence, the cylindrical rod is not centered on the -axis and the interface between the rod and the surrounding Earth formation is nonconformal to the FDTD grid. In this case, conventional FDTD modeling necessitates a staircasing approximation. We consider a cylindrical rod with radius equal to 9 inches and eccentricity equal to 3.5 inches. The formulation described in the previous section for rectangular grids is applicable to cylindrical grids by making and the (local) correspondence of indices , and by incorgrid elements porating periodic boundary conditions on the direction. The computational domain has grid depends on the chosen. The latter varies points, where cm to cm. The cell sizes are from and uniform in the - and - directions with cm. The dipole excitation is the second derivative of a Blackman-Harris pulse [21] centered at 2 MHz. The dipole , where the grid point is located at changes according to the variation on . The field is sampled at . We assume everywhere. Because the conduction current dominates the displacement current in the ) we assume everywhere formation (that is, as well, for simplicity. The reference result is obtained using a

grid points and cm. An open-domain is assumed and hence the FDTD algorithm uses a 10-cell cylindrical perfectly matched layer in the outer boundaries [21], [22], extended to anisotropic media [23]. Fig. 6 shows locally-conformal FDTD results and illustrates the gain in accuracy versus conventional FDTD simulations at same discretization scale. We next consider a problem involving a circular loop transmitter antenna around a perfectly electric conducting circular mandrel that reside inside a circular borehole of radius 12 inches surrounded by an anisotropic conductive Earth formation. We assume the mandrel and antenna radii are equal to 4 inches and 4.5 inches, respectively. The loop antenna is within the plane and centered on the axis. In this case, the 3-D cylindrical grid is adopted to avoid staircasing of the circular geometry of the antenna and mandrel. However, because the circular boundary of the borehole is not centered on the -axis, it follows that the interface between the borehole medium and the surrounding Earth formation is nonconformal to the FDTD grid. This geometry is pertinent to logging-while-drilling sensors used for oil and gas exploration, where the eccentricity arises from mechanical vibrations and/or gravitational pull, and is illustrated, for example, in [24]. The loop antenna is excited with a ramped sinusoidal current [25] at operating frequency MHz. The borehole is filled with an (isotropic) fluid with S/m (oil-based mud). In the FDTD grid, a uniform discretization is utilized along the - and -directions, with cm and 200 grid points utilized along the azimuth direction. We compute the amplitude and phase of the voltage (transimpedance for a unit current excitation) at a second (identical) loop antenna located 30 inches away from the transmitter antenna along the -axis. We first compare the relative errors in the amplitude and phase computed using the proposed locally-conformal FDTD and the conventional FDTD (i.e., with staircasing on the borealong the radial hole boundary) for different cell sizes direction. The relative error is defined as

fine grid with

(31) is the computed result of either the locally-conwhere , and formal FDTD or the conventional FDTD for various is a reference result obtained using conventional FDTD (i.e., the with a much finer grid size. The eccentricity offset offset between the -axis and the borehole axis) is first fixed at 3 inches. The computational domain is discretized using grids, where is varied from in 180 to 104 points. This corresponds to a variation on the vicinity of the borehole interface from cm cm. The cell size is chosen non-uniform to along the radial direction, and gradually increases to a value cm deeper into the Earth formation, as the of discretization is less critical there. The reference result is obtained by a FDTD grid with grid points and cm in the vicinity of the borehole interface. Fig. 7 shows that the proposed locally-conformal

LEE AND TEIXEIRA: LOCALLY-CONFORMAL FDTD FOR ANISOTROPIC CONDUCTIVE INTERFACES

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Fig. 7. Convergence of the relative error E on the phase (top) and amplitude (bottom) of the receiver voltage, for an FDTD implementation using effective conductivities versus a standard FDTD implementation using staircasing, as a function of the radial discretization scale . The anisotropic Earth formation has  and  : S/m, with  .

= 10

Fig. 6. Time-domain scattered field from a cylindrical rod embedded in anisotropic conductive media (see main text for details), and sampled at a point exterior to the cylindrical rod. The top plot shows the field computed using conventional FDTD (with staircasing) for various cell sizes. The plot at the middle shows the same field computed by the proposed locally-conformal FDTD algorithm, showing better convergence for the same cell sizes. The bottom plot shows the zoomed view at a particular time window to further illustrate the gain in accuracy provided by the proposed locally-conformal FDTD algorithm.

algorithm provides a much faster convergence to the reference results. – We next let the offset vary in the interval inches for fixed degrees, and compare the amplitude

=25

1 = 45

and phase results obtained by the proposed locally-conformal FDTD against a finer-grid staircased FDTD, as shown in Fig. 8. In this case, the computational domain is discretized using , where is varied from 122 to 136 points so that cm in the vicinity of the borehole interface for the locally-conformal FDTD. The cm. The finer-grid, staircased FDTD employs results presented in Fig. 8 show very good agreement. Finally, we consider results with fixed offset inches and for different oblique angles . The computational domain for the locally-conformal FDTD results has grid points leading to cm in vicinity of the borehole. The staircased , FDTD simulation has cm there. Fig. 9 illustrates how leading to the amplitude and phase vary according to the oblique angle,

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Fig. 8. Computed phase (top) and amplitude (bottom) of the receiver voltage using an FDTD with effective conductivities and a finer-grid standard FDTD (with staircasing), versus different eccentricity offsets d. The anisotropic Earth formation has  and  : S/m, with  .

= 10

=25

1 = 45

and corroborate once more the good agreement between the locally-conformal FDTD results with that of a finer-grid staircased FDTD result. IV. CONCLUSION AND FURTHER REMARKS We have described a locally-conformal approach for anisotropic conductive interfaces in the FDTD method. The new approach is based on the computation of effective conductivities in non-uniformly filled cells by a quasi-static approximation. Implementation of the new locally-conformal scheme in FDTD simulations involving anisotropic conductive media have yielded improved convergence. Good agreement was observed against conventional FDTD results employing four-times finer grids under a varied range of conditions. As a final note, the approach outlined here can be adapted for the discretization of any inhomogeneous transport coefficient connecting a curl-free driving field to a divergence-free current

Fig. 9. Computed phase (top) and amplitude (bottom) of the receiver voltage using an FDTD with effective conductivities and a finer-grid standard FDTD simulation with staircasing, versus different oblique angles  . The eccentricity inches. The anisotropic Earth formation again has  offset is fixed, d and  : S/m.

10

=

1 =3 = 25

density or, using the language of differential forms [26], [27], the discretization of a Hodge operator connecting a one-form to a two-form. For example, it can be applied to FDTD simulations of heat transfer in media with inhomogeneous anisotropic thermal conductivity, with the heat-current density and the temperature gradient in place of the electric current density and the electric field intensity, respectively. REFERENCES [1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, pp. 302–307, 1966. [2] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain Method, 3rd ed. Boston, MA: Artech House, 2005. [3] J. T. MacGillivray, “Trillion cell CAD-based Cartesian mesh generator for the finite-difference time-domain method on a single-processor 4-GB workstation,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2187–2190, 2008.

LEE AND TEIXEIRA: LOCALLY-CONFORMAL FDTD FOR ANISOTROPIC CONDUCTIVE INTERFACES

[4] W. Yu, “On the solution of a class of large body problems with full or partial circular symmetry by using the finite-difference time-domain (FDTD) method,” IEEE Trans. Antennas Propag., vol. 48, no. 12, pp. 1810–1817, Apr. 2000. [5] T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finitedifference time-domain modeling of curved surfaces,” IEEE Trans. Antennas Propag., vol. 40, no. 4, pp. 357–365, Apr. 1992. [6] S. Dey and R. Mittra, “A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects,” IEEE Trans. Microwave Guided Wave Lett., vol. 7, no. 9, pp. 273–275, Sep. 1997. [7] C. J. Railton and J. B. Schneider, “An analytical and numerical analysis of several locally-conformal FDTD schemes,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 1, pp. 56–66, 1999. [8] M. Chai, T. Xiao, and Q. H. Liu, “Conformal method to eliminate the ADI-FDTD staricasing errors,” IEEE Trans. Electromagn. Compat., vol. 48, no. 2, pp. 273–280, May 2006. [9] N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Trans. Microwave Theory Tech., vol. 45, no. 9, pp. 1645–1648, Sep. 1997. [10] J.-Y. Lee and N.-H. Myung, “Locally tensor conformal FDTD method for modeling arbitrary dielectric interfaces,” Microw. Opt. Technol. Lett., vol. 23, no. 4, pp. 245–249, 1999. [11] W. Yu and R. Mittra, “A conformal finite difference time domain technique for modeling curved dielectric surface,” IEEE Trans. Microw. Wireless Compon. Lett., vol. 11, pp. 25–27, Jan. 2001. [12] I. Zagorodnov, R. Schuhmann, and T. Weiland, “Conformal FDTDmethods to avoid time step reduction with and without cell enlargement,” J. Comp. Phys., vol. 225, pp. 1493–1507, 2007. [13] S. Wang and F. L. Teixeira, “Some remarks on the stability of time-domain electromagnetic simulations,” IEEE Trans. Antennas Propagat., vol. 52, no. 3, pp. 895–898, 2004. [14] Y. Taniguchi, Y. Baba, N. Nagaoka, and A. Ametani, “An improved thin wire representation for FDTD computations,” IEEE Trans. Antennas Propag., vol. 56, no. 10, pp. 3248–3252, 2008. [15] C. J. Railton, D. L. Paul, and S. Dumanli, “The treatment of thin wire and coaxial structures in lossless and lossy media in FDTD by the modification of assigned material parameters,” IEEE Trans. Electromagn. Compat., vol. 48, no. 4, pp. 654–660, 2006. [16] H.-O. Lee and F. L. Teixeira, “Cylindrical FDTD analysis of LWD tools through anisotropic dipping-layered earth media,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 2, pp. 383–388, Feb. 2007. [17] Y.-K. Hue, F. L. Teixeira, L. E. San Martin, and M. Bittar, “Threedimensional simulation of eccentric LWD tool reponse in boreholes through dipping formations,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 2, pp. 257–268, Feb. 2005. [18] J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antennas Propag., vol. 41, no. 7, pp. 994–996, July 1993. [19] G. R. Werner and J. R. Cary, “A stable FDTD algorithm for non-diagonal anisotropic dielectrics,” J. Comp. Phys., vol. 226, pp. 1085–1101, 2007.

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[20] A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett., vol. 31, no. 20, pp. 2972–2974, 2006. [21] F. L. Teixeira and W. C. Chew, “Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 1530–1543, July 2000. [22] F. L. Teixeira and W. C. Chew, “Complex space approach to perfectly matched layers: A review and some new developments,” Int. J. Num. Model., vol. 13, no. 5, pp. 441–455, 2000. [23] F. L. Teixeira and W. C. Chew, “A general approach to extend Berenger’s absorbing boundary condition to anisotropic and dispersive media,” IEEE Trans. Antennas Propagat., vol. 46, no. 9, pp. 1386–1387, 1998. [24] Y.-K. Hue and F. L. Teixeira, “Analysis of tilted-coil eccentric borehole antennas in cylindrical multilayered formations for well-logging applications,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1058–1064, 2006. [25] C. M. Furse, D. H. Roper, and D. N. Buechler, “The problem and treatment of DC offsets in FDTD simulations,,” IEEE Trans. Antennas Propag., vol. 48, no. 8, pp. 1198–1201, Aug. 2000. [26] F. L. Teixeira and W. C. Chew, “Lattice electromagnetic theory from a topological viewpoint,” J. Math. Phys., vol. 40, no. 1, pp. 169–187, 1999. [27] B. He and F. L. Teixeira, “On the degrees of freedom of lattice electrodynamics,” Phys. Lett. A, vol. 336, no. 1, pp. 1–7, 2005.

Hwa Ok Lee received the M.S. degree in electrical engineering from The Ohio State University, Columbus, in 2005, where she is currently working toward the Ph.D. degree. Her current research interests include computational electromagnetics and numerical methods in general.

Fernando L. Teixeira received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1999. From 1999 to 2000, he was a Postdoctoral Research Associate with the Massachusetts Institute of Technology, Cambridge. Since 2000, he has been with the ElectroScience Laboratory and the Department of Electrical and Computer Engineering, The Ohio State University, where he is now an Associate Professor. His current research interests include analytical and numerical techniques for wave propagation and scattering problems. Dr. Teixeira has received many prizes for his research, including the NSF CAREER Award in 2004, the triennial USNC/URSI Booker Fellowship in 2005, and the IEEE MTT-S Outstanding Young Engineer Award in 2010.

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An Auxiliary Differential Equation Method for FDTD Modeling of Wave Propagation in Cole-Cole Dispersive Media Ioannis T. Rekanos, Member, IEEE, and Theseus G. Papadopoulos

Abstract—A method for modeling time-domain wave propagation in dispersive Cole-Cole media is presented. The Cole-Cole model can describe the frequency dependence of the electromagnetic properties of various biological tissues with great accuracy over a wide frequency range and plays a key role in microwave medical imaging. The main difficulty in the time-domain modeling of Cole-Cole media is the appearance of fractional time derivatives. In the proposed method a Padé approximation is employed resulting in auxiliary differential equations of integer order. A finite-difference time-domain method is developed to solve the differential equations obtained. The comparison of analytical and calculated relative complex permittivity values over wideband frequency domain proves the validity of the method. Index Terms—Finite-difference time-domain (FDTD), Cole-Cole model, Padé approximation, dispersive media.

I. INTRODUCTION

T

HE finite-difference time-domain (FDTD) method is an accurate and reliable tool for simulating the electromagnetic wave propagation in dispersive media [1], [2]. Basically, two approaches are followed in FDTD schemes to model dispersive media, both starting from the polarization equation that describes the relation between the electric flux density and the electric field intensity. In the first approach, convolution integrals in the time domain of the electric field and the permittivity are proposed to derive the electric flux density. In the second approach, the polarization equation is transformed from the frequency to the time domain resulting in an auxiliary differential equation (ADE), which is appropriately discretized. Thus, the Maxwell’s curl equations and the ADE form a system that is solved recursively by the FDTD. Both approaches have been applied successfully in the case of Debye and Lorentz dispersive media, where the ADE involves derivatives of integer order that are discretized by means of finite differences. Cole and Cole showed that experimental results lead to the conclusion that Debye and Lorentz models are not accurate for the representation of the frequency dependence of some dispersive media and proposed the empirical Cole-Cole dispersion relation [3]. Actually, the Cole-Cole dispersion relation was found appropriate to model the electromagnetic properties of Manuscript received December 04, 2009; revised April 14, 2010; accepted April 29, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The authors 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.2010.2071365

biological tissues [4], [5], thus it has attracted significant interest when microwave medical imaging applications are considered. Furthermore, the modeling of time-domain wave propagation in Cole-Cole media is of great importance for the study of interactions between microwave pulses and biological tissues. However, the presence of time-domain differentiators of fractional order in the Cole-Cole model makes the implementation of FDTD schemes very difficult, although several techniques have been proposed to deal with this problem. In its original presentation, the proposal of the Cole-Cole model was based on a distribution of relaxation times. By approximating this continuous distribution by a discrete one involving a finite set of relaxation times, Kashiwa et al. proposed the use of as many polarization relations as relaxation times [6]. This results in a set of ADEs that are appropriately weighted in the time-domain computational scheme. Actually, this approach is equivalent to the approximation of the Cole-Cole relative complex permittivity by a sum of Debye terms. Based on the time-domain polarization equation, Torres et al. approximated the fractional derivative, which involves a convolution integral, by a sum of decaying exponentials [7]. This method that is totally developed in the time-domain has also been adopted in [8]. Starting from the frequency-domain representation of relative complex permittivity, Guo et al. utilized the bilinear -transform to obtain a discrete-time polarization equation [9]. In this approach, the fractional order exponents that appear are approximated by polynomials using least squares. As in [6], Kelley et al. proposed the approximation of the Cole-Cole relation by an expansion of Debye terms [10]. In their approach, the estimation of relaxation times, which is a nonlinear optimization problem, was based on particle swarm optimization. Recently, an FDTD implementation of the Cole-Cole dispersion has been presented [11], which, as in [10], utilizes the approximation of the Cole-Cole permittivity by a few Debye terms. In [11], the relaxation times are derived by applying the Nelder-Mead algorithm, which is a heuristic nonlinear optimization method. In the abovementioned methods that employ Debye terms to approximate the Cole-Cole model, the selection of relaxation times and weighting coefficients is a nonlinear problem. Hence, nonlinear optimization techniques are required, which are time consuming to converge, their convergence is not always guaranteed, and they may be trapped in local minima. In this paper, a new FDTD approach to model Cole-Cole media is presented. The method is based on the approximation of the fractional order differentiation term. This term is approximated by a rational function using Padé approximation. As a

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REKANOS AND PAPADOPOULOS: AN AUXILIARY DIFFERENTIAL EQUATION METHOD FOR FDTD MODELING OF WAVE PROPAGATION

result an ADE is derived, which involves time derivatives of integer order. Hence, the application of finite differences within the FDTD framework is easy and straightforward. Moreover, the proposed approach, in contrast to the Debye approximation ones, does not require the solution of any nonlinear optimization problem to estimate Debye relaxation times or other approximation parameters. In the present work, Cole-Cole media with a single relaxation time are considered. A further extension to the case of Cole-Cole media with multiple relaxation times is an issue of significant importance and has already been the subject of a recent letter publication of ours [12]. The idea of employing Padé approximants to model the complex relative permittivity of dispersive media has been previously discussed by Weedon and Rappaport [13]. However, in [13], only Debye, Lorentz and Drude types of media had been considered, where, unlike Cole-Cole media, the time-domain polarization relations do not involve fractional derivatives. The rest of the paper is organized as follows: In Section II, the derivation of the auxiliary differential equation is presented. The FDTD scheme based on the proposed ADE and the corresponding recursive relations are given in Section III. Numerical results that illustrate the validity of the proposed method are presented in Section IV. Finally, the conclusions are discussed in Section V.

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is the phasor of the polarization field, which In (4), in the case of a Cole-Cole medium is given by the polarization relation (5) where . According to inverse Fourier transform, the frequency-do, is transformed into an main term th-order time derivative, i.e., (6) is not an integer, the term However, when the exponent of with , which appears in the polarization relation (5), is transformed, in the time domain, into a fractional derivative, i.e.,

(7) where is the gamma function. As a result, if we take the inverse Fourier transform of the polarization relation (5), we derive the following time-domain differential equation

II. THE AUXILIARY DIFFERENTIAL EQUATION We consider a Cole-Cole dispersive medium with relative complex permittivity given by (1)

and are the optical and the static relative permitwhere tivity, respectively, is the relaxation time, and is the angular is a measure of the frequency. The parameter we obtain the dispersion broadening. We note that for Debye dispersion relation. Usually, the relative complex permittivity (1) is expressed in terms of its real and imaginary part, i.e., , which are given by (2)

(3) where and . The imaginary part (3) gets . The frequency is its maximum value when the relaxation frequency of the medium and indicates the transition region from the static permittivity, , at low frequencies to the optical permittivity, , at high frequencies. In the frequency domain, the relation between the electric and the electric field intensity flux density is given by (4)

(8) The numerical treatment of (8) by means of the FDTD method is in general difficult because the fractional derivative involves time differentiation of a convolution integral. For this reason an alternative approach is proposed, where the objective by means of a rational funcis to approximate the term tion of . For simplicity, we introduce the imaginary variable where is a “characteristic” frequency. Thus, the polarization relation (5) is written as (9) Although there are no restrictions in the selection of , it could , or equal be chosen equal to the relaxation frequency to the central frequency of excitation. is not analytic at (its In (9), the function diverge), which means that a Maclaurin derivatives at expansion does not exist. Furthermore, a Taylor expansion about diverges because is not analytic in the disc with radius ; the disc contains the singular point . Hence, the Padé approximation about is adopted, while, in order to simplify the calculations without loss of generality, we choose . In particular, is approximated by a rational function, i.e.,

(10)

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where and . For the coefficients in the Padé approximacalculation of the and at are required (see Appendix). tion the derivatives of By substituting (10) into (9) the polarization relation takes the following form in the frequency domain

(11)

where (12)

(13)

By applying finite differences, the th order time derivative of , at the time instance ( is the time step), is expressed by the sum (19) In (19), the superscripts of denote time indices, are the parameters related to the finite difference approximation, while , depend on the order of approxthe limits of summation, imation accuracy. For example, if we approximate the second of by means of central differences order derivative , and of second order accuracy, then . Actually, for all time derivatives we adopt central differences with second order accuracy, which results in . Finally, after some manipulation, the polarization is derived from (18) and field at the time instance is given by

Furthermore, the approximated relative complex permittivity derived by the Padé approximation is given by (20) (14)

where (21)

By taking the inverse Fourier transform of (11), we derive the time-domain differential equation

(22)

(15)

with . If we apply finite differences of second order accuracy to the Maxwell’s curl (16) at the time we obtain instance

In fact, (15) is an auxiliary differential equation that approximates (8). Clearly, (15) contains time derivatives of integer order and it is easy to be discretized by means of finite differences.

(23) If we substitute from (20) into (23), the equation for updating the electric field is derived, i.e.,

III. FDTD SCHEME The FDTD scheme adopted in the present work to simulate the wave propagation in a Cole-Cole medium is based on central differences applied to the auxiliary differential equation (15) as well as to the Maxwell’s curl equations (16)

(24)

(17) In what follows, we will consider that the Padé approximation (10) involves polynomials of the same order , a choice clarified in more detail in Section IV. Under this assumption the auxiliary differential equation (15) is written as (18)

A. Steps of the FDTD Scheme Let us assume that the electric and the polarization field are as well as the magnetic field at known till the time instance the instance . Also, to simplify the updating equations for the electric and the polarization field and to reduce the computational burden, terms that are common in (20) and (24)

REKANOS AND PAPADOPOULOS: AN AUXILIARY DIFFERENTIAL EQUATION METHOD FOR FDTD MODELING OF WAVE PROPAGATION

can be calculated once in each iteration. Then, the steps of a single iteration of the FDTD scheme are the following: 1) The auxiliary vector given by

(25) is computed. is actually the common term in (20) and (24). 2) The electric field is updated utilizing the temporary vector , i.e., (26) 3) The polarization field is updated, i.e., (27)

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In order to investigate the computational burden of the proposed scheme, we consider the number of multiplications (or divisions) per field component and computational cell, without taking into account the operations needed for updating the magnetic field. Also, the time independent parameters such as , and are computed once. Thus, from (25), (26), and (27), we conclude that updating the electric multiplications. In and the polarization field requires terms of the Padé order, the number of multiplications equals or . Compared to the number of multiplications needed in [9], the proposed scheme is almost two times faster. The main reason is the introduction of the auxiliary vector , which is common in (26) and (27). Also, the proposed scheme is faster compared to the -term Debye approximation schemes, where the number of multiplications might be equal to , or depending on the particular implementation [11]. Moreover, the Debye-based approaches, unlike the proposed one, require the estimation of relaxation times; a task, which is time consuming, because it involves the solution of a multivariable nonlinear optimization problem.

4) Finally, the updated magnetic field is given by (28) which is derived from (17) by applying finite differences at the time instance . B. Computational Demands From the updating formulae (25)–(28), one can derive the storage demands and the computational burden of the proposed FDTD scheme. In the following, the memory requirements and the number of operations needed will be given on the basis of a single field component evaluation, in a single computational cell. We will examine only the computational demands related to the electric and the polarization field, because the updating relation for the magnetic field (28) is the same in all the FDTD schemes. The relations for updating the electric and the polarization field involve the computation of the auxiliary vector given by (25). It is clear that memory locations are required for storing the previous values of and . Concerning the auxiliary vector , which is used in (26) and (27), only a single memory location is needed for the whole computational domain, because keeping previous values of for further use is not required. As a result, apart from the magnetic field, the storage memory demands of the proposed FDTD scheme involves locations. In terms of the Padé order , the number of memory or , if is even or odd, locations needed is equal to respectively. On one hand, the above memory requirements are actually identical to those associated with the scheme proposed in [9], where the bilinear -transform is employed. On the other hand, the approximation of the Cole-Cole medium by means of Debye terms results in a number of stored variables varying from to , depending on the particular implementation [11]. Hence, if we assume that an th order Padé approximant corresponds to an -term Debye approximation, then the proposed FDTD scheme has almost the same storage require. ments with the Debye-based ones, given that

IV. NUMERICAL RESULTS A. Selection of the Padé Approximation Order When the Padé approximation is adopted, a fundamental question that arises concerns the selection of the orders of polynomials, and . Although, there is no restriction in the selection of the orders, we can achieve significant accuracy in the approximation of the original relative complex permittivity, by choosing appropriately the values of and . In the present study, we are either equal or differ investigate the cases where and by one. To quantify the Padé approximant accuracy, the relative mean square (rms) error of both the real and the imaginary part of as well as of the itself are evaluated. In particular, the rms error between and is given by (29) where the integrations are evaluated along the frequency domain of interest. Obviously, the rms errors of the real part and , and the imaginary part are denoted by respectively. Two examples that illustrate the accuracy of the Padé approximation are examined. The relative complex permittivity of a , and Cole-Cole medium with ps is approximated. For this medium, two different values of ( and ) are considered. In both cases, the approximation accuracy is examined for a wide frequency range, namely from 10 MHz to 100 GHz. The approximation errors ocare presented curred for different values of the orders and and , respectively. in Tables I and II, for From both Tables I and II, it is clear that when the rms approximation error of the relative complex permittivity is higher compared to the case . Furthermore, from (15) it is evident that the order of the ADE, which actually governs the computational demands in both storage and

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TABLE I RELATIVE MEAN SQUARE ERRORS OBTAINED FROM PADÉ APPROXIMATION ;" ; : ) PS, OF " OF A COLE-COLE MEDIUM (" IN THE FREQUENCY RANGE 10 MHZ–100 GHZ FOR DIFFERENT VALUES OF THE POLYNOMIAL ORDERS N AND M

TABLE II RELATIVE MEAN SQUARE ERRORS OBTAINED FROM PADÉ APPROXIMATION OF ;" ; PS, : ) " OF A COLE-COLE MEDIUM (" IN THE FREQUENCY RANGE 10 MHZ–100 GHZ FOR DIFFERENT VALUES OF THE POLYNOMIAL ORDERS N AND M

Fig. 1. Relative mean square errors derived by the Padé approximation of the ;" original relative complex permittivity of the Cole-Cole medium (" ; ps, : ) vs. the Padé order N N M .

Fig. 2. Relative mean square errors derived by the Padé approximation of the original relative complex permittivity of the Cole-Cole medium (" ;" ; ps, : ) vs. the Padé order N N M .

50 = 153

execution time of the FDTD, is equal to . Similar results, not presented here, have been obtained for other values of varying between 0.65 and 0.9. From the above, it is the favorable choice and is concluded that setting in the following, when we simply refer to th order Padé ap-

proximant, we will also mean that . The approximaare illustrated in Figs. 1 tion rms errors versus the order and , respecand 2, for the cases where tively. It can be clearly seen that as the order of the Padé approximant increases, the rms errors decrease. For order

=2

50 = 153

=09

= 50 = 153

( = )

=09

=2

=

=2

= 0 65

= 50 = 153

( = )

= 0 65

=2

=

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Fig. 5. Time variation of the excitation field. Fig. 3. Real and imaginary part of the original, " , and the Padé approximated, "^ , relative complex permittivity of the Cole-Cole medium (" = 2; " = 50;  = 153 ps, = 0:9) for Padé order N = 4.

Fig. 6. Normalized spectrum of the excitation field.

Fig. 4. Real and imaginary part of the original, " , and the Padé approximated, "^ , relative complex permittivity of the Cole-Cole medium (" = 2; " = 50;  = 153 ps, = 0:65) for Padé order N = 4.

and higher, great approximation accuracy is achieved. However, appears to be sufficient because the rms errors obtained for in approximating are below 0.5% ( and for ). The real and the imaginary part of the original and the Padé approxirelative complex permittivity of the mated Cole-Cole medium described earlier are depicted in Figs. 3 and and , respectively. It is noteworthy that 4 for excellent approximation accuracy is achieved in a very wide frequency range from 10 MHz to 100 GHz. B. Validation of the Method After the selection of the order of the Padé approximant and the evaluation of its accuracy in modeling the original Cole-Cole medium, the approximated relative complex permittivity is employed by the FDTD scheme. In order to illustrate the application of the proposed method, we consider an one-dimensional problem, where the Cole-Cole medium occupies the region

, while the rest space is vacuum. A plane wave, which propagates along the positive direction with the electric field polarized in the axis, excites the Cole-Cole medium. The incident wave is a modulated gaussian pulse given by (30) with and central frequency GHz. The time variation of the excitation field is presented in Fig. 5, while its normalized spectrum is shown in Fig. 6. The Cole-Cole medium considered is exactly the same as in the previous numerical investigation, i.e., its characteristic pa, and ps, while two rameters are values of ( and ) are examined. The order , which is an adequately high of the Padé approximant is order, meaning that larger does not improve the accuracy of the approximation significantly, as shown previously. In order to examine the efficiency of the FDTD scheme when the proposed ADE is utilized, the simulated fields are being processed. Our aims are to estimate the relative complex permittivity of the medium as well as to investigate the numerical dispersion. It is well-known that the Fourier transforms of

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the electric field at two distinct positions and inside the Cole-Cole medium are related through the transfer function , i.e., (31) where (32) (33) is the complex wave number. and are comIf the Fourier transforms puted, then, from (31) and (32), the real and the imaginary parts are obtained as of

Fig. 7. Real and imaginary part of the FDTD estimated, "~ , and the Padé approximated, "^ , relative complex permittivity of the Cole-Cole medium (" = 2; " = 50;  = 153 ps, = 0:9), when fourth order auxiliary differential equation is employed.

(34) (35) respectively. By substituting (34) and (35) into (33), we obtain an estimate of the relative complex permittivity, i.e., (36) The abovementioned procedure for estimating the relative complex permittivity has been applied in both cases of the Coleand . It should Cole medium, namely for be mentioned that the FDTD simulated electric field has been . Hence, false recorded at two adjacent FDTD cells interpretation of the results of the multivalued argument func. tion in (35) is avoided, resulting in a reliable estimate of Fig. 7 presents the comparison between the estimate of the relative complex permittivity, , derived by the FDTD simulation . and the Padé approximated one, , for the case when The frequency range presented varies from 100 MHz to 10 GHz, because this range corresponds to the spectral content of the excitation (see Fig. 6). It should be noted that it is pointless to depict the original permittivity in Fig. 7, because it is almost identical to the Padé approximated one. The corresponding reare presented in Fig. 8. Finally, sults for the case when the rms errors between the FDTD estimated and the Padé ap, proximated permittivity are evaluated using (29). For the rms errors obtained are , and . For , the rms errors , and we get are . can be estimated from Finally, the transfer function the Fourier transforms and using (31). By with the analytical one derived comparing the estimate of from (32), we can investigate possible numerical dispersion inis obtained herent in the FDTD scheme. The estimate of

Fig. 8. Real and imaginary part of the FDTD estimated, "~ , and the Padé approximated, "^ , relative complex permittivity of the Cole-Cole medium (" = 2; " = 50;  = 153 ps, = 0:65), when fourth order auxiliary differential equation is employed.

and . Fig. 9 presents the real and the for both , while imaginary part of derived by the FDTD, when the separation distance between the two distinct recording positions of the electric field is equal to and . The corare illustrated in Fig. 10. The responding results for fact that in Figs. 9 and 10 the FDTD and the analytical results for the transfer function are almost identical, even when the sep, shows clearly that no significant aration distance is numerical dispersion is introduced. We have to clarify that the absence of numerical dispersion should not be totally attributed to the efficiency of the proposed FDTD scheme. In particular, numerical dispersion is expected mostly at high frequencies of the propagating pulse. However, in the Cole-Cole medium, high frequencies dissipate rapidly as it is shown in Figs. 9(b) and 10(b).

REKANOS AND PAPADOPOULOS: AN AUXILIARY DIFFERENTIAL EQUATION METHOD FOR FDTD MODELING OF WAVE PROPAGATION

T

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T

Fig. 9. Real and imaginary part of the transfer function (d; ! ) derived from the FDTD simulation by postprocessing electromagnetic field recordings inside the Cole-Cole medium (" = 2; " = 50;  = 153 ps, = 0:9). The separation distance between the two recording positions is (a) d = 101z and (b) d = 1001z .

Fig. 10. Real and imaginary part of the transfer function (d; ! ) derived from the FDTD simulation by postprocessing electromagnetic field recordings inside the Cole-Cole medium with (" = 2; " = 50;  = 153 ps, = 0:65). The separation distance between the two recording positions is (a) d = 101z and (b) d = 1001z .

V. CONCLUSION

ered. Therefore, the proposed modeling of wave propagation in Cole-Cole media could be a valuable tool in the FDTD simulation of interactions between microwave radiation and biological tissues.

The proposed FDTD formulation features a new approach to the problem of electromagnetic wave propagation in Cole-Cole dispersive media. The method circumvents the difficulty that appears due to the presence of the fractional order time derivative in the polarization relation, by utilizing a Padé approximation in the frequency domain. Hence, an auxiliary differential equation is derived to represent the relation between the electric flux density and the electric field intensity. Moreover, the relations for the calculation of the fields are quite simple and easy to implement. In contrast to previously reported methods, which approximate the Cole-Cole relative complex permittivity by means of few Debye terms and use nonlinear optimization techniques to select the appropriate relaxation times, the proposed method approximates the permittivity in a much more straightforward and solid way, without the solution of any nonlinear optimization problem. The numerical results obtained by applying the proposed ADE within the FDTD scheme verify the accuracy of the method even when broadband field excitations are consid-

APPENDIX is a rational funcThe Padé approximant of a function tion whose power series expansion is in agreement with the up to the highest possible order [14]. Taylor expansion of It should be noted that the Taylor expansion might not converge, whereas the Padé approximant does. In its original form the Padé approximant is derived from the Maclaurin expansion of . However, in our case the Padé approximant is obtained , because the derivatives from the Taylor expansion about with at diverge. of If the Taylor expansion of about is given by (A.1)

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with (A.2) then the coefficients of the Padé approximant (10) and are derived by setting

(A.3)

where

. After cross-multiplying (A.3) we obtain

(A.4)

from By equating the coefficients of same powers of , we derive two sets of equations that result in zero until the evaluation of the Padé coefficients, i.e.,

[6] T. Kashiwa, Y. Ohtomo, and I. Fukai, “A finite-difference time-domain formulation for transient propagation in dispersive media associated with Cole-Cole’s circular arc law,” Microwave Opt. Technol. Lett., vol. 3, no. 12, pp. 416–419, Dec. 1990. [7] F. Torres, P. Vaudon, and B. Jecko, “Application of new fractional derivatives to the FDTD modeling of pulse propagation in a Cole-Cole medium,” Microwave Opt. Technol. Lett., vol. 13, no. 5, pp. 300–304, Dec. 1996. [8] J. W. Schuster and R. J. Luebbers, “An FDTD algorithm for transient propagation in biological tissue with a Cole-Cole dispersion relation,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 1998, vol. 4, pp. 1988–1991. [9] B. Guo, J. Li, and H. Zmuda, “A new FDTD formulation for wave propagation in biological media with Cole-Cole model,” IEEE Microwave Wireless Compon. Lett., vol. 16, no. 12, pp. 633–635, Dec. 2006. [10] D. Kelley, T. J. Destan, and R. J. Luebbers, “Debye function expansions of complex permittivity using a hybrid particle swarm-least squares optimization approach,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 1999–2005, Jul. 2007. [11] M.-R. Tofighi, “FDTD modeling of biological tissues Cole-Cole dispersion for 0.5–30 GHz using relaxation time distribution samples-Novel and improved implementations,” IEEE Trans. Microwave Theory Tech., vol. 57, no. 10, pp. 2588–2596, Oct. 2009. [12] I. T. Rekanos and T. G. Papadopoulos, “FDTD modeling of wave propagation in Cole-Cole media with multiple relaxation times,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 67–69, 2010. [13] W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 401–410, Mar. 1997. [14] G. A. Baker and P. G. Morris, Padé Approximants. New York, NY: Cambridge University Press, 1996.

(A.5)

(A.6) of the denominator of the Hence, the coefficients linear Padé approximant are derived from the solution of the equations (A.6). Then the coefficients of the numerator are evaluated from (A.5). REFERENCES [1] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1995. [2] F. L. Teixeira, “Time-domain finite-difference and finite-element methods for Maxwell equations in complex media,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2150–2166, Aug. 2008. [3] K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics I. Alternating current characteristics,” J. Chem. Phys., vol. 9, no. 4, pp. 341–351, Apr. 1941. [4] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues. III. Parametric models for the dielectric spectrum of tissues,” Phys. Med. Biol., vol. 41, no. 11, pp. 2271–2293, Nov. 1996. [5] M. Lazebnik, L. McCartney, D. Popovic, C. B. Watkins, M. J. Lindstrom, J. Harter, S. Sewall, A. Magliocco, J. H. Booske, M. Okoniewski, and S. C. Hagness, “A large-scale study of the ultrawideband microwave dielectric properties of normal breast tissue obtained from reduction surgeries,” Phys. Med. Biol., vol. 52, no. 10, pp. 2637–2656, May 2007.

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.

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, Greece. His research interests include computational methods for wave propagation and inverse scattering problems.

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Numerically Stable Moment Matching for Linear Systems Parameterized by Polynomials in Multiple Variables With Applications to Finite Element Models of Microwave Structures Ortwin Farle and Romanus Dyczij-Edlinger, Member, IEEE

Abstract—Parametric model-order reduction is a very powerful methodology for analyzing large-scale systems with multiple parameters. This paper extends the theory of moment-matching single-point methods to linear systems parameterized by multivariate polynomials. We propose a new algorithm that exhibits high numerical robustness and short runtimes, allows for direction-dependent model orders, and is easy to parallelize. To demonstrate the accuracy and efficiency of the suggested approach, we present the response surfaces of two microwave finite-element models, featuring the operating frequency and material properties as parameters. Index Terms—Reduced order systems, finite element methods.

I. INTRODUCTION

N

UMERICAL methods of electromagnetic (EM) fields computation have in common that the size of the resulting algebraic systems is very large. Nevertheless, thanks to powerful computers and efficient solvers, runtimes for single solutions are often just in the order of minutes. However, in many important applications, e.g., parametric studies, design optimization, or the solution of inverse problems, EM structures need to be characterized not just at one given point but over wide ranges of parameters such as the operating frequency, material properties or geometrical quantities. In such cases, very large numbers of solutions are required, and computer runtime still is a limiting factor. Reduced-order models (ROMs) offer a very attractive remedy, because they are of low dimension and hence very cheap to analyze. While single-parameter methods [1]–[3] are well-known from fast frequency sweeps, multi-parameter approaches [5]–[14] are yet to establish themselves as a mainstream technology. Most modern MOR methods are projection-based, i.e., they restrict the row- and column-spaces of the original systems to suitable subspaces of low dimension; see Section II for a general framework. Specific methods differ mainly in their choice of projection basis: generally, there are single-point (SP) and multi-point (MP) approaches. MP methods [6], [15] are numerically robust and comparatively simple to implement but require

Manuscript received January 19, 2010; revised March 24, 2010; accepted April 20, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The authors are with the Chair of Electromagnetic Theory, Saarland University, Campus D-66123, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2071368

one solution of the original system at each expansion point. Hence they are particularly attractive to use in combination with iterative methods. SP methods, on the other hand, are best-suited for direct solvers: once the factorization of the original system matrix at the expansion point is available, a number of forwardback substitutions and matrix-vector multiplications suffice to generate the ROM. Most SP methods are of moment-matching type, i.e., the transfer function and its leading derivatives are made to coincide with those of the original system at the expansion point. One critical issue with SP methods is numerical stability, because direct computation of higher-order moments [6], [7], [10] is known to become ill-conditioned. While this problem has been solved for the single-parameter case, see [2], [3], and [4] for linear, quadratic, and higher-order polynomial parameterization, SP methods for multiple parameters are at a less mature stage: the authors of [9] propose to discard all mixed-order terms and apply a robust single-parameter algorithm along the coordinate axes only. The method of [11] employs a superspace to the sought projection space, the structure of which allows application of the Arnoldi algorithm, but at highly increased computational costs. Finally, the generalized Arnoldi algorithm of [13], which is stable along one preferred direction, provides a special solution for applications that are broadband in one parameter, e.g., the frequency, but narrowband in all others. To the authors’ knowledge, there are only two SP methods that attempt to resolve the stability issue in a general multi-parameter setting: the approach of [12] reduces the multi-variate system to a series of single-parameter problems but applies to the multi-linear case only, whereas the work of [14] handles polynomial parameterization of arbitrary degree but does not entirely reach the robustness of single-parameter methods. This paper extends the theory of moment-matching MOR [7] to multi-variate polynomial parameterization, including the right-hand side and output functional, as well as two-sided, Petrov-Galerkin-like projections. Our analysis results in an improved algorithm that generalizes [4] to multiple parameters and [12] to higher order. It inherits the numerical robustness of the underlying single-parameter method and avoids the overhead associated with substituting an additional parameter for each higher-order term [10]. The method allows for direction-dependent model orders and outperforms the technique of [14] in memory efficiency and speed.

0018-926X/$26.00 © 2010 IEEE

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Our numerical examples are response surfaces computed by the finite element (FE) method, which feature the operating frequency and material properties as parameters. For the pre-processing required in case of geometrical variables, we refer the reader to [16].

Similarly, the solution vector Taylor series

of (7a) can be expanded in a

(10) Later on, it will prove advantageous to rewrite the series as

II. PRELIMINARIES

(11)

We employ multi-index notation with the usual conventions (1a) (1b)

wherein each sum is a homogeneous polynomial. In the following, we denote the space of homogeneous polynomials of by . Its dimension is given by degree in

(1c)

(12)

(2)

Lemma 1: Given a system of linear equations of the form (7a) and the Taylor series expansion of the solution vector (10), satisfy the recursion the vector-valued Taylor coefficients

(3) (13)

(4) We consider the single-input, single-output (SISO) system

The number of vectors

with

is

(5a) (5b)

(14)

, right-hand side , and output whose system matrix are parameterized by polynomials of maximum functional in scalar parameters degree

Proof of (13): See Appendix A. Proof of (14): By Induction; see [10] Definition 2: The generalized Krylov subspace of order and degree associated with the multi-parameter system (7a), , denotes the space of Taylor coefficients of Lemma 1: (15)

(6)

, , and with mainly interested in the transfer function we assume without loss of generality that with a simplified system , given by

. Since we are , and just deal

Def. 2 is a natural generalization of the nomenclature of [5] reduces to the generalized Krylov space of and [3]: [5] in the multi-linear case and to the higher-order Krylov space of [3] in the single-parameter case. III. PROJECTION SPACES FOR REDUCED-ORDER MODELS

(7a) (7b)

Definition 3: A projection-based ROM

of (7) is defined by (16a)

The Taylor series of

about the origin is defined by (16b) (8) with

wherein the moments

are given by (9)

(17)

FARLE AND DYCZIJ-EDLINGER: NUMERICALLY STABLE MOMENT MATCHING FOR LINEAR SYSTEMS

Herein,

, and

are given by (18a) (18b) (18c)

, and . with trial and test matrices The Taylor series expansions of the reduced-order solution vector and transfer function read

and the corresponding errors

and

(28b) A comparison to (7b) shows that the transfer function the same as that of the original system (7):

is

(29) In analogy to (10) and (13), the Taylor series expansion of and the associated Taylor coefficients are given by

(19)

(30)

(20)

(31)

are given by (21) (22)

The quality of the ROM depends critically on the choice of and . The following Lemma and Corollary address the construction of and the resulting error and generalize earlier results [7] to parameter-dependent right-hand sides: Lemma 4: Given a ROM after Def. 3 with (23) If the trial matrix

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We are now ready to present the main result of this Section: Theorem 6: Given a ROM after Def. 3 with (32) If the trial and test matrices

and

are chosen such that (33a) (33b)

wherein and are Krylov subspaces associated with (7a) and (28a), the ROM matches the leading moments of the origup to order inal model

is chosen such that (34) (24) and the error in the transfer function is of order

is the Krylov space of Def. 2, the ROM matches the wherein leading Taylor coefficients of the original system up to order , in the sense that (25) Proof: See Appendix B. Corollary 5: For the ROM of Lemma 4, the errors (22) are of order , and

(21)

(26) (27) Proof of (26): By plugging (25) into expansion (10). Proof of (27): . Note that Lemma 4 and Corollary 5 hold under the very weak . Loosely speaking, we have achieved assumption (23) on moment-matching up to order (27) solely by an appropriate . selection of the trial space In the following, we will show that the order of the error in the transfer function can be doubled by a judicious choice of . For this purpose, we introduce the the testing space transposed system

, (35)

Proof of (34): See Appendix C. Proof of (35): By plugging (34) into expansion (8). Theorem 6 generalizes the work of [5] to polynomial parameter-dependence and the results of [7] to two-sided and projections and parameter-dependent right-hand sides . By setting and output functionals in (33), it can be seen that the maximum order of the moments a ROM of dimension can match is . and are utilized to conSince the Taylor coefficients and in (24) and (33), and struct the projection matrices because matches the leading moments of the underlying about the origin (34), one might mistransfer function as the Taylor expansion of . This is not the case. take Rather, we have from (16b)

(36) (28a)

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Since and are polynomials in , becomes a rational function of the parameters. Due to the moment-matching property (34), we conclude that the reduced-order transfer is a multi-variate Padé approximation about the function origin [17]. IV. ROBUST COMPUTATION OF PROJECTION SPACES and According to Section III, the projection matrices are to span the linear hull of the multi-variate Taylor coeffiand , which in turn are defined by (13) and (31). cients The structure of these recursions reveals that, once the factorization of the system matrix at the expansion point is available, or is one forward-back suball that is required to compute stitution and a number of matrix-vector multiplications. Hence, single-point ROMs can be constructed very efficiently. However, implementing the recursions (13) and (31) directly would result in severe cancellation and make for an unreliable algorithm. This can easily be seen in the case of linear dependence on a single parameter with constant vectors and : then, (13) and (31) degenerate to the power method, which strongly amplifies components in the directions of the dominant eigenvectors. To overcome this difficulty, we propose to generalize the approach of [12] to the polynomial case. The overall idea is to choose appropriate directions in parameter space, employ proven methods to construct single-parameter ROMs along these directions, and recombine them to a multi-variate ROM. Here are the details: Definition 7: The contraction denotes the linear system with polynomial dependence on a scalar parameter which is obtained from the multi-parameter system by setting . takes the form Lemma 8: The contraction (37a)

(37b)

and satisfy the recursion (41) Proof of (40): Set in the Taylor series (11). Proof of (41): Specialize Lemma 1 to one parameter. Theorem 10: Given the multi-parameter system of (7) denote a set of contraction and its Taylor coefficients , let the Taylor coefficient of order of the directions and with . If is unisolvent in the contraction space of homogeneous polynomials , the uni- and multiand with , variate Taylor coefficients of order , respectively, span the same space: (42) Proof: By (40), every is a superposition of vectors with . On the other hand, unisolvence, as required by the with is representable Theorem, guarantees that every with . by a supersposition of Theorem 10 assures that the projection matrices and of (33) can be computed from the Taylor coefficients of appro. For such single-parameter priately chosen contractions models, robust MOR methods for computing bases of are readily available, e.g., [3], [4]. Equation (41) shows that all contractions require the action of and, in case of two-sided projections with , that , respectively. Hence, just a single matrix factorization of is needed. V. ALGORITHMIC IMPLEMENTATION Alg. 1 presents our prototype method for constructing a unitary trial matrix for a ROM of uniform order according to (33a). The main loop, from Line 3 to Line 15, is over all directions of contraction. To minimize coding and computational efforts, we utilize a hierarchical family of sets of directions, (43)

wherein wherein each lows [12]:

is unisolvent in

. Our specific choice fol(44)

Proof: By plugging into (7) and collecting terms of same power of . Lemma 9: Given the Taylor series expansion of the singleparameter system (37a) about

wherein is a multi-index of dimension . In Line 4, we compute the data (38) of the considered contraction and, in Line of the space 5, we construct a numerically stable basis of Taylor coefficients . For this goal, we employ the WCAWE method [4], because it can handle system matrices and righthand sides of arbitrary polynomial degree. We abbreviate this step by

(39)

(45)

(38)

the coefficients

Note that the total number of basis vectors Alg. 1 is

are given by (40)

computed by

(46)

FARLE AND DYCZIJ-EDLINGER: NUMERICALLY STABLE MOMENT MATCHING FOR LINEAR SYSTEMS

which exceeds the dimension of (14). Hence, many of the vectors do not contribute to the trial matrix ; see Subsection V.A for details. The purpose of the inner loop, starting from Line 7, and the subsequent if-statement of Line 8 is to let only those vectors pass which are known to carry new information. They undergo modified Gram-Schmidt (MGS) orthogonalization in Line 9 and are finally appended to the of (33b) is computed in trial matrix . The test matrix analogous fashion; see Line 6, 11, and 12. Note that, in contrast to the method of [14], no additional auxiliary vectors need to be kept in memory. Alg. 1 does not present the most efficient implementation, but it is conceptionally simple and has the advantage of keeping the WCAWE processes for different directions of contraction decoupled. It is therefore well-suited for parallelization. Algorithm 1 Prototype Implementation—Parallel Version 1: 2: 3: for all 4: Compute 5: 6: 7: for 8: if 9: 10: 11: 12: 13: end if 14: end for 15: end for 16: Compute

do according to (38)

do then

according to (18)

A. Efficiency Improvement—Serial Version Basis vectors that do not contribute to the trial matrix are not completely redundant, because they provide the history needed by recursion (41) and, in consequence, the WCAWE algorithm. The goal here is to compute such vectors cheaply. The improved method is presented in Alg. 2. Inside the main loop, starting from Line (6), we employ incremental sets of contraction directions (47) wherein contains only new directions, which have not been considered before; see Line 7. The key point is that all vectors with belong to the basis of the space of uni-variate Taylor coefficients with of some contraction. By (40), they can also be expressed in terms of multi-variate Taylor cowith , for which a basis is provided by . efficients are linear Hence all for a given contraction direction combinations of the form (48) and the coefficient vectors contraction of the ROM of order

can be computed from the already available (16)–(18)

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rather than that of the original model; see Line 9 to 15. Note that the matrices and vectors corresponding to (38) of the ROM, and , respectively. based contraction are denoted by The procedure is computationally cheap, because and no operations involving original-size matrices, particularly no , forward-back substitutions involving the factorization of are required. On the other hand, processes for different directions are no longer fully decoupled, so that parallelization becomes more complicated. We therefore dub Alg. 2 the serial version. Algorithm 2 Improved Method—Serial Version 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Compute

according to (38)

Compute according to (18) do for for all do according to (38) Compute

Compute

for

end for end for Update end for

according to (38)

do

according to (18)

VI. EXTENSIONS A. Nonzero Expansion Point So far, we have only considered moment-matching about the . However, there are situations when a general origin, may be preferable, e.g., for efficiency expansion point is singular. All that is necessary reasons or when to make the theory of Section III applicable to such cases is to represent in the form (49) B. MIMO Systems The extension to multiple-input multiple-output (MIMO) systems is based on the fact that the -th row of the -th column of the transfer function of the MIMO system (50a)

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(50b) , , with and , is identical to the transfer function of (5), prois the -th column of and the -th column vided that , respectively. This implies that, in the MIMO case, the of spaces in (33) just need to be replaced by the column spaces of and , given by the block vectors (51a)

(51b)

C. Direction-Dependent Choice of Model Order For clarity (see Alg. 1) and to discuss one important line of optimization (see Alg. 2), the method we have presented in this paper is of uniform order , i.e., the ROM order is the same in all directions. However, in many real-world applications, some parameter ranges may be wide and others narrow, so that non-uniform model orders would be more appropriate. It is one great advantage of the proposed method that the implementation of direction-dependent orders is straightforward: since Alg. 1 is based on independent single-parameter ROMs along different directions of contraction, we may set the order of each of them to whatever we deem appropriate. Note, however, that a fully automatic approach will also require an effective adaptive mechanism, which is beyond the scope of this paper; see [18] or ([19], Chapter 5) for the single-parameter case. VII. NUMERICAL EXAMPLES A. Patch Antenna The inset of Fig. 2 shows the structure of a patch antenna as a function of [20]. We consider the reflection coefficient the operating frequency GHz and the relative dielecand tric permittivities of the two substrate layers, . FE discretization with conforming shape functions of third order [21] results in a parameterized system of the form

Fig. 1. Patch antenna. Magnitude of reflection coefficient as a function of fre, (b) quency and the relative electric permittivities " and " . (a) " , (c) " , (d) " . "

=3

=5

=7

=1

Fig. 2. Patch antenna. Magnitude of reflection coefficient at " = 7; " = 7 as a function of frequency. Inset shows structure. Dimensions in mm: a = 47:45; b = 47:45; h = 18:15; l = 13:5; s = 1:75; d = 2:42; e = 2:6; r = 0:64; r = 2:05.

along an outer edge of the parameter domain far away from the . The corresponding error expansion point, for plots, with (53)

(52a) (52b) with degrees of freedom. The output is the can easily be computed. The input impedance, from which , expansion point is set at . According to and the order of the ROM is chosen to be (14), the resulting ROM dimension is 165. Note that, due to the reciprocal structure of (52), the original and transposed systems coincide, so that, according to Theorem 6, one-sided projections are optimal. Fig. 1 shows the magnitude of with as a function of frequency and the relative electric permittivities and . Fig. 2 compares the solutions of the original FE model to those of the present approach and the ROM of [14]

are given in Fig. 3. It can be seen that the errors of the new approach are significantly smaller. Since the projection matrices of the proposed method and [14] are identical in the absence of round-off errors, this improvement is the sole result of higher numerical stability. Minor differences between Alg. 1 and Alg. 2 are due to numerical noise. Table I shows that the difference in ROM generation time between Alg. 1 and Alg. 2 is approximately 88 percent, using non-optimized code, and that both versions are significantly faster than the method of [14]. As far as solution times are concerned, the ROM can be evaluated 258 times per second, which implies a speed-up of more than four orders of magnitude compared to the underlying FE model! For model the surface plots of Fig. 1, based on

FARLE AND DYCZIJ-EDLINGER: NUMERICALLY STABLE MOMENT MATCHING FOR LINEAR SYSTEMS

Fig. 3. Patch antenna. Magnitude of error in reflection coefficient at " ;" as a function of frequency.

7

=7

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=

TABLE I COMPUTATIONAL DATA

evaluations each, we obtain a reduction in computer runtime from more than 47 days for the FE system to approximately 2-1/2 minutes for the ROM. B. Circulator The inset of Fig. 5 presents a circulator with a triangular ferrite post [22]. The relative permeability tensor of the ferrite is given by (54) and and the frequency as We now consider the entries parameters. FE discretization results in a MIMO system of the form

(55a) (55b) The frequency-dependence of the characteristic waveguide impedances is taken into account by a post-processing step; see e.g., [23]. Note that the system features rational parameteriza, we obtain the tion. However, by multiplying (55a) by polynomial structure required by the MOR method of this paper. GHz, ) We set the expansion point at ( GHz, and consider a parameter domain of , and . A ROM of order reduces the dimension of the original FE model, which is ,

Fig. 4. Circulator. Magnitude of transmission coefficients S and S as a function of frequency f and the entries of the relative magnetic permeability , (b)  tensor,  and  . Alg. 1 is used to compute the ROM. (a)  : , (c)  : , (d)  .

0 66

= 1 33

=2

=0

=

to just 252 degrees of freedom. Fig. 4 presents the system response as a function of the parameters. The non-reciprocal behavior of the device can clearly be seen. Note that the sharp resonances are outside the typical range of operation. Each surface plot is based on 40 401 model evaluations. The frequency and is given in Fig. 5. Since response for the system matrices are non-symmetric, Theorem 6 postulates that, for a ROM of fixed size, two-sided projections according to , yield better results than the choice . (33), i.e., Fig. 6 shows that errors (53) decrease by one order of magnitude! The runtimes in Table I confirm the observations of Subsection VII.A: ROM generation using Alg. 2 beats Alg. 1, and both versions are much faster than the method of [14]. Evaluating the ROM is 1600 times faster than solving the underlying FE system. VIII. CONCLUSION In this paper, we have extended the theory of polynomial moment-matching to the multi-parameter case, wherein parameterization includes not only the system matrix but also the input and

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(57) Applying the differential operator

to (57) leads to

(58) Next, we evaluate (58) at mands with side, only terms with tions. We therefore have

. On the left-hand side, only sumremain, whereas, on the right-hand give non-vanishing contribu-

Fig. 5. Circulator. Magnitude of scattering parameters versus frequency at  and  : . Alg. 1 is used to compute the ROM. Inset shows structure of circulator with dimensions in mm.

=1

= 0 37

(59) Solving for

leads to the sought recursion (60)

APPENDIX B Proof of Lemma 4: By induction. For

, we have

(61)

Fig. 6. Circulator. Comparison of ROMs using one sided or two sided projections. Magnitude of error in scattering parameters versus frequency at  and  : . Alg. 1 is used to compute the projection bases.

=1

= 0 37

Since

(62) for some

output vectors. The analysis of the projection spaces has led to a new algorithm that outperforms existing techniques in numerical robustness and runtime. The proposed algorithm allows for direction-dependent model orders and is easy to parallelize. Our future work will focus on error estimation and adaptivity.

, we have

. Plugging (62) in (61) gives (63)

By multiplying (63) by

, the assumption for

is proven: (64)

At the induction step, we have

APPENDIX A Proof of (13): Our starting point is the Taylor series (10). Plugging (10) into (7a) results in

(56) (65)

FARLE AND DYCZIJ-EDLINGER: NUMERICALLY STABLE MOMENT MATCHING FOR LINEAR SYSTEMS

Assuming that the hypothesis holds for we obtain

with

,

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With (75) which is the Petrov-Galerkin condition of ROM (16), and in view of (26) and (73), we arrive at

(66)

(76)

In view of precondition (24), we have (67) REFERENCES Plugging (67) into (66) yields (68) By multiplying (68) by

, we arrive at the desired result, (69)

APPENDIX C Proof of Theorem 6: To proof moment-matching up to order , we have to show that for the difference between the Taylor series expansions of the transfer function of the original and reduced model the following relation holds: (70) Using (21) and (28), we can write

(71) denote the solution vector of the ROM (16) for the Let transposed system (28a). The application of Lemma 4 to the transposed system gives (72) with (73) By plugging (72) into (71), we arrive at (74)

[1] L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 33, no. 9, pp. 352–366, Apr. 1990. [2] P. Feldmann and R. W. Freund, “Efficient linear circuit analysis by Padé approximation via the lanczos process,” IEEE Trans. Comput.Aided Design Integr. Circuits Syst., vol. 34, pp. 639–649, May 1995. [3] Z. Bai and Y. Su, “Dimension reduction of large-scale second-order dynamical systems via a second-order Arnoldi method,” SIAM J. Sci. Comput., vol. 26, no. 5, pp. 1692–1709, 2005. [4] R. D. Slone, R. Lee, and J.-F. Lee, “Broadband model order reduction of polynomial matrix equation using single-point well-conditioned asymptotic waveform evaluation: Derivation and theory,” Int. J. Numer. Meth. Engng., vol. 58, pp. 2325–2342, Dec. 2003. [5] D. S. Weile, E. Michielssen, E. Grimme, and K. Gallivan, “A method for generating rational interpolant reduced order models of two-parameter linear systems,” Appl. Math. Lett., vol. 12, pp. 93–102, Jul. 1999. [6] D. S. Weile and E. Michielssen, “Analysis of frequency selective surfaces using two-parameter generalized rational Krylov model-order reduction,” IEEE Trans. Antennas Propag., vol. 49, pp. 1539–1549, Nov. 2001. [7] P. Gunupudi, R. Khazaka, and M. Nakhla, “Analysis of transmission line circuits using multidimensional model reduction techniques,” IEEE Trans. Adv. Packag., vol. 25, no. 2, pp. 174–180, May 2002. [8] C. Prud’homme, D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. Patera, and G. Turinici, “Reliable real-time solution of parameterized partial differential equations: Reduced-basis output bound methods,” J. Fluids Eng., vol. 124, pp. 70–80, 2002. [9] P. K. Gunupudi, R. Khazaka, M. S. Nakhla, T. Smy, and D. Celo, “Passive parameterized time-domain macromodels for high-speed transmission-line networks,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2347–2354, Dec. 2003. [10] L. Daniel, O. C. Siong, L. S. Chay, K. H. Lee, and J. White, “A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 23, pp. 678–693, May 2004. [11] L. H. Feng, E. B. Rudnyi, and J. G. Korvink, “Preserving the film coefficient as a parameter in the compact thermal model for fast electrothermal simulation,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 24, no. 12, pp. 1838–1847, Dec. 2005. [12] L. Codecasa, “A novel approach for generating boundary condition independent compact dynamic thermal networks of packages,” IEEE Trans. Compon. Pckg. Technol., vol. 28, no. 4, pp. 593–604, Dec. 2005. [13] Y.-T. Li, Z. Bai, Y. Su, and X. Zeng, “Model order reduction of parameterized interconnect networks via a two-directional Arnoldi process,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 27, no. 9, pp. 1571–1582, Sep. 2008. [14] O. Farle, V. Hill, P. Ingleström, and R. Dyczij-Edlinger, “Multi-parameter polynomial order reduction of linear finite element models,” Math. Comp. Model. Dyn. Sys., vol. 14, no. 5, pp. 421–434, 2008. [15] R. Sanaie, E. Chiprout, M. S. Nakhla, and Q.-J. Zhang, “A fast method for frequency and time domain simulation of high-speed VLSI interconnects,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 2562–2571, Dec. 1994.

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[16] R. Dyczij-Edlinger and O. Farle, “Finite element analysis of linear boundary value problems with geometrical parameters,” COMPEL, vol. 28, no. 4, pp. 779–794, 2009. [17] G. A. Baker Jr. and P. Graves-Morris, Padé Approximants, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 1996. [18] Z. Bai, R. D. Slone, W. T. Smith, and Q. Ye, “Error bound for reduced system model by Padé approximation via the Lanczos process,” IEEE Trans. Comput.—Aided Design Integr. Circuits Syst., vol. 18, no. 2, pp. 133–141, Feb. 1999. [19] E. Grimme, “Krylov projection methods for model reduction,” Ph.D. dissertation, Coordinated-Science La., Univ. of Illinois at Urbana-Champaign, Urbana-Champaign, IL, 1997. [20] M. A. G. de Aza, J. A. Encinar, J. Zapata, and M. Lambea, “Full-wave analysis of cavity-backed and probe-fed microstrip patch arrays by a hybrid mode-matching generalized scattering matrix and finite-element method,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 2, pp. 234–242, Feb. 1998. [21] P. Ingelström, “A new set of H(curl)-conforming hierarchical basis functions for tetrahedral meshes,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 106–114, Jan. 2006. [22] M. Koshiba and M. Suzuki, “Finite-element analysis of H-plane waveguide junction with arbitrarily shaped ferrite post,” IEEE Trans. Microw. Theory Tech., vol. 34, no. 1, pp. 103–109, Jan. 1986. [23] J. Rubio, J. Arroyo, and J. Zapata, “SFELP—An efficient methodology for microwave circuit analysis,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 509–516, Mar. 2001.

Ortwin Farle received the Dipl.-Ing. degree and the Dr.-Ing. degree in electrical engineering from Saarland University, Germany, in 2002 and 2007, respectively. He is currently a Senior Research Scientist with the Chair for Electromagnetic Theory at the Saarland University. His main research interests include the theory and application of model order reduction and finite element methods for electromagnetic fields.

Romanus Dyczij-Edlinger received the Dr. techn. degree in electrical engineering from Graz Technical University, Austria, in 1994. He then became a Postdoctoral Fellow with Prof. Jin-Fa Lee at Worcester Polytechnic Institute, MA. From 1995 to 2000, he held industry positions with Motorola CCRL, Schaumburg, IL, and Ansoft Corp., Pittsburgh, PA, respectively. In 2000, he was appointed Professor at Saarland University, Germany, where he has since been heading the Chair of Electromagnetic Theory. His research interests include computational electromagnetics, model order reduction, and mathematical optimization.

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Indoor Channel Spectral Statistics, K-Factor and Reverberation Distance Yochay Lustmann and Dana Porrat

Abstract—The frequency (spectral) statistics of the line of sight channel is adequately described by the Rice distribution. The Ricean K-factor is extracted from a large body of measurements; it is given against terminal separation and carrier frequency in the range 2–18 GHz. A model based on a power law of the terminal separation and the reverberation distance is suggested. The electromagnetic reverberation distance is characterized for two different rooms, it is bigger for the larger room and tends to increase with frequency. The non line of sight spectral and spatial statistics are similar to each other and characterized by the Rayleigh distribution. Index Terms—Electromagnetic propagation, indoor radio communication, multipath channels.

I. INTRODUCTION

I

NDOOR radio propagation enjoys renewed interest due to extensive research of sensor networks, that have applications for entertainment, surveillance, telemedicine and environmental monitoring. Of particular interest in channel characterization is the line of sight (LoS) scenario, where communicating nodes enjoy a high quality link. Radio system design generally relies on models of the propagation environment in the expected areas of deployment. Realistic channel models are needed for a fair comparison of algorithms and parameters during the design process, and for comparing the performance of different radio system. It is generally accepted that line of sight propagation is characterized by the Rice distribution, but a systematic description of the distribution parameters in indoor setting has not been available until now in the public domain literature. This paper quantitatively characterizes the propagation channel in two rooms. We give a simple model of the Ricean K-factor vs. carrier frequency and transmitter-receiver separation. The results span the 2–18 GHz band, encompassing frequencies currently used for cellular and cordless telephones, the Wifi (IEEE 802.11) and Wimax (IEEE 802.16) standards, the entire UWB band, and higher frequencies that may be occupied by future radio systems. With line of sight available between transmitter and receiver, the power received over the direct path follows the free space model. The main cause for received signal distortion are the Manuscript received September 22, 2009; revised March 11, 2010; accepted April 24, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. This research was supported by The Israel Science Foundation (Grant No. 249/06) and the Israeli Short Range Consortium (ISRC). The authors are with the School of Engineering and Computer Science, The Hebrew University, Jerusalem, Israel (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.2010.2071354

multipath, i.e., the diffuse component of the channel. The signal carried over the diffuse component is typically much stronger than the additive noise and interference. The Ricean K-factor equals the ratio of power received over the direct path to the diffuse power and thus characterizes the multipath. Measurements of the K-factor with results between 0 and 7 are reported by a number of papers: [1] measured values of up to 7 at 5.2 GHz, depending on the number of pedestrians in the vicinity of the system. Results from about 2 to 5 at frequencies of 2.4 GHz and 2.6 GHz are given by [2] and [3], note that the setup described in [3] had the two antennas very close to one wall. Collecting significant channel statistics in the spatial domain is difficult because the statistics depend on terminal separation. A spectral-spatial analysis of the Ricean K-factor (i.e., averages over the two dimensions) in [3] was performed over sections shorter than 5 wavelengths of the center frequency, and the authors comment that this choice of area was done empirically, so that the received power over it would not changes significantly. This work presents statistics in the spectral domain, in bands of 2 GHz between 2 GHz and 18 GHz. The similarity of the spatial and the spectral statistics is demonstrated in non light of sight (NLoS) settings, where meaningful spatial statistics can be easily collected. We characterize the dependence of the K-factor on carrier frequency and on transmitter-receiver separation using the reverberation distance [4], also called critical distance [5], a concept taken from the field of acoustics. The reverberation distance is defined as the distance from the transmitter where the direct component is as strong as the diffuse component of the channel response. For a Rice-distributed channel the reverberation distance equals the distance where the K-factor equals one. II. MEASUREMENT ENVIRONMENT AND EQUIPMENT A. The System The setup was based on an Agilent N5230 network analyzer (NA), connected to two omni-directional antennas (Electro-Metrics EM-6865). The network analyzer transmitted sinusoidal waves in the 2–18 GHz band with a frequency step of 1.06 MHz. Each frequency was transmitted for about 0.02 seconds (the reciprocal of the NA intermediate frequency bandwidth), and the relative amplitude and phase, compared to the transmitted signal, were recorded. The SNR of all our LoS measurements was higher than 34 dB. The coherence bandwidth for the LoS phase measurements was between 25 to 85 MHz. The receive antenna (Rx) was placed on a motorized positioner with sub-millimeter accuracy, that was one meter long

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and 20 cm wide. The receiver moved between measurements but was kept immobile during the collection of each channel response. The transmit antenna (Tx) was placed on a cart that was moved to different locations for different measurements, but was immobile during each measurement. Tx and Rx antennas were both maintained 1.25 m above the floor. A computer controlled the location of the receive antenna along the rail, as well as the parameters of the network analyzer and the data collection and management. Measurements were normally performed during nights, when movement of people around the system was minimal. B. Calibration We used a Line of Sight (LoS) measurement with about 45 cm between receiver and transmitter antennas to calibrate the system. The radiation pattern of a biconical antenna is similar to that of a short dipole [6], and its far field boundary is thus about , below the calibration distance of 45 cm over the entire measurement band. This calibration method allows to calibrate out the cable and antenna responses but has the disadvantage of including multipath components. In order to overcome this difficulty we transformed the calibration measurement to the time domain using a rectangular window, cut out the multipath components (by eliminating the small amplitudes that come after the main pulse), and transformed it back to the frequency domain. We got a calibration response that can be used as follows: (1) (2) where is the amplitude and is the phase for the frequency . The distance between Tx and Rx during calibration is denoted , and after the calibration process we consider each transmitted sinusoidal wave as if it had amplitude 1 and phase 0 at from the transmitter. distance C. The Measurement Environment The measurements used in this paper were collected in 2008–2009 in the Ross Building at the Givat Ram campus of the Hebrew University of Jerusalem. We measured in LoS and non line of sight (NLoS) conditions. The LoS measurements were performed in two office rooms as follows. 1) Big room. 9.5 6.5 2.6 m , about 12000 measurements with 47 different cart (Tx) and motorized rail (Rx) positions. 2) Small room. 4 3.5 2.1 m , about 2300 measurements with 8 different cart and motorized rail positions. III. SPATIAL AND SPECTRAL STATISTICS The statistical description of the radio channel, given small-scale fading conditions, is often modeled by the Rice or Rayleigh distributions. The Rice distribution is used in the presence of a dominant component (as common in LoS conditions) and the Rayleigh distribution is typical when there is no dominant component (common in NLoS conditions).

Observing the small-scale fading statistics can be done by sampling the amplitudes and phases of a narrow band signal in an area that is no bigger than a few times the wavelength. In this paper we refer to this as spatial statistics. The small-scale fading phenomenon can also be observed looking at the amplitude and phase distribution of the harmonic components of the channel. We refer to this as spectral statistics. The assumption of identical statistics in the spatial and spectral domains is in fact an assumption of ergodicity, related to the time-frequency ergodicity mentioned by [7]. The LoS channel between sets of directional antennas was found non-ergodic by [8], note that the results there are based on only ten spatial measurements, and that the existence of a line of sight depended strongly on the orientation of hand-held devices. The Rice and Rayleigh distributions can be used to model the small-scale spectral statistics, as shown in Section IV.B, and it is natural to assume that the same statistics apply in the spatial domain. We demonstrate in Section III.B that spatial and spectral statistics of NLoS channel measurements are similar. With a dominant component (as in LoS), the small-scale spatial statistics is not easy to measure because the dominant component itself can change significantly within the sampling area. This causes difficulty because the Rice model assumes a constant dominant component. For spectral statistics this problem does not exist and observing the Rice statistics is simple. We continue with the assumption that the spatial and spectral statistics of the small scale fading are identical also in LoS conditions, so the ability to extract the spectral distribution parameters gives information on the same parameters in space. In other words, we extract the K-factor of the Rice distribution using wideband spectral measurements, and assume it is the same for spatial statistics. A. NLoS Measurements and Analysis For NLoS conditions there is no dominant component, so observing the statistics of the amplitude is simple for both spatial and spectral measurements. In order to test the similarity between the two statistics the following set of measurements was performed. 1) NLOS spatial measurements: The transmitter and the receiver were placed in different rooms of the same building. The receiver moved on a 100 20 cm rectangle using a motorized positioner with 4 mm steps (total of 12801 positions). In each position the frequencies GHz were transmitted and the channel response was measured. the mean NLOS spatial analysis: For each frequency and the variance of the channel’s amplitude were extracted (see Fig. 1). 2) NLoS spectral measurements: Without moving the transmitter or the receiver cart, nine additional measurements were taken in the corners and in the middle of the 100 20 cm rectangle. The measurements were taken with a 1.06 MHz spectral step over the 2–18 GHz band. NLoS spectral analysis: For the nine measurements the mean and the variance of the channel’s amplitude for 2 16–18 GHz) were extracted. In GHz bands (2–4, 3–5

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Fig. 2. on the complex plane. A is the deterministic free space amplitude of the channel. r~ is a random amplitude caused by the multipath components, that has a random phase  . r is the amplitude we measure and 1 is the phase deviation of the measured channel from free space. Fig. 1. Mean and variance of the amplitude of the NLoS measurements. In solid line the spatial measurements, and in dashed line the average of the nine spectral measurements.

Fig. 1 the average (over the nine measurements) of the mean and variance is presented versus the center frequency of each 2 GHz band. B. NLoS Ergodicity Results The moments of the channel’s response calculated over space and over the spectrum can now be compared, as presented in Fig. 1. The resemblance between the two is apparent, except for the low frequencies (3–5 GHz) where the wavelength ( –10 cm) was probably too large to give adequate spatial statistics on the 100 20 cm area. We performed similar NLoS measurements at two other rooms, and got similar resemblance for the mean and variance of the amplitude distributions in the spatial and spectral measurements.1

Knowing that our receiver and transmitter are at fixed points during the measurement, we choose with no further loss of genand denote as erality

IV. ANALYSIS A. Measured Channel Phase

(7)

Assuming free space (FS) conditions, the calibrated channel response is given by (3) (4) (5) where is the frequency of the sinusoidal wave, is the time, is the speed of light, is the distance between Tx and Rx antenna for the calibration measurement, is the distance between Tx and Rx antenna for the measurement and is a propagation delay that corresponds to . In phasor representation we have (6) 1again

we found some discrepancies for the low frequencies.

Fig. 3. Example of the calibrated channel response on the complex plane taken from a LoS measurement of about R = 287:5 cm with center frequency f = 9 GHz and bandwidth 2 GHz. Each black dot represents measured at a different frequency.

Taking into account realistic LoS office conditions, the sinusoidal wave detected at the receiver is the sum of the free space signal and a diffuse component (8) where is a random amplitude and is a random phase. Nois a superposition of all tice that the diffuse component the multipath components, apart from the direct one. with the measured amplitude , Now we can associate (see Fig. 2) and with the phase deviation from free space (9) (10) In Fig. 3, a LoS measurement of is demonstrated. Measuring the amplitude is straightforward and given directly by the NA as it measures the relation between received and transmitted amplitude at each frequency.

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B. Statistical Models 1) The Rice Distribution: Assuming the number of multipath components is sufficiently large, we expect the central limit theshould be almost normal orem to hold, so the distribution of on the complex plane. We can now describe the phase-amplitude joint distribution as a two dimensional Gaussian function, with diffuse parameter and radial symmetry around (14)

Fig. 4. Example of LoS measured phase versus frequency and an appropriate free space simulation according to (12). The estimated time of arrival (T^) for the simulation was calculated using (11). When extracting 1 from the measurements in the circles a modulus correction is needed.

Equation (14) describes the phase-amplitude joint Rice distribution, where the amplitude and phase distribution can be extracted [9], [10]

Measuring the phase deviation from free space is not modulus property. Still we can esas simple because of the using the following algorithm. timate 1) Estimating the wave propagation time : We approximate with great accuracy by applying a Fourier transform on the data received from all frequencies (2–18 GHz), and looking for the peak position in the time domain2

(15)

(16) (17)

(11)

is the modified Bessel Where is the Ricean K-factor, function of the first kind with order zero and erf is the error function:

2) Simulating free space phase versus frequency using (4) (18) (12) 3) Subtracting the phase that was measured by the NA from the simulated one (13) 4) Modulus correction: a) If the value of the subtraction is bigger than is subtracted from the result. b) If the value of the subtraction is smaller than is added to the result. Of course, if the absolute actual phase deviation is bigger than we will essentially get a random result for . This occurs under NLoS conditions, mainly because we cannot estimate the time of arrival properly. Extraction of the phase deviation is illustrated in Fig. 4, where is the vertical distance on the graph between the measured black dots and the simulated solid line. In cases where the absolute vertical distance is bigger than (around 7.125 GHz for example), a correction is needed because of the modulus property. 2In LoS room conditions we can estimate the time of arrival with a very high precision, usually less than 2 ps (much smaller than our shortest wave period of about 55 ps).

2) The Rayleigh Distribution: Given small-scale fading conditions without a dominant component (NLoS condition) we have , so instead of (15) and (16) we have the Rayleigh distribution for the amplitude and a uniform circular distribution ) for the phase (notice that now we have (19) (20) Extracting the diffuse parameter for the Rayleigh distribution : is done using the first moment of the amplitude (21)

C. Extracting the Ricean K-Factor For the Rice distribution, the most common estimation method of the K-factor is based on the first two moments of the channel’s power [11]. We found that for our measured data, the power-based method did not yield adequate results, and sometimes even returned complex numbers. We used our

LUSTMANN AND PORRAT: INDOOR CHANNEL SPECTRAL STATISTICS, K-FACTOR AND REVERBERATION DISTANCE

^, taken from about Fig. 5. Estimates of A using (22). The dots represent A 12000 LoS measurements in the big room, with mean frequency f = 7 GHz. The solid line represents (R )=(R) (5). We got a good agreement also for the other frequencies.

phase measurements in addition to the amplitude to extract the K-factor, and did it in two ways that gave similar results: 1) Phase and Amplitude Method: If we assume that is uniformly distributed, then using (8) the free space power and the diffuse power (2 ) can be extracted simply by calculating the average and variance of

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Fig. 6. Best statistical model for the LoS amplitude (r ) and phase-deviation (1) spectral distribution. The bandwidth is 2 GHz. The Von Mises (VM), Wrapped Cauchy (WC) and Wrapped Laplace (WL) distributions are presented in Appendix A.A. The Nakagami and Log-Normal distributions are presented in Appendix A.B. The Rice distribution is superior over the entire band for the phase and over the band 4–15 GHz band for the amplitude. the Nakagami is better for f = 3 and 16–17 GHz.

(22) (23) (24) uses the time of arrival (11), Another way to extract as can be seen from (5). Fig. 5 shows the estimates of from measurements with GHz. 2) Phase Only Method: Another way to extract the Ricean , denoted K-factor is by finding the standard deviation of . As can be seen in (16), the pdf of the phase-deviation is a symmetric function that depends only on the Ricean K-factor, and can be found by so a one to one relation between calculating the standard deviation of (16) for different values of . The Ricean K-factor can be extracted using a lookup table. V. RESULTS A. Statistical Model 1) LoS: For each LoS measurement we extracted the appropriate parameters of the Rice, Rayleigh, Nakagami and Log-Normal distributions for the amplitude, and the Von Mises, Wrapped Cauchy, Wrapped Laplace and Rice for the phase as described in Sections IV.B2, IV.C1 and Appendix A. The Akaike Information Criterion (AIC) score was calculated for each statistical model as described in Appendix A.C, to test which model had the lowest score. The results of about 14500 measurements (taken with 55 different cart (Tx) and motorized rail (Rx) positions in the big and small rooms) are presented in Fig. 6. The Rice model best describes the phase distribution for all the frequencies. For the amplitude distribution, the Rice model is best for the 4–15 GHz band, and the Nakagami model has the best agreement for and 16–17 GHz.

Fig. 7. Spatial and spectral amplitude histograms for a NLoS measurement with f = 6 GHz. As can be seen from Fig. 1 the mean and variance for the 6 GHz spatial and spectral measurements are almost identical, so we have a good match between the histograms. The Rayleigh and Log-Normal curves for the spectral measurement are also shown.

2) NLoS: For almost all the NLoS measurements the Rayleigh distribution was superior to the Log-Normal one, i.e., it agreed better with the spectral and spatial histograms. The Log-Normal distribution agreed better with a single spatial GHz, see Section III.B. In Fig. 7 the histogram at histogram of one of the spectral measurements is presented together with its Rayleigh and Log-Normal curves. B. K-Factor From each LoS measurement we extracted the spectral K-factor in 2 GHz bands using the phase and amplitude method as described in Section IV.C1. The K-factor is presented in Fig. 8 as a function of the spatial distance between transmitter , after averaging over bins of about 1 cm. and receiver Looking at the results in Fig. 8 we conclude the following: 1) Dependence on terminal separation. The K-factor tends to decrease as the distance between transmitter and receiver increases. This behavior suggests that the determin-

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TABLE I EXTRACTED REVERBERATION DISTANCE (R ) AND SCALING PARAMETER (n) USING MATLAB ‘FIT’ TO (25). f IS THE CENTER FREQUENCY OF THE 2 GHZ BAND USED TO ESTIMATE THE RICEAN K-FACTOR. RMS IS THE ROOT MEAN SQUARE ERROR OF THE K-FACTOR MODEL

Fig. 8. About 12000 LoS K-factor results (dB) over 2 GHz bands taken in the big room and averaged in steps of about 1 cm. The measurements were taken with 47 different cart (Tx) and motorized rail (Rx) positions. For the small room the pattern is similar.

TABLE II PARAMETERS FOR THE SIMPLIFIED MODEL (26) EXTRACTED FROM K-FACTOR RESULTS USING THE MATLAB ‘FIT’ FUNCTION. RMS IS THE ROOT MEAN SQUARE ERROR BETWEEN THE K-FACTOR MODEL AND THE MEASURED RESULTS

Fig. 9. The scaling parameter n(f ) extracted from the K-factor measurements using the model (25). The dashed lines are the average values over frequencies 3–12 GHz given in Table II.

istic LoS component decreases faster than the diffuse . component 2) Dependence on center frequency. For the low to medium frequencies ( 3–12 GHz) the K-factor tends to increase as the center frequency increases. For the high frequencies 12–17 GHz) the trend is reversed and the K-factor ( decreases with center frequency. 1) Modeling the K-Factor: 1) An Empirical Model. The K-factor results can be described by a power law model with frequency dependent and reverberation distance scaling parameter

Fig. 10. The reverberation distance R as extracted from the K-factor measurements using the model from (25). The dashed lines represent R = 2f + , where 2 and are as in Table II.

2) A Simplified Model. A simplified empirical model is applicable in the 3–12 GHz band (26)

(25) Table I lists the empirical values of the reverberation distance and the scaling parameter for the two rooms. Fig. 9 and Table I display the scaling parameter for the big and small rooms. It seems that in the small room the decrease rate of is usually higher than in the big room. This can also be explained by the stronger diffuse influence in the small room.

and are parameters set to fit the results, and where displayed in Table II. These parameters do not depend on frequency. C. Reverberation Distance Fig. 10 and Table I show the estimated reverberation distance for the big and small rooms as function of the center fre. The reverberation distance in the small room is quency

LUSTMANN AND PORRAT: INDOOR CHANNEL SPECTRAL STATISTICS, K-FACTOR AND REVERBERATION DISTANCE

smaller. The dependence of is given in [4]

APPENDIX A COMPARISON OF PROBABILITY MODELS

on the room’s surface area (27)

Using the dimensions of the rooms, we have and m , so we can expect

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m

This is close to the results in Figs. 10 where the average ratio between the reverberation distances is 2.06, with a standard deviation of 0.4. It is also clear from Fig. 10 that the reverberation distance usually increases as the center frequency increases.

A. Phase Distribution Models In order to validate our statistical model for the phase distribution (16) we tested it against three known circular probability density functions. 1) Von Mises (VM)—also known as the circular normal distribution [12]: (28) is the modified Bessel function of order 0. where 2) Wrapped Cauchy (WC) [12]:

VI. CONCLUSION Results from an extensive indoor line of sight measurement campaign in the frequency domain were presented. By extracting the spectral Ricean K-factor for different frequencies and terminal separation, an insight of the indoor multipath diffuse propagation and reverberation distance was given. We found that the average K-factor depends on Tx-Rx dis, where the revertance via a power law beration distance and scaling parameter were extracted. For the central frequency range of 3–12 GHz the scaling parameter was approximately constant, and the reverberation distance increased linearly with frequency. We assume the reverberation distance increase is due to low reflections from the walls for high frequencies, meaning weaker MPCs. The reverberation distance was found to be larger in the bigger room. We attribute this difference to the weaker multipath components (MPCs) there. The direct path’s power does not depend on room geometry or on frequency, but other MPCs travel larger distances in the bigger room, and are thus weaker there. Our results agree with a prediction of [4], namely reverberation distance values that are proportional to the square root of the room’s surface area. When testing the LoS spectral distribution we found that the Rice model best described the phase distribution for all frequencies, and best described the amplitude distribution for the frequency range of 4–15 GHz. In the 2 GHz bands around 3 and 16–17 GHz the Nakagami distribution was found to best describe the amplitude distribution. For the spectral and spatial NLoS distribution we found the Rayleigh model to best describe almost all of the measurements. The spatial-spectral distribution ergodicity assumption was justified using NLoS measurements, where the mean and variance of the amplitudes were found to be similar. To summarize the contributions: • A simple model of the Ricean K-factor that depends on the carrier frequency and on the terminal separation. • A characterization of the electromagnetic reverberation distance in the 3–17 GHz band. • The spectral statistics of indoor channels are described by the Rayleigh and Rice distributions in the 2–18 GHz band, with other distributions (Nakagami and Log-Normal) sometimes suitable in the low and high ends of the band. • A demonstration of the similarity of the spatial and spectral statistics of the channel in NLoS conditions.

(29) 3) Wrapped Laplace (WL) [13]: (30) where and are parameters that were extracted using a . lookup table against B. Amplitude Distribution Models For the amplitude distribution we compared the Rice distribution (15) with the Rayleigh distribution (19), the Nakagami distribution and the Log-Normal distribution, all known to describe the statistics of multipath fading channels. 1) Nakagami: (31) is the Gamma function. where 2) Log-Normal: (32) where and are parameters that where extracted as follows. 1) For the Nakagami distribution the simplest parameters estimation method is based on the first two moments of the received signal power [10] (33) (34) As in the Rice distribution case, we found that for our measured data this method did not always give adequate results. In order to improve the parameters extraction we used our estimate of the Ricean K-factor (35) (36)

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Notice that for NLoS measurements , so according and . Under these to (17) and (35) we have conditions, the Rice and Nakagami distributions coincide with Rayleigh [9]. 2) For the Log-Normal distribution, parameters estimation using the first two moments of the received amplitudes was found adequate [14] (37) (38) C. The Akaike Information Criterion (AIC) In order to examine which probability model best fits our measurements, the Akaike information criterion (AIC), presented by [15] and popularized by [16], was used (39) denotes the histogram of the measured data, where the pdf of the statistical model, is the number of bins in the histogram and is the number of parameters in the statistical model. As described by [16], the AIC is an approximately unbiased estimator of the expected discrepancy, or distance, between probability density functions. ACKNOWLEDGMENT The authors thank Prof. A. Molisch for his helpful comments and useful references. REFERENCES [1] K. I. Ziri-Castro, N. E. Evans, and W. G. Scanlon, “Propagation modelling and measurements in a populated indoor environment at 5.2 GHz,” presented at the Auswireless Conf., 2006. [2] V. Nikolopoulos, M. Fiacco, S. Stavrou, and S. Saunders, “Narrowband fading analysis of indoor distributed antenna systems,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 89–92, 2003.

[3] S. Wyne, T. Santos, F. Tufvesson, and A. Molisch, “Channel measurements of an indoor office scenario for wireless sensor applications,” in Proc. IEEE Global Telecommunications Conf., GLOBECOM’07, Nov. 2007, pp. 3831–3836. [4] J. Andersen, J. Nielsen, G. Pedersen, G. Bauch, and M. Herdin, “Room electromagnetics,” IEEE Antennas Propag. Mag., vol. 49, no. 2, pp. 27–33, Apr. 2007. [5] J. J. Jetzt, “Critical distance measurement of rooms from the sound energy spectral response,” J. Acoust. Society Amer. vol. 61, no. S1, pp. S34–S34, 1977 [Online]. Available: http://link.aip.org/link/?JAS/61/ S34/3 [6] D. Ghosh and T. K. Sarkar, “Design of a wide-angle biconical antenna for wideband communications,” Progr. Electromagn. Res. B, vol. 16, pp. 229–245, 2009. [7] R. Kattenbach and H. Fruchting, “Wideband measurements of channel characteristics in deterministic indoor environment at 1.8 GHz and 5.2 GHz,” in Proc. Personal, Indoor and Mobile Radio Communications, PIMRC’95 and 6th IEEE Int. Symp. on Wireless: Merging onto the Information Superhighway, Sep. 1995, vol. 3, pp. 1166–1170. [8] J. Karedal, A. Johansson, F. Tufvesson, and A. Molisch, “A measurement-based fading model for wireless personal area networks,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4575–4585, Nov. 2008. [9] M. Yacoub, G. Fraidenraich, and J. Santos Filho, “Nakagami-m phase envelope joint distribution,” Electron. Lett., vol. 41, no. 5, pp. 259–261, Mar. 2005. [10] A. F. Molisch, Wireless Communication. New York: Wiley, 2005. [11] L. Greenstein, D. Michelson, and V. Erceg, “Moment-method estimation of the ricean k-factor,” IEEE Commun. Lett., vol. 3, no. 6, pp. 175–176, Jun. 1999. [12] S. R. Jammalamadaka and A. Sengupta, Topics in Circular Statistics. New York: World Scientific, 2001. [13] S. R. Jammalamadaka and T. J. Kozubowski, “New families of wrapped distributions for modeling skew circular data,” Commun. Stat. Theory Methods, vol. 33, no. 9, pp. 2059–2074, 2004. [14] E. Limpert, W. A. Stahel, and M. Abbt, “Log-normal distributions across the sciences: Keys and clues,” BioSci., vol. 51, no. 5, pp. 341–352, May 2001. [15] H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Automat. Control, vol. 19, no. 6, pp. 716–723, Dec. 1974. [16] U. Schuster and H. Bolcskei, “Ultrawideband channel modeling on the basis of information-theoretic criteria,” IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2464–2475, Jul. 2007.

Yochay Lustmann is with the School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel.

Dana Porrat is with the School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel.

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Multipath Fading Measurements at 5.8 GHz for Backscatter Tags With Multiple Antennas Joshua D. Griffin, Member, IEEE, and Gregory D. Durgin, Senior Member, IEEE

Abstract—Multipath fading can be heavy for ultra-high frequency (UHF) and microwave backscatter radio systems used in applications such as radio frequency identification (RFID). This paper presents measurements of fading on the modulated signal backscattered from a transponder for backscatter radio systems that use multiple antennas at the interrogator and transponder. Measurements were performed at 5.8 GHz and estimates of the backscatter channel envelope distributions and fade margins were calculated. Results show that multipath fading can be reduced using multiple transponder antennas, bistatic interrogators with widely separated transmitter and receiver antennas, and conventional diversity combining at the interrogator receiver. The measured envelope distribution estimates are compared to previously derived distributions and show good agreement. Index Terms—Backscatter radio, diversity methods, fading channels, microwave radio propagation, multipath channels, probability, radio frequency identification, RFID.

I. INTRODUCTION ACKSCATTER radio systems are used extensively in applications such as radio frequency identification (RFID) and passive, wireless sensors because backscatter transponders, or radio frequency (RF) tags, can communicate while consuming very little power. In backscatter radio, the RF tag modulates the electromagnetic waves scattered from its antenna(s) by changing the reflection coefficient at the antenna’s terminals. The signal that is scattered by the RF tag is supplied by an interrogator, or reader, and is detected by the reader’s receiver. Unfortunately, the reliability and range of backscatter radio is limited by several effects including multipath fading. To understand and combat multipath in backscatter radio systems, several researchers have reported measurements of both fading on the signal received by the RF tag and tag readability measurements [1]–[7]. Banerjee et al. [8] have presented spatial and frequency diversity measurements for the signal received by the RF tag at 915 MHz. In contrast, Kim et al. [9] have reported measured envelope cumulative distribution functions (CDFs) for the backscattered signal received from the RF tag at

B

2.4 GHz. Their measurements showed that multipath fading on the signal backscattered from the RF tag has different statistics than those encountered in conventional transmitter-to-receiver links [9]. This is because the backscatter channel is a spatial pinhole channel in which each RF tag antenna acts as a pinhole. While the pinholes increase fading on the modulated signal backscattered from the RF tag compared to a conventional transmitter-to-receiver link, adding additional pinholes can lessen their detrimental effect. This improvement is a pinhole diversity gain [10]. Another way to reduce fading in the backscatter channel is with conventional antenna diversity, which attempts to create multiple signals with uncorrelated fading using spatially-separated antennas at the reader. In general, if the signals are selected or coherently combined, fading on the output signal will be reduced. This technique was first introduced to backscatter radio by Ingram et al. [11] with fading simulations using both multiple reader and multiple RF tag antennas. Other researchers [12], [13] have used multiple antennas for this purpose at only the reader; however, none of these papers report multipath fading measurements for multi-antenna RF tags. This paper details multipath fading measurements for RF tags with multiple antennas. Specifically, it expands previously reported non-line-of-sight (NLOS) fading measurements [14] to include LOS multipath fading measurements and conventional diversity combining results for both NLOS and LOS backscatter channels. Section II presents a multipath-fading distribution that applies to the modulated signal backscattered from the RF tag and Section III summarizes the previously discussed NLOS testbed and measurement setup [14], describes that for the LOS measurements, and provides details of backscatter radio signaling. Section IV summarizes the NLOS measurement results, presents the LOS fading measurements, and details diversity combining results in both the NLOS and LOS backscatter channels. Results are presented in terms of envelope distributions and fade margins. The Appendix provides a list of acronyms. II. BACKSCATTER ENVELOPE DISTRIBUTIONS

Manuscript received October 04, 2009; revised February 22, 2010; acceptedMarch 20, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The work was supported in part by the National Science Foundation (NSF) CAREER Grant 0546955. J. D. Griffin was with the ECE Department, Georgia Institute of Technology, Atlanta, GA 30332 USA. He is now with Disney Research, Pittsburgh, PA 15213 USA (e-mail: [email protected]). G. D. Durgin is with the ECE Department, Georgia Institute of Technology, Atlanta, GA 30332 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.2010.2071355

Multipath fading can be characterized by probability distributions and this paper references two distributions to describe fading observed on the backscattered signal from an RF tag. The backscatter first is the envelope distribution for the channel [10], [11]—i.e., the backscatter channel with reader reader receiver antennas—with transmitter, RF tag, and Rayleigh-fading links. This distribution, which can take two forms depending on level of link correlation, has been discussed in detail previously [10] and will not be repeated here. The backscatter second is the envelope distribution of the

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Fig. 1. The block diagram of the backscatter testbed designed and prototyped for microwave multipath measurements. The testbed consisted of laboratory testand-measurement equipment as well as custom direct-conversion receivers and custom RF tags. The testbed can be configured for either monostatic or bistatic operation. Both configurations are shown (modified from [14]).

channel with statistically independent Rician-fading links [15], [16]. It can be expressed

(1)

where is the random channel envelope; is the index of the distribution; and are the envelope variances of the forward and are the Rician and backscatter links, respectively; factors of the forward and backscatter links, respectively; and is a modified bessel function of the second kind with order . In (1), it is assumed that the reader transmitter, reader receiver, and RF tag antennas are adequately separated in space to decorrelate propagation paths in the forward and backscatter links—i.e., link correlation is equal to zero [10]. Expressions backscatter channel envelope disof the general tribution with Rician-fading links have not been derived. See Section IV-C for discussion. III. THE MEASUREMENT SETUP A. The Backscatter Testbed and Measurement Setup This section summarizes the custom testbed for the NLOS measurements discussed previously [14] and expands upon it to include details of the LOS measurements presented in this paper. The testbed could be arranged in either a bistatic or monostatic configuration, as shown in Fig. 1 (modified from [14]). Three custom, linearly-polarized patch antennas each with a broadside gain of approximately 3.8 dBi were used in the testbed. The bistatic configuration employed one patch antenna

to transmit and two to receive; however, in the monostatic configuration, a single patch antenna was used with a microwave circulator to transmit and receive. The transmitted continuous wave (CW) and receiver local oscillator (LO) signals were supplied by a single signal source to allow phase stable measurements. Three custom, coherent, direct-conversion receivers were used to down-convert the modulated backscatter signal received from the RF tag. The receiver for the monostatic configuration had a different front-end than the receivers used in the bistatic configuration allowing it to operate linearly while receiving strong transmitter-antenna reflections returning through the circulator. The baseband I and Q signals from each receiver were digitized and stored in a personal computer for off-line processing. Two custom RF tags were designed for the testbed. The first, the single-antenna tag (STAG), used a single antenna to modulate backscatter and the second, the dual-antenna tag (DTAG), used two antennas. Multipath fading measurements were made with the testbed in its monostatic and bistatic configurations using both the single-antenna and dual-antenna tags for a total of four setup combinations. All four setups were measured under LOS conditions, while only the bistatic setups were used in NLOS conditions. NLOS measurements were not feasible for the monostatic setups because reflections from the transmitter antenna entered the receiver through the circulator limiting the reader’s maximum transmit power. Therefore, the following six channels were measured: the LOS, monostatic single-antenna-tag channel; the LOS, monostatic dual-antenna-tag channel; the LOS, bistatic single-antenna-tag channel; the LOS, bistatic dual-antenna-tag channel; the NLOS, bistatic single-antenna-tag channel; and the NLOS, bistatic dual-antenna-tag channel. The LOS measurements were conducted in room E560 (VLE560) of the Van Leer Building, shown in Fig. 2 (modified from [14]), on the Georgia Institute of Technology Atlanta campus. Although a strong LOS existed between the testbed

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Fig. 2. The bistatic measurement setups for the LOS and NLOS measurements in rooms E560 and E558 of the Van Leer Building at Georgia Tech. Coherent channel samples were taken at 5.79 GHz as a function of RF-tag position (a sample was recorded every 1 cm or approximately every  =5) across the square area with dimension C (modified from [14]).

reader and the RF tag, multipath propagation was still noticeable due to the walls, floor, and ceiling of the room; laboratory desks and workbenches; and other miscellaneous objects located in the laboratory. In these measurements, the RF tag and reader antennas were approximately 178 cm above the floor. In contrast, the NLOS measurements were conducted with the reader transmitter and receiver located in room E560 of the Van Leer Building and the RF tag located in room E558 (VLE558), as shown in Fig. 2. The LOS was blocked by both a sheet-rock wall and a large metal sheet (i.e., the side of a stripline cavity). The RF tag and reader antennas were approximately 86 cm above the floor. In all of the measurements, a 5.79 GHz, unmodulated carrier was transmitted from the reader and modulated by the RF tag with a 31-bit, maximal-length pseudo-random code ( -sequence) [17] at a chip rate of 1 MHz. To measure fading as a function of RF-tag position in an unbiased manner, channel measurements were taken with RF-tag positions distributed evenly over a square area, shown in Fig. 2. A screw-drive linear positioner moved the RF tag in 1 cm increments across a 30 cm 30 cm square area. The testbed was calibrated before each measurement and all measurements are reported relative to their respective calibrations. Details of the calibration procedure and the testbed dynamic range and sensitivity are available in previously published work [14], [16]. B. Backscatter Signaling The signal received from the RF tag is graphically depicted in Fig. 3(a). In a static channel, the modulated carrier is down-con-

verted and the resulting baseband signal can be decomposed into and a time-varying signal that cona constant signal tains the information from the RF tag. Assuming that the RF-tag switches its reflection coefficient between two states for comand , the signals can be represented on an munication, IQ diagram as shown in Fig. 3(b). Here, the signal received by is the vector sum the reader in RF-tag impedance state A of the transmitted carrier (plus other smaller, unmodulated scatand the modulated-backscatter signal, which tered signals) is proportional to the RF-tag antenna load reflection coefficient . Likewise, the received signal in RF-tag impedance state B is the sum of and a signal proportional to . The correand can be written as sponding baseband equation for (2) (3) where and are the complex, baseband coefficients of the forward and backscatter links, respectively, and is the CW tone and , these transmitted from the reader.1 In terms of equations become [19] (4) (5) 1The 1/2 in (2) and (3) simply conserves power when the equations are converted to their passband representations [18].

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with alterations in the RF-tag antenna impedance. Therefore, it and have is always possible that could change such that the same magnitude. In such a situation, an envelope receiver, which can only detect the magnitude of the signals, would not detect the difference between the modulation states; therefore, coherent receivers were used for these measurements. IV. MEASUREMENT RESULTS A. NLOS Measurement Results The NLOS measurements have been discussed in detail previously [14], but are summarized here for comparison with the LOS measurements. NLOS measurements with both the singleantenna tag and dual-antenna tag exhibited deep, rapid fades of up to 40 dB; however, fading in the dual-antenna tag measurement was less severe than that with the single-antenna tag. Thus, a pinhole diversity gain was observed. Furthermore, the measured distribution estimates closely matched the analytical product-Rayleigh distributions discussed previously [10]. B. LOS Measurement Results

In the measurements reported in this paper, was removed from the total received signal with a DC blocking capacitor and , the envelope distribution estimates were calculated using which is proportional to the difference between the complex reflection coefficients.2 One interesting aspect of backscatter signaling is that, aland is fixed, as shown in though the phase between is not. This phase can Fig. 3(b), their phase relative to change with the relative reader and RF-tag positions as well as

1) LOS Spatial Fading Plots: Fig. 4 shows plots of the LOS fading measurements as a function of RF-tag position. Each plot is normalized by the maximum received power. As expected, these plots show much less severe fading than the NLOS measurements. The maximum fade for the bistatic LOS channel was approximately 13 dB while that for the monostatic LOS channel (not shown) was approximately 20 dB. Fading was caused by specular reflections from the concrete block wall behind the measurement area and metal work benches and laboratory equipment on either side. For reference, the largest power received in any of the the LOS bistatic or monostatic measurements was 56 dBm and 83 dBm, respectively.3 2) LOS Fading Distributions: Estimates of the fading CDFs in the LOS channel using the monostatic and bistatic reader configurations are shown in Fig. 5 for both the single-antenna and dual-antenna tags. All of the distribution plots in this paper are normalized by the square root of the average . The average power is defined as power of the distribution where is the envelope probability distribution function and is the channel envelope. For both types of RF tags in Fig. 5, fading in the monostatic channel is more severe than that of the bistatic. The reason for this difference is that using spatially-separated transmitter and receiver antennas at the reader reduces link correlation [10]. While Fig. 5 shows that the monostatic channel has a higher probability of a deep fade than the bistatic channel, it also shows that the monostatic channel has a higher probability of a large envelope. Therefore, large link correlation tends to make fading in the channel swing between extremes because it correlates both destructive interference (fades) and constructive interference (peaks). The fade margins given in Table I, which were calculated from the measured single-antenna and dual-antenna tag CDF estimates, show that pinhole diversity gains do occur in the LOS

2The formulation of V ~ level had been chosen.

3Note that these absolute power values are not normalized by the calibration measurements.

Fig. 3. (a) The passband and baseband signal received from a backscatter RF tag. (b) An IQ diagram showing the components of the backscatter signal received at the reader.

Or, after some manipulation, (2) and (3) can be written [19] (6) (7)

would be different if a reference other than the DC

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Fig. 5. Comparison of the monostatic and bistatic CDF estimates for the (a) single-antenna tag and (b) dual-antenna tag measurements in the LOS channel. The CDF estimates are plotted on axes normalized by the root of the power of each distribution P for unbiased comparisons.

p

C. Discussion Fig. 4. Measured signal power (in dB) received from the (a) single-antenna and (b) dual-antenna tags in the LOS setup. The coloring of each square represents the modulated signal power received from the RF tag at each RF tag (x; y ) position. The power is normalized by the maximum received and the orientation of the x and y axes is given by the LOS setup diagram shown in Fig. 2.

channel; however, they are very small. The fade margin is defined as [14] (8) where is the CDF and is the average power of the distribution. The outage probability is the likelihood has faded that the power received at the reader receiver below by an amount equal to the fade margin—i.e., [14]. The measured monostatic and bistatic CDF estimates for the LOS channel are plotted in Fig. 6 and Fig. 7, respectively, along with the CDF calculated from (1). These figures show that the distribution estimates of the single-antenna and dual-antenna tags (for both the monostatic and bistatic reader configurations) product-Rician CDF calculated approximate the from (1).

Several observations can be made from the LOS measurements outlined in the previous section. First, Fig. 5 indicates that using separate, adequately-spaced reader transmitter and receiver antennas decreases link correlation in the LOS backscatter channel and provides a corresponding reduction in multipath fading. Second, the estimate of the single-antenna tag CDF measured in the bistatic channel and the product-Rician , distribution from (1) are a good match for shown in Fig. 7(a). Third, very small pinhole diversity gains are realized for the LOS channels using the monostatic and bistatic reader configurations as shown by the fade margins in Table I. An unexpected result is that the measured estimates of the dual-antenna tag CDF at each receiver antenna, shown in Fig. 7(b), closely match the product-Rician CDF calculated from (1). One might expect that the strong specular waves scattered from each tag antenna would interfere and worsen the fading distribution compared to that of the single-antenna tag [16] or at least the distribution estimate measured with the dual-antenna tag would differ from that measured with the single-antenna tag. The similarity between the single-antenna and dual-antenna tag distributions can be explained by additional correlation in the LOS backscatter channel. In the NLOS measurements, the high level of multipath propagation experienced by the RF tag resulted in a channel with

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TABLE I FADE MARGINS (IN dB) CALCULATED FROM THE MEASURED, LOS DISTRIBUTION ESTIMATES

Fig. 6. The monostatic CDF estimates for the single-antenna tag and dual-antenna tag in the LOS channel along with the product-Rician CDF calculated . The CDF estimates are plotted on axes from (1) for normalized by the root of the power of each distribution for unbiased comparisons.

Fig. 7. The bistatic CDF estimates for the (a) single-antenna tag and (b) dualantenna tag in the LOS channel along with the product-Rician CDF calculated from (1). The CDF estimates are plotted on axes normalized by the root of the for unbiased comparisons. power of each distribution

a very small spatial coherence distance—i.e., the correlation of the channel as a function of space decreased rapidly [18]. Therefore, it can be assumed that signals received from each tag antenna had very low correlation. This assumption was confirmed by the close match between the measured NLOS distribution estimates and the previously derived analytical product-Rayleigh distributions, which assume that the signals received from each tag antenna are statistically independent [14]. The spatial coherence distance of the LOS channel, however, was much greater than that of the NLOS channel and resulted in correlation between the signals received from the two RF-tag antennas. In other words, the pinholes of the LOS backscatter channel were correlated and adding pinholes offered no additional statistically independent paths over which signals could propagate. In essence, a single phase front propagated from the tag to the reader which made the estimate of the dual-antenna tag distribution resemble that of the single-antenna tag. In a strong LOS channel, RF-tag footprint restrictions will likely not allow adequate antenna spacing to reduce this correlation; however, fading improvements can still be realized using widely separated receiver antennas, as shown in Fig. 5. Since the signal backscattered from each RF-tag antenna is not independent, it is not possible to generalize (1) to the channel as was done with the product-Rayleigh distri-

butions described previously [10]. Even so, the LOS measurements and the arguments above indicate (1) was adequate for tags with one and two antennas in very strong LOS channels. It should be noted that the fading distribution estimates measured at 5.8 GHz are valid at other frequencies. The product-Rayleigh distributions [10] and product-Rician distribution given by (1) are valid in a local area—i.e., the area over which the channel can be accurately represented by a sum of planewaves arriving from the horizon [18]—at any frequency. While the choice of frequency certainly affects propagation characteristics, a local area experiencing some combination of non-specular and specular propagation in the 902–928 MHz band (a popular frequency band for RFID applications in the United States) will follow the same fading distribution as a local area with the same combination of non-specular and specular waves at 5.8 GHz. While the product-Rayleigh [10] and product-Rician distributions may not hold for every situation, they will apply when propagation in the local area of interest is similar to that of the environments measured in this paper. This is not overly restrictive and simply means that, for NLOS channels, propagation in the local area will be composed of a diffuse grouping of nonspecular waves. For LOS channels, local area propagation will be dominated by a single specular wave plus a diffuse group of nonspecular waves.

K = K = 8:5 dB

pP

pP

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TABLE II FADE MARGINS (IN dB) CALCULATED WITH AND WITHOUT MRC FROM THE MEASURED DISTRIBUTION ESTIMATES OF BOTH THE BISTATIC, NLOS CHANNEL AND THE BISTATIC, LOS CHANNEL

No MRC performed. The NLOS fade margins without MRC have been reported previously [14].

Fig. 8. Estimates of the bistatic, NLOS CDFs calculated using MRC compared to previous measurements [14] without MRC for the (a) single-antenna tag and (b) dual-antenna tag measurements. The CDFs are plotted on axes normalized by the root of the power of each distribution P for unbiased comparisons.

p

D. Conventional Diversity Gains in the Backscatter Channel Conventional diversity combining will also provide link gains in the backscatter channel. Maximal ratio combining (MRC) [18] was used to combine the signals received from the singleantenna and the dual-antenna tags in both the NLOS and LOS backscatter channels. The RF-tag reader was assumed to have perfect knowledge of the channel and results are compared to previous results without MRC [14] in terms of CDFs and fade margins in Figs. 8, 9 and Table II, respectively. As shown in Fig. 8, MRC improved all the measured cases, with the most significant improvement observed for the NLOS

Fig. 9. Estimates of the bistatic, LOS CDFs with and without MRC for the (a) single-antenna tag and (b) dual-antenna tag measurements. The CDFs are plotted on axes normalized by the root of the power of each distribution P for unbiased comparisons.

p

channel measurements taken with the dual-antenna tag. The additional antennas of the dual-antenna tag added pinholes doubling the signal pathways in the multipath-rich environment. V. CONCLUSION The multipath fading measurements presented in this paper show that multipath fading exists in both the NLOS and LOS backscatter channels. Deep fades of up to 40 dB were observed on the signal backscattered from the RF tag in the NLOS

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channel while much smaller (up to 20 dB) fades were measured in the LOS channel. Fading in the NLOS measurements followed a product-Rayleigh distribution while that of the LOS channel followed a product-Rician distribution. Multipath fading was reduced using multiple RF tag antennas—i.e., a pinhole diversity gain—in the NLOS channel, but only very small pinhole diversity gains were observed in the LOS channel. Even so, fading in both the NLOS and LOS channels was reduced by using a bistatic reader with widely-spaced transmitter and receiver antennas instead of a monostatic receiver. The benefits of using MRC at the reader receiver were calculated and showed that maximum diversity gains were realized in systems that used multiple RF tag antennas. APPENDIX LIST OF ACRONYMS ADC

analog-to-digital converter.

CDF

cumulative distribution function.

CW

continuous wave.

DAC

digital-to-analog converter.

DTAG

dual-antenna tag.

LOS

line-of-sight.

STAG

single-antenna tag. a backscatter channel with transmitter, RF tag, and receiver antennas.

NLOS

non-line-of-sight.

MRC

maximal ratio combining.

RFID

radio frequency identification. ACKNOWLEDGMENT

The authors would like give a special thanks to R. Pirkl for his expert hardware and data processing advice and J. Duvall for her assistance with the measurements. REFERENCES [1] Auto-ID Centre White Papers from Cambridge, U.K., “UHF band RFID readability and fading measurements in practical propagation environment,” 2005, pp. 37–44. [2] J. Mitsugi and Y. Shibao, “Multipath identification using steepest gradient method for dynamic inventory in UHF RFID,” presented at the Int. Symp. on Applications and the Internet, SAINT Workshops, Hiroshima, Japan, 2007. [3] M. Polivka, M. Svanda, and P. Hudec, “Analysis and measurement of the RFID system adapted for identification of moving objects,” in Proc. 36th Eur. Microwave Conf., Sep. 2006, pp. 729–732. [4] L. W. Mayer, M. Wrulich, and S. Caban, “Measurements and channel modeling for short range indoor UHF applications,” in Proc. 1st IEEE Eur. Conf. on Antennas Propag. EuCAP 2006, Nov. 2006, pp. 1–5. [5] U. Muehlmann, G. Manzi, G. Wiednig, and M. Buchmann, “Modeling and performance characterization of UHF RFID portal applications,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 7, pp. 1700–1706, 2009. [6] A. Lazaro, D. Girbau, and D. Salinas, “Radio link budgets for UHF RFID on multipath environments,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1241–1251, 2009. [7] S. R. Banerjee, R. Jesme, and R. A. Sainati, “Performance analysis of short range UHF propagation as applicable to passive RFID,” in Proc. IEEE Int. Conf. on RFID, Grapevine, TX, Mar. 2007, pp. 30–36.

[8] S. R. Banerjee, R. Jesme, and R. A. Sainati, “Investigation of spatial and frequency diversity for long range UHF RFID,” in Proc. IEEE Antennas Propag. Society Int. Symp., San Diego, CA, Jul. 2008, pp. 1–4. [9] D. Kim, M. A. Ingram, and W. W. Smith, Jr., “Measurements of smallscale fading and path loss for long range RF tags,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1740–1749, Aug. 2003. [10] J. D. Griffin and G. D. Durgin, “Gains for RF tags using multiple antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 563–570, 2008. [11] M. A. Ingram, M. F. Demirkol, and D. Kim, “Transmit diversity and spatial multiplexing for RF links using modulated backscatter,” presented at the Int. Symp. on Signals, Systems, and Electronics, Tokyo, Jul. 24–27, 2001. [12] J. S. Kim, K. H. Shin, S. M. Park, W. K. Choi, and N. S. Seong, “Polarization and space diversity antenna using inverted-F antennas for RFID reader applications,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 265–268, Dec. 2006. [13] A. Rahmati, Z. Lin, M. Hiltunen, and R. Jana, “Reliability techniques for RFID-based object tracking applications,” in Proc. 37th Annu. IEEE/IFIP Int. Conf. on Dependable Systems and Networks (DSN’07), Edinburgh, U.K., Jun. 2007, pp. 113–118. [14] J. D. Griffin and G. D. Durgin, “Multipath fading measurements for multi-antenna backscatter RFID at 5.8 GHz,” in Proc. Int. IEEE Conf. on RFID, Orlando, FL, Apr. 2009, pp. 322–329. [15] M. K. Simon, Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers and Scientists. Norwell, MA: Kluwer Academic Publishers, 2002. [16] J. D. Griffin, “High-frequency modulated backscatter communication using multiple antennas” Ph.D. dissertation, The Georgia Institute of Technology, Atlanta, 2009. [17] R. W. Dixon, Spread Spectrum Systems With Commercial Applications, 3rd ed. New York: Wiley Interscience, 1994. [18] G. D. Durgin, Space-Time Wireless Channels. Upper Saddle River, NJ: Prentice Hall, 2003. [19] J. T. Prothro and G. D. Durgin, “Improved performance of a radio frequency identification tag antenna on a metal ground plane” Master’s thesis, The Georgia Institute of Technology, Atlanta, 2007. Joshua D. Griffin received the B.S. degree in engineering from LeTourneau University, Longview, TX, in 2003, and the MSECE and Ph.D. degrees from the Georgia Institute of Technology (Georgia Tech), Atlanta, in 2005 and 2009, respectively. In 2004, he joined the Propagation Group at Georgia Tech where he researched UHF and microwave propagation for backscatter radio. In 2006, he participated in a research exchange program with the Sampei Laboratory at Osaka University, Osaka, Japan. He joined the Radio and Antennas Group at Disney Research, Pittsburgh, PA, in 2009. His research interests include radio wave propagation, radiolocation, applied electromagnetics, backscatter radio, and radio frequency identification (RFID). Dr. Griffin received the Graduate Research Assistant Excellence Award from the Electrical and Computer Engineering Department at Georgia Tech in 2009. Gregory D. Durgin received the BSEE, MSEE, and Ph.D. degrees from Virginia Polytechnic Institute and State University, Blacksburg, in 1996, 1998, and 2000, respectively. He is an Associate Professor in the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, having joined the faculty in 2003. He spent one year as a Visiting Researcher with the Morinaga Laboratory at Osaka University, Osaka, Japan. He founded the Propagation Group (http://www.propagation.gatech.edu) at Georgia Tech, a research group that studies radiolocation, channel sounding, direction finding, backscatter radio, RFID, and applied electromagnetics. He is an active consultant to industry. He authored Space-Time Wireless Channels, the first textbook in the field of space-time channel modeling. Prof. Durgin is a winner of the NSF CAREER Award as well as numerous teaching awards, including the Class of 1940 Howard Ector Outstanding Classroom Teacher Award (2007), and the 2001 Japanese Society for the Promotion of Science (JSPS) Postdoctoral Fellowship. In 1998, he was co-recipient of the Stephen O. Rice prize for best original journal article in the IEEE TRANSACTIONS ON COMMUNICATIONS.

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Semi-Deterministic Propagation Model for Subterranean Galleries and Tunnels Ludek Subrt, Student Member, IEEE, and Pavel Pechac, Senior Member, IEEE

Abstract—A three-dimensional, semi-deterministic model for wave propagation in subterranean galleries and tunnels is introduced. The model uses a combination of deterministic ray-launching techniques and a stochastic approach to provide efficient predictions that avoid complex calculations of electromagnetic phenomena while preserving the ability to consider reflection, absorption and penetration, as well as diffuse scattering. There is no need to provide a precise definition of the walls in terms of their material electrical parameters and roughness. The wall properties are efficiently described using just three probabilistic parameters that can be calibrated by measurements. Experimental data for three various subterranean scenarios were used to validate the model’s functionality. The scenarios were chosen with regard to validating the model function for different frequencies, cross-sections, lengths and shapes. Index Terms—Electromagnetic propagation, modeling, underground electromagnetic propagation.

I. INTRODUCTION N recent years, a number of works dealing with wave propagation inside subterranean galleries and tunnels have been published [1]–[12]. Subterranean galleries and tunnels are primarily long corridors requiring propagation models that usually differ from other models designed for common indoor scenarios. Some of the published models are derived from waveguide theory [1], [3], [10], because a tunnel can be considered as a special type of a loaded lossy waveguide. Others use the standard deterministic ray-based approach [2], [7]–[9] combined with electromagnetic field theory. Another group of models use parabolic equations to solve electromagnetic wave propagation [5], [12] and the last group of models implements empirical equations [6], [11] to define the specific attenuation inside subterranean galleries. This paper follows the measurements and modeling published earlier in [6], where an empirical model for caves and subterranean galleries was proposed. Furthermore, the model was validated using experimental data from [2]. The empirical model is able to provide results in long straight galleries but cannot take account of changes in gallery profile nor wall material changes along the gallery length. The model also fails if the walls have

I

Manuscript received January 13, 2010; revised April 27, 2010; accepted April 30, 2010. Date of publication September 02, 2010; date of current version November 03, 2010. This work was supported in part by the Czech Ministry of Education, Youth and Sport projects no. OC09075 and MSM 6840770014. The authors are with the Department of Electromagnetic Field, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague CZ-16627, Czech Republic (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2010.2072898

an extremely rough surface and if the gallery profile differs significantly from a regular rectangular shape, which is the usual case for caves. This was the main motivation for developing a new semi-deterministic model. Our model combines deterministic ray-based propagation modeling with a stochastic approach to solving interactions between the rays and the gallery walls. The model, in contrast to classical ray-tracing models, is effectively able to consider diffuse scattering and reflections. It does not require knowledge of the electrical properties of walls or obstacles. The material parameters are replaced by a few probabilistic parameters, which can be obtained from measurements. It allows us to avoid complex deterministic calculations while preserving results adequate for deterministic predictions. The paper is organized into four sections. Section II deals with the new propagation model. All the important parts of the model, as well as the fundamental principles and algorithms, are described in detail in this section. The third section presents the verification of the model function while scenarios descriptions and measurement techniques are briefly described and the calibration of the model is explained. The benefits of the new model are summarized in the last section. II. PROPAGATION MODEL The new 3D semi-deterministic model combines both stochastic and deterministic approaches. This model is partly based on the 2D Motif Model [13], but its features were changed significantly. Let us describe some of the main features of our model. The deterministic part employs ray-based algorithms respecting the exact position of the walls in a scenario while the stochastic approach is used to solve the interactions between the rays and the walls. The stochastic parameters are calibrated by measurement data, which means that Fresnell equations and other complex computations can be replaced with a fast empirical approach while phenomena such as diffuse scattering and reflection are considered. At the same time, there is no need to know the walls’ electrical properties. This is significant since it is problematic to define the material parameters and roughness of the walls. On the other hand, an efficient way to calibrate the model is to measure received power in the straight section with the given cross-section of a tunnel, tune the probabilistic parameters and then predict propagation characteristics for the whole scenario apart from the shape and cross-section of the rest of the tunnel. A. Ray Launching The model is based on the modified Ray-Launching method. Models belonging to this group substitute electromagnetic waves with a high number (an infinite number in an ideal case) of plane waves which are usually represented by their

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Fig. 1. An example of antenna radiation pattern and corresponding launching pattern. (a) Radiation pattern; (b) launching pattern.

Fig. 2. An example of the PRP formation (probability vs. angle). (a) Directional part; (b) omnidirectional part; (c) overall PRP.

directional vectors/rays. If we consider an omnidirectional transmitting antenna we can determine the part of the overall transmitter power carried by each ray as follows:

The above-mentioned overall PRP is created under Phong’s law [14] on the basis of three probabilistic parameters. Phong’s law describes the phenomena caused by an impinging ray of light on a boundary. It has been shown that the reflected light consists of two main components, namely the diffuse reflection (directional part—Fig. 2(a)) and the omnidirectional diffuse part (Fig. 2(b)). If the surface under consideration is perfectly smooth and reflective, Snell’s law holds that only the specular ray is reflected. This assumption cannot usually be satisfied and, in view of this, the diffused component should be also taken into account. Each type of wall in the scenarios is described simply using only three probabilistic parameters—the probability of absorption, reflection and diffuse scattering—which express the three basic phenomena of wave propagation in straight tunnels: reflection, absorption and diffuse scattering. The values of these parameters form the shape of the PRP and depend on the material properties (material constants, width and roughness of the wall for the specific frequency), dimensions and polarization. For greater clarity, all the figures in the paper depict a 2D simplification of the situation. In our model, the three dimensional PRP is always considered. The directional part (Fig. 2(a)) of . The omni-directhe PRP is approximated by the function tional part is expressed by a constant function (Fig. 2(b)). The mathematical expression is given below. C. Probabilistic Parameters of the Model

(1) where

represents power of the individual ray (W), the overall transmitter power (W) and the number of rays launched (-). Rays are launched from the transmitting antenna according to an antenna radiation pattern (Fig. 1(a)) converted into a pattern determining the number of rays that are launched in specific directions (the so-called “launching pattern”, Fig. 1(b)). All the rays then carry the same part of overall power given by the relation (1). When a launched ray intersects a wall, the angle of arrival (AoA) is computed. The subsequent direction of the impinging ray is determined by means of what is referred to as the probability radiation pattern (PRP). The amplitude and phase of the power carried by the rays is not considered and transmitted power is given by the number of rays arriving to the receiver.

When an impinging ray impacts the obstacle, the part of power carried by it is absorbed in the material and the is emitted, i.e. reflected or transmitted through the rest obstacle. The power balance can be expressed as (2) (3) where represents incident power (W), absorbed power emitted power (W), reflected power (W) and (W), power transmitted through the obstacle (W). These relations are taken into account when defining the probability parameters. Let us now define the three basic probabilistic parameters forming the PRP function. Each of the parameters represents a different physical phenomenon and all of them are derived on the basis of relations (2) and (3). Firstly, the probability of absorption of incident power is given by:

B. Probability Radiation Pattern To describe the PRP in more detail, it is a function of the direction of the AoA (both the azimuth and elevation) and probabilistic parameters defining the wall properties. It does in fact determine the probability of the subsequent direction of the impinging ray (expressed in terms of the spatial angle) for each specific AoA (Fig. 2(c)). This implies that a single new ray is generated instead of the impinging ray and therefore the number of rays considered does not increase (in contrast to an absolutely rigorous approach where thousands of rays with different amplitudes would need to be generated). The maximum of the PRP therefore represents the most probable direction of the new ray. Consequently, it helps to decrease the time needed for computation and saves computer memory.

(4) where represents the probability of absorption (-), abincident power (W). It is in fact the sorbed power (W) and ratio of the power absorbed by the material and the overall incirange between 0 and 1. If the value dent power. The values of is set to 1, all power is absorbed (no power is radiated—ray propagation is always terminated). Conversely, if the value is equal to 0, all the power is emitted (reflected or penetrated). The second probabilistic parameter, the probability of the reflection is defined as: (5)

SUBRT AND PECHAC: SEMI-DETERMINISTIC PROPAGATION MODEL FOR SUBTERRANEAN GALLERIES AND TUNNELS

Fig. 3. An example of different settings of normalized PRP (p p = 0; (b) p = 0:5; (c) p = 1.

= 0:1). (a)

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Fig. 5. Determination of the next direction of an impinging ray. (a) Probability radiation pattern; (b) distribution function.

the wall as illustrated in Fig. 2(c). Firstly, the PRP function corresponding to the reflection part is given by:

Fig. 4. An example of different settings of normalized PRP (p (a) p = 0; (b) p = 0:1; (c) p = 1.

= 0:66).

where represents a probability of reflection (-), reflected emitted power (W). It is a ratio of the repower (W) and flected power (proportional to the upper part of area in Fig. 2(c) named “Reflection”) and the overall emitted power (the whole area in the same figure—meaning reflection and penetration tois between 0 and 1, similar to the gether). The range of case of the probability of absorption. 1 means that all power is reflected (Fig. 3(c)), while 0 means that all power passes through the material (Fig. 3(a)). Both above-mentioned cases are extreme and are only mentioned in order to clarify the PRP creation. Finally, the probability of the diffuse scattering is defined as

(6) where

represents the probability of diffuse scattering (-), omni-directionally emitted power (W) and emitted power (W). Graphically, it can be expressed as a ratio of the area of the omni-directional part (Fig. 2(b)) and the area of the whole PRP (Fig. 2(c)). Again, the range is between 0 and 1. If it is set to 0, only specular reflections are considered (Fig. 4(a)). If it is 1, the subsequent direction is independent of the AoA (Fig. 4(c)—the PRP is omnidirectional). has a direct influThe probability of diffuse scattering ence on the shape of the PRP which is approximated by the relations (8) and (10). The power of cosine (-) representing the according to the following rediffuse reflection is given by lation [13]: (7) Based on these three parameters the PRP can be mathematically defined separately for the two hemispheres separated by

(8) elsewhere represents the probability (-), is an angle ranging where from 0 to 180 degrees, is the directional angle determined according to Snell’s law on the basis of the AoA, and , the constant representing the diffuse scattering, is defined as follows: (9) is the Area of directional part of the PRP repwhere (Fig. 2(a)). Secondly, the PRP function corresented by responding to the penetrated part is shaped according to: (See represents the probaequation at bottom of page) where bility (-), is an angle ranging from 180 to 360 degrees, and is the directional angle of the impinging ray. D. Iteration Algorithms When the ray impinges on the wall, the corresponding PRP (Fig. 5) and equivalent distribution function (DF) are generated on the basis of the angle of arrival and the three probabilistic parameters defining the wall. Fig. 5 shows an example of PRP and DF for the following conis equal to 0.2 ditions: angle of arrival of 135 degrees, where and where equal to 0.1. After the creation of PRP (Fig. 5(a)) and corresponding DF (Fig. 5(b)), a random number is generated and the next direction of an impinging ray is determined using the distribution function (Fig. 5(b)). The range of random numbers for DF depends on the requested angle resolution. For the . example we have chosen a range between 0 and In the sample case, a random number in the range between 0 and matches the reflection, a range between and represents the absorption, and between and stands for penetration. If the ray is reflected or penetrated (ranges between and between and ) we have a simple and 0 and immediate method of obtaining the next direction (angle) of an impinging ray. If the ray is absorbed, the iteration is completed and the next ray is launched from the transmitter. Three random numbers (1897, 6684 and 9458) were chosen as an example of the determination of the next direction of an impinging ray. If 1897 is generated, the ray is reflected at an angle of 44 degrees. If a randomly generated number is equal to

for elsewhere

(10)

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Fig. 6. An illustration of the ray travelling randomly from the transmitter by means of PRP.

6684, the ray is absorbed and therefore the current iteration is completed. And finally, if the number is equal to 9458 the ray will pass through the wall and its new angle will be 303 degrees. All three possible results are depicted in Fig. 5. It should be mentioned that if an impinging ray is reflected or penetrated, random number generation is performed twice both for azimuth and elevation. In fact, a completely 3D PRP function is used. After the ray leaves the wall in its new direction, the next cross-section is again computed and the interaction is solved. This is repeated until the ray is absorbed or leaves the scenario. The whole iteration for a single ray is depicted in Fig. 6. The ray is transmitted from the antenna (according to the “launching pattern”), then reflected twice and finally penetrating the tunnel wall as it leaves the tunnel scenario. When a sufficient number of rays have been launched (and the requested precision and range has been achieved), the scenario is divided into a cubical grid and an average signal level in each grid element is computed according to (11). A sufficient number of rays cannot usually be predicted before the simulation has been initiated. The sufficient number is reached when newly launched rays have no significant effect on the results. The only criterion for signal strength determination is the number of rays passing through each element in the grid. If a high number of rays is launched, then the signal level in each grid element is proportional to the number of rays passed through this element. The averaged received power is therefore given by:

(11) where represents averaged received power (W), number of rays passed through the grid element (-), number of launched rays (-) and overall transmitter power (W). E. Model Calibration It is evident that the model is driven by the geometry of the scenario and the probabilistic parameters defining the properties of the walls. Despite the fact that there is an obvious relationship between the physical aspect of the parameters and the actual wall properties, such as material electrical parameters and wall roughness, it would be very complicated to rigorously describe the wall properties in an environment such as a cave. The intention is to calibrate the PRP using empirical data for specific scenarios. The probabilistic parameters can be obtained using computer optimization techniques based on the following algorithm.

Fig. 7. Average profiles (geometrical model) of scenario A (profile A), B (profile B) and C. The profiles including photographs are described in more detail in [6] (A, B) and [2] (C) respectively.

Firstly, a sufficient number of signal level measurement points in the coverage scenario are selected in order to be able to evaluate the prediction performance. Then the prediction is performed based on the initial values of the PRP parameters. The prediction performance can be expressed by means of a classical fitness function:

(12) where is the fitness (-), is the number of measurement points (dB) is the measured attenuation and (dB) of interest, is the predicted attenuation in the given -th point. To find the proper values of the PRP parameters, i.e. to minimize the fitness function (12), an optimization process based on genetic algorithms [15] can then be applied. The prediction calculation is efficient and fast enough to be able to evaluate the fitness function within the process many times and to find the solution within a reasonable period. An example is given in the next section. III. VERIFICATION OF THE MODEL BY MEASUREMENTS A. Scenarios We chose three different scenarios—two subterranean galleries [6] and a long, curved, arched tunnel [2]. All three scenarios were situated in different locations with different profiles and wall properties. Different shapes and material composition were chosen in order to evaluate the functionality of the model. The dimensions of the galleries are much larger than the wavelength of the transmitted signal, thereby satisfying the condition that allows us to use the ray-based propagation model. An average, simplified profile was used during the simulations process (Fig. 7). The profile of one gallery (located in “Prague-Sarka”) is depicted in Fig. 7(a), including its dimensions. It is a semi-circular corridor travelling through sandstone which can be considered as dry (the water content is low and therefore can be ignored). The walls are relatively smooth in comparison with the second gallery (located in “Morina-Amerika”). The second scenario consists of the gallery depicted in Fig. 7(b) which has a rectangular shape with a width of 2.3 m and height of 2.1 m. The corridor passes through limestone and the floor is formed of clay. Both floor and walls are rougher than in the case of the gallery mentioned earlier. In both scenarios, the walls are significantly rougher than the floors and therefore their material properties should be seen to be different.

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TABLE I OPTIMIZED PROBABILISTIC PARAMETERS AND CALIBRATION POINTS

More detailed descriptions of both scenarios, including photos, are given in [6]. The third scenario is a curved, arched tunnel (situated in Berlin, Germany) [2]. The arched tunnel has a semi-circular cross-section with dimensions according to Fig. 7(c). The overall tunnel length is 1079 m. The tunnel shape can be divided into two sections: The first part (up to 350 m from the transmitter) is considered as a straight section and the second part (between 350 m and 800 m) is represented by a left curved section with large radius of curvature. The tunnel walls are formed by prefabricated elements made of dry concrete and therefore needed to be considered as rough. A detailed description of the tunnel can be found in [2]. B. Measurements As mentioned above, measurement results from short straight subterranean galleries [6] and long curved tunnel [2] were utilized to validate the model. 1) Subterranean Galleries (Scenarios A and B): A vertically polarized monopole connected to a 30 dBm portable transmitter was placed in the middle of the gallery profile 1.2 m above the floor level, transmitting a continuous wave at 860 MHz. A halfwave dipole acting as the receiving antenna was moved across the gallery profile to find the maximum signal level at any given location and to avoid the influence of fading caused by multipath propagation. Both characteristics were measured with different distance resolution [6]. In the case of scenario A, the distance step was 1 m unlike scenario B where a 10 m step was used. Comparison of the model prediction with the measured path loss is shown in Fig. 8 as a function of the distance between the transmitter and the receiver in the galleries. Since the model predicts the mean value of the path loss the measured data from scenario A were averaged using running rms window of 30 as in [2]. 2) Long Curved Tunnel (Scenario C): To validate the model on a long curved tunnel scenario the measurements [2] at 945 MHz and 1853 MHz were utilized. For the comparison in Fig. 9 the measured data representing rms values were taken from Figs. 5 and 6 of [2]. C. Model Calibration The general principle of the model calibration allows us to use the same method of the parameters calibration for all three scenarios despite their being different: Two PRPs were

Fig. 8. Measured data and optimized curves for scenarios A and B.

utilized—the first characterized the gallery floor, the second the other walls of the profile. As all scenarios represent a decrease of the signal level over distance, just two points—the measured attenuation at two different distances (see Table I, last column) were utilized to calibrate the PRP parameters. It must be noted that if the wall properties had changed along the path, more PRPs as well as more calibration reference points would have been needed to be introduced accordingly. The size of the cubical grid was set to 40 cm. Genetic evolutionary algorithms [15] were executed with the following settings: the number of generations was set to 40, as well as the number of members in each generation. Individuals were modified by the uniform crossover and a roulette wheel was used as the method of selection for the next generation. The convergence of the optimization was quite fast. The number of necessary evaluations of the fitness function, i.e. the number of signal level predictions performed in the two points, was about 400 for the whole model calibration process. Resultant PRP probabilistic parameters are depicted in Table I. The corresponding model predictions, together with measured characteristics for all scenarios, are shown in Figs. 8 and 9. When we analyze the calibration results in Table I, Profile A and is characterized by a lower probability of absorption . This correhigher values for the probability of reflection sponds to the lower measured attenuation in scenario A in comparison with scenario B. The probability of diffuse scattering is higher for Profile B, which corresponds to our expectations since the walls are more irregular in this case ([6], Figs. 1 is weaker for the and 3). Similarly, the diffuse scattering floor in both scenarios since the floors are significantly smoother than the walls and ceilings. It can be concluded that the optimized model parameters are in close agreement with the expectations based on our visual observations.

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required. This would require the standard processing of all incoming rays meeting the receiver position of interest. REFERENCES

Fig. 9. Measured data and optimized curves for scenario C (1853 MHz curves are displayed 30 dB downward).

Similar observations can also be done for scenario C. It can be observed, that profile C is characterized by low attenuation and low value of ) in contrast to scenarios (high value of A and B which are characterized by high attenuation. It can be seen that a good agreement was obtained for both frequencies even at long distances from the transmitter. IV. CONCLUSION An efficient 3D propagation model for subterranean galleries and tunnels implementing semi-deterministic principles based on a modified ray-launching technique and stochastic approach was presented. There are several advantages in the approach used in comparison with classical ray tracing techniques. Firstly, the model is able to consider all significant physical phenomena: reflection, absorption, penetration and scattering. Secondly, the interaction between rays and obstacles is based on the fast semi-deterministic approach using the PRP concept and therefore complex calculations are avoided. The signal level in each grid element is only proportional to the number of rays passing through the element, which also contributes to the speed of the prediction. Furthermore, wall properties are efficiently described by three probabilistic parameters that can be calibrated by measurements, i.e. the model does not require an exact description of wall roughness, material electrical constants, etc. It was shown that the model is able to be efficiently calibrated by empirical data and to predict the signal strength inside subterranean galleries and tunnels with sufficient precision and low computation time and complexity. There are also limitations of the model—it is not able to predict fast signal fluctuations, e.g. caused by different waveguide modes, and needs to be calibrated by empirical data. In principle, this model is also able to consider wave penetration, i.e. it could also be applied to more complex scenarios where wave propagation through walls must be taken into account. In addition, the model could be enhanced for applications where wideband parameters such as impulse response would be

[1] L. Deryck, “Natural propagation of electromagnetic waves in tunnels,” IEEE Trans. Veh. Technol., vol. 27, pp. 145–150, 1978. [2] D. Didascalou, J. Maurer, and W. Wiesbeck, “Subway tunnel guided electromagnetic wave propagation at mobile communications frequencies,” IEEE Trans. Antennas Propag., vol. 49, pp. 1590–1596, 2001. [3] A. Emslie, R. Lagace, and P. Strong, “Theory of the propagation of UHF radio waves in coal mine tunnels,” IEEE Trans. Antennas Propag., vol. 23, pp. 192–205, 1975. [4] M. Ndoh and G. Y. Delisle, “Propagation of millimetric waves in rough sidewalls mining environment,” in Proc. IEEE VTS 53rd Veh. Technol. Conference, VTC Spring, 2001, vol. 1, pp. 439–443. [5] A. V. Popov and Z. N. Yan, “Modeling radio wave propagation in tunnels with a vectorial parabolic equation,” IEEE Trans. Antennas Propag., vol. 48, pp. 1403–1412, 2000. [6] M. Rak and P. Pechac, “UHF propagation in caves and subterranean galleries,” IEEE Trans. Antennas Propag., vol. 55, pp. 1134–1138, 2007. [7] T. Rautiainen, R. Hoppe, and G. Wolfle, “Measurements and 3D ray tracing propagation predictions of channel characteristics in indoor environments,” in Proc. IEEE 18th Int. Symp. on Personal, Indoor and Mobile Radio Communications, PIMRC, 2007, pp. 1–5. [8] Y. P. Zhang, “Novel model for propagation loss prediction in tunnels,” IEEE Trans. Veh. Technol., vol. 52, pp. 1308–1314, 2003. [9] Y. P. Zhang and H. J. Hong, “Ray-optical modeling of simulcast radio propagation channels in tunnels,” IEEE Trans. Veh. Technol., vol. 53, pp. 1800–1808, 2004. [10] Y. P. Zhang and Y. Hwang, “Theory of the radio-wave propagation in railway tunnels,” IEEE Trans. Veh. Technol., vol. 47, pp. 1027–1036, 1998. [11] Y. P. Zhang, G. X. Zheng, and J. H. Sheng, “Radio propagation at 900 MHz in undergroundcoal mines,” IEEE Trans. Antennas Propag., vol. 49, pp. 757–762, 2001. [12] N. Y. Zhu, F. M. Landstorfer, A. V. Popov, and V. A. Vinogradov, “Numerical study of radio wave propagation in tunnels,” in Proc. 29th Eur. Microwave Conf., 1999, pp. 283–286. [13] P. Pechac and M. Klepal, “Effective indoor propagation predictions,” in Proc. IEEE VTS 54th Veh. Technol. Conf., VTC Fall , 2001, vol. 3, pp. 1247–1250. [14] P. Bui Tuong, “Illumination for computer generated pictures,” Commun. ACM, vol. 18, pp. 311–317, 1975. [15] K. F. Man, K. S. Tang, and S. Kwong, Genetic Algorithms: Concepts and Designs With Disk. New York: Springer-Verlag, 1999. Ludek Subrt (S’08) received the M.Sc. degree in radio electronics from the Czech Technical University in Prague, Czech Republic, in 2008, where he is currently working toward the Ph.D. degree. His current research interests are focused on radiowave propagation and cognitive radio.

Pavel Pechac (M’94–SM’03) received the M.Sc. degree and the Ph.D. degree in radio electronics from the Czech Technical University in Prague, Czech Republic, in 1993 and 1999, respectively. He is currently a Professor in the Department of Electromagnetic Field at the Czech Technical University in Prague. His research interests are in the field of radiowave propagation and wireless systems.

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Communications Broadband Bowtie Dielectric Resonator Antenna Leboli Z. Thamae and Zhipeng Wu

Abstract—Broadband dielectric resonator antenna, formed by carving out notches from cylindrical geometry to form bowtie shape and fed by coaxial probe on one of the notched sides, is presented. The proposed broadband bowtie DRA combines shape deformation with low permittivity resonator to achieve 10 dB impedance bandwidth of up to 49.4% covering frequency range of 4.194–6.944 GHz. The simulated and measured radiation patterns are consistent throughout the operational bandwidth. Index Terms—Bowtie shape, broadband, dielectric resonator antenna (DRA).

I. INTRODUCTION Dielectric resonator antenna (DRA) technology has evolved as an attractive alternative to conventional metallic antennas since its inception in the early 1980’s using basic geometries [1]–[3]. DRAs offer advantages of low cost, small size, light weight, low profile, high radiation efficiency and a variety of feed mechanisms. However, being resonant structures, simple DRAs with low permittivity have an inherently narrow bandwidth of up to 10% [4]. Several bandwidth enhancement techniques such as DRs with multilayers of different materials [5] or air-gap modified structures [6]–[9] have been investigated with observed 010 dB impedance bandwidths ranging from about 25% to 70%. This letter implements the latter concept by forming symmetric notches on the DR to create alternative bowtie geometry with broadband characteristics. Numerical analysis employing CST Microwave Studio (CST MWS) is presented together with practical measurements using the vector network analyzer (VNA) and anechoic chamber to validate the simulations. The concept of bowtie DRA from simple cylindrical DR structure by creating notches to deform its shape is first introduced in Section II. A parametric study investigating the effect of notch dimensions, feed location and DR aspect ratio on the bandwidth of the cylindrical bowtie DR is carried out in Section III. In Section IV, the optimized structure for cylindrical bowtie DRA is prototyped and measured and the experimental observations are then discussed in comparison with results of similar air-gap DRA designs from [6]–[9].

Fig. 1. Geometry of cylindrical bowtie DRA; (a) Top view and (b) Panoramic view.

II. ANTENNA CONFIGURATION The geometry of the bowtie DRA, created by carving out symmetrical notches of depth d and apex angle , from a cylindrical DR of dielectric constant "r , radius r and height h is shown in Fig. 1. The bowtie DR is placed on a 60 mm 2 60 mm ground plane and fed by Manuscript received February 20, 2009; revised May 22, 2009; accepted July 18, 2009. Date of publication September 02, 2010; date of current version November 03, 2010. The authors are with the School of Electrical and Electronic Engineering, University of Manchester, Manchester M60 1QD, U.K. (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.2010.2071332

Fig. 2. Effect of notch depth d on the input impedance of the cylindrical bowtie , and r=h = : . DRA for p

= 8 mm = 90

= 10 12 7

a 50 coaxial probe of height p along the notch apex. The DR material used is the Rogers TMM10i with "r = 9:8. The notch depth can be varied to transform the geometry of the DR into a bowtie modification. It is noted that feed position also varies with d, hence the depth adjustment is expected to have a more distinct effect on the operational impedance bandwidth.

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Fig. 5. Photograph of the optimized cylindrical bowtie DRA prototype.

Fig. 3. The dependence of the input impedance on the probe height p for the : , and r=h = : . cylindrical bowtie DRA at d

= 6 5 mm = 90

= 10 12 7

Fig. 6. Simulated and measured reflection coefficient for the cylindrical bowtie and r=h = : . DRA with d : ,p ,

= 6 7 mm = 7 mm = 90

= 10 12 7

TABLE II COMPARISON OF SEVERAL AIR-GAP DRAS

Fig. 4. The dependence of the input impedance on the apex angle  for the cylindrical bowtie DRA at d : ,p and r=h = : .

= 6 5 mm = 7 mm

= 10 12 7

TABLE I EFFECT OF VARYING THE ASPECT RATIO

Fig. 7. Measured gain of the cylindrical bowtie DRA prototype.

A. Notch Depth

III. PARAMETRIC STUDY CST MWS simulation is used to carry out the parametric study of the bowtie DRA to determine the effect of varying the notch depth d, the probe height p and its location, the apex angle  and the aspect ratio r=h on the operational bandwidth.

The simulation curves of Fig. 2 show how the antenna input impedance varies as the notch depth changes with other parameters fixed. As d increases, the volume of the DR decreases and its shape is modified, the position of the feeding probe is adjusted and less field lines are being confined within the DR. It is noted that for smaller values of d such as d = 4:5 mm, six resonances exists within the 3 to 8 GHz range (3.815, 4.17, 5.345, 6.205, 7.19 and 7.43 GHz) with respective resistances of around 80, 27, 157, 24, 46 and 26 . Increasing d to 6.5 mm shifts the second and fifth resonances to 4.5 GHz and 6.7 GHz with 41.5 and 53 , respectively, and the response

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Fig. 8. Normalized simulation and measurement xz -plane radiation patterns of the optimized cylindrical bowtie DRA at three selected frequencies.

Fig. 9. Normalized simulation and measurement yz -plane radiation patterns of the optimized cylindrical bowtie DRA at three selected frequencies.

between the two resonances is flattening close to 50 realize broadband operation.

impedance to

B. Feed Height and Location The effect of the feed probe height p on the antenna input impedance, when located on the notch apex as in Fig. 1, is demonstrated in Fig. 3. The probe height controls the coupling between the feed and DR, resulting in variations in the input impedance with p = 5 mm exciting only one resonance at 6.82 GHz with mismatched resistance of 109

whereas there are four resonances for p = 9 mm with only one (at 5.1 GHz) close to matching at 44 . The probe height of around p = 7 mm yields much improved impedance matching. Locating the feed along the edge of the bowtie DR to coincide with the y -axis leads to narrower bandwidth response with the fundamental mode being dominant. Interestingly, offsetting the feed away from the y -axis gives rise to potential dual or tri-band operation to be explored in future work. C. Apex Angle The variation of the bowtie DRA input impedance with the apex angle  is illustrated in Fig. 4. The simulation curves show that smaller values of  result in mismatched resonances. For instance, the  = 40 curve demonstrates resonances at 3.98, 4.54, 5.31 and 6.17 GHz with respective resistances of 139, 28, 82 and 13 . The value of  = 100 leads to about six resonances, the third (5.75 GHz) and the fifth (6.85 GHz) having respective resistances of 62 and 46 , which are close to the 50 impedance matching.

D. Aspect Ratio Table I shows the dependence of the simulated bandwidth (BW) as a function of the bowtie DR aspect ratio (r=h) by varying the DR height while keeping the radius fixed. It is noted that increasing h leads to wider impedance bandwidth which saturates around 55% for r = 10 mm. However, a trade-off between the achievable broadband operation and practical DR profile has been considered in the final DR height choice. IV. RESULTS AND DISCUSSION From the above parametric study, optimized DRA notch and feed dimensions with a chosen height of 12.7 mm are found to be d = 6:7 mm, p = 7 mm and  = 90 . A bowtie DRA prototype illustrated in Fig. 5 has been fabricated and tested to validate the simulation results. The simulation and measurement reflection coefficient curves are depicted in Fig. 6 with respective 010 dB impedance bandwidths of 53.7% (4.166–7.226 GHz) and 49.4% (4.194–6.944 GHz). The measured bandwidth is about 4% less than the simulated one but both responses show consistent variations with frequency. Examples of notched DRAs are given in Table II for comparison with the proposed bowtie DRA. The bowtie DRA demonstrates higher bandwidth relative to the first two designs but the next two designs possess more bandwidth due to the fact that they employ special feeding structures to aid the wideband operation while the proposed bowtie DRA is implemented with a simple coaxial probe feed. The measured gain along the 0 angle across the operating frequency band is plotted in Fig. 7, showing variations of about 4.1 to 7.2 dBi. The simulated and measured normalized (maximum = 0 dB) principal

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xz and yz planes radiation patterns for the cylindrical bowtie DRA are given in Figs. 8 and 9 at the frequencies of 4.5, 5.7 and 6.8 GHz. The simulated co-polar radiation patterns for both planes portray reasonable agreement with their measured counterparts for the selected three frequencies. On one hand, the xz -plane patterns demonstrate broadside radiation at the first two frequencies with measured cross-polar levels around 10 dB down on the boresight and the backlobes slightly shifted away from 180 . However, the higher 6.8 GHz xz -plane pattern is slightly distorted, shifting the maximum radiation direction to 050 . On the other hand, the yz -plane patterns are largely symmetrical and broadside at all the three centre frequencies with backlobe radiation of about 010 dB or better. The measured cross-polar levels are more than 10 dB down from the boresight, though they do reach similar levels to co-polar counterparts at some directions. Leakage radiation from the probe feed is the likely cause of the xz -plane pattern distortions and high cross-polar levels away from the boresight in the yz -plane patterns and can be suppressed by using dual-feed techniques as in [10]. V. CONCLUSION Broadband bowtie DRA with symmetrical notches carved from cylindrical DR geometry has been studied both numerically and experimentally. Measurement results from the prototype give impedance bandwidth of about 49%, ranging from 4.194 to 6.944 GHz. From the parametric study, it has been observed that the depth of the notches, which modifies the structural shape and varies the position of the feed, the height of the coaxial probe feed and its location together with the notch apex angle and DR aspect ratio are important in enhancing the impedance bandwidth. ACKNOWLEDGMENT The authors are grateful to Rogers Corporation for their support with the provision of high frequency laminates used for the dielectric resonator antenna.

REFERENCES [1] S. A. Long, M. W. McAllister, and L. C. Shen, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. AP-31, no. 3, pp. 406–412, May 1983. [2] M. W. McAllister, S. A. Long, and G. L. Conway, “Rectangular dielectric resonator antenna,” Electron. Lett., vol. 19, pp. 218–219, 1983. [3] M. W. McAllister and S. A. Long, “Resonant hemispherical dielectric antenna,” Electron. Lett., vol. 20, pp. 657–659, 1984. [4] A. Petosa, A. Ittipiboon, Y. M. M. Antar, D. Roscoe, and M. Cuhaci, “Recent advances in dielectric-resonator antenna technology,” IEEE Antennas Propag. Mag., vol. 40, no. 3, pp. 35–48, Jun. 1998. [5] A. A. Kishk, B. Ahn, and D. Kajfez, “Broadband stacked dielectric resonator antennas,” Electron. Lett., vol. 25, pp. 1232–1233, Aug. 1989. [6] A. Ittipiboon, A. Petosa, D. Roscoe, and M. Cuhaci, in An Investigation of a Novel Broadband Dielectric Resonator Antenna, Baltimore, MD, 1996, pp. 2038–2041. [7] A. A. Kishk, R. Chair, and K. F. Lee, “Broadband dielectric resonator antennas excited by L-shaped probe,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2182–2189, 2006. [8] X.-L. Liang and T. A. Denidni, “H-shaped dielectric resonator antenna for wideband applications,” IEEE Antennas Wir. Propag. Lett., vol. 7, pp. 163–166, 2008. [9] L. N. Zhang, S. S. Zhong, and S. Q. Xu, “Broadband U-shaped dielectric resonator antenna with elliptical patch feed,” Electron. Lett., vol. 44, no. 16, 2008. [10] A. Petosa, A. Ittipiboon, and N. Gagnon, “Suppression of unwanted probe radiation in wideband probe-fed microstrip patches,” Electron. Lett., vol. 35, pp. 355–357, 1999.

Miniaturized Dual-Band CPW-Fed Annular Slot Antenna Design With Arc-Shaped Tuning Stub Meng-Ju Chiang, Tian-Fu Hung, Jia-Yi Sze, and Sheau-Shong Bor

Abstract—The design of a miniaturized dual-band CPW-fed annular slot antenna with arc-shaped tuning stub is proposed. The proposed feeding structure, which includes the arc-shaped tuning stub and a 50- transformer, is connected to the extremity of the CPW fed-line, achieving the dual-band input impedance matching. The return loss of the proposed design exhibits that two wide operating bands are over the bandwidths of 30.8% and 24.0%, respectively. By modifying the angle of the arc-shaped tuning stub, the first-higher order resonant mode can be shifted to the lower frequency band to combine with the fundamental mode of the annular slot, achieving the broadband operation with the bandwidth of 3048 MHz (78.4%). Moreover, the miniaturized design indicates an embedded strip is protruded from the ground plane into a slit, revealing the center frequencies of two resonant bands from 2752 to 1738 MHz (size reduction 60%) and 5022 to 3760 MHz (size reduction 44%), respectively. Two resonant bands of the prototype show the broadside radiation patterns with the maximum peak antenna gains of 3.8 dBi and 5.1 dBi, respectively.



Index Terms—Antenna feeds, coplanar waveguides, multifrequency antennas, Slot Antennas.

I. INTRODUCTION A literature survey shows that the printed slot antenna with the tremendous bandwidth is a promising candidate for realizing broadband or multiple-band to accommodate multi-standard services. For a conventional CPW-fed slot antenna design, the impedance bandwidth can reach about 20%. Several structures have been reported to demonstrate the superior performances of slot antenna for the desired wideband requirements, which include wide square slot [1], bow-tie slot [2], rotated slot [3], and tapered ring slot [4], [5], indicating the impedance bandwidths larger than 100%. Consequently, the hybrid slots design shows an impedance bandwidth of 49% [6]. By using the feeding structure with a widened tuning stub [7] or fork-like tuning stub [8], a square slot antenna for broadband operation achieves the impedance bandwidth of more than 60%. Ali et al. [9] proposed a folded slot with adjacent semicircular slot and a tuning stub to improve the bandwidth, achieving a 42.8% bandwidth, and Liu et al. [10] reported the similar approach to improve the impedance bandwidth of 100%. Simply by protruding four metallic strips into slot, the proposed CPW-fed slot antenna can have an impedance bandwidth larger than 60% [11], [12]. On the other hand, the dual-band operation is in demand for various applications. Since these applications are used simultaneously in many Manuscript received March 14, 2009; revised February 04, 2010; accepted April 12, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. M.-J. Chiang is with the Department of Antenna and Wireless System Integration, HTC Corporation, Xindian City, Taipei County 231, Taiwan (e-mail: [email protected]). T.-F. Hung is with the Department of Electrical and Communications Engineering, Feng-Chia University, Taichung 407, Taiwan (e-mail: [email protected]). J.-Y. Sze is with the Electrical and Electronic Engineering Department, Chung Cheng Institute of Technology, National Defense University, Taoyuan 335, Taiwan (e-mail: [email protected]). S.-S. Bor is with the Department of Electrical Engineering, Feng-Chia University, Taichung 407, Taiwan (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.2010.2071333

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systems, a single antenna that can cover all these bands is needed. In resent years, many types of slot antenna have been developed for dualband operations. The previous researches show that the slot antenna designs are applied to multiple ring slots [13], [14] or modified annular ring [15], inserting composite patches on the backside of the square slot [16], conjunct with a back-patch [17] or slit back-patch [18], implanting a pair of symmetric Y-shape-like slots [19], and utilizing the E -like feeding structure and adding the matching stub at the slot edge and four floating patches on the backside [20]. Moreover, the design presented in [21] shows the dual-band operation is based on an open-circuited stub and the T-mating circuit is also used to match the antenna in both bands. In recent years, there has been a growing research activity on the antenna size for the industrial realization, which not only improves aesthetic appearance, but also facilitates simple design, serviceability, compactness, and robustness. As a result, the need for wireless communication antenna with at least keeping compact size has grown substantially. The literature survey shows that there are several types of miniaturized technologies of slot antenna [22]–[25]. Hong [22] indicated the annular slot antenna by using a loaded capacitor achieve 23.4% size reduction. Ma et al. [23] demonstrated annular slot with a tapered-slot-fed is adequate for ultra-wideband (UWB) applications and Kim et al. [24] reported a meander-type slot antenna with the size about 56% smaller than that of conventional design. Consequently, there is another approach in antenna miniaturization; that is, the two short circuits at the end of the resonant slot are replaced by reactive boundary condition, showing the dimensions of total slot of 0:050 2 0:050 [25]. The work reported in [26] indicates that a pair of the arc-like metal strips is embedded in the annular slot for dual-band operation and achieves 80.6% size reduction. In this communication, the novel feeding structure including of the arc-shaped tuning stub and a 50- transformer is proposed and applied to the dual-band planar slot antenna design on the 0.8-mm FR-4 substrate. By applying the proposed technology presented in Section II for the matching of the dual resonant mode, the prototype reveals the dual-band operation over the bandwidths of 30.8% and 24.0%, respectively. Consequently, Section II also reports the feeding technology with the modified angle of the arc-shaped tuning stub reveals the broadband operation with the bandwidth of 3048 MHz (78.4%). Section III reports the miniaturized design, which is protruded an embedded strip from the ground plane into a slit to reduce two resonant frequencies with size reductions of 60% and 44%, respectively. To the best knowledge of the authors’, the proposed slot antennas are the first to vary the geometric structural parameters, achieving dual-wideband, broadband and miniaturized designs simultaneously. Conclusions are finally drawn in Section IV, along with recommendations for future research. II. DUAL-BAND

AND

BROADBAND CPW-FED ANNULAR SLOT ANTENNA DESIGNS

A. Antenna Configuration Fig. 1(a) shows the schematic of dual-band planar slot antenna on the 0.8-mm FR-4 substrate by using the proposed feeding structure with the arc-shaped tuning stub and a 50- transformer. The ground plane is chosen to be square, and the outer and inner radii of the annular slot are denoted by R1 and R2 , respectively. The prototype including of the ground plane shown in Fig. 1 is 70.0 mm 2 70.0 mm. A 50-

CPW feed-line is used to excite the proposed slot antenna, revealing the line width (Wf ) of the metal strip with two gaps of the distance (g ) between the signal trace and the coplanar ground plane. For the design of 50- CPW feed-line, the dimensions are Wf = 4:2 mm and g = 0:3 mm on the 0.8-mm FR-4 substrate with "r = 4:4. In designing

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Fig. 1. (a) Schematic of the proposed planar annular-ring slot antenna. (b) De: ,R : , tailed dimensions of the feeding structure. R W : : : : ,` ,` ,t ,S , : : : :. ,g ,h ," W

= 19 5 mm = 11 0 mm = 3 0 mm = 9 2 mm = 7 mm = 0 2 mm = 0 1 mm = 4 2 mm = 0 3 mm = 0 8 mm = 4 4

the reference annular slot antenna, the lowest operating frequency ft of the fundamental mode is first evaluated by

ft 

c

1 p"eff c is the speed of light at free space and "eff = 2"r =("r + 1) is the 2 1

(R +R ) 1:8

(1)

effective dielectric constant. The half circumference is the estimated longest current path along the annular slot. In fact, the above formulas have been confirmed to yield acceptably accurate fundamental resonant-band centre frequencies after extensive experiments. Following this design guideline, the reference annular slot antenna is designed on a 0.8-mm FR-4 substrate with R1 = 19:5 mm and R2 = 11:0 mm, respectively. The feeding structure shown in Fig. 1(b) consists of the two tuning sections, which are the arc-shaped tuning stub and 50- transformer. As shown in Fig. 1(b), the arc-shaped tuning stub with the extended angle () and width (t) is linked a 50- transformer, which is connected to the extremity of the CPW fed-line, achieving the dual-band impedance matching. Furthermore, the angle () of the arc-shaped tuning stub is used to control the movable high-band to make dualor broad-band design and the width of the tuning section (Wt ) is used to match the dual-band input impedance. B. Feeding Structure Design The physics of operation mechanisms for the modification of the feeding-structure dimensions are complex. Therefore, the Smith chart results were presented, followed by experimental validations of the particular design. For the dual-band impedance matching, Fig. 2 reveals the measured input impedance on Smith chart for various geometry of feeding structure. Fig. 2(a) plots the measured input impedance on Smith chart when the conventional annular slot excited the fundamental mode at the frequency band from 2505 MHz to 2835 MHz. In order to improvement of the bandwidth, the design shown in Fig. 2(b) reveals

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TABLE I PERFORMANCES OF THE PROPOSED DUAL-BAND AND BROADBAND CPW-FED ANNULAR-RING SLOT ANTENNAS. R W : : : : : : : ,` ,` ,t ,g ,S ,W ,h

= 3 0 mm

= 9 2 mm

= 7 0 mm = 0 2 mm = 0 1 mm

The Ref. design reveals the tuning stub with the width of 4.2 mm

= 19:5 mm, R = 11:0 mm, = 4 2 mm = 0 3 mm = 0:8 mm, " = 4:4

(= W = W = W ) and length of 8.4 mm, respectively.

Fig. 3. The measured and simulated return loss of the reference circular slot antenna and proposed prototypes against frequency.

(`t ) and width (Wt ), indicating two resonant frequency bands of the operating bandwidths of 854 MHz (30.8%) and 1206 MHz (24.0%), respectively. Table I summarizes the corresponding structural parameters for discussions in the following sections and the proposed impedance matching network of the prototype is applied to design of the dual-band slot antenna design, which is reported in the next section.

C. Return Loss and Radiation Patterns

Fig. 2. Measured input impedance on Smith chart for various feeding structures.

an arc-shaped tuning stub with the extended angle ( ) of 60 and width (t) of 0.2 mm is connected to the end of the 50- CPW fed-line. The bandwidth of the prototype is increased, and indicates 25.2% with the central frequency of 2584 MHz. For achieving dual-band operating, a 50- transformer with the length (`s ) of 9.2 mm and width (Ws ) of 3.0 mm shown in Fig. 2(c) is added in the antenna design and the transformer can be used as a series inductor for high-band matching. Consequently, the feeding structure unites the additional patch with the length

In this study, experimental results of impedance and radiation characteristics of the proposed antenna are measured and presented. The S -parameters of the proposed designs are measured after the shortopen-load-through (SOLT) calibration procedures that had been performed by the Agilent 8510C vector network analyzer (VNA). Parallel to the measurements, the prototype shown in Fig. 1 is also theoretically examined by the commercial software package Ansoft HFSS. Fig. 3 shows the simulated and measured return loss curves of the reference circular slot antenna and proposed prototypes against frequency. The reference design, which has the dimensions of the slot mentioned above, is excited the fundamental mode and the first-higher order resonant mode of the annular slot. For the matching of dual-band input impedance, the reference design is added to the proposed feeding structure to accomplish the dual-band operation. When the angle of the extended angle () of the arc-shaped tuning stub is equal to 10 , the dual-band operation achieves the frequency ratio of 1.85. As increasing the angle  from 10 to 60 , the prototype with the frequency ratio of 1.81 indicates two resonant frequency bands of the operating bandwidths of 854 MHz (30.8%) and 1206 MHz (24.0%), respectively. Furthermore, the simulated result, which is marked by the red line with

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Fig. 4. Measured E-plane and H-plane radiation patterns of Ant. 2 ( = 60 ). (a) f c = 2775 MHz; (b) f c = 5026 MHz.

symbols in the red filled circles (), also plots in Fig. 3, revealing excellent agreements with those of the measurements. By continuing with increase of the angle  from 60 to 120 , the broadband design indicates the immense bandwidth of 3048 MHz (78.4%) at center frequency of 3887 MHz. Table I summarizes the extracted results of varying the corresponding structural parameters, showing the tremendously wide frequency ratio from 1.85 to 0. Radiation characteristics of the proposed slot antennas (Ant. 2 and Ant. 3) at operating frequencies have also been studied. Fig. 4(a) and (b) show the measured E-plane (y 0 z plane,  = 90 plane) and H-plane (x 0 z plane,  = 0 plane) radiation patterns of Ant. 2 ( = 60 ) at the frequencies of f c1 = 2775 MHz and f c2 = 5026 MHz and Ant. 3 ( = 120 ) at their frequencies of fc1 = 3000 MHz and fc2 = 4900 MHz, respectively. It is observed that, for the dual impedance bandwidth frequencies of Ant. 2, acceptable broadside radiation patterns are obtained and good cross-polarization characteristics. Those are extremely similar to that of the reference antenna (not shown here) at its fundamental resonant and the first-higher order resonant bands. The cross-polarization level in the H-plane is high at the upper band. This is due to the fact that, to achieve good dual matching, the impedance levels for both resonances can not be too different. Hence, the current nulls in the second band move out the horizontal (shorter) sides of the loop slot [21]. The radiation patterns of Ant. 3 at the frequencies of fc1 = 3000 MHz and fc2 = 4900 MHz reveal the pretty similar E-plane and H-plane radiation patterns to that of Ant. 2. The unique characteristics of the dual-band design are applied to invention of its miniaturized design, which is reported in the next section. III. MINIATURIZED DUAL-BAND ANNULAR SLOT ANTENNA A. Miniaturization of Dual-Band Slot Antenna Fig. 6 shows the schematic of the miniaturized design on the 0.8-mm FR-4 substrate. The dual-band miniaturized operation is based on an

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Fig. 5. Measured E-plane and H-plane radiation patterns of Ant. 3 ( = 120 ). (a) fc = 3000 MHz; (b) fc = 4900 MHz.

Fig. 6. Schematic of the miniaturized designs with a protruded strip from the ground plane into the slot.

open-circuited stub located at the mid point of the loop side opposite to the input CPW feed. It is used to obtain the appropriate equivalent magnetic current distribution in the radiating slots, while still keeping a compact size. The proposed design is used Ant.2 ( = 60 ) with a protruded metal strip from the ground plane into the slit. The embedded metal strip with the length (`d ) and width of 1.0 mm is inserted into the slit with the length (`2 ) and width of 4.0 mm. As shown in Fig. 7, the measured and simulated return loss of the miniaturized designs against frequency indicate that the prototype with `d = 8:4 mm and `2 = 0 mm, revealing two center operating frequencies shifted from 2775 MHz to 2513 MHz and from 5026 MHz to 4303 MHz, respectively. Consequently, the miniaturization shows two bands with the size reductions of 60% and 44% when `d = 17:5 mm and `2 = 11:0 mm, varying on two center frequencies from 2775 MHz to 1738 MHz and 5026 MHz to 3760 MHz, respectively. Moreover, the completed prototype is also analyzed by using HFSS to verify the embedded metal strip

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 11, NOVEMBER 2010

Fig. 7. The measured and simulated return loss of the miniaturized designs against frequency.

Fig. 9. Measured peak antenna gains of the proposed prototypes against frequency.

4.6–7.3 dBi across the operating bandwidth from 2363 to 5411 MHz. The peak antenna gain level of Ant. 3 at its operating band reveals similar with that of Ant. 2. As shown in Fig. 9, the antenna peak gains of the lower operating band of Ant. 5 is measured to be around 3.8 dBi, which is 1.2 dBi lower than that of Ant. 2 in its lower operating band and 5.1 dBi of the upper operating band which is 0.9 dBi lower than that of Ant. 2 in the upper operating band. IV. CONCLUSION The miniaturized dual-band annular slot antenna has been designed and successfully implemented. The proposed feeding structure achieves input impedance matching of the dual-band, exhibiting two wide operating bands over the bandwidths of 30.8% and 24.0%, respectively. The broadband operation with the bandwidth of 3048 MHz (78.4%) reveals the similar E-plane and H-plane radiation patterns to that of dual-band design. The miniaturized design shows the size reductions of two resonant frequencies of 60% and 44%, respectively. Two resonant bands of the miniaturized design show the broadside radiation patterns with the maximum peak antenna gains of 3.8 dBi and 5.1 dBi, respectively. The prototypes demonstrate the potential of the proposed feeding structure for broader bandwidth and miniaturization. ACKNOWLEDGMENT Fig. 8. Measured E-plane and H-plane radiation patterns of Ant. 5. (a) f c = 1738 MHz; (b) f c = 3760 MHz.

The authors would like to thank the National Center for High Performance Computing, Hsinchu, Taiwan, and Ansoft Corporation, Taiwan, for the support of the simulation tools.

REFERENCES effects on the dual-band designs. As shown in Fig. 7, the result of the simulation reveals excellent agreement with that of the measurement. B. Radiation Characteristics Fig. 8 plots the measured E-plane and H-plane radiation patterns of Ant. 5 (`d = 17:5 mm and `2 = 11:0 mm) at 1738 MHz and 3760 MHz, respectively. Two resonant modes shown in Fig. 8 have broadside radiation patterns and similar cross-polarization level in the H-plane with Ant. 2 and Ant. 3. Consequently, the measured peak antenna gains of dual-band (Ant. 2), broadband (Ant. 3) and miniaturized (Ant. 5) designs against frequency are plotted in Fig. 9. The measured results show that Ant. 2 has the maximum peak antenna gains of dual center frequencies about 5.5 dBi and 6.7 dBi with gain variations less than about 1.4 dBi and 1.0 dBi across the operating bandwidth from 2348 to 3202 MHz and 4423 to 5629 MHz, respectively. The peak antenna gain of Ant. 3 is about 6.0 dBi and varies in the range of

[1] J.-W. Niu and S.-S. Zhong, “A CPW-fed broadband slot antenna with linear taper,” Microw. Opt. Technol. Lett., vol. 41, no. 3, pp. 218–221, May 2004. [2] J.-F. Huang and C.-W. Kuo, “CPW-fed bow-tie slot antenna,” Microw. Opt. Technol. Lett., vol. 19, no. 5, pp. 358–360, Dec. 1998. [3] J.-Y. Jan and J.-W. Su, “Bandwidth enhancement of a printed wide-slot antenna with a rotated slot,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 2111–2114, Jun. 2005. [4] T.-G. Ma and C.-H. Tseng, “An ultrawideband coplanar waveguide-fed tapered ring slot antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1105–1110, Apr. 2006. [5] Y.-C. Lee, S.-C. Lin, and J.-S. Sun, “CPW-Fed UWB slot antenna,” in Proc. Asia-Pacific Microwave Conf., 2006, pp. 1636–1639. [6] A. U. Bhobe, C. L. Holloway, M. Piket-May, and R. Hall, “Coplanar waveguide fed wideband slot antenna,” Electron. Lett., vol. 36, no. 16, pp. 1340–1342, Aug. 2000. [7] H.-D. Chen, “Broadband CPW-fed square slot antennas with a widened tuning stub,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1982–1986, Aug. 2003.

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[8] J.-Y. Sze and K.-L. Wong, “Bandwidth enhancement of a microstripline-fed printed wide-slot antenna,” IEEE Trans. Antennas Propag., vol. 49, no. 7, pp. 1020–1024, Jul. 2001. [9] M. Ali and R. Usaha, “A broadband folded-slot antenna with a semicircular extension,” Microw. Opt. Technol. Lett., vol. 41, no. 1, pp. 32–37, Apr. 2004. [10] Y. F. Liu, K. L. Lau, Q. Xue, and C. H. Chan, “Experimental studies of printed wide-slot antenna for wide-band applications,” IEEE Antennas Wireless Propag Lett., vol. 3, pp. 273–275, 2004. [11] X. Ding and A. F. Jacob, “CPW-fed slot antenna with wide radiating apertures,” IEE Proc. Microw. Antennas Propag., vol. 145, no. 1, pp. 104–108, Feb. 1998. [12] J.-Y. Chiou, J.-Y. Sze, and K.-L. Wong, “A broad-band CPW-fed striploaded square slot antenna,” IEEE Trans. Antennas Propag., vol. 51, no. 4, pp. 719–721, Apr. 2003. [13] J.-S. Chen, “Multi-frequency characteristics of annular slot antennas,” Microw. Opt. Tech. Lett., vol. 38, pp. 506–511, Sep. 2003. [14] W.-S. Chen and K.-L. Wong, “Dual-frequency operation of a coplanar waveguide-fed dual-slot loop antenna,” Microw. Opt. Tech. Lett., vol. 30, pp. 38–40, Jul. 2001. [15] T. Lee, J. Jung, H. Lee, and Y. Lim, “Modified annular ring slot antenna for dual WLAN band application,” in Proc. Int. Conf. on Microw and Millimeter Wave Technology (ICMMT), 2008, pp. 410–412. [16] J.-Y. Sze, C.-I. G. Hsu, and S.-C. Hsu, “Dual-broadband multistandard printed slot antenna with a composite back-patch,” IET Microw. Antennas Propag., vol. 2, no. 2, pp. 205–209, Mar. 2008. [17] J.-S. Chen and H.-D. Chen, “Dual-band characteristics of annular slot antenna with circular back-patch,” Electron. Lett., vol. 39, no. 6, pp. 487–488, Mar. 2003. [18] J.-Y. Sze, C.-I. G. Hsu, and S.-C. Hsu, “CPW-fed circular slot antenna with slit back-patch for 2.4/5 GHz dual-band operation,” Electron. Lett., vol. 42, no. 10, pp. 563–564, May 2006. [19] J.-Y. Sze, C.-I. G. Hsu, and S.-C. Hsu, “Design of a compact dual-band annular slot antenna,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 423–426, 2007. [20] C.-J. Wang, J.-J. Lee, and R.-B. Huang, “Experimental studies of a miniaturized CPW-fed slot antenna with the dual-frequency operation,” IEEE Antennas Wireless Propag Lett., vol. 3, pp. 151–154, 2003. [21] D. Llorens, P. Otero, and C. Camacho-Peñalosa, “Dual-band, single CPW port, planar-slot antenna,” IEEE Trans. Antenna Propag., vol. 51, no. 1, pp. 137–139, Jan. 2003. [22] C.-S. Hong, “Small annular slot antenna with capacitor loading,” Electron Lett., vol. 36, no. 2, pp. 110–111, Jan. 2000. [23] T.-G. Ma and S.-K. Jeng, “Planar miniature tapered-slot-fed annular slot antennas for ultrawideband radios,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1194–1202, Mar. 2005. [24] J.-M. Kim, J.-G. Yook, W.-Y. Song, Y.-J. Yoon, J.-Y. Park, and H.-K. Park, “Compact meander-type slot antennas,” in Proc. Antennas and Propag. Int. Symp., 2001, pp. 724–727. [25] R. Azadegan and K. Sarabandi, “A novel approach for miniaturization of slot antenna,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 421–429, Mar. 2003. [26] M.-J. Chiang, J.-Y. Sze, and G.-F. Cheng, “A compact dual-band planar slot antenna incorporating embedded metal strips for WLAN applications,” in Proc. 39th Eur. Microwave Conf., Rome, Italy, Sep. 2009, pp. 221–224.

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Low Profile Spiral on a Thin Ferrite Ground Plane for 220–500 MHz Operation Ioannis Tzanidis, Chi-Chih Chen, and John L. Volakis

Abstract—A small size, low profile spiral antenna (2 thick and 14 in aperture) is described that is backed by a thin ferrite ground plane. The antenna is intended to cover Satellite (UHF TacSat earth to LEO: 243–318 MHz) and enhance Radio (PRC-148, PRC-117F, PSC-5D: 243–512 MHz) communications. We carry out a step-by-step design to achieve the chal; Bandwidth 220–500 lenging goals of RHCP gain 0 dBi at 10 dBi); Low proMHz; VSWR 2:1; Low LHCP (cross-pol) gain ( file (2 ) and desired aperture size 14 . Ground plane ferrite treatment, inductive loading, resistive termination and impedance matching techniques are adopted to raise the gain between 200–300 MHz and achieve the stated at 220 MHz and only goals. As the aperture size is about in height, achieving 0 dBi (from 220–500 MHZ) is quite challenging. Measured and simulated data are presented in the last section leading to impressive conclusions about the capability of the magnetic material coating to overcome the perennial issue associated with the ground plane’s negative effects.

= 0

35

30

Index Terms—Ferrites, ground plane treatment, low-profile, resistive loading, spiral antenna.

I. INTRODUCTION A recent paper [1], showed that volumetric coiling of a spiral antenna can allow for size reduction of about 50% with 015 dBi of total realized gain at 145 MHz using a 600 aperture. Size reduction based on planar zigzagging and meandering has, of course, been known for a while [2]. The work in [1] and [3] presented techniques for spiral miniaturization, including planar (zigzag or square meandering) and volumetric (3D coiling) approaches. It was shown that volumetric spirals enhance gain at lower frequencies, whereas planar meandering shifts the “knee” of the gain curve (at 0 dBi) to lower frequencies. Nevertheless, when a wideband antenna is placed close to a metallic ground plane, low frequency radiation suffers due to low radiation resistance seen by the antenna. Also, for finite ground planes, diffractions from peripheral edges cause pattern distortion. To improve low frequency performance over a ground plane, Kramer et al. [4] and more recently Erkmen et al. [5] used commercial ferrite materials as ground plane coatings. Of course, ferrite materials have been used in industry for many years but primarily as absorbers. Recently, the work in [5] demonstrated that moderately lossy ferrite layers when used as ground plane coatings, can recover or even exceed the gain of the free standing antennas. It was also noted that some material loss can facilitate impedance matching leading to lower return loss at low frequencies [6]. In [7] and [8] it was further noted that low VSWR ( 0), region (II) is an annular aperture (radii: a and b, depth: d), and region (III) is the lower half-space (z < 0). All regions are assumed to be air (" = " ,  =  ).

Fig. 2. Equivalent problem of region (I) based on superposition principle.

cylinder and the infinite plane are perfect electric conductors. The resulting geometry is -symmetric, and we select for our source a -symmetric electric ring current source as

^  ( ^ 0  ( 0 0 ) (z 0 z 0 ) = J r) J( r) = I

where I0 is a constant current and ^ is a unit vector in the -direction. Since the given geometry is azimuthally symmetric, only the T E z modes are excited [9] and the governing equation for the magnetic r) becomes vector potential A (

@ 1 @   @ @

+

p

II. FIELD ANALYSIS Fig. 1 shows an infinitely long conducting cylinder piercing a circular aperture in a conducting plane. A long conducting cylinder of radius a has its axis coincident with the z -axis. It is assumed that the Manuscript received September 15, 2009; manuscript revised March 15, 2010; accepted May 07, 2010. Date of publication August 30, 2010; date of current version November 03, 2010. The authors are with the Department of Electrical Engineering, KAIST, Daejeon, 305-701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2010.2071351

(1)

@ 1 r) = 00 J + k2 0 2 A ( @z 2 

(2)

where k = ! 0 "0 is the wave number in free space. The time dependence, e0i!t , is suppressed. We separate the original problem of Fig. 1 into two parts based on the superposition principle, as shown in r) and the secFig. 2. A solution to (2) consists of the primary Ap ( ondary AI;II;III ( r) potentials, where the primary term is a response  to a current source in Fig. 2(a) and the secondary term is due to the presence of the aperture at z = 0 in Fig. 2(b). We can write the prir) as mary term Ap (

Ap ( r ) = 00 I0 0

0018-926X/$26.00 © 2010 IEEE

1

0

Z1 (< )H1(1) (> ) sin z sin z 0 d (3) H1(1) (a)

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where Z1 (a) = J1 ()N1 (a) 0 N1 ()J1 (a),  = k2 0  2 and Im  > 0. The functions J1 (1) and N1 (1) represent the Bessel functions of the first and second kinds of order one, respectively. The (1) function H1 (1) is the Hankel function of the first kind of order one. In (3), the notation < (> ) designates the smaller (larger) of  and 0 . Since the problem of Fig. 2(b) is source-free, the magnetic vector potential in region (I) is

AI (; z) =

1

0

A~I ( )eiz J 2 (aZ)1+(N)2 (a) d 1

(4)

1

k2 0  2 , Im  > 0 and the Weber transform pair [10]

where  = yields

A~I ( ) =

1

a

AI (; 0)Z1 ()d:

(5)

In region (II), the magnetic vector potential is

AII  (; z ) =

1

am ei (z +d=2) + bm e i (z+d=2) R1 ( m ) 0

m=1

=1

Fig. 3. Magnitude of total electric field E for a ring current (I ) located at  : , z :  with a , b : , d : , and N .

= 1 25

=05

=1

=15

=08

=5

(6)

Similarly, the boundary conditions at z = 0d yield J 1 ( m a) R1 ( m ) = J1 ( m ) 0 N ( a) N1 ( m ): (7) i (d=2) + bm ei (d=2) )I (1) (am e mp 1 m m=1 The parameter m is determined by R1 ( m b) = 0 where m = i (d=2) 0 bp ei (d=2) )[Ap (b) 0 Ap (a)]: (16) = 0ip (ap e k2 0 m2 and Im m > 0. The magnetic vector potential in region (III) is The time-averaged incident power (Pinc ) and the power transmitted through the aperture (Ptran ) are given as Z 1 () III III i (z +d) ~ A (; z) = A ( )e J12 (a) + N12 (a) d: (8) 0 Pinc = 0 i! I0 Re[Ap (r = r ) + AI (r = r )] (17) Applying the Weber transform a (1)Z1 ()d to the E (= 2 1 1 Ptran = ! i!A ) field continuity at z = 0, we obtain 0 m=1 2 m2 N12 ( m b) 0 N12 ( m a) i (d=2) + bm e i (d=2) )4m ( ) (9) m (jam j2 0 jbm j2 ) if k > m (am e A~I ( ) = 2 : (18) m=1  2 i Im [ a b ] if k
2 for the cases considered here. The power transmission behavior shown in Figs. 4 – 6 may have relevance to practical applications particularly when the penetrating cylinder and conducting planes are thick. IV. CONCLUSION Electromagnetic wave penetration into a circular aperture pierced by a long cylinder was investigated. By using the Weber transform and the mode-matching method, we obtained the solution in a fast converging series form. The effects of the plane thickness and source location on the transmission coefficients were also investigated. The formulation based on the mode-matching provides a rigorous, viable, and efficient means to evaluate transmission through a circular aperture pierced by a long cylinder. Our approach based on the Weber transform is rather restricted to the problem possessing azimuthal symmetry such as a constant ring current. However, our theoretical model can be further extended and applied to other practical problems dealing with circular aperture pierced by conducting wires, such as via hole models, antenna feeds or cable systems linking different equipments. Fig. 5. Transmission coefficients versus d=:  I : (a) a  and (b) a : .

=1

=1

= 0 05

= 1:25, z = 0:5 and

where c = b=a and the values of 0mn are tabulated in [12]. First few cutoff wavelengths of the T E0zn modes are listed in Table I. Since all modes are cut off (evanescent) when c = 1:5, T approaches 0 when b= = 1:5 in Fig. 5(a). The transmission power increases as the aperture size becomes larger. Fig. 5(b) illustrates the behavior of T when the cylinder radius is very small (a = 0:05). The variations of T are small as the aperture size (b=) and the aperture depth (d=) change.

REFERENCES [1] J. M. Jin and J. L. Volakis, “Electromagnetic scattering by and transmission through a three-dimensional slot in a thick conducting plane,” IEEE Trans. Antennas Propag., vol. 39, pp. 543–550, Apr. 1991. [2] A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A, vol. 4, no. 10, pp. 1970–1983, Oct. 1987. [3] R. Lee and D. G. Dudley, “Transient current propagation along a wire penetrating a circular aperture in an infinite planar conducting screen,” IEEE Trans. Electromagn. Compat., vol. 32, pp. 137–143, May 1990. [4] E. Zheng, R. F. Harrington, and J. R. Mautz, “Electromagnetic coupling through a wire-penetrated small aperture in an infinite conducting plane,” IEEE Trans. Electromagn. Compat., vol. 35, pp. 295–300, May 1993.

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[5] G. Manara, M. Bandinelli, and A. Monorchio, “Electromagnetic coupling to wires through arbitrarily shaped apertures in infinite conducting screens,” Microw. Opt. Tech. Lett., vol. 13, no. 1, pp. 42–44, Sep. 1996. [6] V. Daniele, M. Gilli, and S. Pignari, “EMC prediction model of a single wire transmission line crossing a circular aperture in a planar screen,” IEEE Trans. Electromagn. Compat., vol. 38, pp. 117–126, May 1996. [7] T. S. Bird, “Exact solution of open-ended coaxial waveguide with center conductor of infinite extent and applications,” IEEE Proc. Microw. Antennas Propag., vol. 134, no. 5, pp. 443–448, Oct. 1987. [8] T. S. Bird, “TE mode excitation of flanged circular coaxial waveguides with an extended center conductor,” IEEE Trans. Antennas Propag., vol. AP-35, pp. 1358–1366, Dec. 1987. [9] D. G. Dudley, Mathematical Foundations for Electromagnetic Theory. Piscataway, NJ: IEEE Press, 1994, p. 168. [10] B. Davies, Integral Transforms and Their Applications, 3rd ed. New York: Springer, 2002, p. 234. [11] COMSOL Multiphysics Version 3.4 COMSOL, 2007. [12] N. Marcuvitz, Waveguide Handbook. New York: McGraw-Hill, 1951, pp. 77–78. Fig. 1. Cross section of a multilayered dielectric cylinder.

An Efficient Asymptotic Extraction Approach for the Green’s Functions of Conformal Antennas in Multilayered Cylindrical Media Jun Wu, Salam K. Khamas, and G. G. Cook

Abstract—Asymptotic expressions are derived for the dyadic Green’s functions of antennas radiating in the presence of a multilayered cylinder, where analytic representation of the asymptotic expansion coefficients eliminates the computational cost of numerical evaluation. As a result, the asymptotic extraction technique has been applied only once for a large summation order . In addition, the Hankel function singularity encountered for source and evaluation points at the same radius has been eliminated using analytical integration. Index Terms—Cylindrical antennas, dyadic Green’s function, method of moments, multilayered media.

I. INTRODUCTION Efficient computation of dyadic Green’s functions (DGF) for antennas in the vicinity of a layered dielectric cylinder has been the focus of several studies in recent years [1]–[7]. The infinite series involved converges slowly however, or even diverge, when the source and observation points are located at the same dielectric interface, that is when 0 =  = ai . A procedure to accelerate the convergence has been introduced in [4], which has been enhanced further in a subsequent study [5]. The expressions developed in [5] can be used to model source and field points that are at the same dielectric interface with a small azimuth separation 1. However, they are singular when  = 0 , hence further investigations have been reported to eliminate the singularity, involving either a hybrid approach [6], or a small argument Hankel function approximation [7]. In the aforementioned studies asymptotic Manuscript received September 23, 2009; revised March 27, 2010; accepted March 31, 2010. Date of publication September 16, 2010; date of current version November 03, 2010. The authors are with the Communications Research Group, Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. (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.2010.2077030

!1

extraction has been considered twice; once for the proper truncation where an n independent asymptotic of the infinite series as n expansion coefficient has been introduced, and secondly as kz for the efficient computation of the Sommerfeld integral. Owing to the rather complex nature of reflections at the dielectric interfaces, the expansion coefficients have been determined using numerical approaches such as extrapolation [5]. Asymptotic DGF expressions have then been achieved using the Sommerfeld identity and the Hankel functions addition theorem [8]. In this article, alternative asymptotic DGF representations are derived using the large order principal asymptotic forms of the Bessel and Hankel functions [9], where simpler non-singular expressions have been obtained together with analytic formulations of the asymptotic expansion coefficients. Use of these expressions saves computation time and provides considerable algebraic simplifications include analytical integration of the Hankel function singularity using the Sommerfeld identity, so that no further measures are needed to handle the singularity. Furthermore, it has been shown that a single asymptotic extraction, for larger n, is sufficient to formulate closed form DGF expressions, which results in a computationally more efficient model. The algorithm has been evaluated using a method of moments (MoM) technique where cylindrically conformal wire antennas have been analyzed. Enhanced convergences of computed input impedances validate the accuracy and efficiency of the proposed formulation.

!1

II. FORMULATION General representations of cylindrical DGF are given briefly in this section based on those reported in [1], [2]. This is followed by the proposed asymptotic DGF formulation, which is fundamentally different from that presented in earlier studies [4]–[7]. A. General DGF Expansions A multilayered cylindrical structure is illustrated in Fig. 1, where each dielectric layer has a permittivity and a permeability of "i and i respectively. When the source and observation points are in the ith layer, the spatial domain dyadic Green’s function for a conformal current element is given by [1]

1

G(r; r ) = 0 8!" 0

0018-926X/$26.00 © 2010 IEEE

i

2

1

01

e0jk

(z

0

1 0 e 01 ^z^ + G z^^ + G ^^

z )

n=

jn(

 )

n=

zz

Gzz ^^ + Gz

z



dkz

(1)

3738

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where

adopted as they lead to significant simplification of the final DGF expressions;

Gzz = ki2 f11 i  @f21 Gz = nk z f11 + j! nkz @ i  @f12 Gz = nk z f11 0 j!" nkz @ 2 2 i  @f12 G = nkki2 f11 0 kz j!" ki2 n @ i 2 @ 2 f22 + jn!i @f21 + kk2i @@ kz  @ i 2 i  @f21 0 n  f11 + j! kz n @ ki = !pi "i ; ki = ki2 0 kz2 : Also [1] f12 = Jn (ki  )H (2) (ki ) n f22 2 (I + Mi+ Ri;i 1Ri;i+1 ) + Hn(2) (ki  )Hn(2)(ki )Ri;i 1 + Jn(ki  )Jn(ki )Ri;i+1 + Jn(ki )Hn(2)(ki  ) 2Mi Ri;i+1 Ri;i 1

(2)

1 ekr n Jn(kr)  2n 2n 2 ekr Yn (kr)  0 n 2n

(3)

0

0

(4)

0

0

0

(5)

2 ekr Hn(2) (kr)  j n 2n

0

@Jn(ki r) = 0 @Hn(2) (ki r)  n Jn(ki r)@r Hn(2) (ki r)@r r (2) Jn(k r1 ) = Hn (kr2 )  r1 n : Jn(k r2 ) Hn(2) (kr1 ) r2

(6)

0

(2)

1;i Ri01;i02

0

2(I 0 Ri

1;i Ri01;i02

0

)

1

0

Ti;i01

(8e)

Useful identities can be then attained as

with Jn (ki ), Hn (ki ) represent the cylindrical Bessel and second type Hankel functions, respectively, Mi6 = (I 0 Ri;i61 1 Ri;i61 )01 ,and Ri;j is the 2 2 2 multiple reflection matrix at the interface of layers i and j , given by [1] 0

(8d)

0

0

= Ri;i 1 + Ti

2n

 1 for larger n, then (8d) reduces to 2 ekr n : Hn(2) (kr)  j n 2n

0

Ri;i01

1 0 j 21 ekr 2n

n

0

Since (ekr=2n)2n

0

0

(8c)

which can be expressed as

0

0

(8b)

Hn(2) (kr) = Jn (kr) 0 jYn(kr)

0

0

f11 f21

n

0

where e is the Euler’s number. The large order Hankel function may be derived using

0

0

(8a)

:

(9a) (9b)

With the aid of (9), asymptotic expressions for the normalized reflection matrix, which is given by (38), can be obtained by the matrix equation (10), shown at the bottom of page. The asymptotic transmission matrix can then be expressed as a

ti;i61

(7)

= rai;i 1 + I

(11)

6

Explicit expressions for the local reflection and transmission matrices,Ri;j and Ti;j , are given in the Appendix. In order to simplify the present formulation considerably, (3)–(6) are presented in a different format compared to those reported in earlier studies.

where the superscript a denotes an asymptotic form. As n increases, local reflections at the boundaries of the ith layer persist while multiple reflections from other layers’ boundaries decline significantly. For instance, in a three layers cylinder it can be shown that

B. Asymptotic DGF Expansions

R3;2

a

When the summation index n is sufficiently large, the large order asymptotic expansion of the Bessel and Hankel functions can be employed. The following principal asymptotic forms [9] have been

a

a1 n (k3 a2 ) = R3a;2 + J(2) Hn (k3 a2 ) a2 2

0 aa12

a

a

t2;3 r2;1 2n

1

0

a

a

r2;3 r2;1

6

:

(12)

6

6

6

0

6

ri;i6

a

t3;2

1 = ("i + "i 1 )( i + i 1 ) ("i 0 "i 1 )(i + i 1 ) + 2"i i k k 0 1 72"i i k k 0 1 j"k ! 2 72"i i k k 0 1 jk ! ("i + "i 1 )(i 0 i 1 ) + 2"i i k k 0 1 or 1 1 = ("i + "i 1 )(i + i 1 ) jk ("i + "i 1 )(i 1 0 i ) + 2"i i k k 0 2"i 1 i 1 72"i i k k + 2"i 1 i 1 !" k 2 k 72"i i k k + 2"i 1 i 1 ("i 1 0 "i )(i + i 1 ) + 2"i i k k 0 2"i 1 i j! k

ri;i61

a

I

2n

6

(10a)

6

6

6

6

6

6

6

6

6

6

6

6

6

: 1

6

(10b)

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 11, NOVEMBER 2010

!1

For simplicity moderately thick cylindrical layers are considered . first, in which the factor (a1 =a2 )2n decays rapidly as n Therefore, a simplified expression for the multiple reflection matrix can be written as

R  R3a;2 a 3;2

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In the present study, an analytic expression for C (kz ) can be deduced from (16) for each of the spatial asymptotic DGF components. In the case of Gazz , the expansion coefficient can be derived as

(13)

a

(14)

Similarly

Ri;i61Ri;i71  a

a

2n

ai01 ai

(

+

Jn (ki ) Hn(2) (ki ai ) a i;i+1 Hn(2) (ki ) Jn (ki ai )

r

0

1

8!"i

01

e0jk (z0z

)

n=1 n=01

( 11

1 n=1 =

01 n=01

(2)

C (kz ) = ki2

@ 0 @z@z 0 2

1) + )

ki2 i

0 i = i0 0 "i = "i

) ( 1+ +1 ) ( +

 Hn kidi  Jn kidi0 (2)

(

(

)

(20a)

1)

(20b)

@ 0 wi0 @z@z 0 2

1

@ 0 wi @z@z 0

Hn(2) (ki di ) : (21) Hn(2) (ki )

2



I

2 kidi rai;i

(16)

(

(2) 0 +1 + Hn (ki  )Jn (ki di01 )

)

rai;i01

g

0jk (z0z ) dkz :

Gazz

1 n=1

=

01 n=01

2

ki2

ejn(0 ) @ 0 0 @z@z 0 Jn ki  Hn @ ki i0 0 wi0 @z@z 0 2

(

+

(17)

)

(2)

(ki )

2

2

2 Hn

1

1

0

(ki  )Jn (ki di01 ) @2 0 (2) ki2 i wi 0 Jn (ki  )Hn (ki di )

0

@z@z

2 e0jk (z0z )dkz : (18)

: (22)

Therefore, Gazz may be obtained by substituting (21) in (18), that is,

(2)

a

!1

1 = (i01 = ("i

)

1)

ki2 i01

+

2 JJnnkikidi00 f21 f22

In earlier studies, no analytical expression has been formulated for the asymptotic expansion coefficient C (kz ), hence it has been determined numerically [5], [7]. As a result, it wasn’t possible to make a direct use of the Sommerfeld identity for integrals such as (18). Instead, the addition theorem was used to replace the infinite summation by the (2) Hankel function H0 (ki   0 ), with the singularity at  = 0 handled using the small argument approximation [7]. Furthermore another for the effilevel of asymptotic extraction has been used as kz cient computation of the Sommerfeld integral in (18).

j0 j

) ( +1 + 1) ( +

+

11 ) dkz + Gzz

(ki )e

(19)

In the same way it can be shown that a fa f11 12 Jn (ki 0 )Hn(2) (ki ) + Jn (ki 0 )Hn(2) a a

C (kz )ejn(0 ) Jn (ki 0 )

2Hn

)

where di = a2i = and di01 = (ai201 )=. As a result, (19) can be written as

where

Gazz

2

2

)

Hn(2) (ki ai ) Jn (ki ai ) Jn (ki ai01 ) Hn (ki ) (2) Hn (ki ai01 )

2 ejn(0 ) ki

2 f 0 fa

(

(

)

+1

(

where two terms of (6) have been eliminated owing to the aforementioned factor of (ai01 =ai )2n . Asymptotic expressions for the DGF can then be obtained by substituting the elements of (16) into the (2)–(5). Hence computationally efficient expansions can be achieved by subtracting the spectral asymptotic expansion elements from, and subsequently adding their Fourier transforms to, the overall DGF components. For example, Gzz can be expressed as

1

(

Jn (ki )

n

r

=

where i = ( i ), wi01 = ( "i+1 ).

(2)

)

(

Hn(2) (ki 0 ) Jn (ki ai01 ) a + Jn (ki 0 ) Hn(2) (ki ai01 ) i;i01

Gzz

( (2)

 Jn(ki0)H (2)(ki) 2 I

2

1

2 HJnn kkii HJnn kkiiaai i i 0 i = i i , i0 "i 0 "i0 = "i "i0 , and wi

R

R

a fa f11 12 a fa f21 22

@ 0 wi0 @z@z 0 @ ki i 0 wi @z@z 0

0 ) Jn (ki ai01 ) + (2) ) Hn (ki ai01 )

(

(15)

a a Numerical computations of (14) may result in i;i61 i;i61 = 0 owing to under-flows in the computations of Bessel functions when n . Therefore, normalized expressions have been employed in which n independent asymptotic matrices are developed as shown in (10), which must then be multiplied by the Bessel and Hankel functions a de-normalization factors given in (34), (35) to obtain i;i61 . Then substituting (14) in (6) provides

!1

ki2 i01

+

With the aid of (8), the following asymptotic identities can be defined

rai;i61rai;i71 : R

2

2 HJnn kkii0

i.e. the multiple reflections matrix asymptotes to the local counterpart for larger n. With no loss of generality, a recursive formula can be expressed as

Ri;i61  Ri;ia 61:

@ 0 @z@z 0

ki2

C (kz ) =

(23)

It can be seen from (23) that Gazz has been decomposed into three terms; the first represents radiation from a source in an infinite homogenous medium while the remaining two correspond to reflections at the boundaries of the ith layer. Further, each term consists of a kz independent coefficient that is multiplied by a Bessel and a Hankel function. Hence, these terms can be integrated analytically using the Sommerfeld identity in conjunction with the Hankel function addition theorem, that is, [8]

e0jk

jr0r j jr 0 r0j

=

0j 2

+1

01

2

dkz e0jk (z0z

1 n=01

)

Jn (ki 0 )Hn(2)(ki )ejn(0 ) : (24)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 11, NOVEMBER 2010

@2 Gaz = 0 @@z 0

The closed form representation of Gazz can then be written as

0jk R 0jk R 0jk R Gazz = ki2 e R + i e Ri + i01 e Ri01 @ 2 e0jk R + wi e0jk R + wi01 e0jk R 0 @z@z 0 R Ri Ri01

2 (25)

= (z 0 z0 )2 + 0 2 + dm2 0 20 dm cos( 0 0 ), m = i, i 0 1, and R = (z 0 z0 )2 + j 0 0 j2 . Thus numerical evalua(2) tion of the improper integral involving H0 (ki j 0 0 j) has been where Rm

avoided. As a consequence of this, simplified asymptotic DGF expansions have been developed by employing asymptotic extraction once only. It should be noted that numerical computation of (17) has been implemented using (22) for the subtracted asymptotic component. This is because the closed form representation of Gazz has been obtained by employing (21), and a valid asymptotic extraction requires the subtraction and addition of the same quantities. The asymptotic expressions developed so far involve two quasi-static images that correspond to local reflections at the boundaries of a moderately thick cylindrical layer. For thinner layers, local reflections at the adjacent boundaries need to be incorporated into the model. This can be achieved by extracting two more quasi static images from the spectral domain Green’s function, which are then added back in the closed form DGF representation. Again taking Gazz as an example therefore, from (22)

Gazz =

1

n=1

01 n=01

Ga =

n=1

n=01

01 1 n=1

bn (kz )ejn(0 ) e0jk (z0z ) dkz

ejn(0 ) kk2i

i

@ + @ 2 Jn (ki 0 )H (2)(ki ) n 0 @@0 @@0 i01 d` ` @2 + @2 +  0 d` @@0 @0 @d` `=i02 2Jn (ki d`)Hn(2)(ki 0 ) i+1 2 2 + d` ` 0 d`@@@0 + @@0 @d`

2

2

`=i

2 @ 2 bn (kz ) 2Jn (ki 0 )Hn(2)(ki d` ) 0 kki2  0 @@0

where

bn (kz ) = Jn (ki 0 )Hn(2)(ki ) +

i01 `=i02

i+1 `=i

(30)

w` Jn (ki d`)

w` Jn (ki 0 )Hn(2)(ki d`):

(31)

Closed form representations of (28)–(30) can then be obtained using (24) in conjunction with the following identity

@ 2 Jn(ki 0 )H (2)(ki ) 2 ki 0 @z@z n 0 i01 @ 2 Jn (ki d`) ki2 ` 0 w` @z@z + 0 2

@ ki2 ` 0 w` @z@z 2 Hn(2) (ki 0 ) + 0 `=i 2Jn (ki 0 )Hn(2)(ki d` ) e0jk (z0z ) dkz

i

2 e0jk (z0z ) dkz

2Hn(2) (ki 0 ) +

i+1

(29)

2

01 n=01

ejn(0 )

`=i02

1

0jk jr0r j

cos( 0 0 ) e jr 0 r0 j = 0 2j

2

2

(26)

and hence from (25)

+1

@2 + @2 dkz e0jk (z0z ) k12 0 @@ 0 @@0 i 01

+1

n=01

Jn (ki 0 )Hn(2)(ki )ejn(0 )

(32)

which gives a unified asymptotic DGF expansion as

0jk R i+1 e0jk R Gazz = ki2 e R + ` R` `=i02 @ 2 e0jk R + i+1 w` e0jk R 0 @z@z 0 R R` `=i02

a

(27)

where

Following a similar procedure, asymptotic expressions for the other DGF components can be accomplished as

@2 Gaz = 0 0 @ 0 @z 1

01

n=1

n=01

bn (kz )ejn(0 ) e0jk (z0z ) dkz

0jk R

` e R` z^^z `=i02 + ki2 cos( 0 0 ) 0jk R i+1 d` e0jk R `  R` ^^ 2 eR + i+1

`=i02 i+1 0 jk R 0jk R 0rr0 e R + w` e R ` `=i02

0 i+1 i02 = ii0022 0+ ii0011 ; i+1 = ii+2 +2 + i+1 0 "i+2 : wi02 = ""ii0011 0+ ""ii0022 ; wi+1 = ""ii+1 +1 + "i+2

2

j ( ; )= 0 4!" i 0jk R 2 e 2 ki R +

G r r0

(28)

:

(33)

It should be noted that employing (32) provides further simplifications in the Ga expression as it eliminates a few highly singular closed-form terms that appear in previous investigations [5]. The spectral domain components have been computed using a deformed path in the complex kz plane of the Sommerfeld integral, where the first term of (17) has been sampled uniformly along that path and approximated into discrete complex images form using the generalized pencil of functions (GPOF) method [10], [11] and then transformed into the spatial domain, as previously reported in [1], [2].

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 11, NOVEMBER 2010

Fig. 2. Input impedance of a  directed dipole of length L when the antenna is placed at dielectric interface, approaching it from either layer i or i .

=1

3741

Fig. 3. Convergence of the input impedance of the  directed printed cylin: , where i refers to the antenna’s layer. drical dipole for L=

=05

=2

III. RESULTS The significance of the proposed solution is demonstrated using an MoM model of a conformal wire dipole, printed on an interface between two cylindrical dielectric layers surrounding a PEC core. Non magnetic dielectric layers have been considered, i.e. i = 0 . Each dipole is divided into eleven segments of equal length 1`. Piecewise sinusoidal expansion and testing functions, given by sin(ki (1` 0 j` 0 `m j))= sin(ki 1`), are employed in a Galerkin’s MoM procedure. Excitation is provided by a delta gap source at the center of the dipole, and the thin wire approximation has been adopted in which the source and observation points are located at 0 =  = a2 , z 0 = a, and z = 0, where a is the wire radius. Following an example reported in [12], the geometry of Fig. 1 is considered assuming the PEC cylindrical core has a radius of a1 = 11:7 cm and is covered by a dielectric substrate with a relative permittivity of "r1 = 2:45 and a thickness of 1 cm with "r2 = "r3 = 1. A  directed printed dipole driven at 6 GHz is modelled at the interface between layers i = 1 and i = 2 using a wire radius of 0.0125 cm. The computed input impedance of this antenna is illustrated in Fig. 2, where it can be observed that good agreement has been achieved compared with the results reported in [12]. The dipole has been modelled approaching the interface from each side, and as expected the input impedance is the same in both cases. This example demonstrates the validity of the suggested approach in the analyses of printed antennas when  = 0 . Fig. 3 illustrates the convergence of the input impedance as a function of the number of terms in the infinite summation of spectral elements for a particular dipole length of L= = 0:5, where it can be seen that the impedance converges using approximately 30 terms when the proposed approach is employed, whereas non convergent results are obtained if the method reported in [1], [2] is implemented directly. This is to be expected when no asymptotic extraction is employed, as has been mentioned in earlier studies [4]–[7]. A  directed dipole of length L= = 0:4 is next modelled at the interface between 1 mm thick dielectric layers "r1 = 4:0, "r2 = 2:45 and "r3 = 1, having a PEC core radius of 12.6 cm at 6 GHz. The input impedance is illustrated in Fig. 4, where it is evident that convergence has been attained using 70 terms irrespective of the direction of approach to the interface. As another validation, the mutual impedance between two z directed current sources has been calculated and compared to that reported in [7] with good agreement as shown in Fig. 5 using the parameters given in the aforementioned article.

Fig. 4. Convergence of the input impedance of a  directed cylindrical dipole printed at the interface of thin dielectric substrates when L= : .

=04

Fig. 5. Mutual impedance between two z directed current sources printed on a two-layer dielectric cylinder when s : .

=05

IV. CONCLUSION A new approach to compute the DGF for antennas in the vicinity of layered cylindrical media has been introduced, where simplified closed form expressions have been obtained and validated for the spatial asymptotic DGF components. These expressions have been

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 11, NOVEMBER 2010

achieved through a single application of asymptotic extraction for larger n. This results in a significant saving in the required computation time as it eliminates the need for another level of asymptotic extraction for larger kz as well as the need to determine the expansion coefficients numerically. Furthermore, the Hankel function singularity when  =  0 has been handled through analytical integration. A simplification in the formulae for the spectral asymptotic expansion elements has also been achieved. The model’s advantage reduces as the radial distance between field and source points increases as well as when the antenna is not located close to a dielectric boundary.

APPENDIX The local reflection and transmission matrices can be defined as [1]

Jn (ki ai01 ) ri;i01 Hn(2) (ki ai01 ) (2) Ri;i+1 = HJnn(k(kiiaai )i ) ri;i+1 Ti;i01 = JnJnk(kiai0ai10)1 ti;i01 (i01) Hn(2) (ki ai ) Ti;i+1 = (2) ti;i+1 Hn k(i+1) ai

Ri;i01 =

where the matrix

(34) (35) (36) (37)

r can be expressed in a convenient form as

() 0  j! 1i+1 Qi ri+1;i = 1 "i+10 1" i+11fii+1 Q  1i+1 fi (") 0  i i +1 j!  "i 1i fi () 0  1 Q 1 i ri;i+1 =  0 " 1iQi i1j!i fi(") i0  

(38a) (38b)

j!

where 

= fi (")fi () 0 (Qi2=!2 ), and

xi @Jn (ki ai ) ki Jn (ki ai )@ (ki ai ) xi+1 @Hn(2) k(i+1) ai 0 k(i+1) Hn(2) k(i+1) ai @ k(i+1) ai n (ki ai ) 1i = ki Jn@J (ki ai )@ (ki ai ) @Hn(2) (ki ai ) 0 ki Hn(2) (ki ai )@ (ki ai )

fi (x) =

Qi2

= (nkz ) k12 0 k2 1 2

i

= n2 !2

"i i 4 ki

(39)

(40)

2

(i+1)

+ "ki+14 i+1 0 "i ki 2+k"2i+1 i+1 (i+1)

i (i+1)

:

(41)

ACKNOWLEDGMENT The authors would like to thank Prof. G. Dural, Middle East Technical University, Ankara, Turkey, for her help in providing reference [1].

REFERENCES [1] Ç. Tokgöz, “Derivation of closed-form Green’s functions for cylindrically stratified media,” M.S. thesis, Dept. Elect. Electron. Eng., Middle East Technical Univ., Ankara, Turkey, Aug. 1997. [2] Ç. Tokgöz and G. Dural, “Closed-form Green’s functions for cylindrically stratified media,” IEEE Trans. Micow. Theory Tech., vol. 48, pp. 40–49, Jan. 2000. [3] J. Sun, C. F. Wang, L. W. Li, and M. S. Leong, “A complete set of spatial domain dyadic Green’s function components for cylindrically stratified media in fast computational form,” J. Electromagn. Waves Appl., vol. 16, pp. 1491–1509, 2002. [4] J. Sun, C. F. Wang, L. W. Li, and M. S. Leong, “Mixed potential spatial domain Green’s functions in fast computational form for cylindrically stratified,” Prog. In Electromag. Res., vol. 45, pp. 181–199, 2004. [5] J. Sun, C. F. Wang, L. W. Li, and M. S. Leong, “Further improvement for fast computation of mixed potential Green’s functions for cylindrically stratified media,” IEEE Trans. Antennas Propag., vol. 52, pp. 3026–3036, Nov. 2004. [6] R. C. Acar and G. Dural, “Numerically efficient analysis of printed structures in cylindrically layered media using closed-form Green’s functions,” in Proc. IEEE Antennas and Propag. Soc. Int. Symp., Jul. 2008, vol. 1, pp. 1–4. [7] S. Karan, V. B. Erturk, and A. Altintas, “Closed-form Green’s function representations in cylindrically stratified media for method of moments applications,” IEEE Trans. Antennas Propag., vol. 57, pp. 1158–1168, Apr. 2009. [8] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: Van Nostrand, 1990. [9] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Washington, DC: Government Printing Office, 1964. [10] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, pp. 229–234, Feb. 1989. [11] Y. L. Chow, J. J. Yang, D. F. Fang, and G. E. Howard, “A closed-form spatial Greens function for the thick microstrip substrate,” IEEE Trans. Microw. Theory Tech., vol. 39, pp. 588–592, Mar. 1991. [12] A. Nakatani, N. G. Alexopoulos, N. K. Uzungolu, and P. L. E. Uslenghi, “Accurate Green’s function computation for printed circuit antennas on cylindrical substrates,” Electromagn., vol. 6, pp. 243–254, Nov./Dec. 1986.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 11, NOVEMBER 2010

3743

A Generalized Asymptotic Extraction Solution for Antennas in Multilayered Spherical Media Salam K. Khamas

Abstract—An efficient model is developed to accelerate the convergence of the dyadic Green’s function’s (DGF) infinite summation when the source and observation points are placed in different layers of a dielectric sphere, thereby expediting computational analysis. The proposed procedure is based on asymptotic extraction principles in which the quasi-static images are extracted from the spectral domain DGF. The effectiveness of the approach is demonstrated in a method of moment model where a microstrip antenna as well as a conformal dipole array have been studied. Index Terms—Dyadic Green’s function, method of moments (MoM), spherical antennas. Fig. 1. A layered dielectric sphere.

I. INTRODUCTION Efficient computation of the dyadic Green’s function for a multilayered dielectric sphere has been investigated in numerous research articles, where several approaches have been proposed to accelerate the convergence of the infinite series such as Watson or Shanks transformations [1]–[3], finite difference approximation [4], a large perfectly conducting (PEC) sphere consideration using Kummer’s transformation [5], as well as incorporating the image theory of a planar structure in the solution [6]. Asymptotic extraction is a well-known procedure that has been extensively used for the efficient computation of the Sommerfeld type integrals in planar and cylindrical media [7], [8]. In a recent study [9], the asymptotic extraction approach has been introduced for spherical media, which has expedited the series convergence considerably when the field and observation points are located in the same layer. In many applications, the DGF needs to be computed when the field and source points are in different layers but still in the vicinity of each other. Examples include; patch antennas excited by a feed in a different layer, stacked arrays, three dimensional antennas, and volumetric arrays that penetrate a dielectric boundary. In this article the asymptotic extraction introduced in [9] has been broadened to consider the problem of source and observation points positioned in different layers. The efficiency of the presented formulation has been confirmed by employing the proposed solution in a moment method model, where structures involve field and source points in adjacent layers have been studied. Two configurations have been considered including a probe-fed circular patch antenna, and a conformal dipole array.

where the superscript f s refers to the layers of field and source points, and 0 , respectively, and uv is the Kronecker delta. It is well-known (fs) that  0s can be expressed in a closed form as [13]

r

r G

G fss (r r ) = I + 1 rr ( 0

)

;

0

0

jk R

0

e

(2)

4R

2

kf

G fs

whereas  es is represented as an infinite summation of spherical eigen modes as shown in the Appendix [12]. The summation can be truncated using a finite number of terms depending on the distance between the field and source points as well as other geometrical factors such as sphere radius and number of layers. When the field and source points are located in the same layer, then using (2) in conjunction with the asymptotic extraction approach [9] can enhance the computations of  fs e significantly. However, when the source and evaluation points are in different layers but in the proximity of each other, the computa(fs) tion efficiency of (1) degrades substantially as  0s is undefined and the asymptotic extraction is valid only for the source and field points within the same spherical layer. Therefore, a large number of summation terms must be added to accomplish convergence, which increases the computation time significantly and necessitates the computation of larger order Bessel and Hankel functions that may produce numerical under flows, or over flows. This article presents a methodology to enhance the computation efficiency of (1) when f = s using an asymptotic extraction technique, where the quasi-static images have been isolated from the infinite expansion. (

)

G

G

6

II. FORMULATION For an antenna radiating next to a multilayered dielectric sphere, the DGF can be represented as a superposition of two components, the (fs) first,  0s , corresponds to radiation in an unbounded media and the (fs) second,  es , accounts for waves scattering owing to the presence of the sphere, i.e., [10]–[12]

G

G

G fse(r r ) = G fss (r r ) fs + G esfs (r r ) ;

0

( 0

)

;

0



(

)

;

0

(1)

Manuscript received February 12, 2010; revised April 21, 2010; accepted April 21, 2010. Date of publication September 02, 2010; date of current version November 03, 2010. The author is with the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail: [email protected]. uk). 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.2010.2071371

A. Asymptotic Expansion Coefficients Fig. 1 illustrates a spherical structure that consists of L layers, where each layer has a permittivity of "i and a permeability of i , which has been assumed to be the same as that of free space, i.e., o . When the source and field points are located in different layers, the asymptotic DGF can be attained by deriving expressions for the transmission and . This can be achieved by emreflection coefficients when n ploying the large order spherical Bessel and Hankel functions principal asymptotic expressions [14], that is

!1

)



(kr)



(

jn kr

(2)

hn

0018-926X/$26.00 © 2010 IEEE

1 2kr(2n + 1) j

2

kr

(2n + 1)

ekr

n+

(3a)

2n + 1 ekr

2n + 1

n0

0

:

(3b)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 11, NOVEMBER 2010

Following a procedure similar to that presented in [9], in which (3) is substituted in the reflection and transmission coefficients given by (18) in [12]. The resultant asymptotic expressions have then been fs fs substituted in the infinite series coefficients Afs M;N ; BM;N ; CM;N and fs DM;N . Therefore it can be shown that when f < s the following expansion coefficients contributions persist as n increases

CNf