IEEE MTT-V055-I01 (2007-01) [55, 1 ed.]

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JANUARY 2007

VOLUME 55

NUMBER 1

IETMAB

(ISSN 0018-9480)

PAPERS

Active Circuits, Semiconductor Devices, and ICs Frequency-Thermal Characterization of On-Chip Transformers With Patterned Ground Shields ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ...... J. Shi, W.-Y. Yin, K. Kang, J.-F. Mao, and L.-W. Li Fully Integrated Differential Distributed VCO in 0.35- m SiGe BiCMOS Technology . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... . G. P. Bilionis, A. N. Birbas, and M. K. Birbas A Low Phase-Noise -Band MMIC VCO Using High-Linearity and Low-Noise Composite-Channel Al Ga N/Al Ga N/GaN HEMTs ..... ... ...... Z. Q. Cheng, Y. Cai, J. Liu, Y. Zhou, K. M. Lau, and K. J. Chen Ultra-Compact High-Linearity High-Power Fully Integrated DC–20-GHz 0.18- m CMOS T/R Switch ........ ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ... Y. Jin and C. Nguyen Analysis of the Survivability of GaN Low-Noise Amplifiers .... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ....... M. Rudolph, R. Behtash, R. Doerner, K. Hirche, J. Würfl, W. Heinrich, and G. Tränkle A 16-GHz Triple-Modulus Phase-Switching Prescaler and Its Application to a 15-GHz Frequency Synthesizer in 0.18- m CMOS . ......... ......... ........ ......... ......... ..... .... ......... ......... ........ ......... ......... Y.-H. Peng and L.-H. Lu A New Envelope Predistorter With Envelope Delay Taps for Memory Effect Compensation .... ........ ......... ......... .. .. ........ ...... S.-C. Jung, H.-C. Park, M.-S. Kim, G. Ahn, J.-H. Van, H. Hwangbo, C.-S. Park, S.-K. Park, and Y. Yang

1 13 23 30 37 44 52

Signal Generation, Frequency Conversion, and Control A Low Phase-Noise Voltage-Controlled SAW Oscillator With Surface Transverse Wave Resonator for SONET Application ..... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... J.-H. Lin and Y.-H. Kao Phase-Noise Reduction of -Band Push–Push Oscillator With Second-Harmonic Self-Injection Techniques .. ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... T.-P. Wang, Z.-M. Tsai, K.-J. Sun, and H. Wang

60 66

Millimeter-Wave and Terahertz Technologies Gold-Plated Micromachined Millimeter-Wave Resonators Based on Rectangular Coaxial Transmission Lines ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ..... E. D. Marsh, J. R. Reid, and V. S. Vasilyev

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Field Analysis and Guided Waves Uniform Electric Field Distribution in Microwave Heating Applicators by Means of Genetic Algorithms Optimization of Dielectric Multilayer Structures ........ ......... ... E. Domínguez-Tortajada, J. Monzó-Cabrera, and A. Díaz-Morcillo Determination of Generalized Permeability Function and Field Energy Density in Artificial Magnetics Using the Equivalent-Circuit Method .... ......... ......... ........ . ........ ......... ........ ..... P. M. T. Ikonen and S. A. Tretyakov Fast Numerical Computation of Green’s Functions for Unbounded Planar Stratified Media With a Finite-Difference Technique and Gaussian Spectral Rules ........ ........ ......... .. A. G. Polimeridis, T. V. Yioultsis, and T. D. Tsiboukis

85 92 100

CAD Algorithms and Numerical Techniques Efficient Modal Analysis of Bianisotropic Waveguides by the Coupled Mode Method . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ...... J. Pitarch, J. M. Catalá-Civera, F. L. Peñaranda-Foix, and M. A. Solano Filters and Multiplexers HTS Quasi-Elliptic Filter Using Capacitive-Loaded Cross-Shape Resonators With Low Sensitivity to Substrate Thickness ...... ......... ........ ......... ......... ........ ......... ....... A. Corona-Chavez, M. J. Lancaster, and H. T. Su Design of Compact Low-Pass Elliptic Filters Using Double-Sided MIC Technology ... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ . M. del Castillo Velázquez-Ahumada, J. Martel, and F. Medina Novel Multifold Finite-Ground-Width CPW Quarter-Wavelength Filters With Attenuation Poles ...... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ...... C.-H. Chen, C.-K. Liao, and C.-Y. Chang Novel Patch-Via-Spiral Resonators for the Development of Miniaturized Bandpass Filters With Transmission Zeros .. .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ . S.-C. Lin, C.-H. Wang, and C. H. Chen Microstrip Realization of Generalized Chebyshev Filters With Box-Like Coupling Schemes ... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ . C.-K. Liao, P.-L. Chi, and C.-Y. Chang Tunable Dielectric Resonator Bandpass Filter With Embedded MEMS Tuning Elements ....... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ... W. D. Yan and R. R. Mansour Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements A Broadband Compact Microstrip Rat-Race Hybrid Using a Novel CPW Inverter .. .. T. T. Mo, Q. Xue, and C. H. Chan

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117 121 128 137 147 154

161

Instrumentation and Measurement Techniques On the Fast and Rigorous Analysis of Compensated Waveguide Junctions Using Off-Centered Partial-Height Metallic Posts ... ......... ......... ........ ......... .. A. A. San Blas, F. Mira, V. E. Boria, B. Gimeno, M. Bressan, and P. Arcioni

168

Microwave Photonics Nonlinear Distortion Due to Cross-Phase Modulation in Microwave Fiber-Optic Links With Optical Single-Sideband or Electrooptical Upconversion .. ......... ........ L. Cheng, S. Aditya, Z. Li, A. Nirmalathas, A. Alphones, and L. C. Ong

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LETTERS

Comments on “Extension of the Leeson Formula to Phase Noise Calculation in Transistor Oscillators With Complex Tanks” . ......... ......... ........ ......... ......... ........ ...... .... ......... ........ ......... ......... ........ ......... . T. Ohira Authors’ Reply ... ......... ........ ......... ......... ........ .... J.-C. Nallatamby, M. Prigent, M. Camiade, and J. Obregon

185 185

Information for Authors .. ........ ......... ......... ........ ......... .......... ........ ......... ......... ........ ......... ......... .

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IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY The Microwave Theory and Techniques Society is an organization, within the framework of the IEEE, of members with principal professional interests in the field of microwave theory and techniques. All members of the IEEE are eligible for membership in the Society upon payment of the annual Society membership fee of $14.00, plus an annual subscription fee of $20.00 per year for electronic media only or $40.00 per year for electronic and print media. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. ADMINISTRATIVE COMMITTEE J. S. KENNEY, President L. BOGLIONI D. HARVEY S. M. EL-GHAZALY J. HAUSNER M. HARRIS K. ITOH

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Editors-In-Chief DYLAN WILLIAMS NIST Boulder, CO 80305 USA Phone: +1 303 497 3138 Fax: +1 303 497 3970 email: [email protected] AMIR MORTAZAWI Univ. of Michigan Ann Arbor, MI 48109-2122 USA Phone: +1 734 936 2597 Fax: +1 734 647 2106 email: [email protected]

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Associate Editors

DANIEL DE ZUTTER ZOYA POPOVIC YOSHIO NIKAWA Universiteit Gent Kokushikan Univ. Univ. of Colorado, Boulder Belgium Japan USA email: [email protected] email: [email protected] email: [email protected] KENJI ITOH JOSÉ PEDRO SANJAY RAMAN Mitsubishi Electronics Univ. of Aveiro Virginia Polytech. Inst. and State Univ. Japan Portugal USA email: [email protected] email: jcp.mtted.av.it.pt email: [email protected] JENSHAN LIN Univ. of Florida USA email: [email protected] M. GOLIO, Editor-in-Chief, IEEE Microwave Magazine G. E. PONCHAK, Editor-in-Chief, IEEE Microwave and Wireless Component Letters

RICHARD SNYDER RS Microwave Company USA email: [email protected] RUEY-BEEI WU National Taiwan Univ. Taiwan, R.O.C. email: [email protected] ALEXANDER YAKOVLEV Univ. of Mississippi USA email: [email protected] T. LEE, Web Master

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Digital Object Identifier 10.1109/TMTT.2007.890647

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

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Frequency-Thermal Characterization of On-Chip Transformers With Patterned Ground Shields Jinglin Shi, Wen-Yan Yin, Senior Member, IEEE, Kai Kang, Jun-Fa Mao, Senior Member, IEEE, and Le-Wei Li, Fellow, IEEE

Abstract—Extensive studies on the performance of on-chip CMOS transformers with and without patterned ground shields (PGSs) at different temperatures are carried out in this paper. These transformers are fabricated using 0.18- m RF CMOS processes and are designed to have either interleaved or center-tapped interleaved geometries, respectively, but with the same inner dimensions, metal track widths, track spacings, and silicon substrate. Based on the two-port -parameters measured at different temperatures, all performance parameters of these transformers, such as frequency- and temperature-dependent maximum available gain ( max ), minimum noise figure (NFmin ), quality factor ( 1 ) of the primary or secondary coil, and power loss ( loss ) are characterized and compared. It is found that: 1) the values of the max and 1 factor usually decrease with the temperature; however, there may be reverse temperature effects on both max and 1 factor beyond certain frequency; 2) with the same geometric parameters, interleaved transformers exhibit better low-frequency performance than center-tapped interleaved transformers, whereas the center-tapped configurations possess lower values of NFmin at higher frequencies; and 3) with temperature rising, the degradation in performance of the interleaved transformers can be effectively compensated by the implementation of a PGS, while for center-tapped geometry, the shielding effectiveness of PGS on the performance improvement is ineffective. Index Terms—Interleaved and center-tapped transformers, maximum available gain, minimum noise figure, pattern ground shields (PGSs), power loss, quality ( ) factor, temperature.

I. INTRODUCTION

S

ILICON monolithic transformers have been widely used in designs of on-chip impedance matching, balun, low-noise amplifier feedback, and other microwave and millimeter-wave components, e.g., such as those introduced in [1]–[4]. Based on different layouts, several types of on-chip silicon transformers and their modeling and optimization were studied over the past

Manuscript received March 23, 2006; revised September 28, 2006. The work of W.-Y. Yin and J.-F. Mao was supported by the National Science Fund for Creative Research Groups under Grant 60521002 via Shanghai Jiao Tong University, by the the Ministry of Education under Doctoral Research Fund Grant 20050248051, by the Shanghai Pujang Talent Project under Grant 05PJ14064, and by the Natural Science Foundation of China under Grant 90607011. J. Shi and K. Kang are with the Institute of Microelectronics, Singapore 117685 (e-mail: [email protected]). W.-Y. Yin and J.-F. Mao are with the Center for Microwave and RF Technologies, School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240 (e-mail: [email protected]). L.-W. Li is with the Department of Electrical Engineering and Computer Science, National University of Singapore, Singapore 119260. 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/TMTT.2006.888934

few years [5]–[11]. Simburger et al. proposed two analytical transformer models for both tapped and stacked geometries [2], and Niknejad and Meyer presented a lumped-element equivalent-circuit model for interleaved planar transformers [5]. More recently, some significant progresses in the development of novel CMOS transformers have been achieved, and among these, the optimized design of a distributed active transformer [12], 30–100-GHz transformers fabricated using SiGe BiCMOS technology for millimeter-wave integrated circuits [13], and a new compact model for monolithic transformers in silicon-based RF integrated circuits (RFICs) [14] are good examples. Similar to silicon-based spiral inductors, silicon-based transformers also suffer from serious power losses at high frequency. The loss mechanisms can be summarized into five categories, which are: 1) skin effects; 2) proximity effects; 3) eddy-currents in the substrate; 4) shunt conduction current flowing in the substrate; and 5) lateral conduction currents flowing in the substrate between the primary and secondary coils. The first three mechanisms are due to time-varying magnetic fields, whereas the remaining two are caused by time-varying electric fields. The losses due to time-varying magnetic fields can be efficiently reduced by increasing the electric conductivity of metals and using a slightly doped substrate with higher resistivity or implementing patterned magnetic shields. In order to reduce losses in the silicon substrate due to the time-varying electric field at a high frequency, one practical way is to employ an appropriate patterned ground shield (PGS), which provides a short terminal to the electric field leaking into the substrate. In some previous studies [15]–[18], it was demonstrated that metal grid shields can be used to effectively reduce mode attenuation of microstrip or coplanar interconnects. For silicon-based spiral inductors, Yue and Wong first presented a PGS spiral inductor model in 1998 [19]. In 2002, Yim et al. further examined the effects of a PGS on the performance of spiral inductors [20]. In particular, novel patterned trench isolation with a floating p/n junction and floating metal poles were also implemented underneath reference spiral inductors [21]. For spiral inductor cases, although the implementation of the PGS may increase the parasitic capacitance, it makes the design more independent of substrate dopant concentrations and can reduce noise coupling into substrate. Therefore, an appropriate choice of the embedding depth of the PGS is very important in the effective implementation of the PGS. However, accurately characterizing frequency- and temperature-dependent PGS transformers has not yet been well conducted. In the design of a transformer, there are multiple choices in its configuration, such as interleaved, center-tapped interleaved, and even stacked

0018-9480/$25.00 © 2006 IEEE

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

geometries. The performance parameters of a transformer, such , self-resonance frequency, as maximum available gain , and power losses are minimum noise figure directly related to its configuration. In this paper, extensive studies are carried out to investigate on-chip (non-)PGS (NPGS) interleaved and center-tapped interleaved transformers at different temperatures, including their frequency- and temperature-dependent maximum avail, minimum noise figure , power loss able gain , and quality factor of the primary or secondary coil. In Section II, the layouts of on-chip interleaved and center-tapped patterned ground shielding transformers are shown. In Section III, some modified temperature-dependent circuit models of these transformers are proposed, and procedures for determining all performance parameters are given. In Section IV, based on the two-port -parameters measured at , , , and different temperatures, the values of factor are extracted and compared. Some conclusions the are finally drawn in Section V. II. ON-CHIP PGS TRANSFORMERS Fig. 1(a) and (b) shows the top view of an interleaved PGS transformer and a center-tapped interleaved PGS transformer fabricated on a silicon substrate using 0.18- m RF CMOS prom, metal cesses with the same internal dimension of m, track spacing of m, track width of and the same PGS implemented as shown in Fig. 1(c) (where the width and spacing of the PGS metal bar are m). The turn numbers of the primary and secondary coils and , respectively. For comparison, are designed to be a group of non-PGS (NPGS) transformers with the same geoas its counterpart was also demetric parameters signed, fabricated, and examined. III. MODIFIED TEMPERATURE-DEPENDENT EQUIVALENT-CIRCUIT MODELS For an interleaved transformer with or without a PGS, its temperature-dependent small-signal equivalent-circuit model can be obtained based on the circuit model given in [22], as shown and in Fig. 2. In Fig. 2(a) and (c), the capacitances can be well treated as temperature-independent elements, due to the constitutive characteristics of silicon oxide, whereas the seof the primary and secondary ries self-resistances , and shunt cacoils, the silicon substrate resistances are all sensitive to the variation of temperpacitances ature. For example, and decrease with temperand increase with temperature. ature, while At a low frequency, the series equivalent self-inductances and ) and series resistances ( and ( ) in Fig. 2(a) can be calculated based on the -parameters converted from the measured -parameters, i.e., (1a) (1b)

Fig. 1. Top and cross-sectional views of on-chip interleaved (Design 1) and center-tapped interleaved (Design 2) PGS transformers (t = 2 m, t = 0:54 m, D = 0:9 m, H = 6:7 m, and D = 500 m). (a) Interleaved PGS transformer: Design 1. (b) Center-taped interleaved PGS transformer: Design 2. (c) Cross section view.

The coupling coefficient is defined as and reactive and the mutual resistive are calculated by [8]

, coupling factors

(3a)

(2a) (2b)

(3b)

SHI et al.: FREQUENCY-THERMAL CHARACTERIZATION OF ON-CHIP TRANSFORMERS WITH PGSs

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The minimum noise figure by [22]

of a transformer is defined (8)

The circuit model of an interleaved NPGS transformer (Design 1) is plotted in Fig. 2(a). At high frequencies, it can be further , simplified as the form shown in Fig. 2(b), where , , and represent the equivalent primary and secondary resistances and inductances, and mutual resistances and inductances between them, respectively (where and ), and (9a) (9b) where (9c) Fig. 2. Small-signal circuit models for interleaved NPGS transformer (Design 1). (a) Circuit model for an interleaved NPGS transformer. (b) Simplified circuit model.

(9d)

where and represent real and imaginary parts of the -parameters, respectively. As pointed out in [8], the mutual resistive factor mainly accounts for the hybrid effects of parasitic capacitances and eddy currents in the silicon substrate. The factor for primary and secondary coils can be evaluated by (4)

(9e)

(9f) From Fig. 2, it can be found that the secondary coils can be determined by

factors of primary and

substrate loss factor

self-resonance factor.

while the self-resonance frequency of the transformer is defined or of the device first becomes purely resistive. at which It is especially known that one of the important performance indicators of a transformer is its maximum available gain, which is defined by [22] (5) where (6a)

(9g) and (6b) or [21] (7a) and (7b)

Equation (9g) indicates that the substrate loss factor will apincreases infinitely. In other words, an proach unity as infinite value of may result in a high factor of the is determined by (9f). primary or secondary coil, and goes to zero or infinitely, will become infiAs nite. This means that making the substrate either short or open can enhance the factor. In this paper, our methodology is to short the substrate by inserting a PGS between the transformer and substrate so as to block electrical fields from penetrating into the silicon substrate.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

Fig. 3. Small-signal circuit models for an interleaved PGS transformer (Design 1).

For an interleaved PGS transformer (Design 1), its circuit model is plotted in Fig. 3. In Fig. 3, there are two paths to the ground, i.e., through the PGS and silicon substrate, re, spectively. We can change the interconnects of , and into a shunt branch of a capacitance and resistance. Similarly, we can also change and into a similar shunt branch. Thus, a simplified circuit model will be obtained, which can be easily compared to that in the NPGS case. The element in Fig. 3 is derived from the , , , , and . and are not only from the silicon substrate, they also and . Hence, the temperature include the effects of coefficients of these elements for a PGS transformer will not completely follow the characteristics of the silicon substrate. For an NPGS transformer of Design 2, its lumped-element equivalent-circuit model is shown in Fig. 4. The elements and represent the series resistance and inductance of the primary (secondary) coil, , , and are utilized respectively; three capacitances to describe the capacitive coupling effect between primary and and are the series resistance secondary coils; , and inductance of the center-tapped metal track; and have the same meanings as indicated above. Following a similar procedure as shown in Fig. 3, the circuit model for a PGS transformer of Design 2 can also be obtained, but suppressed here. The center portion represents the low-frequency model, while the remaining part represents the high-frequency model. At low frequencies, the elements in Fig. 4 satisfy a set of equations as follows: (10a) (10b) (10c) Therefore, (11a) (11b) (11c) (11d)

Fig. 4. Equivalent-circuit model for a center-tapped interleaved NPGS transformer (Design 2).

Since the inductance and resistance depend on both physical and geometric parameters of the structure, and due to the symmetry between the primary and secondary windings, we can assume that (12) where the parameter is given in [23] and its value can be determined using the curve-fitting technique based on the measured data. We then obtain

(13)

(14) (15) At high frequencies, the two-port -parameters of the model in Fig. 4 can be obtained from the extrinsic -parameters, and (16) where represents the -parameter obtained experimentally, is derived using (10a)–(10c). and between Port 1 and Port 2 can The coupling capacitance then be calculated by (17) above is due to the The temperature-dependent change in electric conductivity of metal coils with temperature rising. Over a temperature range from 200 to 900 K, of aluminum, gold, copper, etc. can be determined by [24] S/m

(18)

with the coefficients obtained using the curve-fitting technique, and as summarized in Table I. When the metal trace is made of an alloy, the coefficients vary for different weight ratios of two or three types of pure metals. For example, if the alloy is made of 95% aluminum and 5% , , copper in the weight ratio,

SHI et al.: FREQUENCY-THERMAL CHARACTERIZATION OF ON-CHIP TRANSFORMERS WITH PGSs

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TABLE I COEFFICIENTS FOR DIFFERENT METALS OVER A TEMPERATURE RANGE FROM 200 TO 900 K

Fig. 5. Series resistances of the primary and secondary coils versus frequency = 4 of transformers of Designs 1 and 2, respectively. for

N

and K K . Based on (18), we can easily evaluate the total temperature-dependent series resistances of the primary and secondary metal coils by (19a) where

Fig. 6. Series inductances of the primary and secondary coils versus frequency for a transformer of = 4.

N

a given frequency and a known temperature, the former series resistance of the primary coil is slightly smaller than that in increases with temperature, and it Design 2. Obviously, can be described by a linear equation as follows:

(19b) (19c)

(20) and are two temperature coefficients of the where primary and second coils, respectively, and are determined by

(19d) and ( and ) are the underpass while lengths at metal layer M5 and the primary and secondary coils at metal layer M6, respectively. Fig. 5 shows the calculated of the primary and secondary coils of transformers of Designs 1 and 2 at temperatures of K, K and K, respectively. is It is shown that the rising effect of temperature on easily observable, which will further increase the power loss of metal coils. When the same inner empty dimension, metal track width, track spacing, and turn number are assumed, the total metal track length of the primary or secondary coils in Design 1 is slightly shorter than that in Design 2. Correspondingly, at

(21) K at From Fig. 5, it can be evaluated that GHz, while K at GHz. If the above transformers are made of copper, K GHz, while K at GHz. at is also frequency dependent; such a property Hence, has usually been neglected in most of the literature. (333 K) Fig. 6 shows the extracted series inductances (333 K) of Design 1 with using (2a) and (2b), and where the simulated result is also plotted for comparative purposes.

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TABLE II SILICON RESISTIVITY VALUES AT DIFFERENT TEMPERATURES

TABLE III EXTRACTED CIRCUIT PARAMETERS OF NPGS TRANSFORMER OF DESIGN 1 WITH

N =4

TABLE IV EXTRACTED EQUIVALENT-CIRCUIT PARAMETERS OF TRANSFORMER OF DESIGN 2 WITH

Similar to (20), (where approximately described by

and ) can also be

(22) is the temperature coefficient of the substrate rewhere sistance of silicon, and it was experimentally demonstrated in [25] that the sheet resistance per square of the silicon substrate increases linearly with temperature from a room temperature to as high as 250 C. Based on Arora’s model, the temperature-dependent densities of carrier and mobility in the silicon substrate can be described by [26] (23a)

(23b) (23c) (23d) K

(23e)

N =4

where is the Boltzmann constant, is the unit electron charge is the total dopant concentration in silicon, the in Coulomb, , , , , and have different values parameters for different types of impurities [26], and the resistivity of silicon is given by (24) The coefficient

in (22) is obtained by (25)

Therefore, the temperature-dependent resistivity of silicon is calculated and summarized in Table II, which is very sensitive to the change of temperature. Tables III and IV also further summarize the extracted equivalent-circuit parameters shown in Figs. 2 and 3 for a transformer at different temperatures. The following observawith tions can be made. With respect to the NPGS transformer of Design 1, the imand plementation of a PGS will result in an increase of by a factor of 1.4–2.5, and and will changed slightly. and can be useful for the enhanceThus, the increase of ment of factors of the primary and secondary coils. is not sensitive to the The reactive coupling factor variation of temperature, and it is approximately 0.75 for the transformers of Design 1 with or without PGS. It is found that

SHI et al.: FREQUENCY-THERMAL CHARACTERIZATION OF ON-CHIP TRANSFORMERS WITH PGSs

Fig. 8. Extracted  and 

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for transformer (Design 1) of N = 4.

due to the negligible effect of the PGS at low frequency. Howand ever, as frequency further increases, the values of both of the PGS transformer increases much faster than those of the NPGS case due to the strong capacitive effect that resulted from the PGS. The frequency corresponding to the peak value moves from approximately 10 GHz for the NPGS of transformer to 8 GHz for the PGS case. Fig. 8 shows the extracted mutual resistive and reactive coufor the transformer (Design 1) of pling factors and at room temperature and 373 K. It is noted that does not change significantly at low frequencies because it mainly accounts for the effect of magnetic coupling. On the other hand, is much more sensitive to the the resistive coupling factor . Over the frequency range variation of frequency than that of of a PGS transformer is smaller below 6 GHz, the value of than that of its NPGS counterpart. Fig. 7. Comparison of the extracted and modeled Z -parameters. (a) Comparison of the real part of Z -parameters. (b) Comparison of the imaginary part of Z -parameters (normalized by 1e 9).

0

in Design 2 with , which is slight smaller than its counterpart in Design 1. and The corresponding temperature coefficients in (21) for Design 1 are equal to approximately 3.4 10 K, which is close to that given as above at a low frequency. and (where The shunt substrate resistances ) increase with temperature, and their temperature coeffi, as defined by (25), is equal to 1.01 10 K apcient or , its proximately for the silicon substrate, and for K. temperature coefficient is extracted to be 1.08 10 Fig. 7(a) and (b) shows the comparison of the real and imaginary parts of -parameters for PGS and NPGS transformers at room temperature, respectively. Ex(Design 1) with cellent agreement is obtained between the extracted and simulated -parameters for either a PGS or an NPGS transformer over a wide frequency range. It is interesting to note that as freor of the quency is below 3 GHz, the value of either PGS transformer are almost the same as its NPGS counterpart,

IV. EXTRACTION OF PERFORMANCE PARAMETERS AND DISCUSSION To globally capture the electromagnetic-thermal characteristics of the above transformers, i.e., Designs 1 and 2, four sets m, m, of samples are designed to have and m, fabricated on a 10cm silicon substrate using 0.18- m RF CMOS processes, including: 1) NPGS interleaved transformers; 2) NPGS center-tapped transformers; 3) interleaved transformers with a PGS at M1; and 4) center-tapped transformers with a PGS at M1. The frequency- and temperature-dependent on-wafer two-port -parameters are, respectively, measured using an HP-8510C network analyzer, and Cascade Microtech ground–signal–ground (G–S–G) probes with a temperature-controlled chuck. The chuck temperature can be increased from 223 K to as high as 473 K. In order to reduce the effect of increasing temperature on the probe station during measurements for different temperatures, two-time calibrations at room and higher temperatures are done, respectively. For the NPGS transformer, the open pad structure has been used for deembedding, while for PGS transformers, the open pad together with the PGS was deembedded. The maximum available

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G

Fig. 9. values of transformer (Design 1) with and without a PGS versus frequency at different temperatures, respectively. (a) Design 1 without a PGS. (b) Design 1 with a PGS at M1.

gain, minimum noise figure, power loss, and factor of the primary or secondary coil of transformers are further extracted and subsequently compared. A. Maximum Available Gain Fig. 9(a) and (b) shows the of the interleaved transformers versus frequency at temperatures and K, respectively. In Fig. 9(a) and (b), the temperature is increased from 253 to 298, 333, and 373 K gradually, and the following is shown. does not change 1) As frequency increases, the value of monotonously, and at certain frequency represented by , reaches its minimum. 2) Over the frequency range of 0.5 MHz to approximately GHz in Fig. 9(a), decreases with temperature. However, as frequency increases higher than 8.4 GHz, inthere is significant temperature reverse effect and creases with temperature. 3) Although the implementation of a PGS at the M1 metal layer has little effect on the maxima of , the

G

Fig. 10. values of NPGS transformers (Design 2) versus frequency at different temperatures, respectively. (a) Design 2 without a PGS. (b) Design 2 with a PGS at M1.

curves of in the PGS case at different temperatures become much flatter, as shown in Fig. 9(b). The decrease is mainly contributed by the increase of the conof ductive loss due to imperfect metal coils. Since the silicon conductivity decreases with temperature, the substrate loss will be reduced at a higher temperature, which has a posi. tive effect on A comparison of the implementation of the PGS on is further demonstrated in Fig. 10(a) and (b) for a transformer . In Fig. 10(a), a similar reverse temper(Design 2) with ature effect is observed, as shown in Fig. 9(a). As the frequency GHz, increases with temperature. exceeds On the other hand, it should be noted that the reverse frequency for Design 2 is much higher than that of Design 1. Since the transformers of Designs 1 and 2 are designed to m, have the same geometrical parameters, i.e., m, and m, it is worthwhile to find out the difversus frequency. Fig. 10 shows that ference in terms of of Designs 1 and 2 with at and the

SHI et al.: FREQUENCY-THERMAL CHARACTERIZATION OF ON-CHIP TRANSFORMERS WITH PGSs

Fig. 11. NF of an NPGS transformers of Design 1 versus frequency at different temperatures.

K, respectively. As seen in Fig. 10, when the operating freGHz quency is lower than the cross-point frequency at K, the of Design 1 is larger than that of Design 2. When the operating frequency further increases, however, a of Design 2 becomes reverse effect is observed and the K, GHz. larger than that of Design 1. When Hence, it can be concluded that at a high frequency, stronger electromagnetic coupling between the primary and secondary coils in Design 2 will be expected than that in Design 1; and for Design 2 without PGSs, a similar phenomena can be observed. and can be derived from each other, we Since only give one example, as shown in Fig. 11, to demonstrate of an NPGS the frequency-dependent characteristics of transformer Design 1 at different temperatures. It is obvious does not change monotonously with frequency inthat creasing, which is different from that shown in [22]. This is because the frequency range up to 20 GHz is considered here. The , e.g., inimplementation of a PGS can improve creases by 18%–33% for the interleave case from 4 to 6 GHz, which is not plotted here. B.

Factor

Based on (4), the frequency- and temperature-dependent factor of the primary coil of Design 1 with a PGS is extracted , and the factor of its NPGS counterpart is also with given for comparison, as shown in Figs. 12 and 13. For Design 1, the following is observed. K to 1) Although the temperature rise from K causes a significant decrease in , such negative temperature effects can be compensated, to some extent, using a PGS with an appropriate embedding depth. It is at K is seen that for the PGS Design 1, at K of NPGS still larger than that of Design 1. 2) As indicated in [22], there are also reverse temperature effects, i.e., as frequency exceeds a certain frequency corresponding to the zero temperature coefficient of silicon re-

9

Q

factors of the primary coil of an NPGS transformer of Design 1 Fig. 12. versus frequency at different temperatures.

Q

Fig. 13. factors of NPGS transformers of Designs 2 versus frequency at different temperatures.

sistance and , the factor increases with temperfactor ature. This is because, at higher frequencies, the and is dominated by the silicon substrate resistances , which exhibits a positive temperature coefficient. factor is defined by The relative enhancement in the (26) and in the case of , , , and at and K, respectively. It is apparent that the decrease in the factor of Design 1 can be compensated due to the shielding effectiveness of the PGS embedded at the metal layer of M1. However, in the case of Design 2, there factor that starts is only a slight increase in the curves of the beyond the frequency GHz approximately when the substrate resistance becomes more dominant.

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V. CONCLUSION Frequency-thermal characterization of on-chip transformers with and without PGSs has been carried out in this paper. These transformers are fabricated using 0.18- m RF CMOS processes, where the same inner dimension, metal track width, track spacing, and silicon substrate are. The above results demonstrate that there exist significant differences in the performance parameters between interleaved and center-tapped interleaved configurations. The maximum available gains of interleaved transformers at low frequencies are higher than those of their center-tapped counterparts, but at higher frequencies the situation will be reversed. In an environment of relatively high temperature, the thermal effects on the performance degradation of transformers and even other passive devices must be considered carefully. It should also be emphasized that, in order to enhance the shielding effectiveness of a PGS, its embedding depth must be chosen appropriately. Fig. 14. Power losses versus frequency for NPGS transformer of Design 1 at different temperatures.

ACKNOWLEDGMENT Author J. Shi acknowledges the support and help of the management team, staffs of the Integrated Circuits and Systems (ICS) Laboratory, Institute of Microelectronics, Singapore. REFERENCES

Fig. 15. Power loss versus frequency for NPGS transformers of Design 2 at different temperatures.

C. Power Loss of NPGS transformer of Fig. 14 shows the power loss Design 1 versus frequency at different temperatures, and is defined as (27) It is obvious that the implementation of the PGS can reduce the power loss of the transformer of Design 1 because of its shielding effectiveness. For comparison, Fig. 15 shows the power losses of the NPGS transformer of Design 2 with versus frequency at temperatures of and K, respectively. It is observed that for the transformers of Design 2, the implementation of a PGS has little effect on the reduction of power loss.

[1] D. E. Meharry, J. E. Sanctuary, and B. A. Golja, “Broad bandwidth transformer coupled differential amplifiers for high dynamic range,” IEEE J. Solid-State Circuits, vol. 34, no. 9, pp. 1233–1238, Sep. 1999. [2] W. Simburger, H. D. Wohlmuth, P. Weger, and A. Heinz, “A monolithic transformer coupled 5-W silicon power amplifier with 59% PAE at 0.9 GHz,” IEEE J. Solid-State Circuits, vol. 34, no. 12, pp. 1881–1892, Dec. 1999. [3] I. Aoki, S. D. Kee, D. B. Rutledge, and A. Hajimiri, “Distributed active transformer—A new power-combining and impedance-transformation technique,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 316–331, Jan. 2002. [4] A. Coustou, D. Dubuc, J. Graffeuil, O. Liopis, E. Tournier, and R. Plana, “Low phase noise IP VCO for multi-standard communication using a 0.35 m BiCMOS SiGe technology,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 2, pp. 71–73, Feb. 2005. [5] A. M. Niknejad and R. G. Meyer, “Analysis, design and optimization of spiral inductors and transformers for Si RF IC’s,” IEEE J. Solid-State Circuits, vol. 33, no. 10, pp. 1470–1481, Oct. 1998. [6] J. R. Long, “Monolithic transformers for silicon RFIC design,” IEEE J. Solid-State Circuits, vol. 35, no. 9, pp. 1368–1382, Sep. 2000. [7] Y. K. Koutsoyannopoulos and Y. Papananos, “Systematic analysis and modeling of integrated inductors and transformers in RFIC design,” IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process., vol. 47, no. 8, pp. 699–713, Aug. 2000. [8] K. T. Ng, B. Rejaei, and J. N. Burghartz, “Substrate effects in monolithic RF transformers on silicon,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 377–383, Jan. 2002. [9] J. Hongrui, Z. Li, and N. C. Tien, “Reducing silicon-substrate parasitics of on-chip transformers,” in IEEE Microelectromech. Syst. Conf., 2002, pp. 649–652. [10] T. Biondi, A. Scuderi, E. Ragonese, and G. Palmisano, “Wideband lumped scalable modeling of monolithic stacked transformers on silicon,” in Proc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting, Sep. 12–14, 2004, p. 265. [11] S. J. Pan, W. Y. Yin, and L. W. Li, “Comparative investigation on various on-chip center-tapped interleaved transformers,” Int. J. RF Microw. Comput.-Aided Eng., vol. 14, no. 5, pp. 424–432, 2004. [12] S. Kim, K. Lee, J. Lee, B. Kim, S. D. Kee, I. Aoki, and D. B. Rutledge, “An optimized design of distributed active transformer,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 380–388, Jan. 2005. [13] T. O. Dickson, M. A. LaCroix, S. Boret, D. Gloria, R. Beerkens, and S. P. Voinigescu, “300-100-GHz inductors and transformers for millimeter-wave (Bi) CMOS integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 123–133, Jan. 2005.

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[14] Y. Mayevskiy, A. Watson, P. Francis, K. Hwang, and A. Weisshaar, “A new compact model for monolithic transformers in silicon-based RFICs,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 6, pp. 419–421, Jun. 2005. [15] C. Chien, A. F. Burnett, J. M. Cech, and M. H. Tanielian, “The signal transmission characteristics of embedded microstrip transmission lines over a meshed ground plane in copper/polyimide multichip module,” IEEE Trans. Comp., Hybrids, Manuf. Technol., vol. 17, no. 4, pp. 578–583, Nov. 1994. [16] R. Lowther and S. G. Lee, “On-chip interconnect lines with patterned ground shields,” IEEE Microw. Wireless Compon. Lett., vol. 10, no. 2, pp. 49–51, Feb. 2000. [17] R. D. Luth, V. K. Tripathi, and A. Weisshaar, “Enhanced transmission characteristics of on-chip interconnects with orthogonal gridded shield,” IEEE Trans. Adv. Packag., vol. 34, no. 4, pp. 288–293, Nov. 2001. [18] P. Wang and E. C. C. Kan, “High-speed interconnects with underlayer orthogonal metal grids,” IEEE Trans. Adv. Packag., vol. 27, no. 3, pp. 497–507, Aug. 2004. [19] C. P. Yue and S. S. Wong, “On-chip spiral inductors with patterned ground shields for Si-based RFIC’s,” IEEE J. Solid-State Circuits, vol. 33, no. 5, pp. 743–752, May 1998. [20] S. M. Yim, T. Chen, and K. K. O, “The effects of a ground shield on the characteristics and performance of spiral inductors,” IEEE J. SolidState Circuits, vol. 37, no. 2, pp. 237–244, Feb. 2002. [21] C. A. Chang, S. P. Tseng, J. Y. Chung, S. S. Jiang, and J. A. Yeh, “Characterization of spiral inductors with patterned floating structures,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5, pp. 1375–1381, May 2004. [22] Y. S. Lin, H. B. Liang, T. Wang, and S. S. Lu, “Temperature-dependence of noise figure of monolithic RF transformers on a thin (20 m) silicon substrate,” IEEE Electron Device Lett., vol. 26, no. 3, pp. 208–211, Mar. 2005. [23] Y. Z. Xiong, T. Hui, and J. S. Fu, “Direct extraction of equivalent circuit parameters for balun on silicon substrate,” IEEE Trans. Electron. Devices, vol. 52, no. 8, pp. 1915–1916, Aug. 2005. [24] M. Golio, Ed., The RF and Microwave Handbook. Boca Raton, FL: CRC Press, 2001, A-8. [25] C. Corvasce, M. Ciappa, D. Barlini, F. Illien, and W. Fichtner, “Measurement of the silicon resistivity at very high temperature with junction isolated Van der Pauw structures,” in Instrum. Meas. Technol. Conf., Como, Italy, May 18–20, 2004, pp. 133–138. [26] N. Arora, MOSFET Models for VLSI Circuit Simulation Theory and Practice. New York: Springer-Verlag, 1993, pp. 17–33.

Jinglin Shi received the B.Eng and M.Eng degrees in electronics engineering from Tianjin University, Tianjin, China, in 1993 and 1996, respectively, and the Ph.D. degree from National University of Singapore, Singapore, in 2001. Since September 2000, she has been a Senior Research Engineer with the Integrated Circuits and Systems Laboratory, Institute of Microelectronics, Singapore. Her research interests include modeling and characterization of active and passive devices in BiCMOS and CMOS advance technologies, substrate coupling and device noise, novel design and device optimization for high-frequency applications, and millimeter-wave circuit design.

Wen-Yan Yin (M’99–SM’01) received the M.Sc. degree in electromagnetic fields and microwave techniques from Xidian University (XU), Shaanxi, China, in 1989, and the Ph.D. degree in electrical engineering from Xi’an Jiaotong University (XJU), Xi’an, China, in 1994. From 1993 to 1996, he was an Associate Professor with the Department of Electronic Engineering, Northwestern Polytechnic University (NPU). From 1996 to 1998, he was a Research Fellow with the Department of Electrical Engineering, Duisburg

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University, granted by the Alexander von Humblodt-Stiftung of Germany. Since December 1998, he has been a Research Fellow with the Monolithic Microwave Integrated Circuit (MMIC) Modeling and Packing Laboratory, Department of Electrical Engineering, National University of Singapore (NUS), Singapore. In March 2002, he joined Temasek Laboratories, NUS, as a Research Scientist and the Project Leader of high-power microwave and ultra-wideband electromagnetic compatibility (EMC)/electromagnetic interference (EMI). In April 2005, he joined the School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University (SJTU), Shanghai, China, as a Chair Professor in electromagnetic fields and microwave techniques. He is also the Director of the Center for Microwave and RF Technologies, SJTU. He is a Reviewer for Radio Science, Proceeding of the IEE (Part H), Microwave, Antennas, and Propagaation, as well as international journals. He is an Editorial Board member and Reviewer for the Journal of Electromagnetic Waves and Applications. As a lead author, he has authored over 120 international journal papers, including 15 book chapters. The chapter “Complex Media” is included in the Encyclopedia of RF and Microwave Engineering (Wiley, 2005). His main research interests are in electromagnetic characteristics of complex media and their applications in engineering, EMC, EMI, and electromagnetic (EM) protection, on-chip passive and active MM(RF)IC device testing, modeling, and packaging, ultra-wideband interconnects and signal integrity, and nanoelectronics. Dr. Yin is the technical chair of electrical design of Advanced Packaging and Systems 2006 (EDAPS’06), which is technically sponsored by the IEEE Components, Packaging, and Manufacturing Technology (CPMT) Subcommittee. He is a reviewer for five IEEE TRANSACTIONS.

Kai Kang was born in Xi’an, China, on November 2, 1979. He received the B. Eng degree in electrical engineering from Northwestern Polytechnical University, Xi’an, China, in 2002, and is currently working toward the Ph.D. degree under a joint Ph.D. program between the National University of Singapore, Singapore, and Ecole Supérieure D’électricité, Piers, France. Since 2003, he has been a Research Scholar with the National University of Singapore, where his research interests lie in the area of modeling of passive devices in RFICs and monolithic microwave integrated circuits (MMICs). From 2005 to 2006, he was with the Laboratoire de Génie Electrique de Paris. Since October 2006, he has been a Research Engineer with the Institute of Microelectronics, Singapore. His research interest is modeling of on-chip interconnects and devices.

Jun-Fa Mao (M’92–SM’98) received the B.S. degree from the University of Science and Technology of National Defense, Changsha, China, in 1985, the M.S. degree from the Shanghai Institute of Nuclear Research, Academic Sinica, Shanghai, China, in 1988, and the Ph.D. degree from Shanghai Jiao Tong University, Shanghai, China, in 1992. Since 1992, he has been a faculty member with the Department of Electronic Engineering, Shanghai Jiao Tong University, where he is currently a Professor. He was a Visiting Scholar with the Chinese University of Hong Kong, Hong Kong (1994–1995) and a Post-Doctoral Researcher with the University of Californian at Berkeley (1995–1996). His research interests include the signal integrity of high-speed integrated circuits and microwave circuits. He has authored or coauthored over 100 journal papers (with 17 papers published in IEEE TRANSACTIONS) and has coauthored book. Dr. Mao is a Cheung Kong scholar of the Ministry of Education, China, and associate chair of the Microwave Society of China. He was chair of the IEEE Shanghai Subsection (2004 and 2005) and general chair of Electrical Design of Advanced Packaging and Systems-2006 (EDAPS’06), technically sponsored by the IEEE CPMT Subcommittee. He was the recipient of the Second-Class 2004 National Natural Science Award of China.

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Le-Wei Li (S’91–M’92–SM’96–F’05) received the B.Sc. degree in physics from Xuzhou Normal University (XNU), Xuzhou, China, in 1984, the M.Eng.Sc. degree in electrical engineering from the China Research Institute of Radiowave Propagation (CRIRP), Xinxiang, China, in 1987, and the Ph.D. degree in electrical engineering from Monash University, Melbourne, Australia, in 1992. In July 1992, he was with La Trobe University (jointly with Monash University), Melbourne, Australia, as a Research Fellow. Since November 1992, he has been with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, where he is currently a Professor and Director of the Centre for Microwave and Radio Frequency. From 1999 to 2004, he was involved with the High Performance Computations on Engineered Systems (HPCES) Programme of the Singapore–MIT Alliance (SMA) as an SMA Fellow. He has authored or coauthored 43 book chapters, over 240 international refereed journal papers, 31 regional refereed journal papers, and over 250 international conference papers. He coauthored Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2001). He is an Editorial Board

member for the Journal of Electromagnetic Waves and Applications (JEWA), Electromagnetics, and the “Progress in Electromagnetics Research (PIER)” EMW Publishing book series. He is also an Associate Editor of Radio Science and an Overseas Editorial Board Member of Chinese Journal of Radio Science. His current research interests include electromagnetic theory, computational electromagnetics, radio wave propagation and scattering in various media, microwave propagation and scattering in tropical environment, and analysis and design of various antennas. Dr. Li has been a member of the Electromagnetics Academy, Massachusetts Institute of Technology (MIT), Cambridge, since 1998. He was a recipient of the Best Paper Award presented by the Chinese Institute of Communications for his 1990 paper published in the Journal of China Institute of Communications and the Prize Paper Award presented by the Chinese Institute of Electronics for his 1991 paper published in the Chinese Journal of Radio Science. He was the recipient of a 1995 Ministerial Science and Technology Advancement Award by the Ministry of Electronic Industries, China, and a 1996 National Science and Technology Advancement Award with a medal presented by the National Science and Technology Committee, China. He is on the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.

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Fully Integrated Differential Distributed VCO in 0.35-m SiGe BiCMOS Technology George P. Bilionis, Alexios N. Birbas, and Michael K. Birbas, Member, IEEE

Abstract—We present a fully integrated differential distributed voltage-controlled oscillator implemented in a 0.35- m SiGe BiCMOS technology. The delay variation by a positive feedback tuning technique, adopted from the ring oscillators, is demonstrated as a fine-tuning alternative, which results to an approximately 420-MHz tuning range. The phase noise is 98 dBc/Hz at 1-MHz offset from the 14.25-GHz carrier. An integrated output buffer isolates the oscillator from the measurement equipment. The measured output power is 17.5 dBm and the overall power consumption of the chip is 138.1 mW employing two power supplies of 3.2 and 4.2 V, respectively. Index Terms—Distributed circuits, distributed oscillators, frequency tuning, output buffer, SiGe BiCMOS integrated circuits, transmission lines, voltage-controlled oscillators (VCOs).

I. INTRODUCTION HE NEED for low-cost broadband communication systems results in the adoption of silicon-based technologies for the design of RF components. In this context, SiGe BiCMOS technology holds a prominent position since it combines highperformance HBTs with CMOS digital circuitry, thus reducing the overall system cost. Regarding the RF performance, state-ofthe-art SiGe HBTs offer frequency response, noise figure, and linearity comparable with that of the III/V compound devices; while at the same time, achieving lower cost and higher integration capability [1]. High-frequency performance of passive devices in standard silicon technologies (i.e., spiral inductors and varactors) do not follow the advancement of technology with the same rate as the lithographic scaling or the Ge epitaxy affect active devices. As a consequence, for the case of voltage-controlled oscillators (VCOs), lumped topologies may not be able to achieve the requested oscillation frequencies. This is due to the fact that the passive devices forming their LC tank may exhibit a poor quality factor [2]. Distributed oscillator topologies overcome this inefficiency since their distributed nature allows them to operate at higher frequencies compared to the lumped ones for a specific technology [3]. Kleveland et al. [4] were the first to demonstrate the feasibility of distributed oscillators in silicon-based technologies. They designed a single-ended distributed amplifier in a 0.18- m

T

Manuscript received May 18, 2006; revised September 21, 2006. The authors are with the Department of Electrical Engineering and Computer Technology, Applied Electronics Laboratory, University of Patras, Patras 265 00, Greece (e-mail: [email protected]; [email protected]; mbirbas@ ee.upatras.gr). Digital Object Identifier 10.1109/TMTT.2006.888940

RF CMOS technology and they connected its input with its output, which resulted in a 17-GHz oscillation frequency; but there was no tuning mechanism and the terminations were offchip. Wu and Hajimiri [3] have presented a circuit analysis for the design of single-ended distributed voltage-controlled oscillators (DVCOs) and proposed a new tuning technique; the delay balanced current steering tuning. This circuit analysis does not take into account the finite base resistance and the finite output conductance of the HBTs employed into the design. The last might lead to significant deviation between results and theory for the case of SiGe HBTs operating at the high current regime and at microwave frequencies. The required loop of the oscillator was also closed off-chip, employing large ac coupling capacitors. Guckenberger and Kornegay [5] presented a differential distributed oscillator by designing a close-packed differential distributed amplifier and connecting the input and output with bond wires. As a frequency-tuning technique, they used the current starving method in order to achieve wide tuning range, but the output power varied considerably during frequency tuning. In this study, we present the first fully integrated (no offchip components) differential distributed voltage-controlled oscillator (DDVCO) implemented in a 0.35- m SiGe BiCMOS of 60 GHz and with four metallization technology with an layers available to us. Firstly, we extend the circuit analysis presented in [3] to the differential mode of operation and we furthermore include in our analysis the impact of the finite base resistance, as well as the finite output conductance of the SiGe HBTs. Secondly, we propose the “delay variation by positive feedback tuning” technique [6], used in ring oscillators, as an alternative frequency fine-tuning method. Finally, we present a design methodology, which associates the physical constraints of the design with the circuit parameters, in order to speed up the development and provide insight to the oscillator’s design. This paper is organized as follows. In Section II, we present the circuit analysis and the tuning technique. In Section III, we present the layout considerations. In Section IV the methodology is described and the measured results are presented.

II. CIRCUIT ANALYSIS Similar to the single-ended case [3], a differential distributed oscillator is formed by short circuiting the output of a differential distributed amplifier with its input through ac-coupling capacitors (Fig. 1). The open-loop analysis of the oscillator is derived by performing the small-signal analysis to the associated differential distributed amplifier [7]–[17]. Due to the fact that this analysis is similar to the single-ended case [3], we

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Fig. 1. Schematic diagram of differential distributed oscillator.

1) There is signal superposition at the differential collector line. 2) The differential base line is terminated by its odd-mode characteristic impedance at its far end, as shown in Fig. 1. 3) The differential output “sees” a load equal to the differential-mode characteristic impedance of the base line. The following expression for the differential-mode voltage gain , similar to the one of the single-ended case [3], can be derived:

(2) Fig. 2. Differential-mode half-circuit of one amplification stage.

focus here on its applicability to the differential case where SiGe HBTs are employed. A. Differential-Mode Small-Signal Analysis Let us consider a set of differential amplifiers connected to differential transmission lines according to the separation distances shown in Fig. 1. In order to derive the equation for the differential-mode voltage gain, we also have to consider the differential half-circuit [18] of each amplification stage, such as the one shown in Fig. 2. The relation between the output at the th amplification stage of the differential distributed amplifier is and the input

(1) where

is the small-signal transconductance of the SiGe HBT, is the delay imposed by the MIM capacitors, is the is the differential-mode characteristic angular frequency, impedance of the loaded collector transmission line, is the odd-mode propagation constant of the differential base line, and is the distance shown in Fig. 1. Take into consideration the following.

where is the differential-mode characteristic impedance of the loaded base line, is the number of amplification stages is the odd-mode propagation constant (four in this case), is the separation distance of the loaded collector line, and shown in Fig. 1. We have to note that the differential impedances of the loaded collector and base lines are twice their respective loaded oddmode impedances [11], i.e., (3) (4) B. Effect of the Finite Base Resistance and Output Conductance Transmission line loading is defined as the variation of transmission-line characteristics (characteristic impedance and propagation constant) due to the “absorption” of lumped elements (i.e., resistors and capacitors) by transmission lines. In our case, the odd-mode characteristics of the differential base and collector lines are affected by the lumped elements within the dashed lines in Fig. 2. We extent the concept of the passive absorption to the resistance case, similar to the case of capacitance absorption shown in [3]. Specifically for the differential base line, the small-signal and the base–emitter capacitance affect input resistance

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its odd-mode characteristic impedance and propagation according to the following equations: constant

(5)

(6) , , , and are the odd-mode paramewhere ters of the unloaded differential base line according to the telegrapher’s equations [19]. and the colSimilarly, the finite output conductance affect the odd-mode characterlector-substrate capacitance and the propagation constant of the istic impedance loaded differential collector line according to

Fig. 3. Characteristic impedance versus resistive load. The solid line denotes simulation results and the stars denote theoretical results.

(7)

(8) , , , and are the odd-mode paramewhere ters of the unloaded differential collector line. In order to investigate the impact of transmission line loading due to resistance absorption and to verify (5)–(8), we conduct the following simulation experiment. We simulate a single-ended transmission line with the aid of a commercial electromagnetic simulator [20] in order to extract its -, -, -, and -parameter at 14.2 GHz, employing the methodology presented in [21]. The line is 700- m long, 2.8- m thick, and it has a width of 4 m. Underneath the line there is a 4- m-thick SiO layer and below it there is a 725- m -thick Si substrate with resistivity of 19 cm, and relative di. We moreover connect a variable reelectric constant sistor in the middle of the line and we extract the characteristic impedance and the propagation constant of the loaded transmission line at 14.2 GHz according to the procedure in [21]. Furthermore, we synthesized (5) and (6) using the -, -, -, and -parameter of the unloaded line, the value of the variable resistor, and the length of the line, and we take into account that there is no capacitive loading. The results of this experiment are shown in Figs. 3 and 4. Someone could see that the simulation experiment and theoretical results agree quite well. In contrast to the effect of capacitive loading, which mainly affects the real part of the characteristic impedance and the phase-shift constant of the loaded transmission line [3], the resistive loading mainly affects the imaginary

Fig. 4. Attenuation and phase-shift constants versus resistive load. The solid line denotes simulation results and the stars denote theoretical results.

part of the characteristic impedance and the attenuation constant. This has a profound impact on the design of distributed amplifiers and oscillators when SiGe HBTs operate at high-current densities [22]. In such a case, proper impedance matching at high frequencies is difficult to achieve and there is a degradation of the distributed amplifier gain due to the high loss of the transmission lines. Thus, both loading effects (resistive and capacitive) must be taken into account during the design stage, and special attention is needed for the active device sizing and current biasing. The conclusions of the previously mentioned experiment are not limited to only the single-ended case. The odd-mode operation of a differential transmission line is derived with the same quasi-TEM approach [23], [24] as the single-ended one. As a result, (5)–(8) are also valid for a differential distributed amplifier or oscillator. Finally, the effects of the collector–base depletion capaciand the finite base resistance of the SiGe HBTs tance employed in the design are neglected in our first-order analysis. For the case of the collector–base capacitance, we have to note that the losses of the transmission lines result in a small reverse capacitance can be negain ( -parameter) and, thus, the glected. For the case of the finite base resistance , multifinger SiGe HBTs also limit its impact to the circuit performance (in

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

Fig. 6. Open-loop differential-mode equivalent of the oscillator. The dashed line denotes one of the propagation paths that contribute to the open-loop gain phase. Fig. 5. Schematic diagram of the line synchronization concept.

our case, around 10 ). Thus, for first-order circuit analysis, its effect can be neglected as well.

(13) Substituting (9) to (13) results in

C. Oscillation Condition and Line Synchronization The superposition principle, necessary for the operation of the distributed amplifiers and oscillators, imposes the condition of line synchronization between the collector and base line. Let us consider Fig. 5 where two successive amplification stages are presented. In order to have perfect superposition of the amplified signals at the collector line, the time required for a signal at the base line to propagate through Path 1 and Path 2 must be equal. Similar to the line synchronization condition presented in [3], the previous consideration yields to (9)

By modifying Barkhausen’s criterion [25] for the differential-mode of operation, it is easily proven that, in order to have oscillations, the necessary condition is (10) (11) In our case, this leads to the oscillation condition

(12) where is the large-signal transconductance. Equation (12) states that the overall phase shift of the open-loop gain must be equal to 180 at the frequency of interest. In order to further investigate its effect on the circuit performance, one has to consider one of the propagation paths that contributes to the open-loop gain, such as the one shown in Fig. 6. The phase shift of this propagation path has to be 180 and it is associated with the phase shifts of the differential collector and base lines according to

(14) This is the condition for line synchronization when the delay of the metal–insulator–metal (MIM) capacitors is considered. If this time constant is not taken into account according to (14), then the phase shift of the open-loop gain will be 180 , which is at a lower frequency than expected. Due to the frequency dependency of the transmission line characteristics, as (5)–(8) affirm, the line synchronization condition (9) will be violated and the odd-mode characteristic impedance and propagation constant of the differential transmission lines can be different than the specified ones during the design. D. Delay Variation by Positive-Feedback Frequency-Tuning Technique Several frequency tuning techniques have been presented for the case of distributed oscillators [3], [5]. The idea behind these techniques is to vary the overall delay of the open-loop equivalent circuit, which leads to a change of the oscillation frequency. This variation is achieved by adopting at least one of the three following approaches: • by changing the phase velocity of the loaded transmission lines [3]; • by changing the effective transmission line length that the propagating signal is using during the oscillation [3]; • by changing the delay that the amplification cell contributes to the overall open-loop delay [5]. While the first two cases have been explored, we focus on an enhanced modification of the method used in [5]. This approach, known as the current starvation method, results in large output power deviation across the frequency range. This is due to the fact that the tail current of the differential amplification cell varies with respect to the control voltage and, thus, the differential voltage swing at the output of the amplification cell is not constant. In order to achieve constant output power, we

BILIONIS et al.: FULLY INTEGRATED DIFFERENTIAL DISTRIBUTED VCO IN 0.35- m SiGe BiCMOS TECHNOLOGY

Fig. 7. Application of the delay variation by positive feedback frequency tuning technique to the differential amplification cell.

adopted the “delay variation by positive feedback” frequency tuning technique [6] used in ring oscillators, and we modified every amplification cell of the distributed oscillator accordingly (Fig. 7). The operation of this tuning technique is similar to the case of differential ring oscillators, except that the lumped resistances at the collectors are replaced by transmission lines . The cross-couwith odd-mode characteristic impedance and , pled transistor pair, which consists of transistors exhibits a negative resistance . This resistance is inversely proportional to the tail current of the cross-coupled pair. As the differential control voltage steers current from the differential amplification cell to the cross-coupled pair, its negative , where is the maxresistance changes up to and according to imum transconductance of transistors (15) is the tail current, as shown Fig. 7. As a result, the where delay of the overall differential stage changes with respect to the differential control voltage [6], but the current that contributes to the differential voltage swing is constant, i.e., (16)

Regarding the ratio between the tail currents and , we have to note that although it is clear that a large ratio will result in a wide tuning range, some concerns arise regarding its impact to the line synchronization. Steering a large amount of current between the amplification cell and the cross-coupled pair results to a large deviation of the differential amplifier operating point. As a consequence, the line synchronization condition (9) is violated leading to imperfect superposition of the propagating waves at the differential collector lines. The odd-mode characteristics of the loaded differential transmission lines also change, as denoted by (5)–(8). The last two effects result in phase-noise augmentation due

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Fig. 8. GDSII layout of the DDVCO. The output of the oscillator core is taken at the differential base line.

to impedance mismatching between the terminating resistors and the transmission lines, or even to a lack of oscillation due to violation of (13). In order to avoid considerable synchronization errors between the transmission line pairs, the emitter was set to its minimum value. length of the transistors was chosen with Moreover, the current ratio respect to a tolerable line synchronization error. In our design, m m the above-mentioned ratios are and mA mA. III. LAYOUT CONSIDERATIONS AND DESIGN METHODOLOGY The design of fully integrated distributed oscillators requires the consideration of several physical constraints that the layout design rules, the full integration, and the electromagnetic behavior of the transmission lines impose. These restrictions must be associated with the circuit parameters and limit the range of their possible values. If not properly taken into account at the circuit level of the oscillator design, this can lead to a nonfeasible design. To address this, we present a design methodology that takes into account the previously mentioned considerations. In this context, we furthermore discuss several design aspects with respect to the full integration of the oscillator. A. Oscillator Core Description In Fig. 8, the GDSII layout of the proposed differential distributed oscillator is presented. As we described in Section II and showed in Fig. 1, the differential collector line is connected to the differential base line with the aid of large ac coupling capacitors. Aiming at having large value integrated capacitors with good frequency response, we used MIM capacitors in a parallel connection. For the transmission line modeling, we employed a commercial electromagnetic simulator [20]. We have to note that the choice of the transmission line structure (differential microstrip or coplanar) was made upon the consideration that the design of the oscillator requires transmission

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lines with a balanced tradeoff between high odd-mode characteristic impedance values and low losses at the frequency range of interest. The reason for this consideration lies in the fact that the transmission lines must be “loaded” with high capacitances in order to achieve the required phase shift while occupying reasonable silicon area. The previously mentioned capacitive “loading” effect mainly influences the real part of the characteristic impedance [3]. Furthermore, transmission lines are required to have low losses in order for the oscillator to accomplish large oscillation amplitudes. Electromagnetic simulations revealed that differential coupled coplanar transmission lines exhibit very low characteristic impedance at the frequency of interest due to the thick metal (2.8 m) at the top layer of the process and the thin SiO layer underneath it. Furthermore, differential microstrip lines with the first metallization layer as the finite ground plane exhibit low odd- mode characteristic impedance due to the short vertical distance between the thick signal lines and the finite ground (approximately 2.5 m). As a result, we selected microstrip lines with ground reference to the backplate of the chip, taking into account that the losses in this case can be high, but controllable by the spacing of the signal lines. It is worth noticing that the above considerations are relative to the specific process available to us and, thus, there can be different choices with respect to the performance of the transmission lines and the vertical profile of the technology [3]. Regarding the transmission line bends, it is shown in [5] and [26] that the wave propagation remains unaffected from these discontinuities due to the fact that their dimensions are negligible compared to the wavelength of operation. Thus, 45 and 90 bends do not limit the performance of the transmission lines. However, since in our case the transmission lines were routed in a clockwise manner, a length imbalance between the inner and outer trace of the differential transmission lines will be present, resulting in asymmetry. In order to reduce the length imbalance, the inner trace was designed with a 90 bend at every corner and the outer trace with a 45 bend. An important feature of the proposed design is the fact that the output signal of the oscillator core is extracted from the end of the differential base line and it is fed to the input of the buffer. In this way, the open loop gain bandwidth product is the maximum available. Another aspect of the oscillator core is the internal termination of the differential transmission lines. Large area resistors were used in order to minimize mismatching effects [27] as much as possible. For the terminating resistors at the reverse direction of the differential collector line, special attention was paid to their connection with the power supply. The first metallization layer was used in order to minimize the coupling capacitance between this metal and the differential transmission lines. Finally, all the interconnections between ground, bias, differential tuning, and amplification cells were designed perpendicular to the transmission lines in order to minimize interference. B. Design Methodology From Section III-A, it is clear that the feasibility of the distributed oscillator depends on a large set of physical and elec-

Fig. 9. Design flow for the proposed oscillator. The physical constraints are associated with the electrical parameters at the circuit level in order to attain design feasibility.

trical parameters. Hence, there is need in associating these parameters at the circuit level of development and a design flow for the designer’s guidance. The distance , shown in Fig. 8, is equal to (17)

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Fig. 10. Schematic diagram of the 50- output buffer. The differential input is connected to the end of the differential base line and the 50- outputs to the G–S–G–S–G RF pads.

and links the lengths of the transmission lines with the size of the amplification cell. This must be considered during the circuit design level because the amplification cells must fit between the differential transmission lines. The physical parameters of the design are: 1) the emitter length of the SiGe HBTs used at the differential amplifier of transistors at Fig. 7); 2) , the the amplification cell ( , ; 3) the conductors’ widths of the transmission lines spacing between the conductors of the differential transmission ; and 4) the conductors’ lengths . The lines values of the preceding parameters are bounded by the design rules of the technology and by the electrical performance at , , the worst case. For example, the physical parameters , and of the transmission lines have minimum values of the ones specified from the technology and the maximum ones are calculated by taking into consideration the lower value for the odd mode characteristic impedance that the unloaded transmission lines can have (i.e., 50 ). For a specified set of design parameters, the procedure (Fig. 9) starts with impedance matching of the loaded differential transmission lines over the desired frequency range of oscillation. This is realized by taking into consideration the differential mode of operation of the open-loop equivalent circuit, as defined in Section II, by utilizing the mixed-mode -parameter theory [23] and specifying a threshold value for the reflection coefficients at the four ends of the transmission lines. After the second step of the procedure, the line synchronization condition (9) is verified with respect to the frequency range of interest, according to a maximum specified relative error between the left and right portions of (9). Following that, the physical requirements for the proper placement of the amplification cell in accordance are evaluated. Subsequently, an ac analysis to the distance takes place in order to check if the oscillation condition of the open-loop equivalent is satisfied and, upon confirmation, harmonic-balance and phase-noise analysis are performed with respect to the differential tuning voltage. The proposed design flow is followed by the designer for . Thus, if the frequency a specific current ratio range after the harmonic balance has not been reached, then the ratio must be increased, with respect to the considerations presented in Section II-D. C. Output Buffer Design The extraction of the oscillation signal off-chip might require the application of an output buffer stage. The reason for

this is threefold. Firstly, to provide the standard 50- output impedance in case of the odd-mode characteristic impedance of the differential base line is different than 50 . Secondly, to isolate the oscillator core from the succeeding stage (in our case, the measurement equipment). Thirdly, to increase the output power of the oscillation signal according to the specified levels. For our experimental case, the last reason was not considered due to the fact that we wanted to study the effect of the oscillation amplitude deviation with respect to frequency tuning range. Considerable amplitude deviations mean that the perfect superposition at the differential collector line is not sustained over the frequency tuning range due to line synchronization errors. As a consequence of the above, we designed a multistage output buffer, as described in [28] (Fig. 10) with differential input impedance equal to the differential impedance of the loaded differential base line, output single-ended impedance equal to 50 , high isolation between the differential input and output, and output power equal to the power available from the output of the oscillator core. The last goal is achieved by reducing the bias currents at the differential amplification stages in contrast to the case of a high-speed emitter coupled logic (ECL) output buffer. Furthermore, low-value resistors were used at the collectors of the differential amplifiers. The current sources are implemented by employing an HBT and a polysilicon resistor connected between its emitter and the ground. The voltage at the base of the HBT is controlled by a tunable voltage reference circuitry [29], such as the one shown in Fig. 11. This gives the capability of further reduction of the bias currents at the differential stages, thus performing amplitude attenuation if needed. In this way, considerable voltage deviations of the differential output of the oscillator core can be detected. IV. RESULTS The DDVCO has been implemented in an Austriamicrosystems 0.35- m SiGe BiCMOS technology available to us. A die photograph of the implemented chip is shown in Fig. 12. 0.9 mm and The oscillator core has dimensions 0.9 mm 2 mm due to the overall chip occupies an area of 2 mm fabrication and measurement reasons. The chip is measured with the aid of a microwave probe station. The differential output was probed with a Cascade microwave probe in a ground–signal–ground–signal–ground (G–S–G–S–G) configuration. One of its ports was fed to an Agilent E4440A spectrum analyzer through a microwave cable and the other port was

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Fig. 13. Single-ended output power spectrum of the DDVCO at nominal current bias of the output buffer. Fig. 11. Schematic diagram of the tunable voltage reference circuit.

Fig. 14. Phase noise of the DDVCO.

Fig. 12. Die photograph of the fully integrated DDVCO.

connected to a 50- ac termination resistance. Needle probes were used for the power supply pads, and all the control signals of the chip. The overall power consumption of the oscillator is 138.1 mW, which is in close agreement with the simulation (137.2 mW). The output spectrum of the DDVCO at its center frequency of 14.25 GHz is shown in Fig. 13. The measured power at the input of the spectrum analyzer is 19.9 dBm. Taking into account the insertion loss of the measurement setup, the output power of the buffer is 17.5 dBm. The phase noise is presented in Fig. 14. Due to external interference and the high tuning sensitivity of the oscillator, the oscillation signal is not constant with respect to the frequency. This results to incapability of the spectrum analyzer to capture the carrier frequency with absolute accuracy and perform ideal PM demodulation (a common problem when a high tuning sensitivity VCO is measured). Thus, the phase-noise diagram flattens considerably and the phase-noise measurement is degraded. The measured phase noise in this case is 98 dBc at 1-MHz offset from the carrier. This problem could be possibly addressed by incorporating a high-frequency phase-locked loop (PLL) to perform frequency

Fig. 15. Oscillation frequency versus differential control voltage.

stabilization (the open-loop bandwidth of the PLL must be higher than the phase-noise bandwidth of interest). Fig. 15 presents the oscillation frequency versus the differential control voltage. The tuning range is 13.95-14.37 GHz. It is

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REFERENCES

Fig. 16. Output power versus oscillation frequency. The measurement setup loss (2.4 dB at the frequency range of interest) has not been subtracted.

interesting to note that the frequency tuning follows an inverse exponential behavior similar to the one that the steering current and (Fig. 7) exhibits with respect to the of transistors differential control voltage. Finally, in order to investigate the output power deviations with respect to the oscillation frequency, we measured the single-ended output power in two cases (Fig. 16). The first case was with nominal bias operation of the output buffer and the second one was with reduced bias. The last one was achieved by reducing the control voltage at the tunable voltage reference circuit (Fig. 11). In the case of the reduced bias operation of the output buffer, the maximum power deviation was 0.37 dB. The latter gives a more accurate estimation for the amplitude deviation due to the approximately linear operation of the output buffer stages. V. CONCLUSION A fully integrated DDVCO implemented in a low-cost SiGe of 60 GHz has been BiCMOS technology with a moderate presented. For the first time, the applicability of SiGe HBTs and their impact to the performance of differential distributed oscillators has been analyzed. The “delay variation by positive feedback” frequency tuning technique has been demonstrated for this case. Moreover, we have presented a rigorous and detailed design methodology, which can be used as a design aid from the designer of the distributed oscillator. Experimental results show the feasibility of fully integrated distributed oscillators in commercial low-cost silicon technologies. ACKNOWLEDGMENT The authors would like to thank M. Lobel, J. Kikidis, and T. Franck, all with the Intel Corporation, Copenhagen, Denmark, for helpful discussions and for providing assistance and access into Intel Copenhagen laboratories for the chip measurement.

[1] J. D. Cressler and G. Niu, Silicon Germanium Heterojunction Bipolar Transistors. Norwood, MA: Artech House, 2003, pp. 22–25. [2] I. Bahl, Lumped Elements for RF and Microwave Circuits. Norwood, MA: Artech House, 2003, pp. 58–85. [3] H. Wu and A. Hajimiri, “Silicon-based distributed voltage-controlled oscillators,” IEEE J. Solid-State Circuits, vol. 36, no. 3, pp. 493–502, Mar. 2001. [4] B. Kleveland, C. H. Diaz, D. Dieter, L. Madden, T. H. Lee, and S. S. Wong, “Monolithic CMOS distributed amplifier and oscillator,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 1999, pp. 70–71. [5] D. Guckenberger and K. T. Kornegay, “Design of a differential distributed amplifier and oscillator using close-packed interleaved transmission lines,” IEEE J. Solid-State Circuits, vol. 40, no. 10, pp. 1997–2007, Oct. 2005. [6] B. Razavi, Design of Analog CMOS Integrated Circuits, Int. ed. New York: McGraw-Hill, 2001, pp. 515–518. [7] T. Y. K. Wong, A. P. Freundorfer, B. C. Beggs, and J. E. Sitch, “A 10 Gb/s AlGaAs/GaAs HBT high power fully differential limiting distributed amplifier for III–V Mach–Zehnder modulator,” IEEE J. SolidState Circuits, vol. 31, no. 10, pp. 1388–1393, Oct. 1996. [8] S. Kimura, Y. Imai, and Y. Miyamoto, “Direct-coupled distributed baseband amplifier IC’s for 40-Gb/s optical communication,” IEEE J. Solid-State Circuits, vol. 31, no. 10, pp. 1374–1379, Oct. 1996. [9] H. Suzuki et al., “Very-high-speed InP/InGaAs HBT ICs for optical transmission systems,” IEEE J. Solid-State Circuits, vol. 33, no. 9, pp. 1313–1320, Sep. 1998. [10] B. M. Ballweber, R. Gupta, and D. J. Allstot, “A fully integrated 0.5–5.5 GHz CMOS distributed amplifier,” IEEE J. Solid-State Circuits, vol. 35, no. 2, pp. 231–239, Feb. 2000. [11] H. Ahn and D. J. Allstot, “A 0.5–8.5-GHz fully differential CMOS distributed amplifier,” IEEE J. Solid-State Circuits, vol. 37, no. 8, pp. 985–993, Aug. 2002. [12] Y. Baeyens et al., “InP D-HBT ICs for 40-Gb/s and higher bitrate lightwave transceivers,” IEEE J. Solid-State Circuits, vol. 37, no. 9, pp. 1152–1159, Sep. 2002. [13] H. Shigematsu, M. Sato, T. Hirose, and Y. Watanabe, “A 54-GHz distributed amplifier with 6-Vpp output for a 40-Gb/s LiNbO modulator driver,” IEEE J. Solid-State Circuits, vol. 37, no. 9, pp. 1100–1105, Sep. 2002. [14] S. Masuda, T. Takahashi, and K. Joshin, “An over-110-GHz InP HEMT flip-chip distributed baseband amplifier with inverted microstrip line structure for optical transmission system,” IEEE J. Solid-State Circuits, vol. 38, no. 9, pp. 1479–1484, Sep. 2003. [15] R. Liu, C. Lin, K. Deng, and H. Wang, “Design and analysis of DC-to-14-GHz and 22-GHz CMOS cascode distributed amplifiers,” IEEE J. Solid-State Circuits, vol. 39, no. 8, pp. 1370–1374, Aug. 2004. [16] A. Iqbal and I. Z. Darwazeh, “Analytical expression for the equivalent input noise current spectral density of HBT distributed amplifier based optical receivers,” in IEE Optoelectron. Interfacing at Microw. Freq. Colloq., Apr. 20, 1999, pp. 2/1–2/210, Ref. 1999/045. [17] ——, “Transimpedance gain modeling if optical receivers employing a pin photodiode and HBT distributed amplifier combination,” in IEE Optoelectron. Interfacing at Microw. Freq. Colloq., Apr. 20, 1999, pp. 3/1–3/9, Ref. 1999/045. [18] P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, 4th ed. New York: Wiley, 2001, pp. 224–230. [19] G. Gonzalez, Microwave Transistor Amplifiers-Analysis and Design, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1997, pp. 20–22. [20] Momentum v4.90, Advanced Design System 2005A. Agilent Technol., Santa Rosa, CA, 2005. [21] W. R. Eisenstadt and Y. Eo, “S -parameter-based IC interconnect transmission line characterization,” IEEE Trans. Compon. Hybrids, Manuf. Technol., vol. 15, no. 4, pp. 483–490, Aug. 1992. [22] J. D. Cressler and G. Niu, Silicon Germanium Heterojunction Bipolar Transistors. Norwood, MA: Artech House, 2003, ch. 5. [23] D. E. Bockelman and W. R. Eisenstadt, “Combined differential and common-mode scattering parameters: Theory and simulation,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1530–1539, Jul. 1995. [24] V. K. Tripathi, “Asymmetric coupled transmission lines in an inhomogeneous medium,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, no. 9, pp. 734–739, Sep. 1975. [25] G. Gonzalez, Microwave Transistor Amplifiers-Analysis and Design, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1997, pp. 384–386.

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[26] D. Goren et al., “On-chip interconnect-aware design and modeling methodology, based on high bandwidth transmission line devices,” in Proc. IEEE Design Automation Conf., Anaheim, CA, Jun. 2003, pp. 724–727. [27] “Austriamicrosystems 0.35 m HBT BiCMOS matching parameters datasheet,” Austriamicrosystem, Unterpremstaetten, Austria, 2005. [28] H.-M. Rein and M. Moller, “Design considerations for very-high-speed Si bipolar ICs operating up to 50 Gb/s,” IEEE J. Solid-State Circuits, vol. 31, no. 8, pp. 1076–1090, Aug. 1996. [29] A. W. Buchwald, K. W. Martin, A. K. Oki, and K. W. Kobayashi, “A 6-GHz integrated phase-locked loop using AlGaAs/GaAs heterojunction bipolar transistors,” IEEE J. Solid-State Circuits, vol. 27, no. 12, pp. 1752–1762, Dec. 1992.

George P. Bilionis was born in Pyrgos, Greece. He received the Diploma degree in electrical engineering and computer technology from the University of Patras, Patras, Greece, in 2001, and is currently working toward the Ph.D. degree in electrical engineering and computer science at the University of Patras. He is currently with the Electrical Engineering and Computer Technology Department, University of Patras. His research interests include high-frequency integrated-circuit (IC) design (analog and mixed signal), communication systems, and transmission line and passive components modeling.

Alexios N. Birbas was born in Patras, Greece. He received the Diploma degree in electrical engineering and computer technology from University of Patras, Patras, Greece, in 1985, and the M.S.E.E. and Ph.D. degrees from the University of Minnesota at Minneapolis–St. Paul, in 1996 and 1998, respectively. He is currently a Professor with the Department of Electrical Engineering and Computer Technology, Patras University, Patras, Greece. He has held visiting faculty positions with the Department of Electrical Engineering, University of Minnesota at Minneapolis–St. Paul and with Institut National Polytechnique de Grenoble (INPG), Grenoble, France. He has supervised over ten doctoral students. He has held managerial and consulting positions in the electronics industry. He has authored or coauthored over 50 refereed journal and conference papers. His research interests include electron device modeling, device noise analysis, analog and mixed-signal integrated-circuit (IC) design, and nanoelectronics.

Michael K. Birbas (M’95) was born in Patras, Greece. He received the Diploma degree in electrical engineering and computer technology and Ph.D. degree from the University of Patras, Patras, Greece, in 1985 and 1991 respectively. From 1992 to 1999, he with the Research and Development Manager of Synergy Systems, an electronics-related company that he cofounded. From 1999 to 2002, he was a Research and Development Manager with GIGA Hellas S.A—an Intel company (part of the Optical Products Group), where he was involved with the design and development of 10–40-Gbit/s system solution demonstrators for synchronous digital hierarchy (SDH)/synchronous optical network (SONET) and optical transport network (OTN) applications and of 40-Gbit/s Ser/Des components. Since 2002, he has been with the Applied Electronics Laboratory team, Patras University, where he is involved in the fields of efficient mixed (analog/digital) integrated circuit (IC) and in very large scale integration (VLSI) implementations of soft iterative decoding algorithms with emphasis on turbo-like codes.

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A Low Phase-Noise X -Band MMIC VCO Using High-Linearity and Low-Noise Composite-Channel Al0:3Ga0:7N/Al0:05Ga0:95N/GaN HEMTs Zhiqun Q. Cheng, Yong Cai, Jie Liu, Yugang Zhou, Kei May Lau, Fellow, IEEE, and Kevin J. Chen, Senior Member, IEEE

Abstract—A low phase-noise -band monolithic-microwave integrated-circuit voltage-controlled oscillator (VCO) based on a novel high-linearity and low-noise composite-channel Al0 3 Ga0 7 N/Al0 05 Ga0 95 N/GaN high electron mobility transistor (HEMT) is presented. The HEMT has a 1 m 100 m gate. A planar inter-digitated metal–semiconductor–metal varactor is used to tune the VCO’s frequency. The polyimide dielectric layer is inserted between a metal and GaN buffer to improve the factor of spiral inductors. The VCO exhibits a frequency tuning range from 9.11 to 9.55 GHz with the varactor’s voltage from 4 to 6 V, an average output power of 3.3 dBm, and an average efficiency of 7% at a gate bias of 3 V and a drain bias of 5 V. The measured phase noise is 82 dBc/Hz and 110 dBc/Hz at offsets of 100 kHz and 1 MHz at a varactor’s voltage ( tune ) = 5 V. The phase noise is the lowest reported thus far in VCOs made of GaN-based HEMTs. In addition, the VCO also exhibits the minimum second harmonic suppression of 47 dBc. The chip size is 1.2 1.05 mm2 . Index Terms—Al0 3 Ga0 7 N/Al0 05 Ga0 95 N/GaN high electron mobility transistor (HEMT), monolithic, phase noise, voltage-controlled oscillator (VCO).

I. INTRODUCTION HE AlGaN/GaN high electron mobility transistors (HEMTs), with their high breakdown voltage and high-frequency operation, are promising candidates for power amplifiers to be used in next-generation wireless base stations and military applications [1]–[5]. The AlGaN/GaN HEMTs are also showing good linearity and excellent microwave

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Manuscript received May 11, 2006; revised August 18, 2006. This work was supported in part by the Hong Kong Research Grant Council and National Science Foundation of China under Grant N_HKUST616/04 and by the National Science Foundation of China under Grant 60476035. Z. Q. Cheng is with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, and also with the Department of Information Engineering, Hangzhou Dianzi University, Hangzhou, 310018, China. Y. Cai and J. Liu were with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong. They are now with the Material and Packing Technologies Group, Hong Kong Applied Science and Technology Research Institute Company Ltd., Hong Kong. Y. Zhou was with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong. He is now with Advanced Packaging Technology Ltd., Hong Kong. K. M. Lau and K. J. Chen are with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong (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/TMTT.2006.888942

noise performance [2], [6]–[9], which are attractive not only for low-noise amplifier (LNA) development, but also for low phase-noise voltage-controlled oscillators (VCOs). The VCOs, if monolithically integrated with the power amplifiers, can provide flexible high-power signal sources for a broader range of applications [10]–[12]. The VCOs are also indispensable in fully integrated AlGaN/GaN HEMT transceivers. For improved device performances, advanced device structures are also being investigated. For example, a composite channel high electron mobility transistor (CC-HEMT), which features Al Ga N as the main channel and GaN as the minor channel, was recently demonstrated. CC-HEMTs exhibit enhanced linearity and low noise [13], [14], both of which are favorable for low phase-noise VCOs. In particular, the enhanced linearity enables noise up-conversion factor, which is favorable for a lower phase-noise reduction. In this paper, we demonstrate a fully integrated -band VCO using the CC-HEMT technology that features improved noise characteristics. On-chip interdigitated metal–semiconductor–metal (MSM) varactors and planar spiral inductors form the tunable LC tank. The VCO with a 100- m-wide single HEMT exhibits a tuning range from 9.106 to 9.55 GHz with a maximum output power of 4.2 dBm at a of 5.0 V. Low phase noise of 82 and supply voltage 110 dBc/Hz were obtained at offsets of 100 kHz and 1 MHz, respectively. In addition, the VCO shows excellent second harmonic suppression of 47 dBc. To our best knowledge, this is the highest harmonic suppression reported on AlGaN/GaN HEMT VCOs. II. MONOLITHIC INTEGRATION TECHNOLOGY As shown in Fig. 1, the VCO was designed and fabricated in a fully integrated MMIC process that includes an active HEMT, air-bridge interconnects, spiral planar inductors, metal–insulator–metal (MIM) capacitors (with SiN as the dielectric), and the interdigitated MSM varactors [15]. A. Composite-Channel HEMT The epitaxial structure of the composite-channel HEMT was grown by metal-organic chemical vapor deposition (MOCVD) on a sapphire substrate. The epitaxial layer structure contains a 2.5- m undoped GaN buffer layer, a 6-nm undoped Al Ga N layer, 3-nm undoped spacer, a 21-nm doped 2 10 cm carrier supplier layer, and a 2-nm undoped cap layer. Different from the conventional AlGaN/GaN HEMT,

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Fig. 1. Cross section of the GaN-based HEMT MMIC process.

a thin layer (6 nm) of AlGaN with 5%Al composition is incorporated between the Al Ga N barrier and the GaN buffer, aimed at reducing the alloy scattering at the barrier interface. All the HEMTs have a gate length of 1 m and the fabrication details can be found in [13]. The reduced scattering leads to noise reduction, with a detailed explanation given in [13] and [14]. The HEMT devices exhibit a maximum drain current density of 910 mA/mm and dc extrinsic transconductance of 175 mS/mm. A maximum oscillation frequency of 35 GHz was obtained. An output third-order intermodulation intercept point (OIP3) of 33.2 dBm was obtained at 2 GHz. The minimum noise figure was measured to be less than 3.5 dB up to 10 GHz [14].

Fig. 2. Photograph and high-frequency characteristics of the on-chip spiral inductor.

B. On-Chip Passive Components Since the phase noise of the VCO is predominantly determined by the intrinsic noise of HEMT and the factor of the resonant tank, low-noise HEMT and high- inductors factor and varactors are essential. In order to improve the of inductors, it is essential to reduce the coupling to the GaN buffer layer, which still exhibit nonperfect insulating behavior 10 cm after many with a resistivity in the range of 10 years’ optimization. A low- polyimide dielectric layer (5- m thick) was inserted between the major metal traces (transmission lines and inductor metal) and the GaN buffer to reduce buffer coupling. On-wafer -parameters were measured using Agilent Technologies’ 8722ES network analyzer and Cascade microwave probes. One-port inductors [as shown in Fig. 2(a)] ranging from 1 to 4.5 nH were designed and measured. The pad-only characteristics were measured on the “open” pad pattern [as shown in Fig. 2(b)] to extract the pad’s parasitics. The pad’s parasitics were then deembedded from the overall inductor characteristics by subtracting the -parameters of the “open” pad from the -parameter of the overall inductor. A one-port equivalent-circuit model, as shown in Fig. 2(c), was used to extract the inductance value and factor. Inductance is , and the factor is calculated calculated using using . As shown in Fig. 2(d), a factor of 11 was achieved at 10 GHz in a 2.5-nH inductor. factor of 14 was achieved at a frequency of A maximum 6.5 GHz. The self-resonance frequency of the inductor is over 20 GHz. For a 2.5-nH inductor with metal traces directly on top of the GaN buffer, the factor is approximately 5 at 10 GHz.

Fig. 3. (a) Microphotograph of a fabricated MSM varactor with a close-up view factor as a function of the of the interdigital fingers. (b) Capacitance and varactor’s bias, measured at 10 GHz.

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The MSM planar inter-digitated varactors [15], which factor compared to conventional showed an improved metal–insulator–semiconductor varactors, were also fabricated on the HEMT structure. The varactors were characterized by on-wafer -parameter measurement with a similar deembedding procedure as that used for inductors. In Fig. 3, an MSM

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Fig. 5. Circuit schematics of the X -band VCO.

Fig. 4. Photograph and high-frequency characteristics of the on-chip spiral inductor.

planar varactor’s microphotograph together with its characteristics at 10 GHz were presented. The dimensions of the of 2 m, finger width varactor include finger length of 20 m, finger-to-finger spacing of 1.5 m, and number of fingers of 20. The varactor’s capacitance varies from 2.0 to 0.5 pF when its bias voltage changes from 4 to 5 V. MIM capacitors are made of bottom metal (0.3 m), upper metal (3 m) and SiN dielectric layer (0.2 m). The capacitor [as shown in Fig. 4(a)] and “open” pad], as shown in Fig. 4(b)] for deembedding are designed and measured such as that used for inductors. The capacitance and factor are extracted using

Fig. 4(c) shows the measured result for a 2-pF capacitor according to the above equations. It should be noted that the factor of both the varactor and MIM capacitor are relatively low, between 8–10 GHz, seemingly creating obstacles in achieving low phase noise. However, in our VCO circuit, both of them are in series with the input of the transistor, which presents a small capacitance value and small resistance. As the overall capacitance of series-connected capacitors is dominated by the smaller capacitor, the overall capacitance in the resonant tank is reduced. The overall effective capacitor together with an inductor form an LC tank that resonates in -band. The reduced effective capacitance also results in a higher factor in the effective capacitor according to , even though the resistive loss ( ) remains the same. More discussions are provided in Section III when the design of the VCO is discussed. III. DESIGN OF VCOS The VCO circuit was designed using Agilent Technologies’ Advanced Design System (ADS). The schematic circuit of the

Fig. 6. Stability factor K as a function of the length of the shunt capacitor C .

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with and without

-band VCO is shown in Fig. 5. In the design of VCO, the measured small-signal -parameters of the CC-HEMT was used. A stub was connected between the CC-HEMT’s source and the ground, providing a positive feedback to make the HEMT pf more unstable. A small MIM shunt capacitor was in parallel with the stub in order to shorten the stub’s length and reduce the size of the VCO chip. Fig. 6 shows the simulated stability factor’s ( ’s) variation with the length of stub with or without the shunt capacitor. Without the cathe , the factor is large (not favorable for oscillator pacitor circuits) and the minimum value is larger than 0.2 at a length , the factor is reduced and of 5 mm. With the capacitor length of 1 mm. As a result, the addition less than 0.2 at a not only makes the oscillation more of the shunt capacitor likely, but also helps shorten the length of the feedback stub. The resonant tank was designed and connected to the gate of the CC-HEMT. The resonator circuit includes the spiral inductor , the MIM capacitor , transmission line ( , ), and an interdigitated MSM varactor . The MIM capacitor pF is used to decouple the negative gate bias of the , , and HEMT and the varactor’s control voltage. are applied through off-chip inductors of 10 nH. In the design of the VCO, in order to reduce the phase noise of (tuning frequency divided by the VCO, the tuning gain tuning voltage) of the VCO must be minimized [16]. However, minimizing tuning gain usually results in reduced frequency tuning range. Therefore, the tuning gain was capped limited to 299.5 MHz/V (from small-signal simulation) when the tuning range was specified to be approximately 500 MHz in our design. After tuning the elements’ values (it is mainly inductance)

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

Fig. 8. Measured output spectrum at V

Fig. 7. Microphotograph of the X -band VCO chip. Its size is 1.2

2 1.05 mm .

5 V.

= 5 V, V

= 03 V, and V

=

of the resonant tank in simulation, a large negative resistance is presented at the drain output. An pF and ) connecting the output matching network ( drain port of the CC-HEMT and a 50- load was then designed according to the oscillation conditions as follows: (1) Real

(2)

is the real part of the negative resistance where Real at the drain output of the HEMT and is the and are the real and imaginary imaginary part. parts of the impedance looking into the load network with the 50- load included. is the resonant frequency. As mentioned in Section III, the impedance looking at 9.4 GHz, into the gate of the HEMT is . When yielding a small effective capacitance of 0.108 pF is combined with and in series, the overall capacitance of the resonant loop at the transistor’s input , which indicates that the tuning range of is dominated by will be much less than that of . However, the reduced capacitance improves the factor of the resonant circuit significantly, leading to low phase noise in the -band VCO, as shown in Section IV. The microphotograph of the fabricated -band VCO is shown in Fig. 7. The feedback stub is designed with the meander layout to minimize the chip size. IV. MEASUREMENT RESULTS The VCO was characterized with Agilent Technologies’ E4440A PSA spectrum analyzer. The HEMT’s gate and drain bias were set to 3 and 5 V, respectively, which was experimentally found to be optimal for both phase noise and the tuning frequency range. A typical output power spectrum is of 5 V. The output shown in Fig. 8 at a tuning voltage power at the resonance is 3.3 dBm. In general, the VCO’s phase noise is dominated by two major factors: the thermal noise of the passive components noise of the active (e.g., inductors and capacitors) and the transistor. The thermal noise contribution from the passive elements is minimized by reducing the loss of the passive elements through implementation of thick metal and the insertion of a low- (polyimide) dielectric between the metal traces and

Fig. 9. Oscillation frequency and output power versus the varactor control voltage.

substrate. As for the noise of the AlGaN/GaN HEMTs, it can affect the phase noise of RF/microwave VCOs after frequency up-conversion. As a result, high-linearity HEMTs are preferred for VCO application because they provide a lower noise up-conversion factor. It has also been reported that the AlGaN/GaN HEMTs have a corner frequency in the range of 70–100 kHz [17]. The Hooge parameter of AlGaN/GaN HEMTs have been reported to be in the range of 10 [18], comparable to or slightly lower than that in conventional AlGaAs/GaAs HEMTs. The phase noise realized in this study is approximately 10 dBc/Hz higher than that achieved in an -band VCO using a 0.6- m GaAs MESFET [19]. It can be expected that further improvement in the material quality for suppressing the surface states can lead to competitive noise performance in AlGaN/GaN HEMT VCOs. The output power at the fundamental resonance and the oscillation frequency at a different tuning voltage are plotted in was measured to be 222 MHz/V. Fig. 9. The tuning gain Compared to the previously reported AlGaN/GaN HEMT VCOs [11], the VCO reported here features a smaller active HEMT, a lower supply voltage of 5 V, and eventually lower output power. It should be pointed out that while high-power AlGaN/GaN HEMT VCOs takes advantage of the power-handling capability of AlGaN/GaN HEMTs, low-power VCOs also have their own suitable applications. For example, for a single-chip AlGaN/GaN HEMT transceiver, the low-power VCOs can be used in the receiver front-end. In addition, compared to high-power VCOs, low-power VCOs integrated with amplifier buffers can provide lower frequency pulling.

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The sensitivity of the oscillation frequency upon the load impedance variation is characterized by a pulling figure measurement. The change of oscillation frequency is measured as a function of the phase of the load reflection coefficient for a constant return loss of 12 dB, corresponding to a constant magnitude of 0.25 for the load reflection coefficient. The measurement was carried out using Maury Microwave’s 982B01 load–pull system, and the results are plotted in Fig. 12. The maximum frequency shift from the frequency in the “matched load” situation is 5 MHz. Fig. 10. Oscillation frequency and the phase noise (at 100-kHz and 1-MHz offset, respectively) versus varactor control voltage.

V. CONCLUSION A MMIC -band VCO using Al Ga N Al Ga N/GaN composite-channel HEMTs was designed, fabricated, and characterized. The MMIC features monolithically integrated passive components including planar inductors, MSM varactors, and MIM capacitors. The VCO exhibits phase noise of 82- and 110-dBc/Hz at offsets of 100 kHz and 1 MHz. In addition, the VCO also exhibits a maximum second harmonic suppression of 47 dBc. The low phase noise and high harmonic suppression of the CC-HEMT-based VCOs are attributed to the inherent high linearity and low noise of the CC-HEMT. REFERENCES

Fig. 11. Measured output spectrum of fundament and second harmonic at V = 5 V, V = 3 V, and V = 5 V.

0

Fig. 12. Pulling-figure measurement of the AlGaN/GaN CC-HEMT VCO. V = 5 V, V = 3 V, V = 5 V. The return loss of the load is set to be 12 dB, corresponding to 0 = 0:25:

0

0

j j

0

The phase noise at 100-kHz and 1-MHz offset is characterized within the entire tuning range, as shown in Fig. 10. It is also observed that the second harmonic suppression of 47 dBc was obtained within the entire tuning range. One such example is shown in Fig. 11. To our best knowledge, this is the highest second harmonic suppression reported in VCOs based on GaN HEMTs. The high harmonic suppression and the low phase noise of the VCOs reported here can be attributed to the high linearity and low noise performances of the CC-HEMTs.

[1] K. Kasahara, N. Miyamoto, Y. Ando, Y. Okamoto, T. Nakayama, and M. Kuzuhara, “Ka-band 2.3 W power AlGaN/GaN heterojunction FET,” in Int. Electron Devices Tech. Dig., Dec. 2002, pp. 667–680. [2] K. Joshin, T. Kikkawa, H. Hayashi, S. Yokogawa, M. Yokoyama, N. Adachi, and M. Takikawa, “A 174 W high-efficiency GaN HEMT power amplifier for W-CDMA base station applications,” in Int. Electron Devices Tech. Dig., Dec. 2003, pp. 983–985. [3] Y. F. Wu, A. Saxler, M. Moore, R. P. Smith, S. Sheppard, P. M. Chavarkar, T. Wisleder, U. K. Mishra, and P. Parikh, “30-W/mm GaN HEMTs by field plate optimization,” IEEE Electron Device Lett., vol. 25, no. 3, pp. 117–119, Mar. 2004. [4] J.-W. Lee, L. F. Eastman, and K. J. Webb, “A gallium–nitride push–pull microwave power amplifier,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 11, pp. 2243–2249, Nov. 2003. [5] Y. Chung, C. Y. Hang, S. Cai, Y. Chen, W. Lee, C. P. Wen, K. L. Wang, and T. Itoh, “Effects of output harmonic termination on PAE and output power of AlGaN/GaN HEMT power amplifier,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 11, pp. 421–423, Nov. 2002. [6] W. Lu, V. Kumar, R. Schwindt, E. Piner, and I. Adesida, “DC, RF, and microwave noise performances of AlGaN/GaN HEMTs on sapphire substrates,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 11, pp. 2499–2504, Nov. 2002. [7] J.-W. Lee, A. Kuliev, V. Kumar, R. Schwindt, and I. Adesida, “Microwave noise characteristics of AlGaN/GaN HEMTs on SiC substrates for broadband low-noise amplifiers,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 259–261, Jun. 2004. [8] A. Minko, V. Hoel, S. Lepolliet, G. Dambrine, J. C. De Jaeger, Y. Cordier, F. Semond, F. Natali, and J. Massies, “High microwave and noise performance of 0.17-m AlGaN-GaN HEMTs on high-resistivity silicon substrates,” IEEE Electron Device Lett., vol. 25, no. 4, pp. 167–169, Apr. 2004. [9] Y. Ando, A. Wakejima, Y. Okamoto, T. Nakayama, K. Ota, K. Yamanoguchi, Y. Murase, K. Kasahara, K. Matsunaga, T. Inoue, and H. Miyamoto, “Novel AlGaN/GaN dual-field-plate FET with high gain, increased linearity and stability,” in IEEE Int. Electron Devices Meeting Tech. Dig., 2005, pp. 576–579. [10] H. Xu, C. Sanabria, A. Chini, S. Keller, U. K. Mishra, and R. A. York, “A C -band high-dynamic range GaN HEMT low-noise amplifier,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 262–264, Jun. 2004. [11] V. S. Kaper, R. M. Thompson, T. R. Prunty, and J. R. Shealy, “Signal generation, control, and frequency conversion AlGaN/GaN HEMT MMICs,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 55–65, Jan. 2005.

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[12] C. Sanabria, H. Xu, S. Heikman, U. K. Mishra, and R. A. York, “A GaN differential oscillator with improved harmonic performance,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 7, pp. 463–465, Jul. 2005. [13] J. Liu, Y. G. Zhou, R. M. Chu, Y. Cai, K. J. Chen, and K. M. Lau, “Highly linear Al Ga N/Al Ga N/GaN composite-channel HEMTs,” IEEE Electron Device Lett., vol. 26, no. 3, pp. 145–147, Mar. 2005. [14] Z. Q. Cheng, J. Liu, Y. G. Zhou, Y. Cai, K. J. Chen, and K. M. Lau, “Broadband microwave noise characteristics of high-linearity Ga N/GaN HEMTs,” composite-channel Al GaN N/Al IEEE Electron Device Lett., vol. 26, no. 8, pp. 521–523, Aug. 2005. [15] C. S. Chu, Y. G. Zhou, K. J. Chen, and K. M. Lau, “ -factor characterization of radio-frequency GaN-based metal–semiconductor–metal planar inter-digitated varactors,” IEEE Electron Device Lett., vol. 26, no. 7, pp. 432–434, Jul. 2005. [16] B. Razavi, RF Microelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1998, p. 223. [17] V. S. Kaper, V. Tilak, H. Kim, A. V. Vertiatchikh, R. M. Thompson, T. R. Prunty, L. F. Eastman, and J. R. Shealy, “High-power monolithic AlGaN/GaN HEMT oscillator,” IEEE J. Solid-State Circuits, vol. 38, no. 9, pp. 1457–1461, Sep. 2003. [18] A. Balandin, S. V. Morozov, S. Cai, R. Li, K. L. Wang, G. Wijeratne, and C. R. Viswanathan, “Low flicker-noise GaN/AlGaN heterostructure field-effect transistors for microwave communications,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1413–1417, Aug. 1999. [19] C.-H. Lee, S. Han, B. Matinpour, and J. Lasker, “A low phase noise -band MMIC GaAs MESFET VCO,” IEEE Microw. Guided Wave Lett., vol. 10, no. 8, pp. 325–327, Sep. 2000.

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Zhiqun Q. Cheng received the B.S. and M.S. degrees in microelectronics from Hefei University of Technology, Hefei, China, in 1986 and 1995, respectively, and the Ph.D. degree in microelectronics and solid-state electronics from the Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai, China, in 2000. He is currently a Professor with the Department of Information Engineering, Hangzhou Dianzi University, Hangzhou, China, and a Senior Visiting Scholar with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology (HKUST), Hong Kong. Prior to joining HKUST, he was a Lecturer with the Hefei University of Technology, Hefei, China (1986–1997) and an Associate Professor with the Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai, China (2000–2005). He has carried out research on GaAs HBTs and GaN HEMTs and has been involved with the design of compound semiconductor digital integrated circuits and RF transceivers. He has authored or coauthored 50 technical papers in journals and conference proceedings. His current interests focus on III–V high-power and low-noise devices and circuits for microwave and millimeter applications and high-speed Si and SiGe devices and T/R systems for wireless communications.

Yong Cai was born in Nanjing, Jiangsu Province, China, in 1971. He received the B.S. degree in electronics engineering from Southeast University, Nanjing, China, in 1993, and the Ph.D. degree from the Institute of Microelectronics, Peking University, Beijing, China, in 2003. From 2003 and 2006, he was a Post-Doctoral Research Associate with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, where he was involved with wide-bandgap GaN-based devices and circuits. In August 2006, he joined the Material and Packing Technologies Group, Hong Kong Applied Science and Technology Research Institute (ASTRI) Company Ltd., Hong Kong, where he is a Senior Engineer.

Jie Liu received the B.S. and M.S. degrees in physics from Nanjing University, Nanjing, China, in 2000 and 2003, respectively, and the Ph.D. degree in electrical and computer engineering from the Hong Kong University of Science and Technology (HKUST), Hong Kong, in 2006.. His master’s thesis concerns the Schottky contacts of III-nitride devices. In August 2003, he joined the Department of Electronic and Computer Engineering, HKUST, where he is involved with device technologies and high-frequency characterization techniques of III-nitride HEMTs. In particular, he has focused on the channel engineering of III-nitride HEMTs and developed highly linear composite-channel HEMT and low-leakage current AlGaN/GaN/InGaN/GaN double-heterojunction HEMT. In October 2006, he joined the Hong Kong Applied Science and Technology Research Institute (ASTRI) Company Ltd., Hong Kong, where he is involved with advanced wireless packaging technologies.

Yugang Zhou was born in Hubei Province, China, in 1975. He received the B.S. and Ph.D. degrees in physics from Nanjing University, Nanjing, China, in 1996 and 2001, respectively. From September 2001 to September 2004, he was a Post-Doctoral Research Associate with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology. In September 2004, he joined Advanced Packaging Technology Ltd., Hong Kong. He was mainly involved with MOCVD growth, device fabrication, and device physics of GaN-based heterostructure field-effect transistors (HFETs) prior to September 2004. Since that time, he has been focused on the fabrication of GaN-based high-power LEDs.

Kei May Lau (S’78–M’80–SM’92–F’01) received the B.S. and M.S. degrees in physics from the University of Minnesota at Minneapolis–St. Paul, in 1976 and 1977, respectively, and the Ph.D. degree in electrical engineering from Rice University, Houston, TX, in 1981. From 1980 to 1982, she was a Senior Engineer with M/A-COM Gallium Arsenide Products Inc., where she was involved with epitaxial growth of GaAs for microwave devices, development of high-efficiency and millimeter-wave IMPATT diodes, and multiwafer epitaxy by the chloride transport process. In Fall 1982, she joined the faculty of the Electrical and Computer Engineering Department, University of Massachusetts at Amherst, where, in 1993, she became a Full Professor. She initiated metalorganic chemical vapor deposition (MOCVD) and compound semiconductor materials and devices programs at the University of Massachusetts at Amherst. Her research group has performed studies on heterostructures, quantum wells, strained-layers, and III–V selective epitaxy, as well as high-frequency and photonic devices. In 1989, she spent her first sabbatical leave with the Massachusetts Institute of Technology (MIT) Lincoln Laboratory. From 1995 to 1006, she developed acoustic sensors with the DuPont Central Research and Development Laboratory, Wilmington, DE, during her second sabbatical leave. In Fall 1998, she was a Visiting Professor with the Hong Kong University of Science and Technology (HKUST), Hong Kong, where, in Summer 2000, she joined the regular faculty. She established the Photonics Technology Center for research and development efforts in wide-gap semiconductor materials and devices. In July 2005, she became a Chair Professor of electronic and computer engineering with HKUST. Prof. Lau served on the IEEE Electron Devices Society Administrative Committee and was an editor for the IEEE TRANSACTIONS ON ELECTRON DEVICES (1996–2002). She also served on the Electronic Materials Committee of the Minerals, Metals and Materials Society (TMS) of AIME (American Institute of Materials Engineers). She was a recipient of the National Science Foundation (NSF) Faculty Award for Women (FAW) Scientists and Engineers in the U.S.

CHENG et al.: LOW PHASE-NOISE

-BAND MMIC VCO

Kevin J. Chen (M’96–SM’06) received the B.S. degree in electronics from Peking University, Beijing, China, in 1988, and the Ph.D. degree from the University of Maryland at College Park, in 1993. From January 1994 to December 1995, he was a Research Fellow with NTT LSI Laboratories, Atsugi, Japan, where he was engaged in the research and development of functional quantum effect devices and heterojunction field-effect transistors (HFETs). In particular, he developed device technologies for monolithic integration of resonant tunneling diodes and HFETs on both GaAs and InP substrates for applications in ultrahigh-speed signal processing and communication systems. He also developed the Pt-based buried gate technology that is widely used in enhancement-mode HEMT and pseudomorphic high electron-mobility transistor (pHEMT) devices. From 1996 to 1998, he was an Assistant Professor with the Department of Electronic Engineering, City University of Hong Kong, where he carried out research

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on high-speed device and circuit simulations. In 1999, he joined the Wireless Semiconductor Division, Agilent Technologies, Santa Clara, CA, where he was involved with enhancement-mode pHEMT RF power amplifiers used in dual-band global system for mobile communication (GSM)/digital communication system (DCS) wireless handsets. His work with Agilent Technologies has covered RF characterization and modeling of microwave transistors, RF integrated circuits (ICs), and package design. In November 2000, he joined the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology (HKUST), Hong Kong, as an Assistant Professor and, in 2006, became an Associate Professor. He has authored or coauthored over 140 publications in international journals and conference proceedings. With HKUST, his group has carried out research on novel III-nitride device technologies, III-nitride and silicon-based microelectromechanical systems (MEMS), silicon-based RF/microwave passive components, RF packing technology, and microwave filter design.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

Ultra-Compact High-Linearity High-Power Fully Integrated DC–20-GHz 0.18-m CMOS T/R Switch Yalin Jin and Cam Nguyen, Fellow, IEEE

Abstract—A fully integrated ultra-broadband transmit/receive (T/R) switch has been developed using nMOS transistors with a deep n-well in a standard 0.18- m CMOS process, and demonstrates unprecedented insertion loss, isolation, power handling, and linearity. The new CMOS T/R switch exploits patterned-ground-shield on-chip inductors together with MOSFET’s parasitic capacitances to synthesize artificial transmission lines, which result in low insertion loss over an extremely wide bandwidth. Negative bias to the bulk or positive bias to the drain of the MOSFET devices with floating bulk is used to reduce effects of the parasitic diodes, leading to enhanced linearity and power handling for the switch. Within dc–10, 10–18, and 18–20 GHz, the developed CMOS T/R switch exhibits insertion loss of less than 0.7, 1.0, and 2.5 dB and isolation between 32–60, 25–32, and 25–27 dB, respectively. The measured 1-dB power compression point and input third-order intercept point reach as high as 26.2 and 41 dBm, respectively. The new CMOS T/R switch has a die area of only 250 m. The achieved ultra-broadband performance 230 m and high power-handling capability, approaching those achieved in GaAs-based T/R switches, along with the full-integration ability confirm the usefulness of switches in CMOS technology, and demonstrate their great potential for many broadband CMOS radar and communication applications. Index Terms—Broadband communications, broadband radar, CMOS switch, CMOS transmit/receive (T/R) switch, linearity, on-chip inductor, power handling, RF integrated circuit (RF IC), ultra-wideband (UWB) communications, UWB radar.

I. INTRODUCTION

T

RANSMIT/RECEIVE (T/R) switches are one of the key building blocks in radar and communication systems and their ability to fully integrate with other circuits and operate over very wide bandwidths is needed to enable wideband systems on chip. Most high-performance RF integrated-circuit (RF IC) switches have been implemented in GaAs processes, especially those having bandwidths of multigigahertz and high power-handling capabilities [1]. Silicon-based CMOS technology has fast become one of the most favorable processes for RF ICs due to its low cost and highly integrative capacity.

Manuscript received June 6, 2006; revised September 27, 2006. This work was supported in part by the National Science Foundation and in part by the U.S. Army Corps of Engineers. The authors are with the Electrical and Computer Engineering Department, Texas A&M University, College Station, TX 77843-3128 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/TMTT.2006.888944

Owing to low mobility, high substrate conductivity, low breakthrough voltage, and various parasitic parameters of CMOS processes, it is very challenging to design CMOS switches to achieve low-insertion loss, high isolation, wide bandwidth, and high power handling comparable to their GaAs counterparts [2]. Various CMOS T/R switches have been developed at different frequencies within 900 MHz to 15 GHz [2]–[9]. Fully integrated CMOS T/R switches operating over extremely wide bandwidths up to tens of gigahertz with high linearity and power handling have not yet been reported. As the bandwidths of radar and communication systems are pushed wider or required to cover multibands to address newly emerging applications, the need of these ultra-wideband (UWB) CMOS T/R switches becomes more critical. In CMOS switches, the parasitic capacitances in MOSFETs limit the upper operating frequency and, hence, bandwidth. A 0.25- m CMOS T/R switch operating from 0.45 MHz to 13 GHz has been developed based on synthetic transmission lines implemented using an on-chip coplanar waveguide (CPW) together with the MOSFETs’ capacitances [7]. This T/R switch, however, exhibits insertion loss not as good as its GaAs counterparts due to the high loss of CPW realized on the conductive silicon substrate and the loss associated with the MOSFETs. CMOS switches normally have low power-handling capability as compared to their GaAs counterparts due to low breakthrough voltages and parasitic diodes existing underneath the drain and source of the MOSFET structure [3], [4]. Floating bulk was used in [4], [5], and [9] to keep the parasitic diodes from being forward biased under large input signals, hence, improving the linearity and power handling of CMOS T/R switches. An input 1-dB power compression , of 21 dBm was achieved with a series-shunt point, i.e., topology by resistively floating the bodies of the transistors [5], was obtained using a series MOSFET [9]. A 28-dBm with the bulk floated by an LC tuned network for narrowband applications [4]. The power handling was also improved using an impedance transformer network (ITN) implemented using an external [3] and on-chip [6] LC matching networks. The LC matching network, however, limits the switch’s operating bandwidth due to its relatively strong frequency dependence. Moreover, the ITN causes relatively high insertion loss even though external high- components are used [3]. In this paper, we report on the development of a UWB fully integrated T/R switch, fabricated on a commercial standard 0.18- m CMOS process, with unprecedented performance. The new T/R switch employs an ultra-broadband topology

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Fig. 2. Simplified small-signal equivalent-circuit models for MOSFETs with the bulk and gate floated under: (a) on-state and (b) off-state. “D” and “S” denote the drain and source, respectively.

Fig. 1. Simplified physical structure of nMOS transistors with DNW under on-state.

based on the synthetic transmission-line concept with on-chip spiral inductors. Simultaneously floating and applying negative bias to the bulk or positive bias to the drain are implemented to enhance the linearity and power handling of the switch. The developed CMOS T/R switch exhibits an insertion loss lower than 1 dB from dc to 18 GHz and less than 2.5 dB up to 20 GHz. The measured isolation is between 32–60, 25–32, varies between 25.4–26.2, 22.6–25.4, and 25–27 dB and and 19.8–22.6 dBm from dc–10, 10–18, and 18–20 GHz, respectively. The measured input third-order intercept point (IIP3) reaches as high as 41 dBm. The switch occupies a very 250 m. small die area of 230 m II. CMOS DEVICE AND BROADBAND ENHANCED POWER-LINEARITY T/R SWITCH TOPOLOGIES A. CMOS Device With Deep n-Well and Floating Bulk The developed CMOS T/R switch is implemented using nMOS transistors with deep n-type well (DNW). Fig. 1 shows a simplified geometry of these nMOS transistors with the gate biased so that the devices are operated under the on-state. The DNW separates the bulk of the nMOS transistors from the p-substrate. The p-n junctions between the p-bulk and n regions form a pair of parasitic drain-bulk (or drain) and source-bulk (or source) diodes. With DNW, large resistors can be applied directly to the bulk of nMOS devices, making it floating at high frequencies without latch-up problems that would result in RF ICs consisting of both nMOS and pMOS without DNW. Using DNW thus allows the CMOS T/R switch to be fully integrated with other RF ICs designed using both nMOS and pMOS transistors in a single chip. Floating the bulk forces, the bulk resistances underneath the source and drain junctions open with respect to the ground, leading to a much smaller resistive loss in the conductive p-bulk than with the bulk grounded. Fig. 2 shows simplified smallsignal equivalent-circuit models for the on- and off-states of floating-bulk MOSFETs when the gate is floated with a large rerepresents the resistive loss in the p-bulk between the sistor. source and drain. Using large gatewidths for 0.18- m CMOS within several ohms, thereby resulting devices can produce and represent the gate–source in low loss in the bulk. and gate–drain capacitances due to the overlapping between represents the gate–bulk cathe gate and diffusion areas.

and are the junction capacitances between pacitance. and in the drain–bulk and source–bulk, respectively. Fig. 2(a) represent the distributed capacitances between the inin Fig. 2(b) is the source–drain difversion layer and bulk. fusion capacitance in a multifinger MOSFET. All these parasitic capacitances are on the order of tens of femtofarads for 0.18- m CMOS devices and increase with the device’s gatewidth. B. CMOS T/R Switch Topologies The series-shunt T/R switch with two identical arms is perhaps the most commonly used topology. In this topology, the series and shunt MOSFETs dominate the insertion loss and isolation, respectively. Low insertion loss and high isolation may be achieved by using properly compromised large devices due to their small on-state resistances, which are scaled down approximately as L/W with “L” and “W” denoting the gate length and gatewidth, respectively, in advanced submicrometer CMOS processes. Large devices, however, have significant parasitic capacitances, causing considerable effects to circuit matching and eventually limiting the switch’s bandwidth—especially in the high-frequency regions. These parasitic capacitances are more pronounced in submicrometer CMOS processes. In practical switching circuits, the effects of parasitic capacitances are much more severe than the on-resistances in large submicrometer CMOS devices, particularly at high frequencies. Synthetic transmission lines can be used to alleviate the bandwidth limitation in CMOS T/R switches. A synthetic transmission line can be created by cascading multiple sections of an identical series inductor and shunt capacitor, whose inductance and capacitance can be properly chosen to realize a particular characteristic impedance and velocity. Such a synthetic transmission line approximates a transmission line over a finite bandand cutoff frequency width. The characteristic impedance of an ideal lossless synthetic transmission line can be approximated as (1) and (2) where and represent the inductance and capacitance of the series inductor and shunt capacitor, respectively. The cutoff frequency is thus determined, for given characteristic impedance, by the shunt capacitance.

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Fig. 4. Small-signal equivalent-circuit models of the MOSFETs under: (a) on and (b) off conditions, (c) the T/R switch with ANT-RX on and ANT-TX off, and (d) the ANT-RX on-path.

Fig. 3. Schematics of the CMOS T/R switch implemented using synthetic transmission lines with: (a) negative bias to the bulk and (b) positive bias to the drain of the shunt nMOS devices. ANT, TX, and RX denote the (input) antenna, (output) transmitter, and (output) receiver ports, respectively.

Fig. 3 shows the topologies of the developed CMOS T/R switch. These topologies are identical, except for different bias schemes for the shunt transistors [bulk bias in Fig. 3(a) and drain bias in Fig. 3(b)], used to enhance the switch’s linearity and power handling, which will be discussed in Section II-C, and can be implemented using the same switch. Synthetic and ) and transmission lines, realized using two series ( and ) nMOS transistors, and three series two shunt ( on-chip spiral inductors ( , , and ) are used to achieve , , , a very wide bandwidth. The bias resistors ( , , and ) have large resistances in order to isolate the dc-bias voltages from the RF signals. Large bulk resistors , , , and ) are used to make the transistors ( floating at high frequencies to reduce the substrate loss and increase the switch’s linearity and power-handling capability. Large series and shunt nMOS transistors with gatewidths in the order of several hundred micrometers are used to obtain small on-resistances and, hence, low insertion loss and high isolation for the switch, besides enhancing the linearity and power-handling capability. These devices, although much larger than those used in recently published CMOS T/R switches [5], [6], [8], and [9], still result in extremely wide bandwidth due to the use of the synthetic transmission-line technique. Fig. 4(a) and (b) shows the simplified small-signal equivalent-circuit models of the MOSFETs under on and off conditions deduced from Fig. 2(a) and (b), respectively. The in parallel with the on-model includes the on-resistance , which represents the total capacitance on-capacitance seen at the gate consisting of , , and in Fig. 2(a), consisting of , , the on-state bulk capacitance , and in Fig. 2(a), and the drain–source resistance . The off-model is represented by the off-capacitor , which consists of the total capacitance seen at the gate

consisting of , , and in Fig. 2(b), the off-state bulk consisting of and in Fig. 2(b), the capacitance , and the drain–source resistance drain–source capacitance . Fig. 4(c) shows the small-signal equivalent circuit of the T/R switch assuming the ANT-RX and ANT-TX paths are has a on and off, respectively, considering the fact that below 20 GHz. relatively large impedance as compared to As can be seen, the isolation between the ANT and TX ports is of the shunt mainly determined by the on-resistance . Fig. 4(d) shows the small-signal equivalent MOSFET circuit of the on-path between the ANT and RX ports, where represents the combined off-capacitances of and of , and . C. Linearity and Power-Handling Enhancement Typical CMOS switches have poor linearity and power handling, primarily due to the resultant forward bias of the drain and source parasitic diodes under operation, even with small transient voltage swings. In order to overcome the forward-bias problem and, hence, increasing the switch’s linearity and powerhandling capability, the bulk can be floated and negatively biased simultaneously, as shown in Fig. 3(a). Due to small on-resistance of the series MOSFETs, the source and drain are kept approximately at the same potential under the on-state. The parasitic drain and source diodes are thus always kept reverse biased even when there are strong voltage swings at the drain and source, respectively. However, because the sources of the shunt MOSFETs are grounded, a negative voltage swing on the drain can push the two back-to-back parasitic diodes into a forward-bias region. Consequently, the voltage on the drain is clamped to a certain value in the negative region by these forward-biased diodes, leading to distortion in the output signal and, consequently, degrading the insertion loss, linearity, and power handling. Fig. 5 illustrates three conditions for a shunt MOSFET, which result in different linearity and power-handling capabilities. In Fig. 5(a), the bulk of the MOSFET is grounded directly. When , where the RF voltage swing at the drain is lower than is the forward pinch-on voltage of the parasitic diodes, the

JIN AND NGUYEN: ULTRA-COMPACT HIGH-LINEARITY HIGH-POWER FULLY INTEGRATED DC–20 GHz 0.18- m CMOS T/R SWITCH

Fig. 6. Calculated

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Q of on-chip spiral inductors with and without PGS.

TABLE I SUMMARY OF THE CMOS T/R SWITCH’S DESIGNED PARAMETERS

Fig. 5. Simplified large-signal models for a shunt MOSFET with bulk: (a) grounded, (b) floated, and (c) negatively biased for different linearity and power-handling conditions. The sinusoidal signal is at the drain.

back-to-back parasitic diodes are all forward biased and, consequently, the MOSFET functions as a small forward-biased re. The output voltage of sistor in parallel with the capacitor . Fig. 5(b) the switch is then approximately clamped to shows the MOSFET with the bulk floated through a grounded at the drain would push resistor. A voltage swing reaching the drain parasitic diode forward biased, but the large resistor at the bulk keeps a high impedance between the drain and ground and the source parasitic diode reverse biased, thus improving the power-handling capability. It was reported that the 1-dB comis improved by 2 dB using this technique pression point [5]. Fig. 5(c) demonstrates a new technique to further improve linearity and power handling. It shows the floating bulk of the shunt MOSFET is biased using a negative dc voltage via the bulk . This technique is implemented in the topology bias resistor , shown in Fig. 3(a). Since there is no current flow through the dc potential of the bulk node is kept the same as the negative bias. Therefore, the source parasitic diode is always in the reverse bias and the drain parasitic diode can withstand a strong . Using the negative negative voltage swing to bulk-bias technique can, therefore, lead to larger power handling for CMOS switches than the other two methods shown in Fig. 5(a) and (b). It is noted that, due to no output current required for the negative voltage source, a negative voltage reference can be implemented in fully integrated systems without any noise and stability issues. The linearity and power-handling capability can also be improved by generating a positive dc potential between the drain and bulk of the shunt MOSFETs. This is achieved by applying a positive bias to the drain of the shunt MOSFETs and grounding the bulk resistors, as seen in the topology shown in Fig. 3(b). This positive-drain bias technique is especially attractive when the RF signal entering the receiver or leaving the transmitter port has a positive dc offset because no dc blocks would be needed, thus making it very useful for integration with other RF ICs in a single chip. Using the same CMOS T/R switch, both the nega-

tive-bulk and positive-drain bias techniques give the same measured results for linearity and power handling. III. CMOS SWITCH DESIGN AND FABRICATION The CMOS T/R switch was designed and fabricated using the TSMC 0.18- m CMOS triple-well process [10] with nMOS transistors and on-chip spiral inductors. On-chip inductors in silicon-based RF ICs contribute considerable insertion loss and size because of their limited quality ( ) factor and relatively large size. To achieve a very low insertion loss for the switch, the total resistance of the switch’s on-path, consisting of both on-resistances of the MOSFETs and self-resistances of the on-chip inductors, needs to be designed to be as small as possible. These on-chip inductors were designed using patterned ground shields (PGSs) implemented on the polysilicon layer [11]. PGS implemented on a polysilicon layer gives higher than that using a metal layer due to its relatively low conductivity, which results in less eddy currents. The PGS underneath each spiral prevents or partly prevents the electric fields generated by the current flowing along the spiral from penetrating into the lossy silicon substrate. This not only results in significantly reduced coupling between on-chip inductors through the substrate, but also electrical loss due to the substrate, particularly at high frequencies [12]. On-chip spiral inductors with PGS can, therefore, have high and be located close to each other while keeping sufficient isolation between them, effectively resulting in very low insertion loss and small die area for the switch.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

Fig. 7. Microphotograph of the CMOS T/R switch. The die area without pads is 0.06 mm .

Fig. 9. (a) Measured IIP3 at 5.8 GHz and (b) P bias voltage.

Fig. 8. Measured and calculated: (a) insertion loss and isolation and (b) return loss at the antenna (S ) and TX/RX (S ) ports.

Fig. 6 compares the of the designed 1.5-turn spirals with and without PGS calculated using the electromagnetic was determined using (EM) simulator IE3D [13]. The , where is the input impedance of the lumped-element p equivalent-circuit model of the spiral of the PGS with one port grounded. As can be seen, the inductor is improved up to approximately 37 GHz with a maximum around 20 GHz. The PGS spiral has approximately 0.85-O series resistance below 5 GHz and about 1.67 O at 20 GHz. Several remarks need to be made at this point concerning the CMOS T/R switch design. As implied in (2), small transistors

with 0- and

01.8-V bulk

should be employed to achieve a wide bandwidth for the switch. On the other hand, large series and shunt devices are needed to produce low insertion loss and high isolation, respectively. A tradeoff is thus needed not only in the switch topology, but also in the device size in order to achieve simultaneously an ultra-wide bandwidth along with low insertion loss and high isolation. Particularly for the selected T/R switch topology, the inductance and of the on-chip spirals versus frequency should , the combined off-capacibe optimized together with tances of the series and shunt devices. The spiral inductance and dictate the bandwidth, while the affects the insertion loss. Table I lists the parameters of the designed CMOS T/R switch. All the bias resistors are realized on the polysilicon is aplayer to achieve small layouts. The on-resistance for the employed series and shunt proximately 4 and 11 n field-effect transistor (nFET), respectively. The combined off-capacitance of the series and shunt nFETs is approximately 280 fF. Fig. 7 shows a micrograph of the CMOS T/R switch, including on-wafer RF and dc probe pads, with the TX port terminated by an on-chip 50- resistor. The actual area of the switch is measured only 230 250 m , with the inductors occupying approximately 60% of the chip area. IV. CMOS T/R SWITCH PERFORMANCE Measurements were conducted on-wafer using a probe station, vector network analyzer, and frequency synthesizers. Cali-

JIN AND NGUYEN: ULTRA-COMPACT HIGH-LINEARITY HIGH-POWER FULLY INTEGRATED DC–20 GHz 0.18- m CMOS T/R SWITCH

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TABLE II SUMMARY OF CMOS T/R SWITCHES

bration was performed using a full two-port calibration process and the calibration standards built on the same chip. These calibration circuits have the same RF pads as those in the switch to allow accurate deembedding of the pads. The control and bias voltages were applied directly through dc probes. A.

-Parameter Measurement

Fig. 8 shows the measured and calculated insertion loss, isolation, and return loss of the developed CMOS T/R switch. It exhibits insertion losses of less than 0.7, 1, and 2.5 dB from dc to 10, 10 to 18, and 18 to 20 GHz, respectively. The measured isolation varies from 32 to 60, 25 to 32, and 25 to 27 dB between dc–10, 10–18, and 18–20 GHz, respectively. The measured return losses at the antenna and receiver/transmitter ports are better than 15 and 10 dB from dc to 10 GHz and 10 to 18 GHz, respectively. These results are the same for the three different bias conditions listed in Table I. As can be seen in Fig. 8(a), the measured and calculated insertion losses agree very well to approximately 18 GHz, beyond which the measured insertion loss significantly drops below the simulated result. This discrepancy above 18 GHz has been observed in several chips and is mainly caused by the inaccuracy of the transistors’ available model. We believe that this inaccuracy is due to the parameters of the model, which characterize the silicon substrate loss. The employed transistor model also leads to the discrepancy between the measured and simulated isolation and return loss seen in Fig. 8. B. IIP3 and

Measurement

Fig. 9 shows the measured IIP3 versus bulk bias versus voltage at 5.8 GHz and 1-dB power compression frequency. Since the available large-signal model for the transistors lacks the accuracy when the nFETs are operated at zero source to drain potential with strong voltage swings at the drain, only the measured linearity and power-handling performance are presented. As can be seen in Fig. 9(a), IIP3 can reach as much as 41 dBm by increasing the negative bias voltage. The measurement was performed without bulk bias (bulk bias resistor is grounded) and with bulk bias at 1.8 V. The

with the bulk biased is more than 25 and 20 dBm from 1 to 11.5 and 11.5 to 19 GHz, respectively, reaching 26.2 dBm at around 4 GHz. It is seen that by applying a negative bias to the bulk, the power-handling capability of the switch is increased by approxwere also imately 4 dB. Similar performance for IIP3 and obtained when the drain was biased with a positive voltage, as discussed in Section II-C. C. Comparison to Other CMOS T/R Switches Table II compares the performance of the developed CMOS T/R switch with those recently published. The new switch demonstrates the widest bandwidth, highest frequency, lowest insertion loss, highest isolation, and strongest power handling among the integrated series-shunt CMOS switches [2], [5], [6], [8], and [9]. It also shows wider bandwidth, higher frequency, lower insertion loss, and higher isolation as compared to the series CMOS switch [4]. Moreover, the measured IIP3 is highest among those reported [2], [3], [6], [8], [9]. The developed switch is also the first fully integrated CMOS T/R switch providing very low insertion loss (below 0.7 dB) for UWB communication systems covering 3.1–10.6 GHz. V. CONCLUSION A new fully integrated 0.18- m CMOS T/R switch has been developed with unprecedented performance. The developed CMOS T/R switch represents the first CMOS switch implementing on-chip spiral inductors with PGSs to simulate transmission lines for ultra-broadband performance with miniaturization and simultaneously floating the bulk with negative bulk or positive drain bias to achieve high linearity and power-handling capability. These broadband and high-linearity/power-handling techniques can also be utilized for other kinds of CMOS switches. The developed CMOS T/R switch with a die area less than 0.06 mm , less than 0.7-dB insertion loss, and more than 20-dB return loss and 30-dB isolation from 3.1 to 10.6 GHz also particularly provides the best T/R switch to date for the emerging UWB wireless communication systems operating over that frequency range. The developed CMOS T/R switch with its full integration feature paves the way for full-integration into a complete CMOS system-on-chip.

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REFERENCES

2

[1] M. J. Schindler, M. E. Miller, and K. M. Simon, “DC–20 GHz N M passive switches,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 12, pp. 1604–1613, Dec. 1988. [2] F.-J. Huang and K. K. O, “A 0.5- m CMOS T/R switch for 900-MHz wireless applications,” IEEE J. Solid-State Circuits, vol. 36, no. 3, pp. 486–492, Mar. 2001. [3] ——, “Single-pole double-throw CMOS switches for 900-MHz and 2.4-GHz applications on p-silicon substrates,” IEEE J. Solid-State Circuits, vol. 39, no. 1, pp. 35–41, Jan. 2004. [4] N. Talwalkar, C. Yue, H. Guan, and S. Wong, “Integrated CMOS transmit-receive switch using LC-tuned substrate bias for 2.4-GHz and 5.2-GHz applications,” IEEE J. Solid-State Circuits, vol. 39, no. 6, pp. 863–870, Jun. 2004. [5] M.-C. Yeh, R.-C. Liu, Z.-M. Tsai, and H. Wang, “A miniature low-insertion-loss, high-power CMOS SPDT switch using floating-body technique for 2.4-and 5.8-GHz applications,” in IEEE RFIC Symp. Dig., 2005, pp. 451–454. [6] Z. Li and K. K. O, “15-GHz fully integrated nMOS switches in a 0.13m CMOS process,” IEEE J. Solid-State Circuits, vol. 40, no. 1, pp. 2323–2328, Nov. 2005. [7] Y. Jin and C. Nguyen, “A 0.25- m CMOS T/R switch for UWB wireless communications,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 8, pp. 502–504, Aug. 2005. [8] Z. Li, H. Yoon, F.-J. Huang, and K. K. O, “5.8-GHz CMOS T/R switches with high and low substrate resistances in a 0.18- m CMOS process,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 1, pp. 1–3, Jan. 2003. [9] M.-C. Yeh, Z.-M. Tsai, R.-C. Liu, K.-Y. Lin, Y.-T. Chang, and H. Wang, “Design and analysis for a miniature CMOS SPDT switch using body-floating technique to improve power performance,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 31–39, Jan. 2006. [10] TSMC 0.18- m CMOS process MOSIS Foundry. Marina del Rev, CA, 2005. [11] C. P. Yue and S. S. Wong, “On-chip spiral inductors with patterned ground shields for Si-based RF IC’s,” IEEE J. Solid-State Circuits, vol. 33, no. 5, pp. 743–752, May 1998. [12] J. N. Burghartz and B. Rejaei, “On the design of RF spiral inductors on silicon,” IEEE Trans. Electron Devices, vol. 50, no. 3, pp. 718–729, Mar. 2003. [13] IE3D. Zeland Software, Inc., Fremont, CA, 2005.

Yalin Jin was born in Baoding, China, in 1974. He received the B.S. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 1995, the M.S. degree in electrical engineering from the Chinese Academy of Sciences, Beijing, China, in 1998, and is currently working toward the Ph.D. degree in electrical engineering at Texas A&M University, College Station. From 1998 to 2001, he was with XinWei Telecom Inc., Beijing, China, where he was involved in the design of RF circuits for synchronous code division multiple access (SCDMA) base stations and mobile phones. From 2001 to 2002, he was a Senior Design Engineer with Motorola Electronics Inc., Beijing, China, where he was involved with RF circuits for global system for mobile communication (GSM) cellular phones. During Summer 2006, he interned as an RF IC Designer for multiband orthogonal frequency-division multiplexing (MB-OFDM) ultra-wideband (UWB) integrated transceivers in BiCMOS with Alereon Inc., Austin, TX. His research interests are high-speed ICs and high-frequency RFICs.

Cam Nguyen (F’05) joined the Department of Electrical and Computer Engineering, Texas A&M University, College Station, where he is currently the Texas Instruments Endowed Professor, in December 1990, after working for over 12 years in industry. From 2003 to 2004, he was Program Director with the National Science Foundation (NSF), where he was responsible for research programs in RF electronics and wireless technologies. From 1979 to 1990, he had various engineering positions in industry, including working as a Microwave Engineer with the ITT Gilfillan Company, a Member of Technical Staff with the Hughes Aircraft Company (now Raytheon), a Technical Specialist with the Aeroject ElectroSystems Company, a Member of Professional Staff with the Martin Marietta Company (now Lockheed-Martin), and a Senior Staff Engineer and Program Manager with TRW (now Northrop Grumman). While in industry, he led numerous microwave and millimeter-wave activities and developed many microwave and millimeter-wave hybrid and monolithic integrated circuits and systems up to 220 GHz for communications, radar, and remote sensing. His research group at Texas A&M University currently focuses on CMOS RFICs and systems, microwave and millimeter-wave ICs and systems, and ultra-wideband (UWB) devices and systems for wireless communications, radar, and sensing applications. He has authored over 165 refereed papers, one book, five book chapters, edited three books, and given over 85 conference presentations. He is the founding Editor-in-Chief of Sensing and Imaging: An International Journal, which is published by Springer.

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Analysis of the Survivability of GaN Low-Noise Amplifiers Matthias Rudolph, Senior Member, IEEE, Reza Behtash, Ralf Doerner, Member, IEEE, Klaus Hirche, Joachim Würfl, Member, IEEE, Wolfgang Heinrich, Senior Member, IEEE, and Günther Tränkle, Member, IEEE

Abstract—This paper presents a detailed analysis of the stressing mechanisms for highly rugged low-noise GaN monolithic-microwave integrated-circuit amplifiers operated at extremely high input powers. As an example, a low-noise amplifier (LNA) operating in the 3–7-GHz frequency band is used. A noise figure (NF) 1 8 dB below 2.3 dB is measured from 3.5 to 7 GHz with NF between 5–7 GHz. This device survived 33 dBm of available RF input power for 16 h without any change in low-noise performance. The stress mechanisms at high input powers are identified by systematic measurements of an LNA and a single high electronmobility transistor in the frequency and time domains. It is shown that the gate dc current, which occurs due to self-biasing, is the most critical factor regarding survivability. A series resistance in the gate dc feed can reduce this gate current by feedback, and may be used to improve LNA ruggedness. Index Terms—Amplifier noise, integrated-circuit noise, microwave field-effect transistor (FET) amplifiers, monolithic-microwave integrated-circuit (MMIC) amplifiers, noise, semiconductor device noise.

I. INTRODUCTION

I

T IS well known that GaN high electron-mobility transistors (HEMTs) are excellent candidates for high-power broadband applications in the microwave and millimeter-wave range. However, the power-handling capabilities also can be beneficial in case of low-noise amplifiers (LNAs), e.g., in order to realize extremely linear LNAs. Extracted third-order output intermodulation points (OIP3s) as high as 38 dBm [1] and 43 dBm [2] have been reported. On the other hand, highly rugged LNAs that survive high levels of overdrive input power for a certain time are also desirable for receiver front-ends in various applications. While GaAs-based LNAs typically require the available RF input power not to exceed approximately 20 dBm, it was reported that GaN-based LNAs survived measurements up to 23-dBm [3], 30-dBm [2], almost 31-dBm [4], 37-dBm [5], and 36-dBm continuous wave (CW) [3] and 46-dBm pulsed power

Manuscript received June, 30, 2006; revised August 31, 2006. This work was supported by the German Bundesministerium für Bildung und Forschung through the German Aerospace Center (DLR) under Contract 50 YB 0405. M. Rudolph, R. Doerner, J. Würfl, W. Heinrich, and G. Tränkle are with the Ferdinand-Braun-Institut für Höchstfrequenztechnik, D-12489 Berlin, Germany (e-mail: [email protected]). R. Behtash is with the Ferdinand-Braun-Institut für Höchstfrequenztechnik, D-12489 Berlin, Germany, and also with United Monolithic Semiconductors, D-89081 Ulm, Germany. K. Hirche is with Tesat-Spacecom GmbH & Co. KG, D-71522 Backnang, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2006.886907

[6]. These LNAs offer interesting possibilities for simplified receiver front-end concepts since, e.g., an input protection circuit can be omitted that is required when using conventional technology. While recently several papers presented data proving the ruggedness of GaN-HEMT-based LNAs [2]–[8], to the authors’ knowledge, no investigation has been published on the analysis of the stressing mechanisms when high input powers are applied and how to minimize them by optimized circuit design. In this paper, a detailed analysis based on systematic frequency- and time-domain measurements of HEMTs and LNAs is presented. We use a highly survivable 3–7 GHz GaN-based LNA as an example. This LNA has proven to survive available input powers of 33 dBm for 16 h without damage, and 36 dBm for several hours only lead to a slightly increased gate leakage current [7]. Among the factors causing stress to the device, such as total dissipated power, dc forward gate current, gate, or drain breakdown, we show that dc forward gate current poses the most severe threat to device lifetime, which is in contrast to conventional III–V technologies that are mainly limited by breakdown [9]. It is also shown how the device’s self-biasing under high input drive can be employed to reduce this current by a feedback resistor. Time-domain measurements are presented demonstrating that the high ruggedness of GaN-based LNAs results from the high gate breakdown voltage of these devices. It is thus possible to improve survivability of the LNA by reducing forward gate current, even though this measure causes increased reverse gate voltages. II. GAN FIELD-EFFECT TRANSISTOR (FET) TECHNOLOGY The AlGaN/GaN heterostructure used in this study is grown on a 2-in semi-insulating SiC substrate by metal–organic vapor phase epitaxy. The layer structure consists of a 2.3- m GaN buffer layer, a 3-nm Al Ga N spacer layer, a 12-nm 5 10 cm Si-doped Al Ga N supply layer, a 10-nm Al Ga N barrier layer, and a 5-nm GaN cap layer. An averaged sheet rewas measured using Van-der-Pauw strucsistance of 375 tures on the passivated wafer. The monolithic microwave integrated circuits (MMICs) were realized using the 2-in stepper technology at the Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH), Berlin, Germany, with the following process steps: after the lithography of the ohm level, Ti/Al/Ti/Au was evaporated and capped with a sputtered layer of WSiN as ohmic metallization. Annealing of the contacts at 830 C results in an ohmic contact resistance of 0.4 mm. The mesa structures were realized by reactive ion etching using an Ar/BCl /Cl plasma. 0.4- m electron-beam

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Fig. 1. Circuit schematic of LNA.

Fig. 4. Measurement of LNA, for V = 8 V for both stages, and I = 16 and 24 mA for the first and second stages, respectively. 50- NF and minimum NF NF of the LNA. Broken line: unstressed LNA. Solid lines: LNAs measured after exposure to 33 dBm, 4 GHz for 3 and 16 h, respectively (from [7]).

Fig. 2. Chip photograph of fabricated two-stage LNA.

Fig. 5. Output power of LNA as a function of available input power at 4 GHz in a 50- system. Parameter is the series resistor in the dc path at the gate of the first transistor.

defined Pt/Au contacts were used as Schottky T-gates. Si N metal–insulator–metal (MIM) capacitors and NiCr resistors complete the coplanar MMIC process.

This LNA has been shown to survive up to 33 dBm of available input power at 4 GHz for 16 h without degradation (see Figs. 3 and 4). Increasing the available input power to 36 dBm up to 15 h, a slight irreversible increase in gate leakage current was observed, which caused degradation of the noise performance, while leaving the other electrical properties of the LNA ( -parameters, gain, bias point) unchanged. It is difficult to compare this result with the literature since such information is commonly not revealed [2]–[4], [6]. To the authors’ knowledge, it is the first verification of LNA ruggedness by post-stress noise measurements available in the literature [7].

III. LNA PERFORMANCE

IV. LNA OPERATION UNDER INPUT OVERDRIVE

We investigate a two-stage LNA MMIC based on 4 50 m HEMTs. This amplifier is described in detail in [7]. A circuit schematic is shown in Fig. 1. Fig. 2 presents a chip photograph. V for both stages, It is designed for a supply voltage of mA for the first stage, and mA in case and of the second stage, which corresponds to approximately 12% . Low-noise amplification is achieved in the 3.5–7-GHz frequency range with dB and dB. -param– GHz, eters are shown in Fig. 3. In the range dB is achieved with dB and dB. The lowest noise figure (NF) of 1.4 dB is measured at 6 GHz where the amplifier is matched for minimum noise. This is illustrated by Fig. 4, where the NF of the LNA in a 50- envi). One ronment is plotted together with its minimum NF ( recognizes that for frequencies from approximately 5 to 7 GHz, values, which demonthe 50- results are close to the strates effectiveness of the noise matching.

To study the large-signal behavior of the LNA under input overdrive condition, a circuit variation of the LNA MMIC shown in Figs. 1 and 2 was investigated. This variation does not include the gate bias of the first stage on wafer, i.e., the input coupling capacitance and the dc feed resistor are omitted. This variation gives the freedom to investigate the impact of the gate dc feed network on LNA performance by applying different types externally. We will use this possibility to determine the impact of the value of the feed resistor compared to the on-chip resistance of approximately 6 k . In our setup, the gate bias is applied through a bias-tee with a variable series resistance in its dc path. Fig. 5 shows measured output power as a function of available input power. Since the LNA was not designed for high linearity with its small transistors and low currents, it features an OIP3 of only 26 dBm [7]. Hence, in the case of the measurements presented here, the LNA is already in compression, even for the

Fig. 3. S -parameters before (symbols) and after (lines) stressing the LNA (from [7]).

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Fig. 6. DC-bias currents of the LNA as a function of available input power at 4 GHz. Parameter is the series resistance in the dc feed path of the first LNA stage. (a) Gate current and (b) drain current of the first HEMT. (c) Gate current and (d) drain current of the second HEMT.

lower available input powers, and the – curve is different from the standard case. The output power slightly increases up to a certain point (in our case, approximately 24 dBm) from which it starts to decrease rapidly. This corresponds to the onset of dc gate current, as shown in Fig. 6(a). This current flows if the incident wave causes voltage swings high enough to drive the gate into forward bias. Higher dc feed resistances reduce the slope of the gate current, while the output power decreases faster. The curves for the drain current of the first stage [see Fig. 6(b)] exhibit a similar shape as the curves for the output power. The gate current of the second stage, on the other hand, shows only little variation with available input power [see Fig. 6(c)]. The corresponding drain current is also affected, but its variation is much less pronounced than for the first stage [see Fig. 6(d)]. For all conditions discussed here, the thermal stress due to absolute dissipated power is far too low to cause damage to the LNA. This is due to the low operating voltage, and due to the fact that the HEMTs are driven into saturation, which limits the drain currents. It can also be concluded that the gate of the second HEMT is not exposed to high-voltage swings since it shows only minor variations in gate current. Accordingly, it is the gate of the first stage that is subject to the main stress. This will be analyzed in Section V. V. TIME-DOMAIN ANALYSIS OF A SINGLE HEMT Here, we will investigate which effect causes the main stress on the gate of the first stage when high input powers are applied. In order to do so, a single HEMT is measured on-wafer in a 50- system in the time domain. Although the input and output matching is different from the case where the HEMT is part of an LNA, the dc currents show quite similar behavior, as shown in Fig. 7. The small-signal matching conditions are of minor importance anyway since the HEMT is driven into saturation.

Two effects can cause degradation of the HEMT under RF input drive: forward dc gate current caused by turn-on of the gate diode, and reverse gate current caused by impact ionization during breakdown. As shown in Figs. 6(a) and 7(a), it is possible to reduce the gate current by means of a dc feed resistor. This feedback mechanism can be explained regarding Fig. 1. If the voltage swing at the gate is high enough that it becomes instantaneously forward biased, a dc current will be drawn from the bias supply. This current causes a voltage drop over the series resistor , which reduces the gate bias voltage with being the dc supply voltage. As is a negative voltage, it is reduced further in the presence of gate is reduced, the RF voltage swing current [see Fig. 7(b)]. If is also shifted to lower voltages and, hence, the peaks reaching is reinto positive voltages are reduced. As a consequence, duced. This feedback is not capable of suppressing or delaying the onset of dc-bias current, as seen in the measurement. It sets in at the same available input power for all cases. The reason is that it is the current itself that causes the feedback. Increasing the gate dc feed resistance, therefore, reduces the stress caused by gate current that shifts the gate voltages to more negative values. As seen in Fig. 7(b), the dc voltage can exceed 30 V. Since it can be expected that the peak voltages are significantly below this value, it is a question whether gate breakdown occurs. The maximum negative voltage presented to the gate can be estimated using simple approximations. Given an available that is delivered to the device at , the power corresponding RF peak voltage of the voltage source becomes . The worst case concerning voltage swing is that the gate presents an open circuit to the feedline. This means that the bias voltage is low enough to prevent the voltage from driving the gate into the forward direction; and it means we neglect the gate input capacitance for simplicity. In this case,

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Fig. 8. Gate breakdown measurement under the conditions that the drain was left floating, and V = 0 V. The gate current was limited to 1 mA in the measurement.

0

2

Fig. 7. DC-bias conditions of a 4 50 m HEMT measured in a 50- system. (a) Gate current. (b) Gate voltage. (c) Drain current. Parameter is the series resistance of the gate bias feed. The measurement frequency is 2 GHz.

due to reflection at the open gate, the voltage swing becomes . In this operation condition, it is necessary that the bias voltage is also at in order to keep the whole voltage swing at negative values. Hence, the maximum negative voltage becomes V

(1)

for W, which is the highest available source power we ap. The corresponding plied in this investigation, and V. This value obtained from bias voltage reaches a simplified analysis is close to the measured values that are plotted in Fig. 7(b). It will be shown in the following that the negative peak voltage is also quite well approximated. Fig. 8 presents dc measurement of the gate current as a function of gate–source voltage. From this measurement, it can be V if concluded that the dc gate breakdown occurs at V, but can be extended to approximately 75 V if the drain is left floating. Whether dynamic gate breakdown occurs when RF power is applied to the device cannot directly be seen from the bias point. However, it can be expected that the function of the bias currents and voltages show a kink when breakdown occurs. Hence, the smooth shape of the bias currents and

voltages in Fig. 7 can be taken as an indication that no breakdown occurs. In order to investigate in depth whether the gate is dynamically driven into breakdown, time-domain measurements were GHz for different power levels. These meaperformed at surement results are shown in Fig. 9 for different dc series resistances ranging from 1 to 8 k . The previously discussed bias points shown in Fig. 7 were obtained from the same measurements. Comparison of the trajectories supports the assumption that the feedback due to the gate dc series resistance trades gate current for negative peak voltages. While the available input power is approximately 10 dBm higher for the same gate current of 3.5 mA when increasing the gate resistance from 1 to 8 k , the maximum reverse gate voltage increased from around 20 V to approximately 75 V. It is also observed that the forward dc gate currents are difficult to see from the trajectories because the high reactive currents dominate the behavior and are responsible for the shape of the curves. However, the observation that the gate voltage cannot exceed the forward-bias diode voltage indicates that the gate becomes conductive there. A similar behavior would be expected in the presence of gate breakdown: that the breakdown voltage sets a lower limit to the negative voltage swing. However, such a limit is not observed, even if the gate voltage reaches 75 V. It is worth mentioning that the reason for the curves to slightly tend to negative currents at maximum voltages is attributed to measurement uncertainties affecting the absolute angle of higher harmonics. It must not be confused with the presence of breakdown that would set a hard limit to the maximum reverse voltage. Since breakdown is not the prime degradation mechanism, it can be concluded that LNA ruggedness is, at least at first order, independent of the size of the FET in the input stage. The reason is that the harmful forward gate currents scale with gatewidth. This finding is important since, in the case of GaAs and InP devices, it has been shown that larger devices show higher ruggedness, but this is due to the fact that negative peak voltage is lower for larger devices and, hence, breakdown occurs at higher input powers [9]. Just for the sake of completeness, an output trajectory for the case with a gate dc resistance of 8 k is given in Fig. 10. It is obvious that neither voltage, nor current or dissipated power ever reach critical values. Moreover, as the available input power increases, dc drain current decreases [see Fig. 7(c)], reducing voltage and current swing further.

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Fig. 9. Trajectories at the gate of an HEMT, measured in time domain in a 50- system at 2 GHz. Parameter is the available input power, as indicated in the figures. Different gate dc series resistances are applied. (a) 1 k . (b) 2 k . (c) 4 k . (d) 8 k . The corresponding bias points are given in Fig. 7.

Fig. 10. Trajectories at the drain of an HEMT, measured in time domain in a 50- system at 2 GHz, with a gate series resistance of 8 k . Parameter is the available input power, as indicated in the figure. The data was obtained from the same measurement as shown in Figs. 7 and 9(d).

The time-domain results presented before prove quantitatively that the ruggedness of a GaN-FET-based LNA is mainly the benefit of the high breakdown voltages of the device. The main factor causing stress to the device is the dc gate current, which is easily determined during circuit design and LNA operation. VI. CONCLUSION The ruggedness of GaN-FET-based LNAs has been investigated treating a highly survivable GaN-HEMT LNA MMIC [7] as an example. Its minimum NF measured at 6 GHz is 1.4 dB dB in the 3.5–7-GHz frequency range. An with approximately 20-dB gain, as well as good input and output matching have been achieved. Survivability of this MMIC is characterized by stressing several LNA samples biased for low-noise operation with high input power. These tests were performed at frequencies of 2, 4, and 5 GHz with available powers up to 36 dBm for up to 16 h. This measurement condition exceeded stress tests reported in the literature by approximately 16 h in duration. None of the

circuits degraded during the measurement up to available input powers of 33 dBm. This was verified by post-stress noise, -parameter, and dc measurements, and has proven the high survivability of LNAs in GaN-HEMT MMIC technology. In order to quantify the factors that eventually result in device degradation, measurements of LNA circuits and single FET devices were performed in the frequency and time domains. Regarding the assessment of stressing mechanisms, the following conclusions can be drawn. • Only the first stage of the LNA is subject to stress. While the second stage is driven into saturation, no excessively high drain current or significant gate current are flowing. • Total dissipated power, as well as drain current stay within the safe operation area. • Due to the extremely high breakdown voltages of the GaN FETs, no gate breakdown effect is observed, even if the V. dynamic load line reaches • The main factor causing degradation is dc gate current in the presence of input overdrive. It is also shown that a series resistance in the gate-bias line can improve LNA ruggedness. Due to voltage feedback, the gate current is reduced. This feedback effect reduces the gate-bias voltage, thereby increasing the negative peak voltage. It has been proven by time-domain measurement that, due to the high gate breakdown voltages, this is an excellent approach to optimize the survivability of the LNA. In conclusion, not only are the mechanisms responsible for device stress under high input overdrive identified in this paper, it has also been shown how the ruggedness of the LNA can be improved by a proper choice of the dc gate feed resistor. ACKNOWLEDGMENT The authors would like to thank P. Heymann and J. Schmidt, both with Ferdinand-Braun-Institut für Höchstfrequenztechnik

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(FBH), Berlin, Germany, for performing the noise measurements.

-parameter and

REFERENCES [1] G. A. Ellis, J.-S. Moon, D. Wong, M. Micovic, A. Kurdoghlian, P. Hashimoto, and M. Hu, “Wideband AlGaN/GaN HEMT MMIC low noise amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., 2004, pp. 153–156. [2] S. Cha, Y. H. Chung, M. Wojtowwicz, I. Smorchkova, B. R. Allen, J. M. Yang, and R. Kagiwada, “Wideband AlGaN/GaN HEMT low noise amplifier for highly survivable receiver electronics,” in IEEE MTT-S Int. Microw. Symp. Dig., 2004, pp. 829–832. [3] D. Krausse, R. Quay, R. Kiefer, A. Tessmann, H. Massler, A. Leuther, T. Merkle, S. Müller, C. Schwörer, M. Mikulla, M. Schlechtweg, and G. Weimann, “Robust GaN HEMT low-noise amplifier MMICs for X -band applications,” in Eur. Gallium Arsenide and Other Semiconduct. Applicat. Symp., 2004, pp. 71–74. [4] H. Xu, C. Sanabria, A. Chini, St. Keller, U. K. Mishra, and R. A. York, “A C -band high-dynamic range GaN HEMT low-noise amplifier,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 262–264, Jun. 2004. [5] J. C. De Jaeger, S. L. Delage, G. Dambrine, M. A. Di, F. Poisson, V. Hoel, S. Lepilliet, B. Grimbert, E. Morvan, Y. Mancuso, G. Gauthier, A. Lefrançois, and Y. Cordier, “Noise assessment of AlGaN/GaN HEMTs on Si or SiC substrates: Application to X -band low noise amplifiers,” in Eur. Gallium Arsenide and Other Semiconduct. Applicat. Symp., 2005, pp. 229–232. [6] M. Micovic, A. Kurdoghlian, H. P. Moyer, P. Hashimoto, A. Schmitz, I. Milosavljevic, P. J. Willadsen, W.-S. Wong, J. Duvall, M. Hu, M. Wetzel, and D. H. Chow, “GaN MMIC Technology for Microwave and Millimeter-wave Applications,” in IEEE Compound Semiconduct. Integr. Circuit Symp., 2005, pp. 173–176. [7] M. Rudolph, R. Behtash, K. Hirche, J. Würfl, W. Heinrich, and G. Tränkle, “A highly survivable 3–7 GHz GaN low-noise amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., 2006, pp. 1899–1902. [8] R. S. Schwindt, V. Kumar, O. Aktas, J.-W. Lee, and I. Adesida, “Temperature-dependence of a GaN-based HEMT monolithic X -band low noise amplifier,” in IEEE Compound Semiconduct. Integr. Circuit Symp., 2004, pp. 201–204. [9] Y. C. Chen, M. Barsky, R. Tsai, R. Lai, H. C. Yen, A. Oki, and D. C. Streit, “Survivability of InP HEMT devices and MMIC’s under high RF input drive,” in IEEE MTT-S Int. Microw. Symp. Dig., 2000, pp. 1917–1920. Matthias Rudolph (M’99–SM’05) received the Dipl.-Ing. degree in electrical engineering from the Berlin University of Technology, Berlin, Germany, in 1996, and the Dr.-Ing. degree from Darmstadt University of Technology, Darmstadt, Germany, in 2001. He is currently a Senior Scientist with the Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH), Berlin, Germany. His research is focused on modeling of FETs and HBTs and on the design of power, broadband, and low-noise amplifiers. He authored or coauthored over 40 publications in refereed journals and conferences and Introduction to Modeling HBTs (Artech House, 2006).

Reza Behtash received the Diplom-Ingenieur degree (equivalent to the Masters degree) in electrical engineering from the Technical University Hamburg, Hamburg, Germany in 1999, and the Dr.-Ing. degree from Ulm University, Ulm, Germany, in 2006. From 2000 to 2004, he carried out his dissertation about high-power AlGaN/GaN transistors and integrated power amplifiers at the DaimlerChrysler Research Center, Ulm, Germany. He is currently with the Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH), Berlin, Germany. He is also with United Monolithic Semiconductors, Ulm, Germany. His research interests include processing and characterization of GaN HEMTs and MMICs.

Ralf Doerner (M’97) received the Dipl.-Ing. degree in communications engineering from the Technische Universität Ilmenau, Ilmenau, Germany, in 1990. Since 1989, he has been involved with microwave measuring techniques. In 1992, he joined the Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH), Berlin, Germany. His current research is focused on calibration problems in on-wafer millimeter-wave measurements of active and passive devices and circuits and on nonlinear characterization of microwave power transistors.

Klaus Hirche was born in Esslingen, Germany, in 1956. He received the Dipl.-Ing. degree in electrical engineering from the University of Stuttgart, Stuttgart, Germany, in 1984. In 1984, he joined ANT Nachrichtentechnik (then Bosch Telecom), Backnang, Germany, where he was initially involved in modeling of high-speed bipolar silicon transistors, and then with the design and characterization of InGaAs/InP heterojunction bipolar transistors, InAlAs/InGaAs heterostructure FETs, and InGaAs photodiodes. He was responsible for several research projects concerning opto-electronic components. Since 1999, he has been with Bosch Satcom, and is currently with Tesat-Spacecom GmbH & Co. KG, Backnang, Germany. He has been involved in design, characterization, and reliability evaluation of photodiodes for optical inter-satellite links. Since 2003, he has been responsible for research and development projects on GaN devices for space application.

Joachim Würfl (M’93) received the Ph.D. degree in electrical engineering from the Technical University of Darmstadt, Darmstadt, Germany in 1989. He was involved with the technology and design of high-temperature and high-power GaAs-based devices with the Technical University of Darmstadt. As a Post-Doctoral, he developed micromechanical sensors based on III/V compound semiconductors at the Technical University of Darmstadt. Since 1992, he is heading the Department of Process Technology, Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH), Berlin, Germany, where he has the responsibility for technological design and processing of microwave devices. He manages several projects on microwave power and very high-frequency devices and their technological implementation. His current research interests include high-power GaAs- and GaNbased microwave and millimeter-wave power devices and monolithic integrated circuits. He has authored or coauthored over 100 scientific papers in related fields.

RUDOLPH et al.: ANALYSIS OF SURVIVABILITY OF GaN LNAs

Wolfgang Heinrich (M’84–SM’95) received the Dipl.-Ing., Dr.-Ing., and Habilitation degrees from the Technical University of Darmstadt, Darmstadt, Germany, in 1982, 1987, and 1992, respectively. Since 1993, he has been with the Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH), Berlin, Germany, where he is Head of the Microwave Department and Deputy Director of the institute. He has authored or coauthored over 200 publications and conference contributions. His current research activities focus on MMIC design with emphasis on oscillators, GaAs and GaN power transistors, electromagnetic simulation, and millimeter-wave packaging. Dr. Heinrich served as an IEEE Microwave Theory and Techniques Society (MTT-S) Distinguished Microwave Lecturer (2003–2005). He is chairman of the German IEEE MTT-S/Antennas and Propagation (AP) Chapter (2002–2006).

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Günther Tränkle (M’95) received the Diploma degree from the Technical University Munich, Munich, Germany, in 1981, and the Ph.D. degree from the University of Stuttgart, Stuttgart, Germany, in 1988, both in physics. In 1988, he joined the Walter-Schottky-Institute, Technical University Munich. From 1995 to 1996, he was a Department Head with the Fraunhofer-Institute for Applied Solid-State Physics, Freiburg, Germany. In 1996, he became Head of the Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH), Berlin, Germany. Since 2002, he holds a Chair on microwaves and optoelectronics with the Technical University Berlin, Berlin, Germany. He is currently Chairman of the Section of the Leibniz Association for Mathematics, Natural Science, Engineering. He has authored or coauthored over 200 publications and conference contributions. His current research interests include III/V technology, microwave, and millimeter-wave transistors and circuits, GaN electronics, and high-power diode lasers.

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A 16-GHz Triple-Modulus Phase-Switching Prescaler and Its Application to a 15-GHz Frequency Synthesizer in 0.18-m CMOS Yu-Hsun Peng and Liang-Hung Lu, Member, IEEE

Abstract—A triple-modulus phase-switching prescaler for highspeed operations is presented in this paper. By reversing the switching orders between the eight 45 -spaced signals generated by the 8 : 1 frequency divider, the maximum operating frequency of the prescaler is effectively enhanced. With the triple-modulus switching scheme, a wide frequency covering range is achieved. The proposed prescaler is implemented in a 0.18- m CMOS process, demonstrating a maximum operating frequency of 16 GHz without additional peaking inductors for a compact chip size. Based on the high-speed prescaler, a fully integrated frequency synthesizer is realized. The synthesizer integeroperates at an output frequency from 13.9 to 15.6 GHz, making it very attractive for wideband applications in -band. At an output frequency of 14.4 GHz, the measured sideband power and phase noise at 1-MHz offset are 60 dBc and 103.8 dBc/Hz, respectively. The fabricated circuit occupies a chip area of 1 mm2 and consumes a dc power of 70 mW from a 1.8-V supply voltage. Index Terms—Frequency synthesizers, high-speed operations, phase-locked loops (PLLs), phase-switching prescalers, programmable dividers, triple-modulus frequency division.

I. INTRODUCTION

HE fast-growing market in personal wireless communication has motivated the development of fully integrated transceivers using a cost-efficient CMOS process. As one of the most important building blocks to provide a programmable carrier frequency for signal transmitting and receiving, the frequency synthesizers have been successfully fabricated in deepsubmicrometer CMOS technologies for applications at multigigahertz frequencies [1], [2]. With the limitations imposed on the cutoff frequency of the transistors, it is still a great design challenge to implement CMOS synthesizers operating at frequencies beyond 10 GHz. For phase-locked loop (PLL)-based frequency synthesizers, the only circuit modules operating at the carrier frequency are the voltage-controlled oscillator (VCO) and the prescaler. By employing on-chip LC tanks for the circuit implementations, CMOS VCOs can be realized at very high frequencies [3], [4]. Therefore, the prescalers, especially for the first divider stage,

T

Manuscript received July 21, 2006; revised September 26, 2006. This work was supported in part by the National Science Council under Grant 94-2220-E002-026 and Grant 94-2220-E-002-009. The authors are with the Graduate Institute of Electronics Engineering and Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.886908

are generally considered the speed bottleneck in CMOS frequency synthesizers. In order to achieve the required high-speed operations, a divide-by-two circuit is widely used as the input stage followed by a dual-modulus divider [5], [6]. Though the speed limitations can be effectively alleviated, undesirable spur sidebands appear at an offset frequency apart from the carrier by half of the channel spacing. Alternatively, dual-modulus phaseswitching prescalers [7] are proposed to substitute the input programmable divider, presenting a promising solution to high-frequency synthesizers. However, the potential output glitches associated with the finite transition time and inaccurate timing control of the logic gates may result in miscount in the following divider stages. In order to overcome the limitations of the phase-switching prescaler, a novel switching algorithm is presented in this study. With reversed switching orders between the eight 45 -spaced signals generated by the divide-by-eight circuit, the timing constraint of the prescaler is thus relaxed for glitch-free operations at higher frequencies. In addition, a triple-modulus topology is also employed to provide a wide division ratio and an enhanced frequency covering range. Using a 0.18- m CMOS process, the proposed phase-switching prescaler is implemented, exhibiting a maximum operating speed of 16 GHz. Based on the prescaler design, a 15-GHz frequency synthesizer is realized for demonstration. This paper is organized as follows. Section II describes the proposed triple-modulus phase-switching prescaler including the operation principles and the building blocks. The circuit design of the 15-GHz frequency synthesizer is presented in Section III. The experimental results of the prescaler and the frequency synthesizer are shown in Section IV. Finally, a conclusion is given in Section V. II. TRIPLE-MODULUS PHASE-SWITCHING PRESCALER A. Conventional Phase-Switching Prescalers Typically, the programmable frequency division in a frequency synthesizer is provided by a high-speed dual-modulus prescaler along with a low-speed pulse-swallow counter. Due to the existence of the feedback loop and additional logic gates in the synchronous prescaler, the maximum operating frequency is severely limited. For high-speed applications, a phase-switching scheme has been adopted for the prescalers [8]. A simplified block diagram of the phase-switching prescaler is shown in Fig. 1, where four 90 -spaced phases are generated by the dividers and the output phase is controlled by a 4 : 1

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Fig. 1. Conventional phase-switching prescaler.

Fig. 3. Simulated performance of the phase-switching prescalers with various circuit topologies.

Fig. 2. Waveforms of the phase-switching prescaler for the forward- switching (f =5) and the reverse-switching (f =3) schemes.

multiplexer (MUX). The 4/5 dual-modulus frequency division is achieved by switching the output to the successive phase with a lag of 90 . Fig. 2 shows the associated waveforms to demonstrate the operation of the prescaler. In this timing scheme, the speed limitations on the conventional dual-modulus synchronous dividers can be alleviated. However, it may suffer from output glitches due to the finite transition time and inaccurate timing control of the MUX, especially for high-speed operations. Various circuit techniques were proposed to eliminate the glitch problems by re-timing circuits [8], modified duty cycles [9], and buffers with a reduced unity-gain bandwidth [10]. One of the most effective approaches to extend the time window for glitch-free switching is to reverse the switching orders among the four 90 -spaced phases [11]. The waveforms of the reverse-switching scheme are also shown in Fig. 2, where the time window for correct switching is extended by a factor of three for the 3/4 dual-modulus prescaler. Although the reverse-switching technique provides glitch-free output for dual-modulus frequency division, inaccurate timing control induces phase jitter, which may degrade the output quality of the frequency synthesizer. B. Proposed Phase-Switching Scheme In the design of the phase-switching prescalers, the operating speed is no longer limited by the input divide-by-two circuit. Instead, the propagation delays in the MUX and the decoder become the major concerns in the circuit implementation. For high-speed operations, a prolonged time window for phase switching is desirable to provide a better tolerance such that the nonideal effects including process variation, layout asymmetry, and phase mismatch can be overcome. As indicated in Fig. 2, the reverse-switching scheme effectively extends the time window

Fig. 4. Waveforms of the proposed triple-modulus phase-switching prescaler.

for phase switching. For a dual-modulus prescaler where equally spaced phases are generated by the divider, a maxinput periods is imum switching window equal to provided. However, the frequency covering range decreases as increases. To achieve an extended switching window with sufficient frequency covering range for wideband operations at higher frequencies, a triple-modulus scheme is proposed for the prescaler by controlling the progressive step for the phase switching. The glitch-free switching window and the frequency covering range of the prescaler with various switching schemes are shown in Fig. 3 for performance evaluation. In this study, a triple-modulus phase-switching prescaler is employed for the circuit design, leading to with a frequency covering range of 28.5% and a phase-switching window of six input periods for wideband operations at the -band. The waveforms of the proposed circuit with division ratios of 6–8 are illustrated in Fig. 4. For the design of the control logic, a finite state machine is developed. With – generated by the eight 45 -spaced output phases the asynchronous frequency divider, the state diagram of the triple-modulus prescaler is shown in Fig. 5. To provide a division ratio of 8, the prescaler stays in the previous state without phase switching. On the other hand, the required 7 : 1 and 6 : 1 frequency division are realized by changing the state to the one with a leading phase of 45 and 90 , respectively. Since the triple-modulus operation is achieved by modifying

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Fig. 5. State diagram of the proposed triple-modulus phase-switching scheme.

Fig. 6. Block diagram of the proposed triple-modulus phase-switching prescaler.

the low-speed control logic, it is well suited for high- speed circuit designs in CMOS technologies.

Fig. 7. Schematic of the frequency dividers. (a) Input divide-by-two circuit. (b) Synchronous 4 : 1 frequency divider.

C. Design of the 16-GHz Phase-Switching Prescaler Based on the proposed circuit technique, the triple-modulus phase-switching prescaler is implemented in a 0.18- m CMOS process for demonstration. Fig. 6 shows the block diagram of the prescaler, which is composed of a divide-by-eight circuit, a MUX, a decoder, and control logics. The design considerations of the individual building blocks are described as follows. Frequency Dividers: To alleviate the speed limitations while maintaining a wide input frequency range, a 2 : 1 divider based on current-mode-logic (CML) flip-flops is employed as the input stage. The circuit schematic is shown in Fig. 7(a), where differential output signals are provided at half of the input frequency. For a compact chip size, resistive loads are utilized without peaking inductors. In addition, the tail transistors are removed to minimize the parasitics and to enhance the voltage headroom for high-speed operations. Followed by the input divide-by-two stage, a synchronous frequency divider, as shown in Fig. 7(b), is employed to provide the 4 : 1 frequency division with eight 45 -spaced output phases. Operating at a clock rate half of the input frequency, the 4 : 1 divider is realized by cascading four CML latches with resistive loads. To minimize the loading effect, the outputs of the divide-by-two circuit are buffered by a differential amplifier stage. On the other hand, full-swing buffers are used at the outputs of the 4 : 1 synchronous divider to provide the required signal levels for the MUX, the control logic, and the low-frequency pulse-swallow counter. MUX: With the phase-switching technique, the speed limitations of the frequency dividers are alleviated. Consequently, the maximum operating speed of the triple-modulus prescaler

Fig. 8. Schematic of the 8 : 1 MUX for phase switching.

is determined by the MUX circuit, especially when full logic swing is required at the output. In the proposed circuit topology, the MUX has to complete the output phase switching within six input periods. The schematic of the 8 : 1 MUX is illustrated in Fig. 8 where the desirable phase is selected by one of the four 2 : 1 MUX cells and a tree-type logic gate is adopted to separate the capacitive loadings for a minimum propagation delay. Note that both pseudo-NMOS and pseudo-PMOS topologies are utilized to realize the MUX cells and the logic gates such that the

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Fig. 9. (a) Block diagram and (b) circuit schematic of the modulus control and decoder circuits.

required operating speed can be achieved while maintaining a matched loading for each one of the signal paths. Modulus Control and Decoder: Based on the state diagram in Fig. 5, the modulus control and the decoder are realized by the flipfinite state machine, as shown in Fig. 9(a), where eight flops operating at a clock rate of 1/8 of the input frequency are used to store the current state and only one of the flip-flop outputs goes to high at any designated state. Note that the outputs of – act as the select signals for the 8 : 1 MUX, the flip-flops are employed as the inwhile the delayed signals puts for the combinational logics. In this design, programmable delay cells are added at the outputs of the flip-flops to compensate for the process variation, ensuring correct timing for the phase switching. To minimize the gate delays for high-speed operations, the combinational logics are integrated with the CMOS flip-flops, as shown in Fig. 9(b). For the triple-modulus frequency division, two nonoverlapping control bits and are adopted to enable 7 : 1 and 6 : 1 frequency division, respectively. On the other hand, the phase switching is disabled and the prescaler behaves as a divide-by-eight circuit when both and are low. By monolithically integrating the building blocks, the triplemodulus phase-switching prescaler is designed to operate at -band. For minimum gate delay and an input frequency at phase mismatch, long interconnect wires are prevented and the layout is made as symmetric as possible. With an input signal of 15 GHz, the simulated waveforms of the prescaler for various division ratios are demonstrated in Fig. 10. III. 15-GHZ FREQUENCY SYNTHESIZER The proposed triple-modulus phase-switching prescaler is adopted for the implementation of the 15-GHz frequency synthesizer based on an integer- PLL architecture. Fig. 11 shows the block diagram of the synthesizer, which is composed of a VCO, a programmable frequency divider, a phase/frequency detector (PFD), a charge pump, and a loop filter. With a second-order loop filter, the synthesizer can be treated as a third-order feedback system. Since the property of the PLL is strongly influenced by the loop filter, behavior simulations were employed to optimize the circuit performance. With a

Fig. 10. Simulated waveforms of the proposed prescaler with an input frequency of 15 GHz. (a) Divide-by-six and (b) divide-by-seven operation.

loop bandwidth of 200 kHz in this particular design, the noise in the close-in spectrum is effectively suppressed while maintaining reasonable settling time and spur rejection. -band In order to achieve the required performance for applications, a modified differential Colpitts oscillator [6], as shown in Fig. 12, is adopted as the VCO in the synthesizer deand in the sign. Due to the use of the capacitive divider oscillator topology, the phase noise is effectively reduced by

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Fig. 11. Block diagram of the 15-GHz frequency synthesizer.

Fig. 13. Block diagram and the switching scheme of the programmable divider.

Fig. 14. Circuit schematic of the pulse-swallow counter. Fig. 12. Circuit schematic of the differential Colpitts VCO.

the cyclo-stationary noise effect [12]. For an enhanced operais used as the 1-bit tion bandwidth, the controlled voltage and band selection. By switching the capacitance values of simultaneously, the output of the VCO can be tuned over a wide frequency range without disturbing the startup condition. The continuous frequency tuning of the VCO is achieved by the symmetric varactor structure with differential controlled signals and . The programmable divider is composed of a triple-modulus prescaler and a low-frequency pulse-swallow counter. To have a better understanding on the proposed mechanism for channel selection, a simplified block diagram is illustrated in Fig. 13. , the division ratio can be expressed as For

(1) In this design, five controlled bits for channel selection are used to set , while one bit is assigned to , leading to a division ratio from 194 to 254 for the programmable frequency divider. Fig. 14 illustrates the schematic of the pulse-swallow counter, which is realized in true single-phase clock (TSPC) logic to minimize the power consumption. In conventional circuit designs, the output of the frequency synthesizer is vulnerable to the common-mode noise coupled from the power supply and the substrate. To alleviate the degradation in the phase noise due to this undesirable coupling, a fully differential controlled mechanism is employed. Fig. 15

Fig. 15. Circuit schematic of: (a) the three-state PFD and (b) the differential charge pump.

shows the schematic of the three-state PFD and the differential charge pump circuit. Note that four transistors controlled by the PFD outputs behave as switches to steer the charging currents, while four additional transistors are utilized to minimize the

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Fig. 16. Microphotograph of the fabricated frequency synthesizer.

Fig. 17. Measured input sensitivity curve of the frequency divider.

charge-sharing problem [13] in conventional circuit topologies. In this design, a common-mode feedback (CMFB) circuit with source degeneration is adopted to stabilize the output commonmode voltage with sufficient input dynamic range. As a result, a reduced reference spur level can be achieved at the synthesizer output. IV. EXPERIMENTAL RESULTS The proposed circuits are designed and implemented in a 1P6M 0.18- m CMOS technology. Operated at a 1.8-V supply voltage, the power consumptions of the prescaler and the frequency synthesizer are 40 and 70 mW, respectively. Fig. 16 shows a microphotograph of the fully integrated frequency synthesizer with a chip area of 1 1 mm where the prescaler occupies an active area of 0.4 0.1 mm . The performance of the proposed phase-switching prescaler was first characterized by performing frequency division for various division ratios. Provided a high-frequency signal from an external source, the measured input sensitivity curve of the prescaler is shown in Fig. 17. For an input power level of 0 dBm, an input frequency range from 8.6 to 16 GHz is demonstrated. By integrating the prescaler with the pulse-swallow counter, the programmable divider provides a division ratio from 194 to

Fig. 18. Measured output spectrum and waveform of the programmable divider operating at an input frequency of 16 GHz for: (a) 254 : 1 frequency division and (b) 194 : 1 frequency division.

254 by the 6-bit channel selection. At the highest operating frequency of 16 GHz, the measured output spectra and waveforms of the prescaler are shown in Fig. 18.

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TABLE I PERFORMANCE SUMMARY

Fig. 19. Measured tuning characteristics of the differential Colpitts VCO.

phase noise at 1-MHz offset are 6 dBm and 106.5 dBc/Hz, respectively. The performance of the VCO is evaluated by including the tuning range [14], the figure-of-merit leading to an of 176 dB for the fabricated circuit. The output characteristics of the synthesizer were measured by a spectrum analyzer via on-wafer probing. With a reference frequency of 72 MHz and a charge pump current of 600 A, the measured output spectrum of the fabricated circuit at 14.4 GHz is shown in Fig. 20(a). Note that the sideband spurs appear at an offset frequency equal to the channel spacing with a power level approximately 60 dB below the carrier. The close-in phase noise of the synthesizer output is depicted in Fig. 20(b). At an offset frequency of 1 MHz, the measured phase noise is 103.8 dBc/Hz. The performance of the proposed circuit is summarized in Table I. V. CONCLUSION A triple-modulus phase-switching prescaler has been presented for high-speed operations. Fabricated in a 0.18- m CMOS process, the proposed circuit operates at a maximum input frequency of 16 GHz without output glitches. Based on the prescale circuit, a fully integrated frequency synthesizer has been demonstrated, providing a wideband operation from 13.9 to 15.6 GHz with 61 programmable channels. Fig. 20. (a) Measured output spectrum and (b) close-in phase noise of the frequency synthesizer with an output frequency of 14.4 GHz.

Due to the wide input range provided by the prescaler, the operation frequency of the synthesizer is predetermined by the tuning range of the VCO. Fig. 19 shows the measured tuning characteristics of the differential Colpitts VCO. With the 1-bit controlled signal for band selection, the VCO exhibits an oscillation frequency from 13.9 to 15.6 GHz, indicating an overall tuning range of 11.5%. For the open-loop VCO at an output frequency of 14.34 GHz, the measured output power and

ACKNOWLEDGMENT The authors would like to thank the National Chip Implementation Center (CIC), Hsinchu, Taiwan, R.O.C., for chip fabrication and technical support. REFERENCES [1] C. Lam and B. Razavi, “A 2.6-GHz/5.2-GHz frequency synthesizer in 0.4-m CMOS technology,” IEEE J. Solid-State Circuits, vol. 35, no. 5, pp. 768–794, May 2000. [2] G. C. T. Leung and H. C. Luong, “A 1-V 5.2-GHz CMOS synthesizer for WLAN applications,” IEEE J. Solid-State Circuits, vol. 39, no. 11, pp. 1873–1882, Nov. 2004.

PENG AND LU: 16-GHz TRIPLE-MODULUS PHASE-SWITCHING PRESCALER AND ITS APPLICATION TO 15-GHz FREQUENCY SYNTHESIZER

[3] A. P. Wel, S. L. J. Gierkink, R. C. Frye, V. Boccuzzi, and B. Nauta, “A robust 43-GHz VCO in CMOS for OC-768 SONNET applications,” IEEE J. Solid-State Circuits, vol. 39, no. 7, pp. 1159–1163, Jul. 2004. [4] S. Ko, J.-G. Kim, T. Song, E. Yoon, and S. Hong, “K - and Q- bands CMOS frequency sources with X band quadrature VCO,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2789–2800, Sep. 2004. [5] N. Pavlovic, J. Gosselin, K. Mistry, and D. Leenaerts, “A 10 GHz frequency synthesizer for 802.11a in 0.18-m CMOS,” in Proc. 30th Eur. Solid-State Circuits Conf., Sep. 2004, pp. 367–370. [6] Y.-H. Peng and L.-H. Lu, “A Ku-band frequency synthesizer in 0.18- m CMOS technology,” IEEE Microw. Wireless Compon. Lett., to be published. [7] J. Craninckz and M. S. J. Steyaert, “A 1.75-GHz/3-V dual-modulus divided-by-128/129 prescaler in 0.7-m CMOS,” IEEE J. Solid- State Circuits, vol. 31, no. 7, pp. 890–897, Jul. 1996. [8] N. Krishnapura and P. R. Kinget, “A 5.3-GHz programmable divider for HiPerLAN in 0.25-m CMOS,” IEEE J. Solid-State Circuits, vol. 35, no. 7, pp. 1019–1024, Jul. 2000. [9] X.-P. Yu, M.-A. Do, J.-G. Ma, and K.-S. Yeo, “A new 5 GHz CMOS dual-modulus prescaler,” in IEEE Circuits Syst. Int. Symp., May 2005, vol. 5, pp. 5027–5030. [10] C.-C. Ng and K.-K. M. Cheng, “Ultra low power 2.4-GHz 0.35 m CMOS dual-modulus prescaler design,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 2, pp. 75–77, Feb. 2005. [11] S. Keliu, S.-S. Edgar, S.-M. Jose, and S. H. K. Embabi, “A 2.4-GHz monolithic fractional-N frequency synthesizer with robust phase-switching prescaler and loop capacitance multiplier,” IEEE J. Solid-State Circuits, vol. 38, no. 6, pp. 866–874, Jun. 2003. [12] R. Aparicio and A. Hajimiri, “A noise-shifting differential Colpitts VCO,” IEEE J. Solid-State Circuits, vol. 37, no. 12, pp. 1728–1736, Dec. 2002. [13] B. Terlemez and J. P. Uyemura, “The design of a differential CMOS charge pump for high performance phase-locked loops,” in Proc. Int. Circuits Syst. Symp., May 2004, pp. 561–564. [14] J. Kim, J.-O. Plouchart, N. Zamdmer, M. Sherony, Y. Tan, M. Yoon, R. Trzcinski, M. Talbi, J. Safran, A. Ray, and L. Wagner, “A power- optimized widely-tunable 5-GHz monolithic VCO in a digital SOI CMOS technology on high resistivity substrate,” in Proc. Int. Low-Power Electron. Design Symp., Aug. 2003, pp. 434–439.

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Yu-Hsun Peng was born in Hsinchu, Taiwan, R.O.C., in 1982. He received the B.S. degree in communication engineering from National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 2004, and the M.S. degree in electronics engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2006. His research interest includes the development of frequency synthesizers for RF applications.

Liang-Hung Lu (M’02) was born in Taipei, Taiwan, R.O.C., in 1968. He received the B.S. and M.S. degrees in electronics engineering from National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 1991 and 1993, respectively, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 2001. During his graduate study, he was involved in SiGe HBT technology and monolithic-microwave integrated-circuit (MMIC) designs. From 2001 to 2002, he was with IBM, where he was involved with low-power and RF integrated circuits for silicon-on-insulator (SOI) technology. In August 2002, he joined the faculty of the Graduate Institute of Electronics Engineering and the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C., where he is currently an Associate Professor. His research interests include CMOS/BiCMOS RF and mixed-signal integrated-circuit designs.

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A New Envelope Predistorter With Envelope Delay Taps for Memory Effect Compensation Sung-Chan Jung, Hyun-Chul Park, Min-Su Kim, Gunhyun Ahn, Ju-Ho Van, Hoon Hwangbo, Cheon-Seok Park, Member, IEEE, Sung-Kil Park, and Youngoo Yang, Member, IEEE

Abstract—We present a new linearization method for high-power amplifiers, using an envelope predistorter (EPD) including envelope delay taps and control circuits, for memory effect compensation. The lower and upper third-order intermodulation (IM3 ) components, generated by the EPD, can be separately controlled for their magnitudes and phases by the additional memory effect compensation circuits. By using experimental results for a high-power (30-W peak envelope power) class-AB amplifier, further linearity improvement was also demonstrated using the proposed EPD. For a two-tone signal with a tone spacing of 20 MHz, the proposed EPD, with only a single delay tap, cancelled the lower and upper IM3 components by 20.84 and 18.17 dB, while the conventional EPD, with no envelope delay tap, cancelled them by 11.67 and 8.50 dB, respectively. Index Terms—Adjacent channel leakage power ratio (ACLR), envelope delay tap, envelope predistorter (EPD), linear power amplifier, linearity, memory effect.

I. INTRODUCTION

D

UE TO explosive subscriber increases and increases in various emerging services, high spectral efficiency becomes more and more important for wireless communication systems. Modulation formats with high spectral efficiency, such as quadrature phase-shift keying (QPSK), 16- or 64-quadrature amplitude modulation (QAM), have been extensively used. The systems, using these modulation formats, require highly linear and efficient signal transmission to maintain communication quality and to suppress the unwanted side effects caused by too much dc power consumption. To obtain high efficiency, power amplifiers, used in digital communication systems, operate near the saturation region. However, power amplifiers, which operate near the saturation region, generate a spectral regrowth that causes distortion of the magnitude and phase of the signal. A number of research efforts have reported results addressing analyses or linearization techniques for intermodulation (IM) distortion.

Manuscript received August 18, 2006; revised October 2, 2006. This work was supported by the Korean Government (MOEHRD) under Korea Research Foundation Grant KRF-2005-041-D00564. S.-C. Jung, H.-C. Park, M.-S. Kim, G. Ahn, J.-H. Van, H. Hwangbo, C.-S. Park, and Y. Yang are with the School of Information and Communication Engineering, Sungkyunkwan University, Suwon 440-746, Korea (e-mail: [email protected]; [email protected]). S.-K. Park is with the Wave Electronics Company Ltd., Suwon 441-814, Korea. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2006.886909

Among the various linearization techniques, the predistortion method places a distortion generator in front of the power amplifier. The distortion generator provides an opposite transfer function to the power amplifier. Hence, the composite transfer functions of the predistorter and power amplifier become linear, i.e., the IM distortions generated by the predistorter and power amplifier cancel each other. Compared with the well-known feedforward technique, the predistortion method has a relatively high efficiency because of additional output loss and dc power consumption of the auxiliary amplifier in the feedforward method. However, the performance of the predistortion method with respect to bandwidth and linearity improvement are limited [1], [2]. The memory effect is one of the main reasons for the predistorter to have a limited linearization performance. The lower and upper IM distortion terms have different magnitudes and/or relative phases in the presence of the large memory of the highpower amplifiers in conjunction with multiple distortion generation mechanisms. The influence of the memory effect can be partially reduced in the power amplifier using some compensation techniques [3]–[6]. In this paper, we propose a new envelope predistorter (EPD) with envelope delay taps for memory effect compensation. Using a simplified amplifier model, a representative generation mechanism of the memory effect was analyzed. Based on the analysis results, add-on memory effect compensation circuits were devised. The operational principles of the EPD with envelope delay taps were also derived. The memory effect compensation circuits for the EPD were composed of a delay line and baseband variable gain amplifiers (VGAs). The proposed EPD and 30-W peak envelope power (PEP) power amplifier at 2.14-GHz band were implemented for experimental validation. Cancellation performances of the were measured using a two-tone signal of 20-MHz tone spacing. The adjacent channel leakage power ratio (ACLR) performance, using a four-carrier down-link wideband code division multiple access (WCDMA) signal, was also measured and analyzed at a center frequency of 2.14 GHz. Experimental results for linearity improvement using the proposed EPD will be presented in comparison with the simple EPD without the envelope delay tap. II. NONLINEARITY GENERATED BY THE MEMORY EFFECT The memory effects in high-power amplifiers appear as an imbalance between the lower and upper IM tones in their magnitudes and phases for two-tone excitation. Therefore, the memory effects of a main amplifier significantly affect the performance of general predistorters. Many previous research efforts have

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was a center frequency as . We where , the impedances around the funassumed damental frequency band were then regarded to be equal as . Generally, the load impedance of a well-matched power amplifier at a fundamental frequency band can be a pure real value by virtue of impedance matching. Equations (2) and (3) can then be simplified as (4) (5)

Fig. 1. Spectral representation for the output signal for a nonlinear system under two-tone excitation.

is a load resistance at a fundamental frequency. where On the other hand, the second-order IM and harmonic voltage components across the load are calculated as

(6) (7) (8) Fig. 2. Simplified amplifier model used for the analysis.

where

been reported that analyze the generation mechanism or accurately measure the effects [7]–[12]. Fig. 1 shows a frequency-domain representation of the nonlinear response of a nonlinear system under two-tone excitation. components, i.e., and terms, are The mainly generated by the third-order nonlinearities. In addition, or harmonic terms ( and the second-order IM term ) can regenerate the components after being modulated with the fundamental components. This regeneration process, which is strongly related to the memory effect, makes a crucial impact on the output performance. To analyze the regeneration process, a simplified amplifier model was utilized, as illustrated in Fig. 2. It had only a drain current source as a nonlinear component and frequency-dependent load impedance, which had only a memory with its imaginary part for simplicity. The nonlinear drain current was represented using a Taylor-series expansion as

and . From the assumption . of narrow tone spacing, term. Equations (7) Equation (6) depicts a low-frequency and (8) show high-frequency second-order harmonic terms. As already mentioned, the second-order IM and harmonic regeneration in conjunction with terms can be involved in the fundamental terms. The regeneration process happens with and the second-order nonlinearities, such as in (1). The regenerated voltages via term, which is more significant, are given by

(9)

(1) The voltage components across the load were mainly term. When the two-tone input signal generated by the , the terms were was applied as found as follows: (10)

(2)

(3)

terms from (9) We additionally have two lower and upper and (10), respectively. They have different phases from the main terms in (4) and (5). In particular, the lower and upper terms, generated from the low-frequency IM term, have negative polarities with respect to each other. Therefore, the total signals can be acquired by a vector sum of the terms in (4)

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Fig. 4. Conceptual diagram of a predistorter with memory effect compensation capability.

III. EPD WITH ENVELOPE DELAY TAPS A. Circuit Configuration Fig. 3. Vector representation of the total IM terms. (a) IM terms after adding the terms generated from the second-order harmonic signals. (b) Eventual lower and upper IM terms after adding the terms generated from the second-order IM signal.

and (9) for the lower and (5) and (10) for the upper , respectively. terms after Fig. 3(a) shows the phase deviation of both adding the terms generated from the second-order harmonic terms, which have signals. Finally, we can have the total different magnitudes and phases between the lower and upper terms, after adding the terms generated from the low-frequency second-order IM term, as shown in Fig. 3(b). As we see in the analysis results, the complex load impedances in the low- and high-frequency second-order IM and harmonic bands result in an imbalance between the lower and upper terms in their magnitudes and phases through the secondorder nonlinearities of the device. Therefore, the memory effect gets worse as the operational class of the amplifier goes down from A to B or C because of an increase of the crossover distortion, which is mainly the second-order nonlinearity. A well-designed bias network could mitigate the memory effect by terminating the second-order IM and harmonic terms so that they have very low impedance. Large tantalum or electrolytic capacitors of about several microfarads at the bias netvoltage. work have been employed to terminate the In addition, a quarter-wave bias feed line has been used because of its low-loss characteristics, as well as its capability of terminating high-frequency second-order harmonic voltages. However, it has been very difficult to completely compensate for the memory effects of the high-power transistors for several reasons. Some of these reasons are the considerable internal parasitic/pre-matching components, high thermal time constant, very low optimum load impedance, and significant crossover distortion because of class-AB operation.

As described, the lower and upper signals of high-power amplifiers generally have different magnitudes and relative phases because of the memory effect. It is very difficult to signals of the power amplifier completely remove the using predistorters because general memoryless predistorters do not have a capability to track the memory effect of the main amplifier. Therefore, better performance could be achieved if the memory effect compensation circuit is added to the conventional memoryless predistorter. Fig. 4 shows the conceptual diagram of the predistorter with memory effect compensation capability. With the help of an additional memory effect compensation circuit, the overall predistorter has a better cantones, as depicted in cellation capability for the unequal Fig. 4. For the concept, shown in Fig. 4, a proposed practical EPD with envelope delay taps for memory effect compensation is shown in Fig. 5. It is composed of an envelope detector, baseband variable-gain amplifiers (VGAs), IM generation circuits, and the memory effect compensation circuit. The memory ef) fect compensation circuit has envelope delay taps ( and baseband VGAs. If one delay tap is considered, the envelope signal is divided and passed through two paths: one is directly passed to the VGAs and supplied to an adder, which is for a basic predistorter operation, and another is passed to VGAs and supplied to an adder for memory after a time delay of effect compensation. The tones, generated by the delayed envelope signal, have different relative phases from the terms generated by the punctual envelope signal. A summation of the tones results in an imbalance on both the magrespective nitudes and phases, as analyzed in Section II. B. Assessment for the Memory Effect Compensation For an easy assessment of memory effect compensation, the circuit blocks of the EPD system were modeled using simplified numerical blocks, as presented in Fig. 6. For two-tone excitation and frequency components, the RF signals of of

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Fig. 6. Operational diagram of the EPD with an envelope delay tap.

Fig. 5. Circuit diagram of the EPD with envelope delay taps for memory effect compensation capability.

and and the low-frequency signals of are expressed as follows:

and (11) (12) (13) (14)

where is the second-order coefficient of the nonlinear transfer function of the envelope detector with a low-pass filter. and are separated for in-phase (I) and quadrature (Q) paths and controlled by the baseband VGAs. These VGAs are modeled using a simple multiplier. They are added to each other by the I/Q adders. The output signals after the I/Q adders are given by

(17) where is the second-order cross-modulation coefficient of can also be obtained with a the IM generation circuits. similar form. The fundamental components having a coefficient in (17) can be ignored since . of is obtained after summing The output signal and . It is then simplified as

(15) (16) where

and

are voltage gains of the VGAs. and . and from (15) and (16) are applied to the IM generators where the predistorted output signals are generated with the fundamental input signals. One of the predistorted signals, i.e., is obtained as

(18) where and

is

, and

in (18) are provided as follows:

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Fig. 7. Amplitude response of IM tones as a function of M and M . (a) Lower and (b) upper IM tones. A = A = 1;  = 5 nS, and a tone spacing = 10 MHz.

(19)

Fig. 8. Phase response of IM tones as a function of M and M . (a) Lower and (b) upper IM tones. A = A = 1;  = 5 nS, and a tone spacing = 10 MHz.

(20) Equations (18)–(20) show that the magnitudes and phases for the lower and upper components are different functions and gains of the VGAs. Based on the of the time delay components were calculated equations, the lower and upper with specific values of the parameters such as a tone spacing of of 5 nS, and a fixed and to 1. Fig. 7 shows 10 MHz, levels in decibels with various and the distributions of in the memory effect compensation circuit. It represents tone over the various a very different distribution of each and values. Using the same condition, very different phase through the various and distributions of each values can be also observed from Fig. 8(a) and (b). To investigate the effect of the time-delay variation, the signals were calculated again with varying from 12.5 to and were fixed to 2 and 2 for this case, re25 nS. variation is shown in Fig. 9. The spectively. The plot of the component are 3.74 dB in magnivariations of the tude and 73.7 in phase [see Fig. 9(a)] while the

component varies 0.61 dB in magnitude and 30.4 in phase [see Fig. 9(b)]. Consequently, it was verified that the magnitones can be septudes and phases of the lower and upper arately controlled using five parameters of and . Therefore, the predistorter with the envelope delay taps has better tracking capability to the memory effect of the main power amplifier. IV. EXPERIMENTAL RESULTS For experimental validation of the proposed EPD, a 30-W PEP class-AB amplifier using Freescale’s LDMOSFET MRF21030 was fabricated. Measured lower and upper ACLRs at the 5-MHz offsets for the fabricated amplifier are presented using a 2.14-GHz four-carrier down-link WCDMA signal (a peak to average power ratio of 9.82 dB) in Fig. 10. Asymmetry of approximately 1–2 dB between the lower and upper ACLR levels is maintained. Agilent’s signal generator of ESG4438C and spectrum analyzer of 8564E were used for the measurements.

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Fig. 11. Comparative PSDs for two-tone excitation.

Fig. 9. IM characteristics for the  sweep. (a) Lower and (b) upper IM components. A = A = 1; M = 2; M = 2; and a tone spacing = 10 MHz.

0

Fig. 12. Comparative PSDs using a four-carrier down-link WCDMA signal at an output power of 35 dBm.

Fig. 10. ACLR characteristics of the fabricated power amplifier.

On the basis of the circuit configuration and simulation results, the overall predistortion system was implemented that linearized the main amplifier. The envelope detector was designed and implemented using a Schottky diode and lumped passive low-pass filter. Baseband VGAs were fabricated using generators a high-speed analog multiplier. Cartesian type were fabricated using a varactor diode, 3-dB hybrid coupler, and a Wilkinson combiner, as shown in Fig. 5. A coaxial delay line was employed to make up an envelope delay tap. After an adjustment of the five parameters ( and ) for the two-tone input, the maximum cancellation for

signals was achieved. Fig. 11 plots power spectral denthe sities (PSDs) for the original output of the amplifier, the output after linearization using the simple EPD (no envelope delay tap), and the output after linearization using the proposed EPD (one envelope delay tap). The cancellation levels, at an output power of 39 dBm using the proposed EPD, are significantly better for component (20.84 versus 11.67 dB) and the both the lower component (18.17 versus 8.50 dB) compared with upper the simple EPD, as indicated in Fig. 11. In the same way, experiments were performed using a fourcarrier down-link WCDMA signal. The ACLR levels were measured and compared for the three cases: without EPD (original amplifier output), using the simple EPD, and using the proposed EPD. From Fig. 12, the original lower and upper ACLR levels of the power amplifier are 39.67 and 41.50 dBc at an output power of 35 dBm. The simple EPD improves them by 7.83 and 5.83 dB, respectively. In addition, the proposed EPD further improves them by 2.33 and 2.67 dB more. The noise floor of the spectrum analyzer was approximately 87 dBm. The ACLR improvements, according to the output power levels, are compared between the simple and proposed EPDs in Fig. 13. The

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Fig. 13. ACLR improvements through the various output power levels.

superior performance of the proposed EPD supports the correct operation of the proposed method for memory effect compensation. V. CONCLUSION One of the fundamental mechanisms of memory effects for high-power amplifiers was analyzed. Based on the analysis, a new EPD, with envelope delay taps for memory effect compensation, was proposed. The circuit configuration and operational principles were explained and analyzed using simplified functional representations for the circuit blocks. The analysis results of the proposed EPD using two-tone excitation were presented. The proposed EPD is capable of controlling the lower components separately in their magnitudes and and upper phases, while the simple EPD only has a capability to control components at the same time. both of the The EPD, including the memory effect compensation circuits, was built on the basis of the analysis and simulation. Experiments, using two-tone and four-carrier WCDMA down-link signals at 2.14 GHz, have proven the feasibility of the proposed EPD for linearization of high-power class-AB amplifiers. The memory effects of the RF power amplifiers have been one of the most critical concerns to those who are developing or studying linearization techniques. Therefore, the simple memory effect compensation technique, proposed in this paper, will be very effective on the performance improvement of the predistorters based on analog/RF circuits.

[4] B. Kim, Y. Yang, J. Cha, Y. Y. Woo, and J. Yi, “Measurement of memory effect of high-power Si LDMOSFET amplifier using two-tone phase evaluation,” in 58th IEEE ARFTG Conf. Dig., San Diego, CA, Nov. 2001, pp. 159–167. [5] W. Woo and J. S. Kenney, “A predistortion linearization system for high power amplifiers with low frequency envelope memory effects,” in IEEE MTT-S Int. Microw. Symp. Dig., 2005, pp. 1545–1548. [6] M. A. Nizamuddin, P. J. Balister, W. H. Tranter, and J. H. Reed, “Nonlinear tapped delay line digital predistorter for power amplifiers with memory,” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, pp. 607–611. [7] J. H. K. Vuolevi, T. Rahkonen, and J. P. A. Manninen, “Measurement technique for characterizing memory effects in RF power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 8, pp. 1383–1389, Aug. 2001. [8] Y. Yang, J. Yi, J. Nam, B. Kim, and M. Park, “Measurement of two-tone transfer characteristics of high-power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 568–571, Mar. 2001. [9] F. M. Ghannouchi, H. Wakana, and M. Tanaka, “A new unequal threetone signal method for AM–AM and AM–PM distortion measurement suitable for characterization of satellite communication transmitters/ transponders,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 8, pp. 1404–1407, Aug. 2001. [10] C. F. Campbell and S. A. Brown, “Application of the unequal two-tone method for AM–AM and AM–PM characterization of MMIC power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., 2001, pp. 103–106. [11] A. A. Moulthrop, C. J. Clark, C. P. Silvia, and M. S. Muha, “A dynamic AM/AM and AM/PM measurement technique,” in IEEE MTT-S Int. Microw. Symp. Dig., 1997, pp. 1455–1458. [12] N. Suematsu, Y. Iyama, and O. Ishida, “Transfer characteristic of IM3 relative phase for a GaAs FET amplifier,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2509–2513, Dec. 1997.

Sung-Chan Jung was born in Seoul, Korea, in 1973. He received the B.S. degree in electronic engineering and the M.S. and Ph.D. degrees in information and communication engineering from Sungkyunwan University, Suwon, Korea, in 1998, 2000, and 2006, respectively. He is currently a Post-Doctoral Researcher with the MCS Laboratory, Sungkyunkwan University. His current research interests include design of high-power amplifiers, linearization techniques, and efficiency enhancement techniques for the base stations and mobile terminals.

Hyun-Chul Park was born in Seoul, Korea, in 1980. He received the B.S. degree in information and communication engineering from Sungkyunkwan University, Suwon, Korea, in 2006, and is currently working toward the M.S. degree in information and communication engineering at Sungkyunkwan University. His research interests include design of high linear and efficient handset power amplifiers with HBT or CMOS processes.

REFERENCES [1] J. Yi, Y. Yang, M. Park, W. Kang, and B. Kim, “Analog predistortion linearizer for high-power RF amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2709–2713, Dec. 2000. [2] C. H. Park, F. Beauregard, G. Carangelo, and F. M. Ghannouchi, “An independently controllable AM/AM and AM/PM predistortion linearizer for cdma2000 multi-carrier applications,” in IEEE MTT-S Int. Microw. Symp. Dig., 2001, pp. 53–56. [3] J. H. K. Vuolevi, J. Manninen, and T. Rahkonen, “Memory effects compensation in RF power amplifiers by using envelope injection technique,” in IEEE MTT-S Int. Microw. Symp. Dig., 2001, pp. 257–260.

Min-Su Kim was born in Seoul, Korea, in 1978. He received the B.S. degree in electronic engineering from Incheon University, Incheon, Korea, in 2005, and is currently working toward the M.S. degree at Sungkyunkwan University, Suwon, Korea. His current research interests include transmitter and linearization techniques.

JUNG et al.: NEW EPD WITH ENVELOPE DELAY TAPS FOR MEMORY EFFECT COMPENSATION

Gunhyun Ahn was born in Seoul, Korea, in 1978. He received the B.S. degree in electronic engineering from Anyang University, Anyang, Korea, in 2005, and is currently working toward the M.S. degree in information and communication engineering at Sungkyunkwan University, Suwon, Korea. His research interests include Doherty amplifier design and linearization techniques.

Ju-Ho Van was born in Iksan, Korea, in 1982. He is currently working toward the B.S. degree in information and communication engineering at Sungkyunkwan University, Suwon, Korea. His research interests include high-efficiency switching-mode power amplifiers.

Hoon Hwangbo was born in Cheonan, Korea, in 1976. He received the B.S. degree in electrical engineering and M.S. degree in information and communication engineering from Sungkyunkwan University, Suwon, Korea, in 1999 and 2001, respectively, and is currently working toward the Ph.D. degree at Sungkyunkwan University. His research interests include memory effects and modeling of the microwave packaging devices.

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Cheon-Seok Park (M’02) was born in Seoul, Korea, in 1960. He received the B.S. degree in electrical engineering from Seoul National University, Seoul, Korea, in 1988, and THE M.S. and Ph.D. degrees in electrical AND electronic engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1990 and 1995, respectively. He is currently a Professor WITH the School of Information and Communication Engineering, Sungkyunkwan University, Suwon, Korea. His research interests include design of RF power amplifiers, linearization techniques, and efficiency enhancement techniques.

Sung-Kil Park was born in Gyeongbuk, Korea, in 1967. He received the B.S. degree in electronic engineering from Busan University, Busan, Korea, in 1990, and the M.S. and Ph.D. degrees in electrical and electronic engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1992 and 1998, respectively. From 1998 to 2000, he was with Samsung Electronics, Suwon, Korea. Since May 2000, he has been with the Wave Electronics Company Ltd., Suwon, Korea, where he is currently the Director of the Research Center. His research interests include design of high-power amplifiers, linearization techniques, and efficiency enhancement techniques for base stations.

Youngoo Yang (S’99–M’02) was born in Hamyang, Korea, in 1969. He received the Ph.D. degree in electrical and electronic engineering from the Pohang University of Science and Technology (Postech), Pohang, Korea, in 2002. From 2002 to 2005, he was with Skyworks Solutions Inc., Newbury Park, CA, where he designed power amplifiers for various cellular handsets. Since March 2005, he has been with the School of Information and Communication Engineering, Sungkyunkwan University, Suwon, Korea, where he is currently an Assistant Professor. His research interests include power-amplifier design, RF transmitters, RF integrated-circuit (RFIC) design, and modeling of high-power amplifiers or devices.

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A Low Phase-Noise Voltage-Controlled SAW Oscillator With Surface Transverse Wave Resonator for SONET Application Jon-Hong Lin, Student Member, IEEE, and Yao-Huang Kao, Member, IEEE

Abstract—A surface transverse wave (STW) resonator-based oscillator was developed in response to SONET OC-48 application. To meet the low jitter objective, a high- STW resonator was designed and fabricated in this study. The residual phase-noise measurement techniques are used to evaluate the feedback oscillator components, such as the loop amplifier, STW resonator, and electronic phase shifter, which can play important roles in determining the oscillator’s output phase-noise spectrum. The oscillator’s white phase-noise floor is 170 dBc/Hz for carrier-offset frequency greater than 1 MHz. The oscillator’s phase-noise level is 67 dBc/Hz at a 100-Hz carrier offset. Both low close-in phase-noise and low white phase-noise floor makes the oscillator meet low jitter requirement. The electronic frequency tuning range exceeds 200 ppm. The oscillator provides 13.5 dBm of output power and consumes 65 mA from 5–V power supply.

Fig. 1. Block diagram of a feedback loop oscillator.

+

Index Terms—Phase noise, SONET, surface acoustic wave (SAW), voltage-controlled oscillators (VCOs).

I. INTRODUCTION

I

N HIGH-SPEED digital communication systems, a clock recovery circuit is used for data integrity. The clock is usually extracted from a phase-locked-loop circuit with a low jitter voltage-controlled oscillator (VCO). Due to the availability of high frequency and high-quality (high- ) resonator, the VCOs were mostly fabricated at 622 MHz either by the fourth harmonic of 155-MHz crystal oscillator (VCXO) or directly by the 622-MHz voltage-controlled saw oscillator (VCSO) [1], [2]. Both cases suffer from the degradation factor dB on the phase noise as applications to of OC-48 at 2488.32 MHz. Seldom studies on 2488 MHz have been presented [3]–[5]. In this study, a highly stable VCSO with a surface transverse wave (STW) resonator working directly at 2488.32 MHz is developed. The high- STW resonator on quartz was demonstrated in 1987 [6]. Its advantages over the conventional Rayleigh waves are the very high velocity and low propagation loss [7]. The wave velocity of the STW is approximately 5000 m/s. High wave velocity makes the

Manuscript received December 29, 2005; revised September 10, 2006. This work was supported by the ftech Company and by the Southern Taiwan Science Park Administration under Contract 92-1001-B062-001. J.-H. Lin is with the Institute of Communication Engineering, National Chiao-Tung University Hsin-Chu, Taiwan 30050, R.O.C. (e-mail: [email protected]). Y.-H. Kao is with the Department of Communication Engineering, Chung-Hua University, Hsin-Chu, Taiwan 300, R.O.C. (e-mail: yhkao@chu. edu.tw). Digital Object Identifier 10.1109/TMTT.2006.888575

inter-digit transducer (IDT) wider and then slightly reduces the requirement of the photolithography process. A coupled-mode resonator is carefully designed to accommodate the request of wide band tuning and low phase-noise applications. The unloaded quality factor equal to 5500 was realized in this study. It is essential to choose the proper elements to achieve low phase noise. Here, the residual phase noise in each module is carefully evaluated. Residual noise is the noise added to a signal when the signal is processed by a two-port device and it is composed of both AM and FM components. The residual phase-noise measurement techniques [8] are employed to evaluate the components of the feedback loop oscillator, such as the loop amplifier, STW resonator, and electronic phase shifter, which can play important roles in determining the oscillator’s output phase-noise spectrum. Agilent’s E5503B phase-noise measurement system was used to measure the residual phase noise of each component and the phase noise of the VCSO. The phase noise of this oscillator is approximately 67 dBc/Hz at 100-Hz offset. The measured white phase-noise floor of the oscillator is 170 dBc/Hz. Both low close-in phase-noise and low white phase-noise floor makes the oscillator meet low jitter requirement. The electronic frequency tuning range exceeds 200 ppm. The oscillator provides 13.5 dBm of output power and consumes 65 mA from 5-V power supply. High output power makes the oscillator be able to drive emitter-coupled logic (ECL) circuits directly without using an extra buffer amplifier. II. OSCILLATOR DESIGN It is noted that the oscillator with a one-port surface acoustic wave (SAW) resonator suffers from large parasitic capacitance from IDTs. Here, the architecture with a two-port resonator forming a feedback loop is chosen as shown in Fig. 1.

0018-9480/$25.00 © 2006 IEEE

LIN AND KAO: LOW PHASE-NOISE VOLTAGE-CONTROLLED SAW OSCILLATOR WITH STW RESONATOR FOR SONET APPLICATION

It consists of a single-loop amplifier, an electronic phase shifter, a lumped-element reactive Wilkinson power splitter, a lumped-element reactive phase adjusting, and a two-port STW resonator. The resonator acts as a short circuit with zero phase shift at the desired frequency. No output buffer amplifier is used because it may degrade the oscillator’s white phase-noise floor. The oscillation starts as the closed-loop gain satisfies Barkausen’s criteria. During design phase, the open-loop gain is evaluated by breaking the loop at the appropriate plane with equal input and output impedances, as noted by line AB in Fig. 1. Here, the impedances are 50 , seen from network analyzer measurement. The impedance of each module is actually set to 50 for convenience. This approach has the advantage that the noise characteristics of the individual component as measured in an open-loop configuration have a direct bearing on the closed-loop phase noise of the oscillator. To achieve low insertion loss, high frequency, and highfactor, the SAW resonator with STW is employed. The STW is a shear wave with very high velocity and energy trapping. It can reduce the diffraction of the shallow bulk wave into the substrate and lead to the decrease in device insertion loss and increase in the resonator . The width of the transducer is approximately 0.5 m. The overlap aperture is approximately 250 m. This larger transducer width also makes it possible to mass produce the resonator with acceptable yield. Since the STW does not associate volume charge with propagation, its propagation loss is small [7] To achieve the proper turnover temperature, the ST-cut quartz 90 , Euler angles (0 , 132, 75 , 90 ), is employed as the substrate of this resonator, which has the turnover temperature approximately at 45 C. This feature makes the oscillator without extra temperature compensation circuit work well in the real environment. It simplifies circuit design and lowers the cost. The resonator is detailed in Fig. 2(a) and (b). The die size is 1.8 mm 1.2 mm. Two shorted reflectors (90 fingers) are placed outside the input and output IDTs (100 pairs). A shorted grating with three fingers is placed between the input and output IDTs. The resonant modes formed by input and output IDTs are coupled just as two coupled parallel LC resonators. The coupling is carefully tuned by the central grating. This results in a two-mode response such as that shown in Fig. 2(c). Due to the grounding grating, the insertion loss is reduced to 4–5 dB, which is much smaller than that of 10–15 dB in a conventional SAW or STW delay line. [5] The approximate linear phase change obtained within the 3-dB bandwidth equals to 1.713 10 rad/Hz. The upper and lower limits of the phase change are above 90 . The is equal to 1537. estimated loaded factor The group delay is approximately 1.713 10 rad/Hz. High insertion loss out of the passband is revealed in Fig. 2(d). The spurious out of passband is suppressed under 30 dB. The center frequency is trimmed to 2488.32 MHz. The residual phase noise is as shown in Fig. 2(e). It is measured by applying the drive power approximately the same as the power in the steady-state oscillation condition. The corner of the flicker noise is out of the scope of this measurement. The HBT monolithic amplifier is selected as the loop amplifier because of low noise figure and high dynamic range. The dB is at 17 dBm and the bandwidth is 4 GHz. Its bandwidth

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Fig. 2. (a) Photograph and (b) block diagram of the STW resonator. (c) Phase and insertion loss responses from 2.486 to 2.490 GHz. (d) Insertion loss from 2.468 to 2.508 GHz. (e) Residual phase noise of STW resonator.

was properly selected to prevent high second harmonics. The nominal gain of 17 dB is much greater than that required to overcome the total loop losses to insure the stable oscillation. The magnitude of gain variation over temperature is approximately

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Fig. 5. Total phase shift and the open-loop gain at the oscillation frequency.

Fig. 3. Residual phase noise of loop amplifier.

Fig. 4. Residual phase noise of electronic phase shifter.

0.005 dB C and this feature can prevent the AM–PM noise caused by the temperature variation. The residual phase noise is shown in Fig. 3. The noise floor is approximately 170 dBc/Hz with a flicker noise corner at 17-kHz offset. Since the white phase-noise floor will be raised by a resistive attenuator in the feedback loop, an unequal Wilkinson power splitter is employed to adjust the excess loop gain instead of a resistive attenuator [9]. To save the volume, the circuit is realized by the lump reactive components instead of a transmission line. The electronic phase shifter is used to tune the oscillation frequency. The electronic phase shifter is constructed with inductors and two silicon tuning diodes using a T-circuit structure. The phase noise and tuning linearity will be affected by the tuning diodes. High residual phase noise of the tuning diodes will degrade the phase noise of the VCO. The residual phasenoise performance is shown in Fig. 4. With proper selection of varactor diodes, the high tuning linearity and low residual phase noise are achieved at the same time.

The phase shift of the loop amplifier and power splitter is approximately equal to 80 and 90 , respectively. The electronic phase shifter is approximately 40 . Since the total rad, another 10 is phase shift around the loop must be required, which is from the loop phase adjust constructed with fixed lumped reactive components. The frequency dependences of the total phase shifter and open-loop gain seen from the reference plane A-B line indicated in Fig. 1 are shown in Fig. 5. V Curves and are the total phase shift with and V, respectively. Curves and are the reV and V, respectively. spective open-loop gain with The group delay is approximately 1.74 10 rad/Hz. As compared to Fig. 2(c), we see that the SAW resonator dominates the change of the total phase shift. The slight increase in group delay may be from the tunable phase shifter with varactors. The oscillation frequency is predicted at the zero-crossing point with enough gain margins at approximately 2 dB. This gives us the benefit of low flicker noise from the amplifier without deep gain compression. The tuning bandwidth is approximately equal to the resonator’s 1-dB bandwidth. It is approximately from 2487.85 to 2488.85 MHz. III. OSCILLATOR PERFORMANCES The performances of the oscillator with an STW resonator are measured. The narrow and wide scan of the output spectrum and relative levels of the harmonic are shown in Fig. 6(a) and (b), respectively. Since the STW resonator do not have a second harmonic response and the bandwidth of the loop amplifier is limited under 4 GHz, the second harmonics of the oscillator is suppressed below 58 dB, as shown in Fig. 6(b), without any output low-pass filter. The tuning characteristic is shown in Fig. 7(a) with 200-ppm range and good linearity. The frequency dependence on temperature is illustrated in Fig. 7(b). The turnover temperature is approximately 45 C, which is mainly determined by the STW resonator. The frequency discriminator method was used to measure the phase noise of the VCSO [11], [12]. The delay line frequency discriminator is implemented with Agilent’s E5503B phase noise measurement system and a delay line. Unlike the phase detector method, the frequency discriminator method does not require a second reference source phase locked to the device-under-test (DUT). This makes the frequency discriminator method extremely useful for measuring sources that are difficult

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Fig. 6. (a) Measured output spectrum for the 2488.32-MHz and (b) harmonic spectrum.

to phase lock, especially for the free-running VCOs that are drift quickly. The delay line frequency discriminator converts short-term frequency fluctuations of the source into voltage fluctuations that can be measured by a baseband analyzer. The conversion is a two-part process, first converting the frequency fluctuations into phase fluctuations, and then converting the phase fluctuations into voltage fluctuations. The output power of the DUT is divided between a delay line and the local oscillator input of a double-balanced mixer. The frequency fluctuations to phase fluctuations transformation takes place in the delay line. As the frequency changes slightly, the phase shift incurred in the fixed delay time will change proportionally. The delay line converts the frequency change at the line input into the phase change at the line output when compared to the undelayed signal arriving at the mixer in the second path. The double-balanced mixer, acting as a phase detector, transforms the instantaneous phase fluctuations into voltage fluctuations. With the two signals 90 out of phase, the IF voltage output of the mixer is proportional to the input phase fluctuations. The voltage fluctuations can then be measured by the baseband analyzer and converted to phase-noise units. A 312-ns delay line implemented by an ANDREW LDF5–50A low-loss coaxial cable was used in the measurement. Long enough delay time is needed to ensure the sensitivity of measurement system. The phase noise of the oscillator is measured as shown in Fig. 8. The measured parameters of this VCSO and the specifications of the other commercial products are summarized in Table I.

Fig. 7. Dependence of the oscillation frequency on: (a) tuning voltage and (b) temperature.

near the The spectral shape in curve 1 indeed arrears carrier. The intersection point with the curve is around 50-kHz offset. The phase noise is 153 dBc/Hz at 100-kHz offset, which is lower than those indicated in Table I. To analyze the shaping behavior of the close loop, the residual phase noises are examined. For comparison, the residual phase noises in components are also presented in Fig. 8. It reveals that the noise from the STW resonator (curve 2) is dominant at approximately 10 dB above those from the amplifier (curve 3) and phase shifter (curve 4). The phase shifter has the same order of magnitude as that in the loop amplifier. The system’s floor is also indicated as shown by curve 5, which is much lower than the measured items. As referred to Fig. 5, the magnitude of the loop gain under steady state is assumed to one with a rather wide bandwidth, at least 500 kHz at 2488.32 MHz. The phase is assumed linear with slope or group delay , which roughly equals to 1.74 10 rad/Hz in our case, within the limited bandwidth.

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frequency of 2.48 GHz to avoid the degradation of phase noise due to frequency multiplication. Two-coupled modes are designed to achieve wide tuning. Trimming the central grating while keeping the steep phase change enhances the tuning bandwidth of resonator. The tuning capability achieves 200 ppm. In comparison with the other commercial products, the phase-noise performance of this study is better than 8 dB at 100-kHz offset frequency. The white phase-noise floor is approximately 170 dBc/Hz in this study. The phase-noise near carrier is confirmed to follow the prediction of Leeson’s model. It is concluded that the behavior of phase noise is dominated by the residual noise of the SAW resonator and is shaped by the important factor of group delay.

ACKNOWLEDGMENT Fig. 8. Measured phase noise for the 2488.32-MHz STW oscillator.

TABLE I MEASURED RESULT FOR THE VOLTAGE-CONTROLLED STW OSCILLATOR AND COMPARISON WITH THE OTHERS

The authors would like to thank Dr. L. Wu, ftech Company, Tainan, Taiwan, R.O.C., for fabricating the STW resonator and Dr. B. Temple, Agilent Technologies, Spokane, WA, for the measurement of phase noise.

REFERENCES

Hence, the normalized open-loop gain is written as , is the offset frequency from the center frequency. where The closed-loop gain is then obtained as Closed-loop gain

According to [9], the power spectral density (PSD) of phase noise can be shaped by multiplying the square of the closed-loop gain to the residual phase noise. Here, the shaping factor for [10]. As near to the carrier with the PSD is small , the shaping appears as . It is concluded that the phase noise is indeed shaped from the residual noise term, which is originated from the high- resby the onator. The calculated results are in good agreement with the measurements. IV. CONCLUSIONS A STW-based oscillator was designed and fabricated in this study. The resonator is operated directly at the specific

[1] O. Ishii, H. Iwata, M. Sugano, and T. Ohshima, “UHF AT-CUT crystal resonators operating in the fundamental mode,” in IEEE Int. Freq. Control Symp., 1998, pp. 975–980. [2] N. Nomura, M. Itagaki, and Y. Aoyagi, “Small packaged VCSO for 10 Gbit Ethernet application,” in IEEE Int. Ultrason., Ferroelectr., Freq. Control Symp., 2004, pp. 418–421. [3] B. Fleischmann, A. Roth, P. Russer, and R. Weigel, “A 2.5 GHz low noise phase locked surface transverse wave VCO,” in IEEE Int. Ultrason., Ferroelectr., Freq. Control Symp., 1989, pp. 65–69. [4] I. D. Avramov, “Very wide tuning range, low-noise voltage controlled oscillators using ladder type leaky surface acoustic wave filters,” in IEEE Ultrason., Ferroelectr., Freq. Control Symp., 1998, pp. 489–496. [5] C. E. Hay, M. E. Harrell, and R. J. Kansy, “2.4 and 2.5 GHz miniature, low-noise oscillators using surface transverse wave resonators and a SiGe sustaining amplifier,” in IEEE Ultrason., Ferroelectr., Freq. Control Symp., 2004, pp. 174–179. [6] T. L. Bagwell and R. C. Bray, “Novel surface transverse wave resonators with low loss and high ,” in Proc. IEEE Ultrason. Symp., 1987, pp. 319–324. [7] H. Ken-ya, Surface Acoustic Wave Devices in Telecommunications. Berlin, Germany: Springer-Verlag, 2000. [8] G. K. Montress, T. E. Parker, and M. J. Loboda, “Residual phase noise measurements of VHF, UHF, and microwave components,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 41, no. 5, pp. 664–679, Sep. 1994. [9] T. E. Parker and G. K. Montress, “Precision surface acoustic wave (SAW) oscillator,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 35, no. 3, pp. 342–364, May 1988. [10] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE, vol. 54, no. 2, pp. 329–330, Feb. 1966. [11] D. Scherer, “Design principles and measurement of low phase noise RF and microwave sources,” presented at the Hewlett-Packard RF and Microw. Meas. Symp. and Exhibition, Hasbrouck Heights, NJ, 1979. [12] C. Schiebold, “Theory and design of the delay line discriminator for phase noise measurement,” Microw. J., pp. 103–112, Dec. 1983. [13] I. D. Avramov, O. Ikata, T. Matsuda, and Y. Satoh, “High-performance surface transverse wave based voltage controlled feedback oscillators in the 2.0 to 2.5 GHz range,” in Proc. IEEE Int. Freq. Control Symp., 1998, pp. 519–527. [14] J. H. Lin and Y. H. Kao, “A low phase noise voltage controlled saw oscillator with surface transverse wave resonator for SONET application,” in IEEE Int. Asia–Pacific Microw. Conf., Suzhou, China, 2005, vol. 2, pp. 973–976.

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LIN AND KAO: LOW PHASE-NOISE VOLTAGE-CONTROLLED SAW OSCILLATOR WITH STW RESONATOR FOR SONET APPLICATION

Jon-Hong Lin (S’04) was born in Tainan, Taiwan, R.O.C., in 1970. He received the B.S. degree in electrical engineering from National Cheng-Kung University, Hsin-Chu, Taiwan, R.O.C., in 1994, the M.S. degree in communication engineering from National Chiao-Tung University, Hsin-Chu, Taiwan, R.O.C., in 2001, and is currently working toward the Ph.D. degree in communication engineering at National Chiao-Tung University.

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Yao-Huang Kao (M’76) was born in Tainan, Taiwan, R.O.C., in 1953. He received the B.S., M.S., and Ph.D. degrees in electronic engineering from National Chiao-Tung University, Hsin-Chu, Taiwan, R.O.C., in 1975, 1977, and 1986, respectively. In 1986, he joined the Department of Communication Engineering, National Chiao-Tung University. He is currently with the Department of Communication Engineering, Chung-Hua University, Hsin-Chu, Taiwan, R.O.C., where he is a Professor. He has also been a Visiting Scholar involved with research in nonlinear circuits with the University of California at Berkeley (1988) and Bell Communication Research (Bellcore) (1989). He is also a Technical Consultant for RF circuits for both industry and government institutes. His current research interests involve nonlinear dynamics and chaos, high-speed optical communications, and microwave and CMOS RF circuit designs.

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Phase-Noise Reduction of -Band Push–Push Oscillator With Second-Harmonic Self-Injection Techniques To-Po Wang, Student Member, IEEE, Zuo-Min Tsai, Student Member, IEEE, Kuo-Jung Sun, and Huei Wang, Fellow, IEEE Abstract—A low phase-noise -band push–push oscillator using proposed feedback topology is presented in this paper. The oscillator core was implemented in a 0.18- m CMOS process. By using a power splitter and a delay path in the feedback loop connecting the output and current source of the oscillator, a part of the oscillator output power injects to the oscillator itself. With the proper phase delay in the feedback loop and high transconductance of the current source, a low phase-noise oscillator is achieved. The amplitude stability and phase stability are analyzed, the phenomena of the phase-noise reductions are derived, and the device-size selections of the oscillator are investigated. The time-variant function, impulse sensitivity function, is also adopted to analyze the phase-noise reductions of the second-harmonic self-injected push–push oscillator. These theories are verified by the experiments. This self-injected push–push oscillator achieves low phase noise of 120.1 dBc/Hz at 1-MHz offset from the 9.6-GHz carrier. The power consumption is 13.8 mW from a 1.0-V supply voltage. The figure-of-merit of the oscillator is 188.3 dBc/Hz. It is also the first attempt to analyze the second-harmonic self-injected push–push oscillator. Index Terms—Current source, delay line, figure-of-merit, high- resonator, self-injection.

I. INTRODUCTION HE increasing demands on wireless data communication have motivated the development of RF front-end circuits toward tens of gigahertz. Being a crucial component in wireless systems, voltage-controlled oscillators (VCOs) impose restrictions on both active and passive devices for the technology of choice. As the CMOS feature size advances to deep submicrometer, CMOS fundamental oscillators at frequencies up to millimeter-wave range were reported [1], [2]. However, it usually requires expensive process technology, while delivering low output power. In order to implement low-cost oscillators for high-frequency applications, a cross-coupled push–push oscillator using a 0.25- m CMOS technology was proposed to achieve an output frequency twice as high as the fundamental frequency with good fundamental rejection, even though the of the transistors output frequency is higher than the [3]–[5]. Though demonstrating high oscillating frequency and

T

Manuscript received January 30, 2006; revised August 3, 2006. This work was supported in part by the National Science Council under Contract NSC 93-2752-E-002-002-PAE, Contract NSC 93-2219-E-002-024, and Contract NSC 93-2213-E-002-033. The authors are with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.886912

acceptable output power, the push–push VCO phase noise is usually limited due to lack of high- on-chip inductors. To overcome the poor phase noise in a free-running oscillator, low phase-noise oscillators usually use the high- resonators to stabilize the oscillating signals for reducing the noise components [6]–[9]. With a high- factor, the VCO will be more stable because the phase fluctuation and frequency fluctuation approximate zero. Injection lock with external stable signal source is another technique used in Si-based circuit design such as frequency dividers [10]–[12] and quadrature generation [13]. With an external injection low-noise signal, the locking range and phase noise of the injection-locked oscillator is associated with the injection signal strength and the factor of the resonator of the VCO. The larger the injection signal amplitude, the wider the locking range, and the better the phase noise will be. The feedback-loop technique is applied in lasers [14], [15] or used to stabilize the oscillators [16]–[18]. The regular self-injection-locked oscillator [19]–[21] is shown in Fig. 1(a) where the self-injection signal has the same frequency as the oscillator. The oscillator output signal goes through the circulator, and then into the input port of the power divider. Part of the oscillator output signal feeds back to the circulator as the self-injection signal. In the feedback loop, the delay cable, high- factor resonator, or amplifier, may be used. The phase noise is reduced after self-injection locking as compared to the phase noise of the free-running oscillator while satisfying the stability conditions. In this paper, a proposed feedback topology is presented. Part of the oscillator output power injects to the current source directly rather than going through the circulator and returning to the oscillator output port. Under the stable oscillation condican be transformed tions, the feedback push–push signal to a stable signal with larger amplitude to improve the oscillator phase noise. The larger signal injection can be achieved beof the current source cause of the larger transconductance rather than the small-signal injection in the reported literatures [19]–[21]. The phase noise of the push–push oscillator using the proposed feedback topology shown in Fig. 1(b), without a circulator and an external amplifier, rivals the phase noise of the push–push oscillator using the regular feedback topology with a circulator shown in Fig. 1(a). The analysis procedures in this paper are based on the methods proposed in [20]–[27]. However, it is the first attempt to analyze the amplitude and phase stability, as well as phase-noise reduction for the proposed second-harmonic self-injected push–push oscillator. This paper is organized as follows. In Section II, the second-harmonic self-injected oscillator topology is presented,

0018-9480/$25.00 © 2006 IEEE

WANG et al.: PHASE-NOISE REDUCTION OF

-BAND PUSH–PUSH OSCILLATOR WITH SECOND-HARMONIC SELF-INJECTION TECHNIQUES

Fig. 1. Setups for: (a) the regular self-injection-locked oscillator and (b) the proposed self-injected push–push oscillator.

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Fig. 2. Circuit topology of the self-injected push–push oscillator.

the stability conditions of amplitude fluctuation and phase flucand push–push signal tuation of the fundamental signal are analyzed, and the phenomena of the phase-noise are also derived reductions for the push–push signal in detail. Section III gives the circuit design method and device-size selections. Section IV gives experimental results and characterization. Section V gives the conclusions. II. STABILITY CONDITIONS AND PHASE-NOISE ANALYSIS OF THE PROPOSED CIRCUIT TOPOLOGY A. Amplitude Stability Analysis The circuit topology of the second-harmonic self-injected push–push oscillator is shown in Fig. 2. The oscillator consists of a cross-coupled pair – , a current source , a power splitter, and a tunable delay path containing a delay-line cable and a tunable phase shifter. The fundamental frequency of the oscillator is . In order to analyze the behavior of the second-harmonic self-injected oscillator easily, Fig. 3 shows the simplified oscillator model consisting of an representing the tank loss, a LC tank, a conductance , and the mildly nonlinear transconfeedback signal to ). An oscillator’s noise can be modeled ductances ( by a noise-current source or a complex noise admittance [24] shown in Fig. 3. physically reprephysically sents the oscillator amplitude fluctuations and represents the oscillator phase fluctuations. This complex noise is used for the phase-noise reduction analysis. admittance For the amplitude and phase stability analysis, the dynamic equation for this oscillator shown in Fig. 3 is derived as

(1)

Fig. 3. Simplified push–push oscillator model.

The free-running voltages ( and ) with harmonic terms are assumed time variant and are written as

(2) (3) where and are the fundamental and second harmonic amplitudes of oscillation, respectively. is the amplitude mismatch of the fundamental signals for and . is the amplitude mismatch of the second and The and harmonic signals for are the instantaneous phases of fundamental and second harmonic amplitudes of oscillation, respectively.

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For the self-injected push–push oscillator, part of the output signal feeds back to the current source. The current source transwith the mildly nonlinear transconductance forms the feedback signal to a larger current format . With the equivalent parallel resistance for the second harmonic , the feedof the tank is produced back-signal amplitude crossing the tank . This tank amplitude may be larger generated by the than the second harmonic amplitude – if the path loss in the feedback cross-coupled pair rather than the loop is small. This situation is low-power injection in [20] and [21]. The dynamic amplitude behaviors are not considered in [20] and [21] due to the low-power injection. However, the dynamic amplitude behaviors of this study are needed to be considered . because of the large-signal injection For the push–push oscillator in Fig. 3, the differential fundamental signals are almost canceled in the feedback path. Only the second harmonic signal without amplitude mismatch is considered in the feedback path. Other high-order harmonic signals are also neglected due to the relative small amplitudes of oscillations. Therefore, the feedback signal can be written as (4) where is the amplitude-attenuation factor of the delay path is the constant phase parameter. Substitute (2)–(4) into and (1) and separate the real and imaginary parts of the fundamental and second harmonic signal , respectively. The signal real and imaginary parts of the dynamic equations for fundacan be written as mental signal

In order to simplify these equations, the amplitude, phase, and phase derivatives are assumed to vary slowly in (5) and (7). Phase is also assumed to vary slowly in (6) and (8). Therefore, the amplitude and phase dynamic equations (5)–(8) become (9) (10)

(11)

(12) The in (11) is the quality factor defined as of the LC tank. is the equivalent parallel resistance of the tank . For the amplitude stability analfor the fundamental signal ysis, perturbations are characterized as and , where and are the steady-state solutions for (10) and (12), respectively. and are also the amplitude and phase fluctuations of the oscillator, respectively. If these fluctuations are small, (10) and and , respectively, and (12) can be linearized around become (13)

(5) (14) (6) The real and imaginary parts of the dynamic equations for the second harmonic signal can be derived as

(7)

For the stable amplitude, the amplitude fluctuations (13)–(14) for fundamental signal and second harmonic signal need to approach zero, respectively. The stable condition for , which means the tank loss (13) is is smaller than . In order to let (14) approach zero, is zero by assuming that the second harmonic frequency is stable. Equation (14) can then be simplified as

(15) is similar to in (15). In order to let (15) approach zero, the denominator and numerator need to be simultaneously greater than zero or smaller than zero. The conditions approxiand mate

(8)

(16)

WANG et al.: PHASE-NOISE REDUCTION OF

where integer.

-BAND PUSH–PUSH OSCILLATOR WITH SECOND-HARMONIC SELF-INJECTION TECHNIQUES

is the phase delay shown in Fig. 2, and

is an

B. Phase-Stability Analysis For the phase-stability analysis, (9) and (11) are used to anand alyze the phase stability of the fundamental signal for the self-injected push–push second harmonic signal oscillator, respectively. Perturbations are characterized as and , where are the steady-state are the phase fluctuations. Equations solutions and (9) and (11) can then be written as (17)

(18) From (17), the phase fluctuation of the fundamental signal equals zero. Therefore, fundamental frequency will be phase stable at . In (18), assuming the phase fluctuations are small, and is zero due to amplitude stable, then (18) can be linearized around , and it becomes (19) In order to stabilize the phase, the phase fluctuation (19) needs to approach zero. The condition is

(20)

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is the feedback phase delay shown in Fig. 2, and where is an integer. From (19), with a larger transconductance of current source, a larger equivalent parallel resistance of the tank for the fundamental signal , a lower amplitude-attenof the delay path, smaller amplitude imbaluation factor for the second harmonic signal, and higher second ance , the phase fluctuation of the second harmonic amplitude decreases with time more quickly. Thereharmonic signal fore, the oscillator will be phase stable. C. Phase-Noise Reduction of the Self-Injected Push–Push Oscillator For the phase-noise reduction analysis of the push–push , the dynamic equations for the amplitude and signal phase of the self-injected push–push oscillator including the , shown in Fig. 3, complex noise admittance are (21) and (22), shown at the bottom of this page. and in (21) and (22) physically represent the amplitude fluctuation and phase fluctuation of the second har, respectively. Assume (21) and (22) are permonic signal and , where turbed by are the steady-state solutions for (21) and (22), and are the amplitude and phase fluctuations of the second harmonic for the oscillator, respectively. If these fluctuations signal and are small, (21) and (22) can be linearized around become (23) and (24), shown at the bottom of this page. The spectral characteristic of the amplitude and phase fluctuations can be obtained by Fourier transforming (23) and (24). Equations (23) and (24) then become (25) and (26), shown at the bottom of the following page. presents the variables in the transformed or The tilde spectral domain, and is the noise frequency measured relative to the carrier. In (25), the first term on the right-hand side represents the AM-to-AM noise, and the second term represents the conversion of PM noise to AM noise. In (26), the first term on the right-hand side represents the PM-to-PM noise, and the

(21)

(22)

(23)

(24)

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second term on the right-hand side represents the conversion ation (i.e., phase noise) becomes of AM-to-PM noise. For a practical oscillator, the amplitude fluctuation is significantly attenuated by the amplitude limiting (31) mechanism. Therefore, the AM-to-PM noise conversion is negligible in the phase-noise analysis and focus here will be on PM-to-PM noise. As a result, the phase fluctuations (26) of the which can be reduced further by increasing the transconducpush–push oscillator can be arranged as tance of the current source, increasing the equivalent for the fundamental signal parallel-resistance of the tank , reducing the amplitude-attenuation factor of the (27) for second delay path, reducing amplitude imbalance harmonic signal, and increasing second harmonic amplitude where , half the 3-dB bandwidth of the os. From (31), the closer to the carrier frequency, the lower cillator tank circuits. The power spectrum of the self-injected the phase noise can be obtained. However, the phase-noise , improvement will be limited if the free-running oscillator alpush–push oscillator’s fluctuations is given by represents an ensemble average. The ready has good phase noise. This is because a low phase-noise where the notation power spectral density of the oscillator’s phase fluctuation (i.e., oscillator typically has a small half the 3-dB bandwidth of the or a high quality oscillator tank circuits phase noise) is then of the LC tank. From (31), at the frequencies far factor from the carrier, the phase noise reduces to that of free-running oscillator noise properties (28)

(32)

When the oscillator is in the free-running state without selfinjection feedback current . The transconductance can be assumed zero in (28) because the in (28) deals with the feedback signal shown in Fig. 3. The free-running phase noise can be written as

Approximate to : 2) Phase-Delay Loop When the oscillator is in the self-injection state with the phase, the value of the phase-delay loop is set at delay loop , where is an integer. From (15), (16), approximately (19), and (20), the amplitude and phase fluctuations increase with time, and the amplitude and phase difference are unstable. The phase noise becomes worse than that in the free-running state.

(29)

Substituting (29) into (28), the phase fluctuation (i.e., phase noise) of the oscillator becomes

(30) The different phase delays ( in (30) are considered separately as follows. Approximate to : While 1) Phase-Delay Loop the oscillator is in the self-injection state with the phase-delay loop, , shown in Fig. 2, approximates to , where is an integer. From (16) and (20), the oscillator is stable. From (30), the power spectral density of the oscillator’s phase fluctu-

D. Noise Upconversion in Second-Harmonic Self-Injected Push–Push Oscillator The Hajimiri’s time-varying phase-noise theory [26] is also adopted to analyze the second-harmonic self-injected push–push oscillator. The impulse sensitivity function (ISF) is dimensionless with period , which describes how much phase shift results from applying a unit impulse. Since ISF is a periodic function with period , it can be written in a Fourier series (33) where the terms are the real-value coefficients, and is the phase of the th harmonic. With the Fourier coefficients in (33),

(25)

(26)

WANG et al.: PHASE-NOISE REDUCTION OF

-BAND PUSH–PUSH OSCILLATOR WITH SECOND-HARMONIC SELF-INJECTION TECHNIQUES

Fig. 4. ISFs of push–push signals (2f ) for the free-running push–push oscillator without feedback and the stable self-injected push–push oscillator with 45 phase delay (  ).

0

the relationship of the device noise corner frequency and corner in the phase spectrum can be written as [26] the (34) is the where is the twice of the dc value of the ISF and noise uproot mean square value of the ISF. From (34), the conversion can be significantly reduced by minimizing . Since is twice the dc value of the ISF over a period, reduction of the region can be achieved by minimizing phase noise in the the dc value of the ISF [26], [27]. In order to obtain the dc values of the ISFs, the ISFs are calculated using the approximate analytical method in [26]. Although this method is an approximation, it is the easiest to use and rapidly develops important insights into the behavior of an oscillator [26], [27]. Fig. 4 shows the calculated ISFs of for the free-running push–push the push–push signals oscillator without feedback and the stable second-harmonic self-injected push–push oscillator with 45 phase delay ( ). The absolute dc values for the ISFs of the free-running push–push oscillator without feedback and the stable second-harmonic self-injected push–push oscillator with 45 phase delay ( are calculated as 0.0031 and 0.002, respectively. The absolute ISF dc value of the free-running push–push oscillator is 1.55 times larger than the other one. This means that the free-running push–push oscillator has a noise upconversion factor, leading to a worse phase larger noise in the region. From time-varying phase-noise theory [26], the square of the dc value of the ISF is proportional to the region. Therefore, from the ISF calcuphase noise in the region of the free-running lation, the phase noise in the push–push oscillator is approximately 3.8 dB worse than the self-injected push–push oscillator with 45 phase delay. III. CIRCUIT DESIGN A. Oscillator Topology and Phase-Delay Loop Consideration The push–push oscillator core was implemented in a commercial 0.18- m bulk CMOS technology. The circuit

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schematic of the -band self-injected push–push oscillator has been shown in Fig. 2. The oscillator core is composed of an LC tank and a cross-coupled pair with each nMOS of 32 fingers and total gatewidth of 80 m. The larger CMOS devices with lower flicker noise were used in the oscillator design. The total is 160 m. gatewidth of the current source initiates oscillation at the fundamental freThe negativequency across the output of the cross-coupled pair along with the harmonic components. The output node is located in the middle of the inductor, behaving as a virtual ground for the differential mode and an open circuit for the common mode with respect to the cross-coupled pair. The out-of-phase fundamental components are cancelled, while the in-phase harmonic components sum up at this node. As a result, an enhanced component at the frequency twice of the LC resonant frequency appears as the oscillator output. For the self-injected push–push oscillator, a part of the output signal is extracted from the power splitter, then passing through the tunable delay-line path and returning to the gate of the current source. For a stable oscillator, the amplitude fluctuations (15) and phase fluctuations (19) need to approach zero. The amplitude- and phase-stability conditions (16) and (20) have to be satisfied. From (16) and (20), the phase delay shown in Fig. 2 has to be in the range (35) where is an integer. From (30), the phase noise can be further reduced near the carrier frequency by using the high-transconcurrent source with 160- m gatewidth, and ductance using the low-loss delay path with low amplitude-attenuation . A push–push oscillator using a regular self-injecfactor tion delay path was also designed for comparison. B. MOS Device Size Consideration In the oscillator design, the device size has to be considered. The active devices have influences on the oscillator performances such as phase noise, power consumption, oscillating frequency, and the figure-of-merit. In CMOS design, the MOSFET noise) at low frequencies generally exhibits flicker noise ( dea spectral density of the input referred voltage noise scribed in [28, p. 343] (36) where is the process-dependent constant, is the oxide is the gatewidth, and is the gate length. The capacitance, voltage noise is inversely proportional to the frequency , and . Increasing the gate area the MOS gate area with constant and fixed bias results in lower noise. However, this will trade off with power consumption and noise. In order to reduce the noise contribution to the the self-injected push–push oscillator, the 160- m gatewidth nMOS serves as the current source for lower flicker noise ( noise) consideration. For the cross-coupled pair in the oscillator, the simulated performance with different 0.18- m nMOS gate area and different bias conditions are shown in Table I. From this

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TABLE I SIMULATED PERFORMANCE OF THE FREE-RUNNING CROSS-COUPLED OSCILLATORS WITH DIFFERENT DEVICE SIZES AND BIAS CONDITIONS

table, using a larger gate-area device in the oscillator cross-coupled pair, lower oscillator phase noise, better figure-of-merit, and higher output power can be obtained under the same bias conditions and the same device-size current sources. However, using a larger gate-area device will consume a little more dc power. Typically, an oscillator operating at low frequency has better phase noise than that of an oscillator operating at high frequency. A low phase-noise oscillator can be achieved by increasing the power consumption. Therefore, there is a tradeoff between oscillator phase noise and power consumption. In order to evaluate the performance of an oscillator, the figure-of-merit considering the phase noise, carrier frequency, and power consumption is widely adopted in the oscillator papers, e.g., the figure-of-merit is used in [29]–[31], [33], and [39]. The performance of an oscillator evaluated by the figure-of-merit , carrier frequency , including the phase noise , and dc power consumption offset frequency can be defined by

Fig. 5. Simulated phase noise of the free running state, unstable state of selfinjection, and stable state of self-injection for the push–push signal (2f ) of this oscillator.

C. Inductor Consideration mW

(37)

The first and third terms of (37) represent the contributions of phase noise and power consumption to figure-of-merit, respectively. In (37), the values of the figure-of-merit and phase noise are denoted with a negative sign, and the power consumption is denoted with a positive sign. Therefore, the figure-of-merit is linearly proportional to the phase noise, and the figure-of-merit mW . The phase is inversely proportional to the noise has greater and more direct impact on figure-of-merit than does for a fixed operating frethe power consumption . As a result, from (37) and Table I, using larger quency gate-area devices in the cross-coupled pair turns out to be a better design approach in this study. Fig. 5 shows the simulated phase noise of the free-running state, unstable state of self-injection, and stable state of self-injection for the push–push signal of the oscillator with a total 160- m gatewidth current source and a cross-coupled pair with each total 80- m gatewidth nMOS device. The phase noise of the free-running push–push oscillator is improved by using the proposed self-injection feedback loop under stability conditions.

The oscillator uses on-chip inductors in the LC tank, and an external RF choke shown in Fig. 2 is used to isolate supply from the feedback path for the push–push signal voltage for the flexibility consideration. The push–push signal generated from the cross-coupled pair will go to the oscillator output port where it is almost not affected by the supply because of the high-impedance RF choke. voltage For a single-chip self-injected push–push oscillator in the future, from [4] and [5], in order to obtain a high impedance for the second harmonic signal looking toward the bias , the bias circuits of the push–push oscillator consist of coplanar waveguide (CPW) an RF bypass capacitor and a . For an -band push–push osfor the push–push signal cillator design shown in Fig. 2, the bias circuit can be implemented with a 4-pF on-chip bypass capacitor and a 5-nH on-chip miniature 3-D inductor [37]. For the 5-nH on-chip inductor, the measured factor is 6.2 at 10 GHz, and the self-resonance frequency is 16 GHz [37]. Table II gives the simulated free-running oscillator phase-noise data at 1-MHz offset with respect to different factors of a 5-nH on-chip RF-choke inductor. From this table, the phase noise degrades only 0.43 dB (from 116.17

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TABLE II SIMULATED FREE-RUNNING OSCILLATOR PHASE NOISE WITH RESPECT TO -FACTORS OF THE 5-nH ON-CHIP RF-CHOKE INDUCTOR

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to 115.74 dBc/Hz) when the factor reduces by 10 (from 14 to 4). In addition, the oscillator phase noise with respect factors of the 0.7-nH on-chip inductor of the to different LC tank used in this design is also investigated. The simulated free-running oscillator phase noise degrades 6.7 dB (from 117.9 to 111.2 dBc/Hz) when the factors of the 0.7-nH inductor reduce by 10 (from 14 to 4). As a result, the phase noise of the self-injected push–push oscillator is dominated by the factor of the 0.7-nH on-chip inductor of the LC tank. The simulated factor of the 0.7-nH on-chip inductor is 14.5 at 10 GHz in this design. For an oscillator with on-chip RF choke in the future, the simulated phase noise of the free-running push–push oscillator with fully integrated on-chip RF choke approximates 116 dBc/Hz at 1-MHz offset, and this value is similar to that of the free-running push–push oscillator with external RF choke in this design.

Fig. 6. Chip photograph with the chip size 0.61 mm push–push oscillator.

2 0.63

mm of the

IV. MEASURED RESULTS OF THE SELF-INJECTED PUSH–PUSH OSCILLATORS Fig. 6 shows the photograph of the fabricated 0.18- m push–push CMOS oscillator with a chip size of 0.61 mm 0.63 mm including the testing pads. On-wafer probing was performed to characterize the performance of the oscillator. The phase-noise characteristic of the oscillator is measured using the Agilent E4448A spectrum analyzer. The free-running phase noise can be measured directly from the oscillator output without a power splitter and a feedback path or from one of the power splitter’s output ports while the other splitter’s output port is connected with the 50- load. The measured free-running phase noise is similar for these two cases. The free-running phase noise is 116.1 dBc/Hz at 1-MHz offset from the carrier, as shown in Fig. 7. The is 1.0 V, and the dc power oscillator supply voltage consumption is 13.8 mW. The oscillator is self-injected by a part of its output power through the off-chip power splitter, delay-line cable, and tunable phase shifter with the total phase shown in Fig. 2 approximating 45 . From (35), delay the oscillator is in the stable range. The measured output power is 14 dBm at 9.6-GHz oscillating frequency after calibrating the path loss, and the fundamental rejection is 45 dB. Fig. 8 shows the measured phase noise from 100 kHz to for the free-running 10 MHz of the push–push signals push–push oscillator and the stable second-harmonic self-injected push–push oscillator with 45 phase delay .

Fig. 7. Measured phase noise of the free-running push–push oscillator at a 9.6-GHz output frequency. ( 116.1 dBc/Hz at 1-MHz offset from the carrier. The resolution bandwidth is 91 kHz.)

0

From this figure, the phase-noise corner frequency is formed at the offset frequency around 1.1 MHz in the self-injected push–push oscillator. The measured phase noise of the self-injected push–push oscillator with 45 phase delay is improved approximately 4 dB at 1-MHz offset from the free-running push–push oscillator. The dotted line repreportion for the sents the calculated phase noise in the self-injected push–push oscillator with 45 phase delay. The calculated phase-noise reduction is approximately 3.8 dB in the region. The measured phase-noise reduction from 100 kHz to 1 MHz is in agreement with the calculated phase-noise difference. In addition, the measured good 45-dB fundamental rejection for the second-harmonic self-injected push–push oscillator is because of the small amplitude mismatch of the fundamental signals for and and small

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Fig. 9. Measured, simulated, and calculated phase noise with respect to the total phase delays of the self-injected push–push oscillator.

Fig. 8. Phase noise of the free-running push–push oscillator and the secondharmonic self-injected push–push oscillator with 45 total phase delay.

amplitude mismatch of the second harmonic signals and . From (30), the small amplitude misfor of the second harmonic signals and the large match are also the factors second-harmonic oscillation amplitude resulting in the low phase-noise oscillator. At the same bias condition, the output power is 16.5 dBm with a phase noise of 119 dBc/Hz at 1-MHz offset for the stable self-injected oscillator using a regular delay path and a circulator. However, if the proposed oscillator is unstable when the total shown in Fig. 2 approximates 170 not in phase-delay the stable range described in (35), the measured phase noise of the unstable self-injected oscillator is 111 dBc/Hz at 1-MHz offset from the carrier. The phase noise degrades approximately 9.1 dB compared to that of the stable self-injected oscillator ( 120.1 dBc/Hz at 1-MHz) with the same power consumption. Fig. 9 shows the measured, simulated, and calculated phase noise at 1-MHz offset with respect to the total phase delay illustrated in Fig. 2 of the self-injected push–push oscillator. The simulation tool is Agilent’s Advanced Design System (ADS), and the function blocks of the power splitter and phase shifter can be found in the “System-Passive” branch. region is obtained by The calculated phase noise in the estimating the phase-noise reduction with usage of the ISF. The measured, simulated, and calculated phase noises are better than 116.1 dBc/Hz of the free-running state. In region (I) of Fig. 9, the phase delay is from 0 to 90 satisfying the derived amplitude stability condition (16) and derived phase stability condition (20) simultaneously. The measured and simulated phase noises are both better than 116.1 dBc/Hz of the free-running state. In region (II), the total phase delay is from 90 to 270 . These phase delays do not satisfy both the amplitude stability condition (16) and phase stability condition (20). The measured and simulated phase noise is worse than 116.1 dBc/Hz of the free-running state, and a worst 107.5-dBc/Hz phase noise occurs at the total phase delay near 180 in this region. These results can also be observed in Section II-C.2. In region (III), the total phase delay is from 270 to 360 satisfying only the phase stability condition (20). The measured

Fig. 10. Measured output frequencies and output powers with respect to the supply voltages (V ) of the self-injected push–push oscillator.

and simulated phase noise is almost worse than that of the freerunning state. The total phase delay including the probes, power splitter, delay-line cable, and tunable phase shifter are measured by using the HP8510 network analyzer. While a 3-dB attenuator is inserted in the feedback loop to defined in (30), increase the amplitude-attenuation factor the measured phase noise degrades 2.9 dB at 1-MHz offset in the stable self-injection state. When a 6-dB attenuator is inserted in the feedback loop, the phase noise degrades 5.1 dB at 1-MHz offset in the stable self-injected state. While increasing of the current source by only raising the transconductance its gate voltage, the oscillator phase noise improved up to 2.8 dB at 1-MHz offset in the stable self-injection state. These behaviors can be observed from the derived equations (30) and (31). There is a 3-MHz oscillating frequency difference between the free-running state and the self-injected state under the same bias condition. This is due to the fact that the oscillator output loadings of these two cases are slightly different. Fig. 10 shows the measured output frequencies and output powers with respect of the self-injected push–push to the supply voltages oscillator. The frequency of the second-harmonic self-injected push–push oscillator is tuned by adjusting the supply voltages . For the output power higher than 15 dBm, the tuning . While range is 97 MHz by adjusting the supply voltage tuning the frequency, the total phase delay can be adjusted to the stable region by tuning the phase shifter to keep low phase noise. The figure-of-merit described in (37) of the free-running push–push oscillator and the

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-BAND PUSH–PUSH OSCILLATOR WITH SECOND-HARMONIC SELF-INJECTION TECHNIQUES

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TABLE III PERFORMANCE SUMMARIES OF THE RECENTLY REPORTED SI-BASED OSCILLATORS AROUND 10 GHZ WITH THIS STUDY

stable self-injected push–push oscillator are 184.3 and 188.3 dBc/Hz, respectively. Table III compares the recently reported Si-based oscillators around 10 GHz with this study. It is observed that the phase noise and figure-of-merit of the free-running push–push oscillator can be improved by using the proposed second-harmonic self-injection technique. For the proposed second-harmonic self-injected push–push of the current oscillator, with a higher transconductance source, a higher feedback current can be can improve obtained. Increasing the feedback current the phase noise because of the increased oscillation amplitude crossing the tank , where is the equivalent parallel resistance of the tank for the self-injection signal. However, the phase-noise improvement will slow approaches the oscillator down when the tank amplitude shown in Fig. 2. supply voltage Since the proposed self-injection topology does not require a circulator, it has the potential for integrating on a single chip in the future. The bias circuit can be implemented with a 4-pF on-chip bypass capacitor and a 5-nH on-chip miniature 3-D inductor [37] for the RF choke. In order to implement a single-chip self-injected push–push oscillator in the future, the delay path variation due to process variation has to be taken into account. An on-chip tunable delay line can be used to alleviate the path variation [38], but an additional process is required. In practical applications, the temperature-independent reference such as bandgap-reference circuits can be used to alleviate the temperature variation.

V. CONCLUSION The design of the low phase-noise second-harmonic self-injected push–push oscillator has been presented. The low phase noise can be achieved when the total phase delay approximates to the stable region . In addition, with higher transconductance of the current source, lower amplitude-attenuation factor of the for tunable delay path, smaller amplitude mismatch and , and larger the second-harmonic signals in , a better oscillator second-harmonic oscillation amplitude phase noise can be obtained. Moreover, using larger device gate area in the oscillator cross-coupled pair, a better oscillator phase noise and figure-of-merit can also be achieved. Furthermore, this second-harmonic self-injected push–push oscillator without a circulator or an amplifier in the feedback path can achieve low phase noise of 120.1 dBc/Hz at 1-MHz offset and good figure-of-merit of 188.3 dBc/Hz. ACKNOWLEDGMENT The authors would also like to thank M.-F. Lei, National Taiwan University, Taipei, Taiwan, R.O.C., for helpful discussion. The authors also thank Prof. George D. Vendelin, Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C., for editing this paper’s manuscript. The chip was fabricated by TSMC, Hsinchu, Taiwan, R.O.C., through the Chip Implementation Center (CIC), Taiwan, ROC.

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REFERENCES [1] H. Wang, “A 50-GHz VCO in 0.25-m CMOS,” in IEEE Int. SolidState Circuits Conf. Tech. Dig., Feb. 2001, pp. 372–373. [2] L. M. Franca-Neto, R. E. Bishop, and B. A. Bloechel, “64-GHz and 100-GHz VCOs in 90-nm CMOS using optimum pumping method,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2004, pp. 444–445. [3] C. C. Chang, R. C. Liu, and H. Wang, “A 40-GHz push–push VCO using 0.25-m CMOS process,” in Asia–Pacific Microw. Conf. Dig., 2003, vol. 1, pp. 73–76. [4] R. C. Liu, H. Y. Chang, and H. Wang, “A 63-GHz VCO using a standard 0.25-m CMOS process,” in IEEE Int. Solid-State Circuit Conf. Tech. Dig., Feb. 2004, pp. 446–447. [5] P. C. Huang, M. D. Tsai, H. Wang, C. H. Chen, and C. S. Chang, “A 114-GHz VCO in 0.13-m CMOS technology,” in IEEE Int. SolidState Circuit Conf. Tech. Dig., Feb. 2005, pp. 404–406. [6] Y. T. Lee, J. Lee, and S. Nam, “High-Q active resonators using amplifiers and their applications to low phase-noise free-running and voltage-controlled oscillators,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2621–2626, Nov. 2004. [7] S. W. Park, “Theoretical verification on the effect of an additional DR in push–push FET DROs,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 466–468, Nov. 2003. [8] K. Hosoya, K. Ohata, T. Inoue, M. Funabashi, and M. Kuzuhara, “Temperature- and structure-parameters-dependent characteristics of V -band heterojunction FET MMMIC DROs,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 347–355, Feb. 2003. [9] F. X. Sinnesbichler, “Hybrid millimeter-wave push–push oscillators using silicon–germanium HBTs,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 422–430, Feb. 2003. [10] H. R. Rategh, H. Samavati, and T. H. Lee, “A CMOS frequency synthesizer with an injection-locked frequency divider for a 5-GHz wireless LAN receiver,” IEEE J. Solid-State Circuits, vol. 35, no. 5, pp. 780–787, May 2000. [11] S. Verma, H. R. Rategh, and T. H. Lee, “A unified model for injectionlocked frequency dividers,” IEEE J. Solid-State Circuits, vol. 38, no. 6, pp. 1015–1027, Jun. 2003. [12] H. R. Rategh and T. H. Lee, “Superharmonic injection-locked oscillator as low power frequency divider,” in VLSI Circuits Symp. Tech. Dig., 1998, pp. 132–135. [13] A. Mazzanti, P. Uggetti, and F. Svelto, “Analysis and design of injection-locked LC dividers for quadrature generation,” IEEE J. Solid-State Circuits., vol. 39, no. 9, pp. 1425–1433, Sep. 2004. [14] K. Hsu and S. Yamashita, “Single-polarization generation in fiber Fabry–Perot laser by self-injection locking in short feedback cavity,” J. Lightw. Technol., vol. 19, no. 4, pp. 520–526, Apr. 2001. [15] S. Yamashita and G. J. Cowle, “Single-polarization operation of fiber distributed feedback lasers by injection locking,” J. Lightw. Technol., vol. 17, no. 3, pp. 509–513, Mar. 1999. [16] H. Kwon and B. Kang, “Linear frequency modulation of voltage-controlled oscillator using delay-line feedback,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 6, pp. 431–433, Jun. 2005. [17] T. Banky and T. Berceli, “Investigations on noise-suppression effects of nonlinear feed-back loops in microwave oscillators,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2004, pp. 2015–2018. [18] J. E. Rogers and J. Long, “A 10-Gb/s CDR/DEMUX with LC delay line VCO in 0.18-m CMOS,” IEEE J. Solid-State Circuits., vol. 37, no. 12, pp. 1781–1789, Dec. 2002. [19] K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE, vol. 61, no. 10, pp. 1386–1410, Oct. 1973. [20] H. C. Chang, “Stability analysis of self-injection-locked oscillator,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 9, pp. 1989–1993, Sep. 2003. [21] H. C. Chang, “Phase noise in self-injection-locked oscillators-theory and experiment,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 9, pp. 1994–1999, Sep. 2003. [22] B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits., vol. 39, no. 9, pp. 1415–1424, Sep. 2004. [23] H. C. Chang, X. Cao, M. J. Vaughan, U. K. Mishra, and R. A. York, “Phase noise in externally injection-locked oscillator arrays,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 11, pp. 2035–2042, Nov. 1997. [24] H. C. Chang, X. Cao, U. K. Mishra, and R. A. York, “Phase noise in coupled oscillators: Theory and experiment,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 604–615, May 1997. [25] H. C. Chang, E. S. Shapiro, and R. A. York, “Influence of the oscillator equivalent circuit on the stable modes of parallel-coupled oscillators,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 8, pp. 1232–1239, Aug. 1997.

[26] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. [27] J. Choi and A. Mortazawi, “Design of push–push and triple-push oscillators for reducing 1=f noise upconversion,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3407–3414, Nov. 2005. [28] Y. Tsividis, Operation and Modeling of the MOS Transistor, 2nd ed. Boston, MA: Wiley, 1981. [29] N. Fong, J. Plouchart, N. Zamdmer, D. Liu, L. Wagner, C. Plett, and N. Tarr, “Design of wideband CMOS VCO for multiband wireless LAN applications,” IEEE J. Solid-State Circuits., vol. 38, no. 8, pp. 1333–1342, Aug. 2003. [30] S. Ko, H. D. Lee, D. W. Kang, and S. Hong, “An X -band CMOS quadrature balanced VCO,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2004, pp. 2003–2006. [31] N. J. Oh and S. G. Lee, “11-GHz CMOS differential VCO with backgate transformer feedback,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 733–735, Nov. 2005. [32] L. Jia, J. G. Ma, K. S. Yeo, and M. A. Do, “9.3–10.4-GHz-band crosscoupled complementary oscillator with low phase-noise performance,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1273–1278, Apr. 2004. [33] Z. Gu, B. Bartsch, A. Thiede, R. Tao, and Z. G. Wang, “Fully integrated 10 GHz CMOS LC VCOs,” in 33rd Eur. Microw. Conf., 2003, pp. 583–586. [34] S. Li, J. Kipnis, and M. Ismail, “A 10-GHz CMOS quadrature LC-VCO for multirate optical applications,” IEEE J. Solid-State Circuits., vol. 38, no. 10, pp. 1626–1634, Oct. 2003. [35] W. Z. Chen, C. L. Kuo, and C. C. Liu, “10 GHz quadrature-phase voltage controlled oscillator and prescaler,” in Eur. Solid-State Conf., 2003, pp. 361–364. [36] D. Axelrad, E. D. Foucauld, M. Boasis, P. Martin, P. Vincent, M. Belleville, and F. Gaffiot, “A multi-phase 10 GHz VCO in CMOS/SOI for 40 Gbits/s SONNET OC-768 clock and data recovery circuits,” in IEEE RFIC Symp., 2005, pp. 573–576. [37] C. C. Tang, C. H. Wu, and S. I. Liu, “Miniature 3-D inductors in standard CMOS process,” IEEE J. Solid-State Circuits, vol. 37, no. 4, pp. 471–480, Apr. 2002. [38] D. Kuylenstierna, A. Vorobiev, P. Kinner, and S. Gevorgian, “Ultrawide-band tunable true-time delay lines using ferroelectric varactors,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 2164–2170, Jun. 2005. [39] H. Kim, S. Ryu, Y. Chung, J. Choi, and B. Kim, “A low phase-noise CMOS VCO with harmonic tuned LC tank,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, pp. 2917–2924, Jul. 2006.

To-Po Wang (S’05) was born in Hsinchu, Taiwan, R.O.C., in 1975. He received the B.S. degrees in mechanical engineering and in electrical engineering from National Sun Yat-Sen University, Taiwan, R.O.C., in 1998, the M.S. degree in communication engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 2000, and is currently working toward the Ph.D. degree at the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. From 2000 to 2003, he was with the BENQ Corporation, Taipei, Taiwan, R.O.C., where he was engaged in mobile phone research. His research interests are in the areas of RF and millimeter-wave integrated circuits (ICs) in CMOS, SiGe BiCMOS, and compound semiconductor technologies.

Zuo-Min Tsai (S’01) was born in MaioLi, Taiwan, R.O.C., in 1979. He received the B.S. degree in electrical engineering and Ph.D. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2001 and 2006, respectively. His research interests are the theory of microwave circuits.

WANG et al.: PHASE-NOISE REDUCTION OF

-BAND PUSH–PUSH OSCILLATOR WITH SECOND-HARMONIC SELF-INJECTION TECHNIQUES

Kuo-Jung Sun was born in Tainan, Taiwan, R.O.C. He received the B.S. degree in electrical engineering from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1991, and the Masters degree in business and administration (MBA) and M.S. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1997 and 2005, respectively. His research interests include the design of RF integrated circuits (RFICs) and mixed-signal circuits.

Huei Wang (S’83–M’87–SM’95–F’06) was born in Tainan, Taiwan, R.O.C., on March 9, 1958. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1980, and the M.S. and Ph.D. degrees in electrical engineering from Michigan State University, East Lansing, in 1984 and 1987, respectively. During his graduate study, he was engaged in research on theoretical and numerical analysis of electromagnetic radiation and scattering problems. He was also involved in the development of mi-

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crowave remote detecting/sensing systems. In 1987, he joined the Electronic Systems and Technology Division, TRW Inc. He has been an MTS and Staff Engineer responsible for monolithic-microwave integrated-circuit (MMIC) modeling of computer-aided design (CAD) tools, MMIC testing evaluation, and design, and then became the Senior Section Manager of the Millimeter-Wave Sensor Product Section, RF Product Center, TRW Inc. In 1993, he visited the Institute of Electronics, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., to teach MMIC-related topics. In 1994, he returned to TRW Inc. In February 1998, he joined the faculty of the Department of Electrical Engineering, National Taiwan University, as a Professor. He was elected the first Richard M. Hong Endowed Chair Professor of National Taiwan University in 2005. Dr. Wang is a member of Phi Kappa Phi and Tau Beta Pi. He is a Distinguished Microwave Lecturer for the 2007–2009 term. He was the recipient of the Distinguished Research Award of the National Science Council, R.O.C. (2003–2006).

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Gold-Plated Micromachined Millimeter-Wave Resonators Based on Rectangular Coaxial Transmission Lines Eric D. Marsh, Member, IEEE, James Robert Reid, Member, IEEE, and Vladimir S. Vasilyev

Abstract—Resonators based on microfabricated rectangular coaxial (recta-coax) transmission lines are presented. The resonators were fabricated using a multilayered nickel electroplating process that enables the fabrication of three-dimensional structures. As a method to improve performance, gold was plated on the resonators using an electroless process. A model for predicting the effect of the plating on a resonator quality ( ) factor is presented and verified by measurements. Measured resonators show that the plating can increase the factor of nickel resonators by over 50% at frequencies above 44 GHz. Utilizing gold plating, 50resonators are demonstrated at 44 and 60 GHz with unloaded factors of 213.1 and 242.3, respectively.



Index Terms—Micromachining, millimeter-wave resonators, factor, transmission line resonators.

I. INTRODUCTION

Fig. 1. Three-dimensional model of recta-coax transmission line showing the four design dimensions.

TABLE I COMPARISON OF DIFFERENT METALS FOR USE WITH RECTA-COAX LINES

R

ECTANGULAR coaxial (recta-coax) transmission lines offer significant advantages for the design of integrated millimeter-wave circuits when compared to microstrip or coplanar waveguide transmission lines [1]. As shown in Fig. 1, recta-coax lines consist of a rectangular center line suspended in air and enclosed on all sides by a ground plane. Since recta-coax lines are shielded, lines and components can be closely spaced and even pass over each other with minimal crosstalk, allowing for complex, yet compact signal routing networks. The shielding also makes the performance of the recta-coax lines independent of the substrate material, allowing the substrate to be chosen based on optimum cost and/or active device performance and allowing recta-coax designs to be reused in different surroundings and on different substrates without additional simulations. Finally, recta-coax lines are TEM lines, which make them nondispersive and, therefore, useful over a very broad frequency range (dc to over 200 GHz). Recta-coax lines have traditionally been fabricated utilizing a single metal such as nickel [1], [2] or copper [3]–[5]. However,

Manuscript received July 17, 2006; revised September 27, 2006. This work was supported in part by the Air Force Office of Scientific Research under Grant LRIR 92SN04COR, and in part by the Center for Advanced Sensor and Communications Antennas. E. D. Marsh and J. R. Reid are with the Antenna Technology Branch, Air Force Research Laboratory, Hanscom AFB, MA 01731 USA (e-mail: [email protected]; [email protected]). V. S. Vasilyev is with S4 Inc., Burlington, MA 01803 USA (e-mail: vladimir. [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/TMTT.2006.888947

neither of these metals is ideal. Nickel lines suffer significant losses due to the relatively poor conductivity of nickel as compared to gold, copper, or silver (see Table I for a comparison of these metals), while both nickel and copper oxidize, resulting in degraded performance over time. Gold has often been used for integrated transmission lines due to its relatively good conductivity and high corrosion resistance. However, gold is a poor structural material because it is very malleable. One potential solution is to plate nickel or copper lines with a thin gold layer. For the nickel lines, a thickness of 1–2 m would be ideal, while for copper lines, a very thin 0.1–0.2- m layer would be preferred. In this study, we demonstrate the plating of gold on nickel recta-coax transmission lines. The plating is done in individual steps to provide uniform coating on all surfaces of the resonators. A one-step plating process is shown to deposit approximately 0.5 m of gold and increase the quality ( ) factor of 50 , 60-GHz resonators by 48% over pure nickel resonators, while a two-step plating process results in a film factor by approximately 0.8- m thick and improves the 55% over a pure nickel resonator. A model for calculating the resonator factor as a function of design dimensions and plated metal thickness is also presented. The model is shown to

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MARSH et al.: GOLD-PLATED MICROMACHINED MILLIMETER-WAVE RESONATORS BASED ON RECTA-COAX TRANSMISSION LINES

accurately predict the of the gold plating.

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factor of the resonators and the effects

II. DESIGN A model for calculating the resonator quality ( ) factor has been developed. In this model, the characteristic impedance is first calculated based on the line geometry. The line loss is then calculated using the characteristic impedance and the material properties. Finally, the resonator factor is calculated from the line loss. Using this model, four resonators were realized at two characteristic impedances and at two frequencies. A. Characteristic Impedance The characteristic impedance of a transmission line can be calculated as (1) where and are the inductance and capacitance per unit m/s is the phase velocity length of the line and of the line, which is equal to the speed of light in vacuum for an air core TEM transmission line. Chen [6] provides the following formula for calculating the line capacitance for lines where the (the case for the designs in this inner conductor is thin paper):

Fig. 2. Magnitude of a time-varying field applied to the surface (right-hand side of figure) of a conductor will decrease exponentially as it propagates inward. For a thin conductor on top of a thick conductor, the skin depth  is defined as the depth into the conductor where the magnitude of the field has been decreased to a value of 1=e from its magnitude at the surface.

metal. This situation is illustrated in Fig. 2. For the two-metal case, the skin depth is then calculated as (5)

(2) where defined in Fig. 1.

where and are defined in Fig. 2, and is the skin depth of the plated conductor calculated using (4). The value of can be calculated as

pF/m, and the remaining values are (6)

B. Attenuation Constant The attenuation constant is the sum of the conductor loss and the dielectric loss . Since the dielectric in these lines is air, the dielectric loss can be neglected. The conductor attenuation constant is calculated using Wheeler’s incremental inductance method [7]. Without derivation, the attenuation constant can be calculated with (3) where , , and are the surface resistance, skin depth, and permeability of the conductor, respectively, is the phase veis the change in characteristic impedance when locity, and . The skin depth all walls of the conductor are receded by of a conductor is defined as the depth into the conductor where of its magnithe field amplitude has decreased to 36.8% tude at the surface of the conductor. When a single conductor is used, this can be calculated as

where is the skin depth of the second conductor calculated using (4). For single conductor systems, the surface resistance is calculated as (7) where is the conductivity of the metal and is the skin depth calculated by (4). However, for a two-metal system, the surface resistance needs to take into account the fact that the currents flow in both metals. In this case, the surface resistance is calculated as (8) where and are the surface resistances of the two conductors calculated using (7). The model also accounts for surface roughness by adding a factor to the attenuation constant as given by Edwards [8]

(4)

(9)

for angular frequency and conductor conductivity . However, for a plated metal where the outer conductor is less than threshold occurs in the second one skin depth thick, this

is the attenuation constant calculated in (3) and where is the rms surface roughness of the conductor. For all of the calculations in this study, a value of m was used.

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TABLE II DIMENSIONS OF THE FOUR RESONATOR DESIGNS WITH LOSS AND CALCULATED FOR NICKEL RESONATORS

Q FACTOR

This value was measured from early samples of the recta-coax lines with an optical surface profilometer. C.

Factor

For short-circuited quarter-wavelength transmission line resonators, the unloaded at the resonant frequency can be calculated as [7] Fig. 3. Three-dimensional solid model of resonator type B from Table II.

(10) where is the attenuation constant, calculated here with (3) and is the wavelength at the resonant frequency. (9), and D. Resonator Simulations Four quarter-wavelength transmission line resonators were designed. For all of the designs, the thickness of the center conductor was set to 30 m. The resonator widths were chosen to create 50- resonators (designs A and B) or 75- resonators (designs C and D) and the resonator lengths were chosen to create resonators with center frequencies of 44 GHz (designs A and C) or 60 GHz (designs B and D). The resulting resonator attributes are given in Table II. The resonators were capacitively coupled, and the width of the coupling gap was tuned using Ansoft’s High Frequency Structure Simulator (HFSS) to achieve critical coupling. Full-wave electromagnetic simulations of the designs were performed using Ansoft’s HFSS [9]. The solid models of the devices included the etch access holes in the resonators, one physical attribute of the resonators not accounted for in the analytical model. Fig. 3 shows a three-dimensional view of the final resonator design with the outside ground made partially transparent so that the enclosed center conductor is visible. As can be seen, the feed region is supported only by a post directly resonator is supported only below the probe pad, while the at the grounded end. With traditional microelectromechanical system (MEMS) devices, long cantilevers such as this would be subjected to significant deformation due to residual stresses; however, thicker materials mitigate this problem and measurements of nickel cantilevers in a test structure show negligible deflection. The resonators were simulated using nickel, gold, and copper to determine the effect of the material on the device property. The simulations for nickel and gold provide bounds for the gold-plated nickel structures, as it was not feasible to simulate a gold-nickel bi-layer structure.

Fig. 4. SEM photograph showing the fabricated resonators.

CA, and fabricated in their EFAB process [10].1 The process used to fabricate these resonators had 26 nickel layers and allowed structures up to 296- m tall to be realized. After fabrication, individual die sites were shipped to the AFRL for measurement and gold plating. Fig. 4 shows the fabricated resonators on the die. Electroless gold plating of the nickel resonators was carried out by a process of immersion in a water-based solution. Processing was done one die at a time. Each die was cleaned at room temperature for 60 s with Micro 210 cleaner. Next, the nickel was activated and any surface oxide removed using an etch in diluted (10%) hydrochloric acid at room temperature for 30 s. The gold deposition was performed using a water-based cyanidecontaining sodium gold sulfree buffer solution pH fite, Na Au SO , ethylenediamine, and potassium fluoride at 70 C 2 C for 10–25 min under constant agitation. Finally, the die was washed in deionized water and dried in air at 115 C. Some specimens were double plated with one intermediate electrical testing between the plating stages. IV. MEASUREMENT

III. FABRICATION The resonators were fabricated under a foundry contract. Designs were submitted by the Air Force Research Laboratory (AFRL), Hanscom AFB, MA, to Microfabrica Inc., Van Nuys,

The -parameters of the fabricated resonators were measured using an Agilent E8361A vector network analyzer and Cas1[Online].

Available: http://www.microfabrica.com

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cade Infinity I67-A-GSG-150 probes over the frequencies of 27–67 GHz with data points every 0.025 GHz. A two-port shortopen-load-thru (SOLT) calibration was performed using Cascade ISS 101-190 standards so that the reference plane was at the probe tips. Once the -parameters were obtained from simulation and from measurement, the data was processed to obtain the unloaded of the resonators using the critical points method from Chua and Mirshekar-Syahkal [11]. In this method, the one-port -parameters are converted to input impedance and plotted on the real-imaginary plane. The two critical points are the points where the reactance is at a maximum and minimum and and . The frequencies the corresponding frequencies are and are obtained from the point where the resonant loop intersects with itself. From these four frequencies, the resonant can be determined as frequency (11) The unloaded

factor

is found with (12)

where the term is determined through a series of calculations , and will not be shown here. Weng et al. show that using for low- resonators, should be neglected [12]. However, they did not provide a numerical definition of low versus high . In this study, the method using as in (12) was chosen, and the mean value of from 234 measurements was 0.985. The frequencies of the critical points have a direct effect on the value of the extracted . Due to this, the spacing of the measured data points (here, 0.025 GHz) causes a digitization of the extracted values if uncorrected. In order to extract more accurate critical points, a cubic spline curve was fit to the test data, specifically on the resonant loop. After this correction, a more Gaussian distribution of the extracted values was noted, as was expected. All of the factors reported in this paper represent the average of measurements taken on up to ten resonators. A total of six different dice were used for measurement and testing purposes. Of these dice, one was measured and no plating was done. One of the dice was plated once before any measurements were taken, then the resonator factors were measured, and the die was used for thickness measurements in a scanning electron microscope (SEM) equipped with a focused ion beam (FIB). Two of the dice were measured, plated one time, and then measured again, and the last two dice were measured, plated one time, measured, plated a second time, and then measured again. As a result, up to ten separate resonators (two resonators of each type per die) were measured before any plating was done. Up to ten separate resonators were also measured after one plating step was done, and up to four different resonators were measured after two plating steps were done. Due to damaged probe pins, it was not always possible to measure all of the resonators. Reported values for the unplated devices all have measurements from at least nine separate devices. Values reported for devices plated once have at least seven separate devices. Values reported for devices plated twice have as few as one measured device. It

Fig. 5. Measured results from a 50- 44-GHz nickel resonator (design A). (a) Magnitude (in decibels) versus frequency. (b) Smith chart.

should be noted that the probe pins were damaged exclusively during testing and not during fabrication. V. RESULTS AND DISCUSSION A Smith chart and real-imaginary plot of measured one-port -parameters from a 50- 44-GHz nickel resonator (design A) are shown in Fig. 5. The plots shown are typical for these resonators. The resonance loop is centered at 44 GHz and can easily be seen in Fig. 5(a). In the desired state of critical coupling, the resonant loop passes through the center of the Smith chart. The chart shows that the resonator is quite close to being critically coupled. The gold-plating process was successful in plating only the nickel structures and not plating the alumina substrate. Early in the process, there was concern that the gold plating would not coat the interior surface of the resonators because of poor circulation of the reagents from the interior of the surface to the solution outside the ground planes. Therefore, the thickness of the gold film on the interior surface was experimentally determined using an SEM equipped with a FIB. This process requires cutting a hole into the top nickel metal, and is thus a destructive

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Fig. 7. Data from modeling (solid and dotted lines), HFSS simulations (open data points), and measurement (solid data points with error bars of one standard deviation) of 50- resonators. Test data is from designs A and B with solid nickel and nickel plated once with gold.

TABLE III COMPARISON OF MODELED AND MEASURED

Q

Fig. 6. (a) Utilizing a FIB, an opening was cut into the top surface of the resonator. (b) Zooming in on the region of the cut shows the gold on both the top and bottom surfaces of the nickel.

measurement. As a result, the measurement was performed on only one die. Fig. 6 shows a cross section of a hole cut into the top ground plane of one of the resonators. The gold film can be seen on both the top and bottom of the metal. The thickness on the bottom of the metal is measured to be 0.46 m by normalizing the thickness of the nickel to the 8- m-thick nominal design value. The gold on the top surface of the ground plane includes both the plated gold and a thin gold layer that was deposited to facilitate imaging in the SEM. The device measured in this test was plated once. The surface roughness of the gold film was measured on two die and found to be 103.5 and 141.5 nm. This value is approximately twice that found for the bare nickel. It should be noted, however, that these measurements were taken on the top surface of the die, and not on the interior of the coaxial lines. Fig. 7 plots the calculated unloaded- values for 50- 44and 60-GHz resonators (designs A and B) for nickel, gold, copper, and gold-plated nickel resonators. The gold-plated

nickel resonators are calculated for a plating thickness of 0.46 m based on the measurement from the FIB. These calculated values are compared with HFSS simulations of nickel, gold, and copper resonators. Measured values are also plotted for the nickel resonators and for nickel resonators that have been plated once with gold. The measured values are the values for all measured resonators and the error average bars show one standard deviation. The model predicts a slightly than the measured devices. This could be explained higher by the model not accounting for the etch access holes that are present in the solid model for the simulation, and in the fabricated structures, or due to higher surface roughness than

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Q

Fig. 8. Percentage improvement in the resonator plating depth for two different frequencies.

Q factor as a function of the

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TABLE IV IMPROVEMENTS DUE TO GOLD PLATING AND FITTED METAL THICKNESS VALUES

This plating approach can also be applied to provide a corrosion resistant surface to copper resonators. In this case, a thin coating less than 0.25- m thick would be used because the gold coating will increase the loss of the copper lines. At, 30 GHz, a 0.25- m-thick gold surface would increase the loss of copper lines by less than 6.7%. It remains to be shown if this thin coating would be sufficient to prevent long-term corrosion of the copper. VI. SUMMARY

Q

Fig. 9. Measured factor for resonators before plating, after they were plated once, and then after they were plated a second time.

the value used. The etch access holes are necessary for the microfabrication process, and likely have a slightly negative effect on the performance of the resonators. The test data comparing values for all of the the measured resonant frequencies and resonator designs are provided in Table III. Utilizing the model, the effect of gold plating on the nickel resonators can be predicted. This is done for designs A and B in Fig. 8. It is clear from the figure that for millimeter-wave frequencies, the plating depth only needs to be 1–1.5- m thick for the resonators to have essentially the same performance as solid gold resonators, and even a 0.5- m-thick plating has a significant impact on the performance of the resonators. Fig. 9 shows the measured factors for resonators with no plating, after one plating process, and after two plating processes. Table IV details the percent improvement for one and two platings over an unplated resonator. Fitting this measured improvement percentage to the calculated values provides an estimated gold thickness of 0.54 and 0.8 m. The fitted value after one plating is consistent with the 0.46- m-thick value measured with the FIB.

Gold plating of nickel recta-coax resonators has been shown to be feasible after the resonators have been fabricated and released. Gold plating of the nickel structures has been shown to improve the resonator factor by over 50% at frequencies above 44 GHz. This translates to the line loss being decreased by over 33%. Utilizing gold plating, nickel core resonators can deliver losses comparable to gold transmission lines, while maintaining the strength of nickel, and gaining the corrosion resistance of gold. A model for predicting the resonator factor has been presented. This model includes the effect of the plating thickness on the performance of the resonator. This model is validated through measurements of resonators at two frequencies and two impedances. The technique presented here should be applicable to a variety of recta-coax structures such as transmission lines, filters, and couplers. ACKNOWLEDGMENT The authors thank R. Cortez, Sensors Directorate, Air Force Research Laboratory, Hanscom AFB, MA, for her work on determining the gold-plating thickness by using a FIB system and SEM. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, Department of Defense, or the U.S. Government. REFERENCES [1] J. Reid, E. Marsh, and R. Webster, “Micromachined rectangular coaxial transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, pp. 3433–3442, Aug. 2006. [2] E. Brown, A. Cohen, C. Bang, M. Lockard, G. Byrne, N. Vendelli, D. McPherson, and G. Zhang, “Characteristics of microfabricated rectangular coax in the band,” Microw, Opt. Technol. Lett., vol. 40, pp. 365–804, Mar. 2004. [3] J.-B. Yoon, B.-I. Kim, Y.-S. Choi, and E. Yoon, “3-D construction of monolithic passive components for RF and microwave ICs using thick-metal surface micromachining technology,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 279–288, Jan. 2003.

Ka

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[4] I. L. Garro, M. Lancaster, and P. Hall, “Air-filled square coaxial transmission line and its use in microwave filters,” Proc. Inst. Elect. Eng. —Microw., Antennas, Propag., vol. 152, no. 3, pp. 155–159, Jun. 2005. [5] D. Filipovic, Z. P. K. Vanhille, M. Lukic, S. Rondineau, M. Buck, G. Potvin, D. Fontaine, C. Nichols, D. Sherrer, S. Zhou, W. Houck, D. Fleming, E. Daniel, W. Wilkins, V. Sokolov, and J. Evans, “Modeling, design, fabrication, and performance of rectangular -coaxial lines and components,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, pp. 1393–1396. [6] T.-S. Chen, “Determination of the capacitance, inductance, and characteristic impedance of rectangular lines,” IRE Trans. Microw. Theory Tech., vol. MTT-9, no. 9, pp. 510–519, Sep. 1960. [7] D. Pozar, Microwave Engineering. Reading, MA: Addison-Wesley, 1990. [8] T. Edwards, Foundations for Microstrip Circuit Design, 2nd ed. New York: Wiley, 1992. [9] Ansoft High Frequency Structure Simulator. ver. 9.0, Ansoft Corporation, Pittsburgh, PA, 2004. [Online]. Available: www.ansoft.com [10] A. Cohen, G. Zhang, F. Tseng, F. Mansfield, U. Frodis, and P. Will, “EFAB: Rapid low-cost desktop micromachining of high aspect ratio true 3-D MEMS,” in IEEE Int. MEMS Conf., Jan. 1999, pp. 244–251. [11] L. Chua and D. Mirshekar-Syahkal, “Accurate and direct characterization of high-Q microwave resonators using one-port measurement,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 978–985, Mar. 2003. [12] P. Weng, L. Chua, and D. Mirshekar-Syahkal, “Accurate characterization of low-Q microwave resonator using critical-points method,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 349–353, Jan. 2005. Eric D. Marsh (M’03) received the Bachelor’s degree in electrical engineering from Iowa State University, Ames, in 1999, and the Master’s degree in electrical engineering from the Air Force Institute of Technology, Dayton, OH, in 2004. In 1999, he joined the United States Air Force as a Developmental Engineer. Since 2004, he has been with the Sensor’s Directorate, Air Force Research Laboratory, Hanscom AFB, MA, where he performs basic research on micromachined millimeter-wave transmission lines and components.

James Robert Reid (S’94–M’97) received the B.S.E.E. degree from Duke University, Durham, NC, in 1992, and the M.S.E.E. and Ph.D. degrees from the Air Force Institute of Technology (AFIT), Dayton, OH, in 1993 and 1996, respectively. Since 1997, he has been with the Air Force Research Laboratory (AFRL), Hanscom AFB, MA, where he conducts research into the application of micromachining and MEMS to front-end antenna technology. His current research is in the areas of micromachined transmission lines, RF MEMS switches and varactors, millimeter-wave phase shifters, and switching networks. Dr. Reid is a member of Sigma Xi.

Vladimir S. Vasilyev received the M.S. degree in chemical engineering from the Ural Polytechnical Institute, Yekaterinburg, Russia, and the Ph.D. degree in quantum electronic materials engineering from the Mendeleyev’s Institute of Chemistry and Technology, Moscow, Russia. He is currently a Senior Researcher with S4 Inc., Burlington, MA, where he works on site with the Sensors Directorate, Air Force Research Laboratory, Hanscom AFB, MA. His current research concerns the design and simulation of sensors and the development of novel fabrication processes for microwave components and MEMS. Dr. Vasilyev is a member of the American Association for Crystal Growth and the Materials Research Society.

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Uniform Electric Field Distribution in Microwave Heating Applicators by Means of Genetic Algorithms Optimization of Dielectric Multilayer Structures Elsa Domínguez-Tortajada, Juan Monzó-Cabrera, and Alejandro Díaz-Morcillo, Member, IEEE

Abstract—In this paper, the design of a dielectric multilayer around a clay sample is presented to achieve a uniform electric field distribution over that sample. This structure is located within a multimode microwave-heating oven and is designed by means of genetic algorithms. The permittivity and geometric values for the sample surrounding layers are the selected parameters to be optimized in order to minimize the ratio between the typical deviation and the absolute value of the electric field. The results demonstrate the improvement of the electric field uniformity over the sample using dielectric layers. Index Terms—Dielectric properties, electric field uniformity, genetic algorithms, microwave energy, microwave heating, multimode applicator.

I. INTRODUCTION HE ELECTRIC field distribution in a multimode applicator is a combination of all the modes excited in a given frequency range, and usually this distribution is not uniform. If the considered materials have homogeneous properties, the field uniformity is directly related to the temperature distribution in the sample. Therefore, one of the most important and desired characteristics in microwave heating processes is the uniformity of the electric field distribution over the material to be heated because the quality of the process depends on it [1]–[3]. Different methods have been used in order to obtain uniform heating patterns in certain regions. For example, in [4], the sample is placed on a turning tray and a method to simulate the heating process inside domestic microwave ovens is presented. Mode stirrers are used in [5]–[8] to modify the field distribution inside a multimode cavity. Several studies on these elements prove their good working on multimode applicators, mainly for low-loss materials. In [9], a procedure to find out the position, structure, and permittivity of certain materials placed inside the applicator is proposed. The parameters of these structures have an effect on the field distribution and they are chosen in order to obtain temperature uniformity over the sample. A different way of increasing the uniformity is detailed in [10] and [11]. In these studies, the applicator is fed by means of

T

Manuscript received May 5, 2006; revised July 27, 2006. This work was supported in part by the Fundación Séneca under Project 00700/PPC/04. The authors are with the Departamento de Tecnologías de la Información y las Comunicaciones, Technical University of Cartagena, Cartagena 30202, Spain (e-mail: [email protected]; [email protected]; alejandro.diaz@upct. es). Digital Object Identifier 10.1109/TMTT.2006.886913

a combination of different sources and mode matching is used to analyze the designs. However, this modal analysis requires canonical structures, i.e., rectangular samples. Likewise, the use of slotted guides as a feeding system [12] can significantly improve the uniformity [13]–[15], obtaining different field distributions depending on the slots’ location. In [16], the use of two generators and a magic tee adapter for achieving good impedance matching is described. By controlling the signal strength of each source, a better uniformity than using only one magnetron is obtained as well. The dimensions and geometry of the cavity also have an influence on the field uniformity over the sample. If planar samples must be uniformly heated, a meander applicator can be used [17]. In [18], a single mode modified cylindrical cavity is designed in order to uniformly vulcanize samples of rubber. In this study, several dielectric layers surrounding the sample are used to obtain electric field uniformity. This differs from the procedure carried out in [19] since, in that study, these dielectric layers were employed in order to optimize load matching. Previous studies [20]–[22] use this configuration as a matching technique as well, and good results were obtained there with efficiencies above 95%. Other studies use dielectric susceptors as a form of active packaging in order to modify temperature profiles when using microwave heating [23]. In [24], Mechenova and Yakovlev emphasize the complexity that implies the optimization of the field pattern and propose optimization methods based on commercial programs that implement the electromagnetic simulation. In this study, a genetic algorithm is presented with the purpose of correctly designing the dielectric and geometric properties of the materials of the structure that will enclose the sample. The application of genetic algorithms to different electromagnetic problems is described in [25]. Several examples, such as [26]–[28], show that it is particularly effective in designing specific electromagnetic systems. II. THEORETICAL STUDY A. Problem Description The procedure followed in this study is similar to the one described in [19]; nevertheless, it is summarized below. The problem to be solved consists of a 24 24 24 cm metallic rectangular applicator. The multimode cavity is fed by means of a WR340 waveguide centered at the upper wall of the oven and mode around 2.45 GHz, which is one of excited with the the industrial–scientific–medical (ISM) designated electroheat frequencies [29].

0018-9480/$25.00 © 2006 IEEE

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The product inside the cavity is clay with relative permit. The clay sample has been chosen to tivity study real industrial dielectric materials that present high values of dielectric constant and loss factor. Although the method can deal with any number of layers, in this study, the sample is surrounded by a maximum of two dielectric layers, whose geometry and permittivities are designed for optimum electric field uniformity. The genetic algorithm is managed in MATLAB and it uses the CST Microwave Studio commercial software to perform the electromagnetic calculations. CST Microwave Studio applies the finite integration technique and the perfect boundary approximation to solve the considered electromagnetic problems [30]. This full-wave simulator can work with noncanonical structures, although in this study, only rectangular samples are considered. In this study, the electric field uniformity in the sample must be evaluated and maximized. Therefore, the evaluation function in the genetic algorithm involves the typical deviation of the electric field in the sample surface normalized by the averaged value of the electric field strength in the sample

Fig. 1. Layout of sample and multilayer structure inside the applicator.

TABLE I GENETIC-ALGORITHM PARAMETERS

(1) where are the points considered in the upper and lower surfaces of the sample. The uniformity control is constrained to the surfaces since a fast electric field drop inwards the sample is assumed due to the consideration of high losses. Equation (1) must be minimized with the genetic algorithm because the typical deviation considers the electric field variability and, besides, the electric field values on the sample must be maximized. The genetic algorithm is designed to maximize functions and, for this reason, the evaluation function has been computed as (2) where is a positive value higher than the maximum value that the function can achieve and is used to identify the has been set to 2. This value fitness value. In this case, was assumed high enough since all the simulation values for were always around 1. Fig. 1 illustrates the scenario considered in the CST Microwave Studio environment. There, two dielectric layers with variable thickness and permittivity values surround the sample. The waveguide feeding the cavity is also represented. A PTFE tray is included to consider real situations in which transportation or supporting systems are needed. B. Genetic-Algorithm Implementation Genetic algorithms are used to obtain the optimum design in terms of uniformity that, in this particular case, implies obtaining the dielectric and geometric properties of the materials that make up the dielectric structure that encloses the sample. First, it is necessary to define the design parameters, in this case, and , with and their range of variation. A combination of these design parameters constitutes an individual

that is assessed by using the evaluation function that returns as a result the normalized typical deviation (1). The genetic algorithm is implemented as [19] details. There, the CST full-wave simulator was validated as well. III. RESULTS AND DISCUSSION Three different configurations for the dielectric multilayer structure have been considered and optimized by means of the genetic algorithm. In all the evaluated configurations, the input power at the waveguide feeding has been set to 1 W. The simplest structure considered consists of a single dielectric layer surrounding the sample. More complex structures take and, conseinto account an additional dielectric layer quently, the sample is enclosed by two dielectrics. As a particular case of this last configuration, the internal layer has been forced to behave as air in order to simulate a situation in which dielectric contact must be avoided and, as a result, the configuration is sample–air–dielectric. The design parameters, included as input parameters in the optimization process, are the dielectric properties and the thickness of each layer. The variation ranges for the genetic algorithm design parameters have been , and limited for all scenarios as cm. Table I shows the genetic-algorithm configuration that has been established after a study of the problem behavior. Firstly, the evaluation of the system without dielectric layers has been considered to obtain the electric field distribution in

DOMÍNGUEZ-TORTAJADA et al.: UNIFORM ELECTRIC FIELD DISTRIBUTION IN MICROWAVE HEATING APPLICATORS

Fig. 2. Electric field over the clay sample and inside the applicator for the configuration without dielectric layers.

Fig. 3. Genetic-algorithm evolution in the one-layer design.

the sample. This study is useful to verify the improvement in uniformity achieved with the optimized layers. Fig. 2 shows the computed electric field pattern over the sample. From this result, one can conclude that the electric field is mainly concentrated on two hot spots by the sides of the sample, which leads to an uneven electric field pattern. In this case, the average of the electric field in the sample is 105.7 V/m and the normalized typical deviation is 0.771. For the one-layer configuration, Fig. 3 shows the genetic-algorithm evolution and the convergence of all the individuals of the population to fitness values close to the best one. In order to better appreciate the minimization process, we depict the trend instead of , which is maximized by the genetic algorithm and depends on a not optimized constant such as . The optimized parameters are, in this case, and cm. With these features, the multimode structure yields a normalized typical deviation equal to 0.221 and the electric field mean is 228.9 V/m. Thus, the electric field value has been increased from 105.7 to 228.9 V/m and the typical deviation has decreased from 81.5 to 50.6 V/m. Consequently, that structure really achieves better

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Fig. 4. Electric field over the clay sample and inside the applicator for the onelayer configuration.

Fig. 5. Genetic-algorithm evolution in the two-layer design.

uniformity and a higher electric field average than the structure without layers. Fig. 4 depicts the electric field inside the applicator. If the power efficiency is defined as the ratio between the absorbed power by the sample and the incident power into the cavity, a deeper analysis on this aspect can be done. In fact, the power efficiency for the configuration without layers is 31.8%. By adding the optimized dielectric layer. the power efficiency increases up to 62.85% and, consequently, it represents an enhancement in the absorbed energy by the sample. Next, the electric field uniformity is analyzed for the design with two dielectric layers. The best value for the normalized typical deviation achieved is 0.1284 and the average of the electric field in the sample is 168.2 V/m with cm, and cm. Fig. 5 depicts the evolution of the genetic algorithm. There, it can be observed that a good optimization has been achieved in the second generation; nevertheless, there are no more improvements in the following generations. The electric field in the sample in this two-layer structure is smaller than in the single-layer design, but it is more uniform.

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Fig. 6. Electric field over the sample and inside the applicator for the two-layer configuration.

Fig. 7. Genetic-algorithm evolution in the two-layer design considering air the internal one.

Fig. 6 shows the electric field for the upper surface plane. A high-quality uniform field over the sample can be observed. In this case, the power efficiency reaches 37.15%, which is better than in the case of using no external dielectric. On the other side, the layers also absorb energy; the internal mold absorbs 7.6% and the external one absorbs 32.4%. This points out that a compromise solution between uniformity and efficiency would be the best design strategy. The last design analyzed here consists of two layers, considering air material for the internal one, as previously explained. For this configuration, the genetic-algorithm evolution is depicted in Fig. 7, where the best value for the normalized typical deviation is 0.1887 and the average of the electric field in the sample is 154.5 V/m. A progressive optimization for the best individual can be observed along the first five generations. The best design in this case consists of the following values cm, for the design parameters: and cm. This design shows better electric field uniformity when compared to the design without layers,

Fig. 8. Electric field over the sample and inside the applicator for the air dielectric configuration.

+

Fig. 9. Spectrum of the magnetron employed in the experience.

but the optimum two-layer design is better regarding the uniformity and the electric field strength in the sample. Fig. 8 illustrates the electric field distribution in the upper surface of the sample. From these results, one can conclude that there is a good optimization evolution, but the normalized typical deviation for this configuration is worse. This is consistent with the reduction in the degrees of freedom in the design. IV. EXPERIMENTAL VALIDATION In order to validate the simulation procedure and to check the changes in the electric field pattern using dielectric layers, the following tests have been performed. A 32 32 32 cm cavity has been fed by means of a magnetron with the spectrum illustrated in Fig. 9. The spectrum has been measured by means of the Rhode ZVRE spectrum analyzer. It can be observed from this measurement that the magnetron oscillates at 2.475 GHz instead of the assumed 2.45 GHz for these types of devices. Therefore, all simulations in this section were carried out at the former frequency.

DOMÍNGUEZ-TORTAJADA et al.: UNIFORM ELECTRIC FIELD DISTRIBUTION IN MICROWAVE HEATING APPLICATORS

Fig. 10. (a) Thermography of the wood sample. (b) Computer simulation with the same configuration.

The cavity was loaded with a 15.1 15.2 1.7 cm piece of . This sample was located inwood with side the cavity leaving 8.8 and 9 cm from the walls of the applicator and it was irradiated during 46 s at a 600-W power level. The thermographic camera CYCLOPS PPM was used in order to obtain the temperature patterns across the sample. The CST Microwave Studio simulator was used in order to compare those temperature profiles with the electric field distribution. Fig. 10(a) shows the temperature distribution at the wood sample, while Fig. 10(b) depicts the electric field distribution at 2.475 GHz. Similar patterns can be observed in both images. However, the differences between both figures might be due to thermal migration during and after microwave irradiation since the thermography was taken 90 s after the generator stopped due to the time employed in opening the cavity. After a cooling period, the sample was covered with a . 15.1 15.2 1.1 cm PTFE layer with It was irradiated during 62 s at a 600-W power level. By using the thermographic camera, the temperature pattern shown in Fig. 11(a) was obtained. Fig. 11(b) illustrates the electric field distribution computed by the electromagnetic simulator. Once again, the distribution of the main lobes is very similar in both images. The thermography was taken 60 s after the generator stopped.

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Fig. 11. (a) Thermography of the wood sample using the PTFE layer. (b) Computer simulation with the same configuration.

By comparing experimental data in Figs. 10 and 11, one can observe that the insertion of a dielectric mould highly changes the spatial distribution of the electric field and, accordingly, the temperature patterns. It must be remarked that the proposed procedure for obtaining uniform electric field distributions is, at the moment, difficult to implement since it requires very precise material permittivity values. However, the experimental validation shows that this approach is accurate enough to correctly predict the temperature patterns and the changes that the dielectric layers may produce on them. V. CONCLUSION The use of dielectric layers surrounding a high-loss dielectric product to achieve electric field uniformity for multimode microwave-heating ovens has been presented. A genetic algorithm optimization procedure has been used in order to obtain the optimum values of the dielectric molds around the sample, namely, permittivity and thickness. The results for three configurations show that the electric field uniformity is better using dielectric layers. The best performance was reached by the two-layer structure. However, a relationship among the typical deviation, the mean electric field

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intensity in the sample, and the power efficiency of the process is perceived and, depending on the microwave heating application, a compromise among these three parameters must be established. When comparing this technique with mode-stirring results obtained in [7] and [8], one can conclude that this technique is capable of avoiding edge overheating and, thus, it is a better alternative for producing a more uniform electric field distribution when high-loss materials must be heated. The uniformity results obtained here, together with those obtained in [19] for matching purposes, show that this technique is actually a potential method to solve ordinary problems in microwave heating. Finally, it must be remarked that despite the fact that the technique described in this paper shows good promise, the practical implementation requires very precise material manufacturing to obtain the designed permittivity values. Future research is envisaged in order to design this kind of multilayer structures with commercial materials.

REFERENCES [1] A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating. London, U.K.: Peregrinus, 1986. [2] T. V. Chow-Ting-Chan and H. C. Reader, Understanding Microwave Heating Cavities. London, U.K.: Artech House, 2000. [3] G. Roussy and J. A. Pearce, Foundations and Industrial Applications of Microwave and Radiofrequency Fields, Physical and Chemical Processes. New York: Wiley, 1996. [4] P. Kopyt and M. Celuch-Marcysiak, “FDTD modelling and experimental verification of electromagnetic power dissipated in domestic microwave ovens,” J. Telecommun. Inf. Technol., no. 1, pp. 59–65, 2003. [5] J. Monzó-Cabrera, “Estudio del secado asistido por microondas en los materiales laminares,” Ph.D. dissertation, Dept. Telecommun. Eng., Polytech. Univ. Valencia, Valencia, Spain, 2002. [6] J. Monzó-Cabrera, J. M. Catalá-Civera, P. Plaza-González, and D. Sánchez-Hernández, “A model for microwave-assisted drying of leather: Development and validation,” J. Microw. Power Electromagn. Energy, vol. 39, no. 1, pp. 53–64, 2004. [7] P. Plaza-Gonzalez, J. Monzó-Cabrera, J. M. Catalá-Civera, and D. Sánchez-Hernández, “Effect of mode-stirrer configurations on dielectric heating performance in multimode microwave applicators,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1699–1706, May 2005. [8] ——, “New approach for the prediction of the electric field distribution in multimode microwave-heating applicators with mode stirrers,” IEEE Trans. Magn., vol. 40, no. 3, pp. 1672–1678, May 2004. [9] K. A. Lurie and V. V. Yakovlev, “Method of control and optimization of microwave heating in waveguide systems,” IEEE Trans. Magn., vol. 35, no. 3, pp. 1777–1780, May 1999. [10] J. Pitarch, A. J. Canós, F. L. Peñaranda-Foix, J. M. Catalá-Civera, and J. V. Balbastre, “Synthesis of uniform electric field distributions in microwave multimode applicators by multifeed techniques,” in 9th Int. Microw. High-Freq. Heating Conf., Loughborough, U.K., Sep. 1–5, 2003, pp. 221–224. [11] J. M. Catalá-Civera, J. Pitarch, M. Contelles-Cervera, F. PeñarandaFoix, and J. V. Balbastre, “On the possibilities of multifeeding techniques to improve the electric field uniformity in multimode microwave applicators,” in 10th Int. Microw. High-Freq. Heating Conf., Modena, Italy, Sep. 12–15, 2005, pp. 360–363. [12] S. Keller, G. Roussy, B. Maestrali, and J. M. Thiebaut, “Experimental and theoretical study of waveguide slot radiators,” in 6th Int. Microw. High-Freq. Heating Conf., Fermo, Italy, Sep. 8–12, 1997, pp. 321–325. [13] J. A. Cebrián-Gascón, E. de los Reyes, and D. Sánchez-Hernández, “Use of commercial electromagnetic simulators for uniformity enhancement of slotted waveguide-fed microwave heating applicators,” in 7th Int. Microw. High-Freq. Heating Conf., Valencia, Spain, Sep. 13–17, 1999, pp. 225–228.

[14] V. V. Stepanov, “The multielement excitation for improvement of uniformity microwave energy distribution for partly filled rectangular microwave chamber,” in 7th Int. Microw. High-Freq. Heating Conf., Valencia, Spain, Sep. 13–17, 1997, pp. 173–176. [15] S. Stanculovic, L. Feher, and M. Thumm, “Optimization of slotted waveguides for 2.45 GHz applicators,” in 9th Int. Microw. High-Freq. Heating Conf., Loughborough, U.K., Sep. 1–5, 2003, pp. 313–316. [16] G. Roussy and N. Kongmark, “Improving electric field distribution in a microwave heating device,” J. Microw. Power Electromagn. Energy, vol. 35, no. 4, pp. 253–257, 2000. [17] S. Frandos, “Numerical modeling of heating thin dielectric sheets with a microwave applicator of meanders type,” in 7th Int. Microw. HighFreq. Heating Conf., Valencia, Spain, Sep. 13–17, 1999, pp. 187–190. [18] J. M. Catalá Civera, S. Giner Maravilla, D. Sánchez Hernández, and E. de los Reyes, “Pressure-aided microwave rubber vulcanization in a ridged three-zone cylindrical cavity,” J. Microw. Power Electromagn. Energy, vol. 35, no. 2, pp. 92–104, 2000. [19] E. Domínguez-Tortajada, J. Monzó-Cabrera, and A. Díaz-Morcillo, “Load matching in microwave-heating applicators by means of genetic algorithms optimization of dielectric multilayer structures,” Microw. Opt. Technol. Lett., vol. 47, no. 5, pp. 426–430, Dec. 2005. [20] J. Monzó-Cabrera, A. Díaz-Morcillo, J. L. Pedreño-Molina, and D. Sánchez-Hernández, “A new method for load matching in multimodemicrowave heating applicators based on the use of dielectric-layer superposition,” Microw. Opt. Technol. Lett., vol. 40, no. 4, pp. 318–322, Feb. 2004. [21] J. Monzó-Cabrera, J. Escalante, A. Díaz-Morcillo, A. MartínezGonzález, and D. Sánchez-Hernández, “Load matching in multimode microwave-heating applicators based on the use of dielectric layer moulding with commercial materials,” Microw. Opt. Technol. Lett., vol. 41, no. 5, pp. 414–417, Feb. 2004. [22] J. Monzó-Cabrera, E. Domínguez-Tortajada, A. Díaz-Morcillo, J. M. Catalá-Civera, and D. Sánchez-Hernández, “On the possibilities of load matching in microwave applicators using dielectric layer superposition around the sample,” in 9th Int. Microw. High-Freq. Heating Conf., Loughborough, U.K., Sep. 1–5, 2003, pp. 309–312. [23] M. Sato, “Recent development of microwave kilns for industries in Japan,” in Proc. 3rd Microw. Radio Freq. Applicat. World Congr., Sydney, Australia, Sep. 2002, pp. 281–289. [24] V. A. Mechenova and V. V. Yakovlev, “Efficiency optimization for systems and components in microwave power engineering,” J. Microw. Power Electromagn. Energy, vol. 39, no. 1, pp. 15–28, 2004. [25] Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms, ser. Microw. Opt. Eng.. New York: Wiley, 1999. [26] S. Chakravarty and R. Mittra, “Design of microwave filters using a binary-coded genetic algorithm,” Microw. Opt. Technol. Lett., vol. 26, no. 3, pp. 162–166, Aug. , 200. [27] D. Suckley, “Genetic algorithm in the design of FIR filters,” Proc. Inst. Elect. Eng.—G: Circuits, Devices, Syst., vol. 138, no. 2, pp. 234–238, Apr. 1991. [28] S. P. Panthong and S. Jantarang, “3G mobile wireless routing optimization by genetic algorithm,” in IEEE Can. Elect. Comput. Eng. Conf., Bangkok, Thailand, May 4–7, 2003, vol. 3, pp. 1597–1600. [29] A. C. Metaxas, Foundations of Electroheat: A Unified Approach. New York: Wiley, 1996. [30] “CST Microwave Studio Manual, HF Design and Analysis,” ver. 5th, CST, Darmstadt, Germany, 2004.

Elsa Domínguez-Tortajada was born in Valencia, Spain, in June 1976. She received the Dipl. Ing. degree in telecommunications engineering from the Universidad Politécnica de Valencia, Valencia, Spain, in 2001, and the Ph.D. degree from the Universidad Politécnica de Cartagena, Cartagena, Spain, in 2005. In 2002, she joined the Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena, where she is currently an Associate Lecturer. She currently combines her teaching and researching activities in different areas, which cover microwave-assisted heating processes, microwave applicators design and optimization, and numerical techniques in electromagnetics.

DOMÍNGUEZ-TORTAJADA et al.: UNIFORM ELECTRIC FIELD DISTRIBUTION IN MICROWAVE HEATING APPLICATORS

Juan Monzó-Cabrera was born in Elda (Alicante), Spain, in January 1973. He received the Dipl. Ing. and Ph.D. degrees in telecommunications engineering from the Universidad Politécnica de Valencia, Valencia, Spain, in 1998 and 2002, respectively. From 1999 to 2000, he was a Research Assistant with the Microwave Heating Group (GCM). In 2000, he joined the Departamento de Teoría de la Señal y Radiocomunicaciones, Universidad Politécnica de Cartagena, Cartagena, Spain, as an Associate Lecturer. He is currently an Associate Lecturer with the Departamento de Tecnologías de la Información y Comunicaciones, Universidad Politécnica de Cartagena. He has coauthored over 40 papers in referred journals and conference proceedings. He holds several patents regarding microwave heating industrial processes. He is a Reviewer for several international journals. His current research areas cover microwave-assisted heating and drying processes, microwave applicator design and optimization, and numerical techniques in electromagnetics. Dr. Monzó-Cabrera is a member of the Association of Microwave Power in Europe for Research and Education (AMPERE), a European-based organization devoted to the promotion of RF and microwave energy.

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Alejandro Díaz-Morcillo (S’95–M’02) was born in Albacete, Spain, in 1971. He received the Ingeniero (Ms. Eng.) and Doctor Ingeniero (Ph.D.) degrees in telecommunication engineering from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 1995 and 2000, respectively. From 1996 to 1999, he was a Research Assistant with the Department of Communications, UPV. In 1999, he joined the Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena (UPCT), Cartagena, Spain, as Teaching Assistant, where, since 2001, he has been an Associate Professor. He leads the Electromagnetics and Matter Research Group, UPCT. His main research interests are numerical methods in electromagnetics, electromagnetic compatibility (EMC), and industrial microwave heating systems.

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Determination of Generalized Permeability Function and Field Energy Density in Artificial Magnetics Using the Equivalent-Circuit Method Pekka M. T. Ikonen, Student Member, IEEE, and Sergei A. Tretyakov, Senior Member, IEEE

Abstract—The equivalent-circuit model for artificial magnetic materials based on various arrangements of broken loops is generalized by taking into account losses in the substrate or matrix material. It is shown that a modification is needed to the known macroscopic permeability function in order to correctly describe these materials. Depending on the dominating loss mechanism (conductive losses in metal parts or dielectric losses in the substrate), the permeability function has different forms. The proposed circuit model and permeability function are experimentally validated. Furthermore, starting from the generalized circuit model, we derive an explicit expression for the electromagnetic field energy density in artificial magnetic media. This expression is valid at low frequencies and in the vicinity of the resonance also when dispersion and losses in the material are strong. The currently obtained results for the energy density are compared with the results obtained using different methods. As a practical application example, we calculate the quality factor of a microwave resonator made of an artificial magnetic material using the proposed equivalent-circuit method and compare the result with a formula derived for a special case by a different method known from the literature. Index Terms—Artificial magnetic materials, circuit model, energy density, permeability function.

I. INTRODUCTION

RTIFICIAL electromagnetic media with extraordinary properties (often called metamaterials) attract increasing attention in the microwave community. One of the widely studied subclasses of metamaterials are artificial magnetic materials operating in the microwave regime, e.g., [1]–[8]. For example, broken loops have been used as one of the building blocks to implement double-negative (DNG) media [9], [10], artificial magneto-dielectric substrates are today considered as one of the most promising ways to miniaturize microstrip antennas [11]–[17], and several suggestions have lately been proposed to use artificial magnetic resonators, e.g., in filter design (see, e.g., [18] and [19]).

A

Manuscript received May 5, 2006; revised August 28, 2006. This work was supported under the frame of the European Network of Excellence Metamorphose and by the Academy of Finland and TEKES under the Center-of-Excellence Program. The authors are with the Radio Laboratory/SMARAD Centre of Excellence, Helsinki University of Technology, Espoo FI-02015 TKK, 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/TMTT.2006.886914

The extraordinary features of metamaterials call for careful analysis when studying the fundamental electromagnetic quantities in these materials. Recently, a lot of research has been devoted to the definition of field energy density in DNG media [20]–[23]. The authors of [20], [21], and [23] derived the energy density expression starting from the macroscopic media model and writing down the equation of motion for polarization (electric charge) or magnetization density in the medium. Furthermore, the complex Poynting theorem was used to search for expressions having the mathematically correct form to be identified as energy densities. Following the terminology presented in [23], we call this method the “electrodynamic method.” In [22], one of the authors of this paper used another method. Starting from the material microstructure, an equivalent-circuit representation was derived for the unit cell constituting specific artificial dielectric and magnetic media. Lattices of thin wires and arrays of split-ring resonators were considered in [22]. The stored reactive energy and, furthermore, the field energy density, were calculated using the classical circuit theory. The authors of [23] later called this method the “equivalent-circuit method.” Though these two methods apply to media with the same macroscopic permeabilities and permittivities, they are fundamentally very different, as will be clarified later in more detail. Moreover, the final expressions for the field energy density in artificial magnetic media given in [22] and [23] differ from each other. One of the motivations of this study is to clarify the reasons for this difference. Here we concentrate only on artificial magnetic media and set two main goals for our work, which are as follows. • To understand the differences and assumptions behind the electrodynamic and equivalent-circuit methods when deriving the field energy density expressions. We verify using a specific example (magnetic material unit cell) that, in the presence of nonnegligible losses, one should always calculate the stored energy at the microscopic level. • To generalize the previously reported equivalent-circuit representation for artificial magnetic media [22]. The generalized circuit model takes into account losses in the matrix material. It is shown that this generalization forces a modification to the widely accepted permeability function used to macroscopically describe artificial magnetic media. The generalization has a significant importance, as it is shown that, in a practical situation, matrix losses strongly dominate over conductive losses. The proposed circuit representation and permeability function are experimentally validated. We measure the magnetic polarizability of a small magnetic material sample and

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compare the results with those given by the proposed analytical model and the previously used model. The results given by the proposed model agree very well with the measured results, whereas the old model leads to dramatic overestimation for the polarizability. Using the generalized circuit model, we derive an expression for the field energy density in artificial magnetic media. This expression is compared with the results obtained using the electrodynamic method in [23], and reasons for the differences are discussed. As a practical application example, we calculate the quality factor of a microwave resonator made of artificial magnetic material using the equivalent-circuit and electrodynamic methods. The differences between the final expressions are discussed. II. ELECTRODYNAMIC METHOD VERSUS EQUIVALENT-CIRCUIT METHOD It is well explained in reference books (e.g., [24] and [25]) that, for the definition of field energy density in a material having nonnegligible losses, one always needs to know the material microstructure. First of all, the reactive energy stored in any material sample is a quantity that can be measured. When the material is lossless, no information is needed about the material microstructure for this measurement. Indeed, we can measure the total power flux through the surface of the sample volume1 and, since there is no power loss inside, we can use the Poynting theorem to determine the change in the stored energy. This is the reason why the field energy density in a dispersive media with negligible losses can be expressed through the frequency derivative of macroscopic material parameters. In the circuit theory, the same conclusion is true for circuits that contain only reactive elements. It is possible to find the stored reactive energy in the whole circuit knowing only the input impedance of a two-port [26]. Simple reasoning reveals that in the presence of nonnegligible material losses, the above described “black box” representation and direct measurement are not possible. Without knowing the material microstructure or the circuit topology, we do not know which portion of the input power is dissipated and which is stored in the reactive elements. Thus, the energy stored in lossy media cannot be uniquely defined by only utilizing the knowledge about the macroscopic behavior of the media [24], [25]. When defining the energy density in a certain material using the electrodynamic method, one first writes down the equation of motion for charge density or for magnetization in the medium using the macroscopic media model [21], [23]. We stress that this equation is the macroscopic equation of motion, containing the same physical information as the macroscopic permittivity and permeability. Further, the complex Poynting theorem is used to identify the mathematical form of the general macroscopic energy density expressions. Having the form of these expressions in mind, one searches for similar expressions in the equation of motion and defines them as energy densities. The problem of the electrodynamic method is the fact that it only utilizes the knowledge about the macroscopic behavior of the media, which, as explained above, in not enough. The 1In

practice, this measurement can be done, for example, by positioning a material sample inside a closed waveguide and measuring the S and S parameters.

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Fig. 1. Broken loop loaded with an infinitesimally small lumped circuit.

aforementioned difficulty is avoided in the equivalent-circuit method [22]. Based on the known microscopic medium structure, one constructs the equivalent circuit for the unit cell of the medium. Careful analysis is needed to make sure that the circuit physically corresponds to the analyzed unit cell. After this check, the stored reactive energy and the corresponding field energy density can be uniquely calculated using the classical circuit theory. Next, we illustrate the difference between the electrodynamic and equivalent-circuit methods using a specific case of brokenloop composites. Consider the broken loop shown in Fig. 1. Following an example given in [25], we load the loop with an electrically infinitesimally small circuit consisting of lumped elements. Let us assume that the additional inductance and capaci. tance are chosen to have values A simple check reveals that, in this case, the input impedance of the load circuit is frequency independent and purely resistive: . When the loop is electrically small, it can be represented as a resonant contour and the total loss resistance reads , where is the loss resistance due to the finite conductivity of the loop material. Next, we consider two different broken loops. The first loop is made of silver and loaded with the circuit shown in Fig. 1. The second loop has no loading circuit and is made of copper, thus, it has much higher conductive losses, as compared to the silver ring. If the load resistor is chosen so that the total resistance for the silver loop equals of the unloaded loop made of copper, the macroscopic descriptions of media formed by these two different loops are exactly the same. Thus, the electrodynamic method predicts the same value for the reactive energy stored in these two media. Inspection of Fig. 1 clearly indicates, however, that this is not the case. There is some additional energy stored in the load inductance and capacitance, which is invisible on the level of the macroscopic permeability description. Proper definition of the stored energy must be done at the microscopic level, which is possible with the equivalent-circuit method. III. EQUIVALENT-CIRCUIT METHOD: BRIEF REVIEW OF EARLIER RESULTS A commonly accepted permeability model as an effective medium description of dense (in terms of the wavelength) arrays of broken loops, split-ring resonators, and other similar structures reads (1) (see, e.g., [1], [2], [4], and [8].) In (1), is the amplitude factor is the undamped angular frequency of the zeroth pole pair (the resonant frequency of the array), and

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Fig. 2. (a) Magnetic material sample in the probe magnetic field of a tightly wound long solenoid. (b) Equivalent-circuit model (losses in the matrix material are not taken into account).

Fig. 3. (a) Magnetic material sample in the probe magnetic field of a tightly wound long solenoid. (b) Equivalent-circuit model taking into account losses in the matrix material.

is the loss factor. The model is obviously applicable only in , the perthe quasi-static regime since, in the limit meability does not tend to . At extremely high frequencies, materials cannot be polarized due to inertia of electrons, thus, a [24]. However, (1) physically sound high-frequency limit is gives correct results at low frequencies and in the vicinity of the resonance. This is the typical frequency range of interest, e.g., when utilizing artificial magneto-dielectric substrates in antenna miniaturization [13], [16], [17]. The other relevant restriction on the permeability function is the inequality [24]

incident magnetic field. The electromagnetic field energy density in the material was found to be [22]

(2) valid in the frequency regions with negligible losses. Physically, the last restriction means that the stored energy density in a passive linear lossless medium must always be larger than the energy density of the same field in vacuum. Macroscopic model (1) violates restriction (2) at high frequencies, which is another manifestation of the quasi-static nature of the model. In the vicinity of the magnetic resonance, the effective permittivity of a dense array of split-ring resonators is weakly dispersive, and can be assumed to be constant. In [22], the energy density in dispersive and lossy magnetic materials was introduced via a thought experiment. A small (in terms of the wavelength or the decay length in the material) sample of a magnetic material [described by (1)] was positioned in the magnetic field created by a tightly wound long solenoid having inductance , Fig. 2(a). The inclusion changes the impedance of the solenoid to (3) The equivalent circuit with the same impedance was found to be that shown in Fig. 2(b), [22] with the impedance seen by the source

(6) An important assumption in [22] and in this paper is that the current distribution is nearly uniform over the loop. This means that the electric dipole moment created by the excitation field is negligible as compared to the magnetic moment. The aforementioned assumption also sets the validity range for the equivalent-circuit method. The inclusions must be small compared to the wavelength so that higher order modes are not induced in the loops and the medium can be described using a single resonant circuit. In [22], only losses due to nonideally conducting metal of loops were taken into account, and losses in the matrix material (substrate material on which metal loops are printed) were neglected. It will be shown below that neglecting the matrix losses can lead to severe overestimation of the achievable permeability values. IV. GENERALIZED EQUIVALENT-CIRCUIT MODEL AND PERMEABILITY FUNCTION Losses in the matrix material (typically a lossy dielectric laminate) can be modeled by an additional resistor in parallel with the capacitor. Indeed, if a capacitor is filled with a lossy dielectric material, the admittance reads (7) where the latter expression denotes a loss conductance. Thus, the microscopically correct equivalent-circuit model is as shown in Fig. 3(b). The impedance seen by the source can be readily solved as follows: (8)

(4) The macroscopic permeability function corresponding to this model reads

which is the same as (3) if (5) The aforementioned equivalent-circuit model is correct from the microscopic point-of-view since the modeled material is a collection of capacitively loaded loops magnetically coupled to the

(9)

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Comparing (1) and (9), we immediately notice that (1) is an insufficient macroscopic model for the artificial material if the losses in the host matrix are not negligible. A proper macroscopic model correctly representing the composite from the microscopic point-of-view is

(10) Equation (9) is the same as (10) if Fig. 4. (a) Schematic illustration of the metasolenoid. (b) Finite-size metasolenoid approximated as a magnetic ellipsoid.

and the permeability function takes the form

(16) (11) . The macroscopic Above we have denoted permeability function of different artificial magnetic materials can be conveniently estimated using (10), as several results are known in the literature for the effective circuit parameter values for different unit cells, e.g., [2], [6], and [8]. For the use of (10), it is important to know the physical nature of the equivalent loss resistor . If losses in the matrix material are due to a finite conductivity of the dielectric material, the complex permittivity reads (12) where is the conductivity of the matrix material. Thus, we see from (7) that the loss resistor is independent from the frequency and can be interpreted as a “true” resistor. Moreover, in this case, the permeability function is that given by (10). However, depending on the nature of the dielectric material, the loss mechanism can be very different from (12), and in other situations, the macroscopic permeability function needs other modifications. For example, let us assume that the permittivity obeys the Lorentzian type dispersion law

(13) is the angular frequency of the electric resonance, where is the amplitude factor, and is the loss factor. Moreover, we assume that the material is utilized well below the electric res. With this assumption, the permittivity onance, thus, becomes (14) We notice from (7) that, in this case, the equivalent loss resistor becomes frequency dependent as follows: (15)

where is a real-valued coefficient depending on the dielectric material. For other dispersion characteristics of the matrix material, the permeability function can have other forms. V. EXPERIMENTAL VALIDATION OF THE PROPOSED CIRCUIT MODEL AND PERMEABILITY FUNCTION Here, we present experimental results that validate the generalized equivalent-circuit model and the corresponding macroscopic permeability function. The measurement campaign and the experimental results are described in detail in [8]. For the convenience of the reader, we briefly outline the main steps of the measurement procedure. The measured artificial magnetic particle, metasolenoid, is schematically presented in Fig. 4(a). In that figure, and denote the cross-sectional dimensions of the broken loops, is the separation between the loops, is the strip width, and is the width of the gap in the loops. In [8], the effective permeability of a medium densely filled with infinitely long metasolenoids was derived in the form (17) where is the volume filling ratio, is the cross section area of the broken loop, and the total effective impedance was presented in the form (18) For the experimental validation, a finite-size metasolenoid was approximated as an ellipsoid cutoff from a magnetic medium described by (17) [see Fig. 4(b)], where is the longitudinal length of the metasolenoid sample. The magnetic polarizability of the and the effective permeability of the medium are ellipsoid related using the classical formula for the polarizability of an ellipsoid (e.g., [27]). On the other hand, permeability is defined using the equivalent circuit [see Fig. 3(b)]. Thus, the magnetic polarizablity of the measured sample contains all the relevant

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data for validating both the proposed equivalent circuit and the permeability function. The field amplitude scattered by an electrically small material sample was measured by positioning the sample inside a standard rectangular waveguide. In addition to this, the scattered field amplitude was analytically calculated enabling the from the measured results [8]. Though it extraction of is not explicitly mentioned in [8], substrate losses were taken into account when analytically calculating the magnetic polarizability of the sample. The authors used (18) to define the total impedance of the metasolenoid unit cell, however, complex permittivity was used when calculating the effective capacitance. Thus, the equivalent circuit of the unit cell used to analyze the measured sample is the proposed circuit shown in Fig. 3(b).2 It can easily be verified using the data presented in [8] that the following expression for the effective impedance [derived using the circuit in Fig. 3(b)] exactly repeats the analytical estimation for the magnetic polarizability of the sample:

(19) . The analytically calculated [ where given by (19) is used in (17)] and measured magnetic polarizabilities are shown in Fig. 5. The measured and calculated key parameters are gathered in Table I. The polarizability and permeability values in Table I are the maximum values. The measured results agree rather closely with the analytical calculations when the proposed model is used. The slight difference in the resonant frequencies, and the slightly lowered polarizability values in the measurement case are most likely caused by limitations in the accuracy of the manufacturing process. The implemented separation between the rings is probably slightly larger than the design value. This lowers the effective capacitance and is seen as a weakened magnetic response and a higher clearly resonant frequency. Moreover, the measured indicates that the effect of the lossy glue used to stack the rings is underestimated in the analytical calculations. In the analytwas used for the total loss tanical calculations, would accurately gent [8]. A loss-tangent value produce the measured polarizability values, and the bandwidths curve) would visually coincide. (defined from the If the matrix losses are neglected in the circuit model [ in (19)], the analytical calculations lead to dramatically overestimated polarizability and permeability values. It is, therefore, evident that the proposed generalization of the circuit model and the permeability function have a significant practical importance. Though the model has been validated using a specific example, we can conclude that matrix losses can strongly dominate over conductive losses in structures where the unit cells are closely spaced. This is physically well understandable since, in this case, the electric field is strongly concentrated inside the substrate. 2Please

note that the unit cell of the metasolenoid contains two broken loops [8]. In the quasi-static regime the structure can be described using the total effective current (the sum of the currents in two adjacent loops) flowing in the unit cell. Thus, the use of the equivalent circuit in Fig. 3(b) is justified, just like for a “double split-ring particle” [3] in the fundamental mode.

Fig. 5. Analytically calculated (proposed model) and measured magnetic polarizabilities.

TABLE I ANALYTICALLY CALCULATED AND MEASURED PROPERTIES OF THE METASOLENOID SAMPLE

VI. ELECTROMAGNETIC FIELD ENERGY DENSITY Following the approach introduced in [22], we will next generalize the expression for the energy density in artificial magnetics using the experimentally validated circuit model. In the time–harmonic regime, the total stored energy reads [notations are clear from Fig. 3(b)]

(20)

(21) Using the notations in (11), the stored energy can be written as (22) The inductance per unit length of a tightly wound long solenoid is , where is the number of turns per unit length

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and is the cross-sectional area. The relation between the curinside the solenoid is . rent and magnetic field Thus, the stored energy in one unit-length section of the solenoid reads

(23) from which we obtain the expression for the electromagnetic field energy density in the artificial material sample (24) We immediately notice that if there is no loss in the matrix maand ), then and (24) reduces terial ( to (6).

Fig. 6. Magnetic field energy density given by different expressions. ! is the resonant frequency of the metasolenoid medium.

A. Comparison with the Results Obtained Using the Electrodynamic Method

than the energy density in vacuum (when there are no losses, , this nonphysical behavior takes place at frequencies where the quasi-static model is still valid, but restriction (2) is violated) [23, Fig. 1(a) and (b)]. This behavior is avoided with the result obtained using the equivalent-circuit method since that approach is based on the microscopic description of the medium, which is always in harmony with the causality principle. Fig. 6 depicts the normalized magnetic field energy density in a medium formed by the metasolenoids introduced in Section V (in Fig. 6, “EC” denotes the equivalent-circuit method and “ED” denotes the electrodynamic method). The amplitude and the loss factors have been estimated using factor (11) and the data presented in [8]. In this particular example, the energy densities given by (6) and (24) are practically the same over the whole studied frequency range (the result given by (6) is not plotted in Fig. 6 since that curve visually coincides with that given by the equivalent-circuit method). This is due mask the effect of to the fact that large values of and and in (6) and (24). The results given by the electrodynamic method and the classical expression (26) also visually coincide. However, as was mentioned above, the energy density expression given by the electrodynamic method predicts the same nonphysical behavior as the classical expression. The field energy density is smaller than the energy density in vacuum at frequen. cies

The above derived result differs from the result found in [23] and obtained using the electrodynamic method

(25) The procedure and the underlying assumptions to obtain (25) have been briefly reviewed in Section II. The classical expression for the magnetic field energy density in media where absorption due to losses can be neglected reads [24], [25] (26) in It is seen that in the presence of negligible losses [ (1)] the energy density result given by (25) is the same as the result predicted by the classical expression (26). However, (24) predicts a different result. The authors of [23] use this fact to state that the result obtained using the electrodynamic method is more inherently consistent than the result obtained using the equivalent-circuit method. The equivalent-circuit method is known to give a perfectly inherently consistent result for the energy density in dielectrics obeying the general Lorentzian type dispersion law [22]. The general Lorentz model is a strictly causal model. This is, however, not the case with the modified Lorentz model (1). As is speculated already in [22], the reason for the difference in the results obtained using (24) and (26) in the small-loss limit is related to the physical limitations of the quasi-static permeability model (1). Thus, when (1) is used as the macroscopic medium description, (24) should be also used in the presence of vanishingly small losses. The electrodynamic method, though being inherently consistent with the classical expression, predicts nonphysical behavior at high frequencies. At high frequencies, the energy density given by the electrodynamic method is smaller

VII. PRACTICAL APPLICATION EXAMPLE Here, we present a specific application example for the established theory and study the differences between the equivalent-circuit method and the electrodynamic method using this example. We consider an electrically small microwave resonator made of an artificial magnetic material, and calculate the quality factor of the resonator using the two methods. The quality-factor estimation of artificial magnetic resonators has a strong practical importance as several suggestions have been recently made to use such resonators, e.g., in filter design (see, e.g., [18] and [19]).

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To be able to compare the results from the two methods, we first consider (1) as the macroscopic medium description since substrate losses are not taken into account in [23]. After the comparison, we use (10) as the medium description, and calculate the quality factor for the case when the substrate losses are taken into account.

dielectrics that are modeled by the general Lorentz dispersion model, both macroscopic electrodynamic method and the microscopic circuit method give the same correct result for the quality factor. This shows that the problem of the electrodynamic method is indeed in the noncausal nature of the quasistatic model of broken loop composites.

A. Quality Factor Using the Equivalent-Circuit Method

C. Quality Factor Taking Into Account Substrate Losses

The power lost in the unit volume of a medium described by (1) can be derived based on the circuit presented in Fig. 2(b) and the result reads

When substrate losses are taken into account, the equivalentcircuit method yields the following expression for the power dissipated in the medium:

(27) (32) Above we have assumed that radiation losses in the quasi-static regime are negligible compared to the ohmic losses. The energy stored in the same unit volume of the medium reads

Furthermore, the stored energy reads (33)

(28) and for the quality factor we get and for the quality factor we get (34) (29) which, as expected, is the quality factor of a series resonant circuit.

We immediately notice that if there is no loss in the matrix material, (34) reduces to (29). VIII. CONCLUSION

B. Quality Factor Using the Electrodynamic Method The electrodynamic method also yields (27) when calculating the power loss (see [23, eqs. (26) and (27)]). However, the stored energy in this case reads (30) leading to the following expression for the quality factor: (31) When comparing the quality factors obtained using the two methods, we notice that both of them behave in a physically sound manner when losses in the medium are low and further decrease. We remind here that in the low-loss limit (26) can also be used to determine the energy stored in the resonator. Further, if we use (27) to calculate the dissipated power [now is small in (27)], the resulting quality factor coincides with (29). However, if losses become significant, (26) is not valid anymore, but the quality factor obtained using the electrodynamic method also behaves in a nonphysical manner. The quality factor increases with increasing strong losses. This nonphysical behavior is avoided with the result obtained using the equivalent-circuit method. It is interesting to note that when the general (causal) Lorentz model [ replaces in the nominator of (1)] is used, the quality factor, obtained using the electrodynamic method, is given by (29) [23, eqs. (25) and (28)]. Thus, for instance, for artificial

In this paper, we have explained differences between recent approaches used to derive field energy density expressions for artificial lossy and dispersive magnetic media. The equivalentcircuit model of broken loops and other similar structures has been generalized to take into account losses in the dielectric matrix material. It has been shown that a modification is needed to the macroscopic permeability function commonly used to model these materials in the quasi-static regime. Moreover, depending on the nature of the dominating loss mechanism in the matrix material, the permeability function has different forms. The proposed circuit model and the modified permeability function have been experimentally validated, and it has been shown that, in a practical situation, matrix losses can dramatically dominate over conduction losses. Using the validated circuit model, we have derived an expression for the electromagnetic field energy density in artificial magnetic media. This expression is also valid when losses in the material cannot be neglected and when the medium is strongly dispersive. The results have been compared to recently reported alternative approaches. As a practical application example, we have calculated the quality factor of a microwave resonator made of an artificial magnetic material using the proposed method and a different method introduced in the literature. ACKNOWLEDGMENT The authors would like to acknowledge and thank Prof. C. Simovski, State University of Informational Technology, Mechanics, and Optics at St. Petersburg, St. Petersburg, Russia, for useful discussions.

IKONEN AND TRETYAKOV: DETERMINATION OF GENERALIZED PERMEABILITY FUNCTION AND FIELD ENERGY DENSITY IN ARTIFICIAL MAGNETICS

REFERENCES [1] M. V. Kostin and V. V. Shevchenko, “Artificial magnetics based on double circular elements,” in Proc. Bianisotrop., Périgueux, France, May 18–20, 1994, pp. 49–56. [2] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [3] R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B, Condens. Matter, vol. 65, pp. 1 444 401–1 444 406, Apr. 2002. [4] M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J., B, vol. 28, no. 3, pp. 263–269, Jul. 2002. [5] A. N. Lagarkov, V. N. Semenenko, V. N. Kisel, and V. A. Chistyaev, “Development and simulation of microwave artificial magnetic composites utilizing nonmagnetic inclusions,” J. Magn. Magn. Mater., vol. 258–259, pp. 161–166, Mar. 2003. [6] B. Sauviac, C. R. Simovski, and S. A. Tretyakov, “Double split-ring resonators: Analytical modeling and numerical simulations,” Electromagnetics, vol. 24, no. 5, pp. 317–338, 2004. [7] J. D. Baena, R. Marqués, F. Medina, and J. Martel, “Artificial magnetic metamaterial design by using spiral resonators,” Phys. Rev. B, Condens. Matter, vol. 69, pp. 014 4021–014 4025, Jan. 2004. [8] S. I. Maslovski, P. Ikonen, I. A. Kolmakov, S. A. Tretyakov, and M. Kaunisto, “Artificial magnetic materials based on the new magnetic particle: Metasolenoid,” Progr. Electromagn. Res., vol. 54, pp. 61–81, 2005. [9] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, no. 18, pp. 4184–4187, May 2000. [10] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [11] R. C. Hansen and M. Burke, “Antenna with magneto-dielectrics,” Microw. Opt. Technol. Lett., vol. 26, no. 2, pp. 75–78, Jul. 2000. [12] S. Yoon and R. W. Ziolkowski, “Bandwidth of a microstrip patch antenna on a magneto-dielectric substrate,” in IEEE AP-S Int. Symp., Columbus, OH, Jun. 22–27, 2003, pp. 297–300. [13] H. Mossallaei and K. Sarabandi, “Magneto-dielectrics in electromagnetics: Concept and applications,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1558–1567, Jun. 2004. [14] M. K. Kärkkäinen, S. A. Tretyakov, and P. Ikonen, “Numerical study of a PIFA with dispersive material fillings,” Microw. Opt. Technol. Lett., vol. 45, no. 1, pp. 5–8, Apr. 2005. [15] M. E. Ermutlu, C. R. Simovski, M. K. Kärkkäinen, P. Ikonen, S. A. Tretyakov, and A. A. Sochava, “Miniaturization of patch antennas with new artificial magnetic layers,” in 2005 IEEE Int. Antenna Technol. Workshop, Singapore, Mar. 7–9, 2005, pp. 87–90. [16] K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 135–145, Jan. 2006. [17] P. M. T. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric substrates for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 6, pp. 1654–1662, Jun. 2006. [18] J. Garcia-Garcia, F. Martín, F. Falcone, J. Bonache, I. Gil, T. Lopetegi, M. A. G. Laso, M. Sorolla, and R. Marqués, “Spurious passband suppression in microstrip coupled line bandpass filters by means of split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 9, pp. 416–418, Sep. 2004.

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[19] J. Garcia-Garcia, F. Martín, F. Falcone, J. Bonache, J. D. Baena, I. Gil, E. Amat, T. Lopetegi, M. A. G. Laso, J. A. M. Iturmendi, M. Sorolla, and R. Marqués, “Microwave filters with improved stopband based on sub-wavelength resonators,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 1997–2006, Jun. 2005. [20] R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A, vol. 299, pp. 309–312, Jul. 2002. [21] T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B, Condens. Matter, vol. 70, pp. 205 1061–205 1067, Nov. 2004. [22] S. A. Tretyakov, “Electromagnetic field energy density in artificial microwave materials with strong dispersion and loss,” Phys. Lett. A, vol. 343, pp. 231–237, Jun. 2005. [23] A. D. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B, Condens. Matter, vol. 73, pp. 165 1101–165 1107, Apr. 2006. [24] L. D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media, 2nd ed. Oxford, U.K.: Pergamon, 1984. [25] L. A. Vainstein, Electromagnetic Waves (in Russian), 2nd ed. Moscow, Russia: Radio i Sviaz, 1988. [26] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1999. [27] A. Sihvola, Electromagnetic Mixing Rules and Formulas, ser. Electromagn. Waves. London, U.K.: IEE Press, 1999. Pekka M. T. Ikonen (S’04) was born on December 30, 1981, in Mäntyharju, Finland. He received the M.Sc. degree (with distinction) in communications engineering from Helsinki University of Technology, Espoo, Finland, in 2005, and is currently working towards the Ph.D. degree at Helsinki University of Technology. He is a member of the Finnish Graduate School of Electronics, Telecommunications, and Automation (GETA). His current research interests include the utilization of electromagnetic crystals in microwave applications and small antenna miniaturization using magnetic materials. Mr. Ikonen was the recipient of the IEEE Antennas and Propagation Society (IEEE AP-S) Undergraduate Research Award (2003–2004), the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Undergraduate Scholarship (2004–2005), and the URSI Commission B Young Scientist Award in the URSI XXVIII General Assembly (2005).

Sergei A. Tretyakov (M’92–SM’98) received the Dipl. Engineer-Physicist, the Candidate of Sciences (Ph.D.), and the Doctor of Sciences degrees from the St. Petersburg State Technical University (Russia), in 1980, 1987, and 1995, respectively, all in radiophysics. From 1980 to 2000, he was with the Radiophysics Department, St. Petersburg State Technical University. He is currently a Professor of radio engineering with the Radio Laboratory, Helsinki University of Technology, Espoo, Finland. He is also a Coordinator of the European Network of Excellence Metamorphose. His main scientific interests are electromagnetic field theory, complex media electromagnetics, and microwave engineering. Prof. Tretyakov served as chairman of the St. Petersburg IEEE Electron Devices (ED)/Microwave Theory and Techniques (MTT)/Antennas and Propagation (AP) Chapter (1995–1998).

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Fast Numerical Computation of Green’s Functions for Unbounded Planar Stratified Media With a Finite-Difference Technique and Gaussian Spectral Rules Athanasios G. Polimeridis, Student Member, IEEE, Traianos V. Yioultsis, and Theodoros D. Tsiboukis, Senior Member, IEEE

Abstract—A computational methodology for the calculation of Green’s functions for unbounded planar stratified media is proposed in this paper. Unlike several techniques that are based on analytical calculations in the spectral domain, this one is based on a previously proposed finite-difference discretization of the corresponding one-dimensional differential equation and the resulting spectral matrix problem. A smart approximation of the radiation boundary condition, based on the mathematical methodology of Gaussian spectral rules and the associated construction of an optimal nonuniform grid in the unbounded space, succeeds in maintaining the tridiagonal nature of the matrix spectral problem, providing a highly efficient means of computing the desired multilayer Green’s function. Index Terms—Finite-difference technique, Gaussian spectral rules, Green’s functions, stratified media.

I. INTRODUCTION NTEGRAL-EQUATION methods constitute one of the most powerful classes of methods for the computational analysis of integrated microwave circuits and multilayered printed antennas [1]–[4]. The rigorous analysis of geometries in a layered medium with an integral-equation-based method of moments (MoM) [5] requires the computation of the corresponding Green’s functions, which are traditionally represented by the Sommerfeld integrals in the spatial domain, and by corresponding closed-form expressions in the spectral domain. Several techniques for an efficient computation of these functions have been proposed in this context, a detailed presentation of which can be found in [6]. It is interesting to note that there is a variety of competing methods, some of them focusing on more or less analytical techniques, while others are oriented toward primarily computational methodologies. For instance, such a computational approach has been proposed for the evaluation of electromagnetic Green’s functions

I

Manuscript received June 20, 2006; revised September 9, 2006. This work was supported in part by the Greek General Secretariat of Research and Technology under Grant PENED 03ED936. The authors are with the Telecommunications Division, Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, 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/TMTT.2006.886919

in unbounded planar layered media [6]. This particular approach is based on the replacement of the analytical forms of the spectral-domain Green’s function with discrete ones, obtained from a finite-difference solution of the governing differential equations for the corresponding magnetic vector potential components [6]. This formulation is then combined with an eigenvalue analysis of the resulting finite-difference matrix to represent the vector potential components as a sum of pole-residue terms, reminiscent of the extracted propagating wave contributions in the discrete complex image method (DCIM) [7]. The pole-residue form of the Green’s function spectrum allows the analytic evaluation of the Sommerfeld integrals [6], thus yielding a finite sum of Hankel functions as the spatial form for the Green’s function. The only problem with this method is that, unlike what is the case in the well-established problem of shielded multilayer structures (multilayered media with infinite transverse dimensions, and with both top and bottom ground planes) [8], in the case of the unbounded problem, the tridiagonal character of the finite-difference matrix is destroyed due to the necessary enforcement of the radiation condition. The existence of a full submatrix of the auxiliary variables in [6] weakens the computational efficiency of the proposed algorithm. In this paper, we propose an algorithm for the implementation of the radiation condition via optimal nonuniform second-order finite-difference representation, which efficiently overcomes this mismatch and maintains the tridiagonal character of the corresponding matrix. The algorithm is based on the development of an appropriate Gaussian spectral rule using a Padé–Chebyshev approximation of the Neumann-to-Dirichlet (ND) map [9], [10]. This approach makes it possible to avoid the direct rational operator approximation [6], which results in nonlocal operator boundary conditions that destroy the sparse structures of finite-difference methods. This Gaussian spectral rule methodology could be conceived as a counterpart to Gaussian quadrature integration for the solution of a differential equation, which provides optimal nonuniform finite-difference grids. The merit of the proposed approach, apart from the fact that the tridiagonal property is preserved, is that it provides exponential convergence of the error as the number of points increases, ultimately resulting in highly accurate representations with just three points in the radiation condition approximation.

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POLIMERIDIS et al.: FAST NUMERICAL COMPUTATION OF GREEN’S FUNCTIONS FOR UNBOUNDED PLANAR STRATIFIED MEDIA

To demonstrate the implementation of the Gaussian spectral concept in the computational calculation of Green’s function, for the sake of simplicity, we focus here on vertical dipole excitation. The extension to horizontal dipoles can be considered straightforward, although a detailed treatment is a matter of future investigation. This paper is organized as follows. The proposed absorbing boundary condition (ABC) via optimal nonuniform grid is presented in Section II. In Section III, we incorporate the optimal grid in the interior uniform finite-difference grid with a suitable conjugation scheme. Numerical results are presented in Section IV and a comparison with [6] shows the optimality of the proposed algorithm. Additional important mathematical issues related to the Padé–Chebyshev approximation are discussed in the Appendix.

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and is the where . sine of the incidence angle of the wave on the plane Moreover, we assume that only the so-called propagating modes are present in the spectrum of the solution, hence, with . Instead of the direct implementation of the approximate condition, one can modify the following ordinary second-order difin the ferential equation for the vector potential spectral domain [6] (3) in such a way that it would be easier to solve and for the new solution would still well approximate (2) [12]. Bearing this in mind, we consider the standard coordinate stretching [13]–[15]

II. MODIFIED ABC VIA OPTIMAL GRID The essence of the computational technique presented in [6] and [8] is to discretize the differential equation for the magnetic vector potential representing the Green’s function under investigation with a finite-difference technique. The resulting discrete model in the spectral domain is then expressed in a matrix form, which is tridiagonal and very easy to solve in the case of bounded planar layered media. However, in the most interesting case of unbounded structures, the radiation condition has to be enforced, in the form of an ABC. It turns out that the efficiency of the finite-difference scheme strongly depends on how the radiation condition at the truncated boundary of the interior grid is enforced. More precisely, if the truncation boundary is set at a point and , the exact form of the unbounded domain is in the region the radiation or impedance boundary condition [6] is (1) which clearly originates from the differential equation itself. Instead of using (1) to substitute for the derivative term in the finite-difference scheme, a pole-residue approximation of the inverse square root function was implemented in [6] by means of a fitting technique [11]. The specific local ABC introduces a full block submatrix, of dimension equal to the number of terms in the pole-residue approximation, at one or both corners of the finite-difference matrix along the main diagonal [6]. The presence of these blocks violates the tridiagonal nature of and weakens the general efficiency of the novel algorithm introduced in [6]. Here, we propose a modified ABC scheme via an optimal nonuniform finite-difference grid, which with an additional remaintains the tridiagonal property of duction factor of more than ten for the approximation of the inverse square root function. The impedance boundary condition (1), with a slight modification, takes the following form: (2)

otherwise

(4)

, which is Next, we define a new function equal to if and take to be real. We could simply in the region over the consider a fictitious “extension” of truncation point. The equation that this new function satisfies can be written in divergence form as (5) where if otherwise

(6)

Equation (5) is a standard divergence equation with disconand are continuous tinuous coefficients; . across the interface at The transformed Helmholtz equation (5) becomes diffusive and remains the same as the original (3) (absorbing) for elsewhere. From now on we will approximate the solution of (5) and drop the bars over all symbols. The impedance for condition (2) is then written as (7) and the error functional for the discrete absorbing condition is (8) where is the discrete solution of (5) and denotes the finitedifference analog of the derivative of . Now, using a second-order finite-difference approximation of the diffusive part of (5) [12], we want to approximate the ABC and get an explicit expression for the error functional (8) in terms of the steps of the finite-difference discretization. For this , we consider purpose, using the change of variables

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(5) for ditions:

with the following boundary con-

We symmetrize the system (10) and (11) by defining a and the following tridiagonal “scaled” variable system of equations is obtained:

(9) Let us approximate the solution of this problem by a nonuniform three-point finite-difference scheme. In such a scheme, the numerical solution is defined at primary nodes

(14) where is the Kronecker delta and the coefficients are related to the cell sizes via

with (15)

and and the finite-difference derivatives are defined at dual nodes where the

and is assumed to be infinite. We define real symmetric tridiagonal matrix

with and

(16)

We denote the step sizes by and let be its eigenvalues and first components of its eigenvectors, respectively. Using the eigendecomposition of , and after some algebraic manipulation, the discrete impedance function can be expressed as a Padé approximation of order [16],

and and solve the discretized problem

(10) with the boundary conditions (11) It is important to note that we pursue something more than a solution of (10) and (11). In fact, we are interested in determining the locations of points or, equivalently, the cell sizes , providing minimal error (8) for any possible [9], [10]. The analytical ND map of (10) and (11) at can be defined by the following expression:

(17) first, then synHence, the aim is to find the parameters and, finally, compute from its entries. thesize matrix We require that (17) will be an as accurate as possible approxwithin a given spectral interval imation of . can be determined from the The unknown parameters minimization of error (13). It can be shown that the most appropriate norm is based on a Padé–Chebyshev approximant [16], which is equivalent to setting the following corresponding moments equal to zero:

(12) where is the well-known impedance function. Hence, we seek the minimization of the error (13) is the discrete ND map of the same boundary value where problem. An explicit expression for this function via a solution of (10) and (11) has to be found in terms of the unknown cell , and this expression will be matched to the analytsizes ical solution.

(18) . Such an approximation exhibits expofor nential reduction of the error with respect to the order of the approximation. To solve the problem, it is easier to reformulate (18) in terms of Chebyshev polynomials and simply require that all th-order Chebyshev moments of the residual function are zero, i.e., (19)

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where is the transformed spectral parameter in the Chebyshev interval from 1 to 1. After some algebraic manipulation (see the Appendix), (19) is written in the semidiscrete form (20) for

, where

(21) Fig. 1. Optimal grid for  = 10.

are the unknown parameters and (22)

is a function of a simple form (see the Appendix). From a mathematical point-of-view, (20) could be thought of as the set of equations to produce a Gaussian integration rule. In this context, and are the points and weights, respectively, of the integration rule that is constructed. Moreover, instead of solving the nonlinear set of (20), it is much more efficient to use the Lanczos algorithm [17]. In this case, the algorithm refers to the space of polynomials, thus forming the orthogonal polynomials with respect to the measure of integration. It can be also are exactly the eigenvalues of the proven that the numbers tridiagonal Lanczos matrix that is generated. from (21) From the values of and , we obtain and and, due to (17) and the orthogonality property , we directly compute and

(23)

We then have to solve an inverse spectral problem, i.e., syntheand size the tridiagonal matrix (16) from its eigenvalues first components of its eigenvectors . This can be done by applying the standard Lanczos algorithm for the diagonal matrix [17]. Finally, the primary and secondary cell lengths are directly computed, due to (15), via the recursion

and

Fig. 2. Spectral distributions of the relative error (as a function of L) for various grids.

error. We have to note that the stability of the grid optimization scheme, which results in alternating points for the potential and its derivative, requires proper reorthogonalization in the application of Lanczos algorithms. It is well known from the corresponding theory that reorthogonalization is necessary and proves to be sufficient to guarantee stability of the Lanczos algorithm [17]. Moreover, in the logarithmic scale, the points and cannot be seen. Finally, in Fig. 2, we present the specfor various tral distributions of the relative error grids. As expected, the error decays exponentially versus until the roundoff error level is reached.

(24) III. CONJUGATION SCHEME where . This simple recursion essentially completes the calculation of the optimal nonuniform grid. is shown in Fig. 1. An example of an optimal grid for Apparently, the grid density increases close to the origin (truncation boundary) to provide exponential convergence of the

In Section II, we obtained a modified ABC via an optimal nonuniform grid for the diffusive part of (5). The main objective of this paper is the implementation of such a grid in the interior uniform scheme (14) in order to maintain the tridiagonal character of the finite-difference matrix . Bearing this in

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mind, we rewrite the equations on the truncation boundary for the two grids [6]

(25) and

(26) where, in (26), we keep the notation of Section II to make clear that this variable is not the electric or magnetic potential, but it can be considered as an auxiliary variable of the enforced radiation condition in a way similar to the case of the variable in [6]. Equations (25) and (26) together with the conjugation condition [10]

Fig. 3. Finite-difference matrix via rational function fitting.

(27) collapse to just one three-term equation

(28) and . where Eventually, (10), (14), and (28) show that computation of all the unknowns is reduced to solving just one tridiagonal system. IV. NUMERICAL RESULTS The following numerical example involves the application of the proposed modification to the existing technique for the derivation of closed-form Green’s functions for unbounded planar layered media. In order to demonstrate the improvement achieved, the case of a vertical-magnetic dipole located above a conductor-backed four-layer dielectric [6] was considered. The thickness of each layer (from bottom to top) was taken to be and . The corresponding relative permittivities were and . The magnetic , where moment of the dipole was set to is the wavenumber in free space. The dipole is above the top layer. Two different situated at a distance of planes of observation were considered, the plane of the source and the plane of the top interface of the . The number of finite-difference layered substrate , while the grid points (uniform grid) was set to number of auxiliary variables associated with the enforcement terms for the case of of the radiation condition was for the optimal tridiagonal scheme rational fitting and proposed in this paper. Figs. 3 and 4 demonstrate the difference

Fig. 4. Finite-difference matrix via optimal grid.

in the character of the associated matrices, i.e., the substitution submatrix with an equivalent tridiagonal of a full block. It is clear that we avoid the destruction of the tridiagonal character of the finite-difference matrix with an additional reduction factor of 18 for the approximation of the radiation condition. In such a way, we improve the efficiency of the eigenvalue decomposition of , which determines the complexity of the novel method proposed in [6]. We have to mention here that the method presented in this paper applies as well for the case of a

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Fig. 5. Magnitude of the vector potential versus distance from the dipole for the case of a vertical magnetic dipole above a four-layer conductor-backed dielectric medium (z = z = 0:7 ).

Fig. 7. Magnitude of the vector potential versus distance from the dipole for the case of a vertical magnetic dipole above a four-layer conductor-backed dielectric medium (z = 0:6 ; z = 0:7 ).

Fig. 6. Phase of the vector potential versus distance from the dipole for the case of a vertical magnetic dipole above a four-layer conductor-backed dielectric medium (z = z = 0:7 ).

Fig. 8. Phase of the vector potential versus distance from the dipole for the case of a vertical magnetic dipole above a four-layer conductor-backed dielectric medium (z = 0:6 ; z = 0:7 ).

dipole inside the dielectric slab without any modification. The only thing needed is to relocate the only nonzero element of the excitation vector [6], which indicates the location of the dipole. Next, in Figs. 5–8, the calculated magnitude and phase of the vector potential for the two planes of observation and the are shown. The results are in interval very good agreement, a fact that validates the proposed modified ABC via optimal nonuniform grid. Finally, we have to say a few things about the amount of work needed, in terms of CPU time, for the approximation of the radiation condition. The algorithm of the vector fitting [6] for the evaluation of the auxiliary variables is iterative, requiring approximately 30–40 iterations in order to converge. Moreover, the CPU time needed for a single iteration is approximately 4–5 times more than the CPU time needed for the implementation of the Gaussian spectral-rule-based ABC. Hence, besides the

higher accuracy and the maintenance of the tridiagonal character of the finite-difference matrix, the proposed ABC requires significantly less CPU time. V. CONCLUSION A highly efficient finite-difference technique for the computation of Green’s functions for unbounded planar stratified media has been presented. The use of the quite recently investigated concept of Gaussian spectral rules in the context of differential equations has been utilized to provide a highly accurate spectral approximation of the radiation boundary condition in terms of an additional optimal nonuniform grid, naturally conjugated to the piecewise uniform grid within the layers of the planar structure. This kind of approach not only exhibits exponential accuracy in the approximation of the radiation condition, but also provides a tridiagonal spectral matrix problem, which is solved very fast. The overall analysis is eventually straightforward and

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easy to apply and has the distinctive advantage of avoiding elaborate complex plane integrations that may be involved in existing analytical techniques. The introduction of Gaussian spectral rules in the context of electromagnetics and the associated construction of an optimal ABC are also noteworthy byproducts of the proposed analysis.

we shall obtain the analogous expression for the approximating rational function (17)

(35)

APPENDIX The function under consideration possesses the integral expansion

It follows from (31) and (35) that the formal condition on for a rational function (17) to be the Padé–Chebyshev approximant to (29) is

(29) which means that this function is of a Markov kind. What we Padé–Chebyshev approxwant is to construct the imant (19). For this reason, it is convenient to introduce the and the inverse Zhukovsky function Zhukovsky function . Using the Chebyshev series for a simple pole [18], we obtain (30) where

with

(the prime symbol denotes that the term for is to be divided by 2). By exploiting (29), (30), and the orthogonality of , we compute, after some algebraic manipulation, the th Chebyshev coefficient of (29) as follows: (31) where (32) After some changes of variables, we embody the singular behavior of the kernel in the Chebyshev weight function and get

(33) with

, . On the other hand, if we set

, and

(34)

(36)

REFERENCES [1] J. R. Mosig, , T. Itoh, Ed., “Integral equation techniques,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures. New York: Wiley, 1998. [2] D. M. Pozar, “A rigorous analysis of a microstrip feed patch antenna,” IEEE Trans. Antennas Propag., vol. AP-33, no. 10, pp. 1045–1053, Oct. 1985. [3] D. R. Jackson and N. G. Alexopoulos, “Gain enhancement methods for printed antennas,” IEEE Trans. Antennas Propag., vol. AP-33, no. 9, pp. 976–987, Sep. 1985. [4] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media—Part I: Theory,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 335–344, Mar. 1990. [5] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1983. [6] V. I. Okhmatovski and A. C. Cangellaris, “A new technique for the derivation of closed-form electromagnetic Green’s functions for unbounded planar layered media,” IEEE Trans. Antennas Propag., vol. 50, no. 7, pp. 1005–1016, Jul. 2002. [7] M. I. Aksun, “A robust approach for the derivation of closed-form Green’s functions,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 5, pp. 651–658, May 1996. [8] A. C. Cangellaris and V. I. Okhmatovski, “New closed-form Green’s function in shielded planar layered media,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2225–2232, Dec. 2000. [9] V. Druskin and L. Knizhnerman, “Gaussian spectral rules for the threepoint second differences: I. A two-point positive definite problem in a semi-infinite domain,” SIAM J. Numer. Anal., vol. 37, no. 2, pp. 403–422, 1999. [10] V. Druskin and L. Knizhnerman, “Gaussian spectral rules for second order finite difference schemes,” Numer. Algorithms, vol. 25, no. 1, pp. 139–159, 2000. [11] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” Tech. Rep., 1997, IEEE paper PE-194-PWRD-0-11-1997. [12] S. Asvadurov, V. Druskin, M. N. Guddati, and L. Knizhnerman, “On optimal finite-difference approximation of PML,” SIAM J. Numer. Anal., vol. 41, pp. 287–305, 2003. [13] W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations in stretched coordinates,” Microw. Opt. Technol. Lett., vol. 7, pp. 599–604, Sep. 1994. [14] W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett., vol. 15, pp. 363–369, Aug. 1997. [15] F. L. Teixeira and W. Chew, “PML-FDTD in cylindrical and spherical grids,” Microw. Opt. Technol. Lett., vol. 7, pp. 285–287, Sep. 1997. [16] G. A. Baker and P. Graves-Morris, Padé Approximants. London, U.K.: Addison–Wesley, 1996. [17] B. N. Parlett, The Symmetric Eigenvalue Problem. Philadelphia, PA: SIAM, 1998. [18] S. Paszkowski, Computational Applications of Chebyshev Polynomials and Series. Moscow, Russia: Nauka, 1983.

POLIMERIDIS et al.: FAST NUMERICAL COMPUTATION OF GREEN’S FUNCTIONS FOR UNBOUNDED PLANAR STRATIFIED MEDIA

Athanasios G. Polimeridis (S’01) was born in Thessaloniki, Greece, in 1980. He received the Diploma degree in electrical and computer engineering from the Aristotle University of Thessaloniki (AUTH), Thessaloniki, Greece, in 2003, and is currently working toward the Ph.D. degree at the AUTH. His research interests include computational electromagnetics, with emphasis on the development and implementation of integral-equation-based algorithms.

Traianos V. Yioultsis was born in Yiannitsa, Greece, in 1969. He received the Diploma degree (with honors) in electrical engineering and Ph.D. degree in electrical and computer engineering from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1992 and 1998, respectively. From 1993 to 1998, he was a Research and Teaching Assistant with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki. From 2001 to 2002, he was a Post-Doctoral Research Associate with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign. Since 2002, he has been a Lecturer with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki. His current interests include the analysis and design of microwave circuits and antennas with fast computational and optimization techniques and the modeling of complex wave propagation problems.

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Theodoros D. Tsiboukis (S’79–M’81–SM’99) received the Diploma degree in electrical and mechanical engineering from the National Technical University of Athens, Athens, Greece, in 1971, and the Ph.D. degree from the Aristotle University of Thessaloniki (AUTH), Thessaloniki, Greece, in 1981. From 1981 to 1982, he was with the Electrical Engineering Department, University of Southampton, Southampton, U.K., as a Senior Research Fellow. Since 1982, he has been with the Department of Electrical and Computer Engineering (DECE), AUTH, where he is currently a Professor. He has served in numerous administrative positions including Director of the Division of Telecommunications, DECE (1993–1997) and Chairman, DECE (1997–2001). He is also the Head of the Advanced and Computational Electromagnetics Laboratory, DECE. He has authored or coauthored eight books and textbooks including Higher-Order FDTD Schemes for Waveguide and Antenna Structures (Morgan & Claypool, 2006). He has authored or coauthored over 125 refereed journal papers and over 100 international conference papers. He was the Guest Editor of a special issue of the International Journal of Theoretical Electrotechnics (1996). His main research interests include electromagnetic-field analysis by energy methods, computational electromagnetics (finite-element method (FEM), boundary-element method (BEM), vector finite elements, MoM, FDTD method, ADI-FDTD method, integral equations, and ABCs), metamaterials, photonic crystals, and inverse and electromagnetic compatibility (EMC) problems. Prof. Tsiboukis is a member of various societies, associations, chambers, and institutions. He was the chairman of the local organizing committee of the 8th International Symposium on Theoretical Electrical Engineering (1995). He was the recipient of several awards and distinctions.

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Efficient Modal Analysis of Bianisotropic Waveguides by the Coupled Mode Method Jaime Pitarch, José M. Catalá-Civera, Member, IEEE, Felipe L. Peñaranda-Foix, Member, IEEE, and Miguel A. Solano, Member, IEEE

Abstract—A modal analysis based on the coupled mode method is performed for the resolution of partially filled waveguides with a general bianisotropic medium. The calculation of the generalized telegraphist’s equations can be made by using two different strategies: direct and indirect formulations. In this paper, the indirect formulation of the method is extended to bianisotropic materials and exposed in detail. The convergence of both formulations is compared and it is demonstrated that the indirect formulation converges much faster than the direct formulation, which means that fewer modes are required for a given precision. Results of some particular cases of the constitutive relationships, like dielectric, magnetic, ferrites, and omega materials are shown and compared with the bibliography or commercial electromagnetic simulators. Good agreement is found, which demonstrates the validity and reliability of the method. Index Terms—Coupled mode analysis, nonhomogeneous media, scattering matrices, waveguide discontinuities.

I. INTRODUCTION

B

IANISOTROPIC media [1]–[8] are the most general linear media, and have received considerable attention in the last few years. Electrodynamics of bianisotropic media is developing very extensively. Due to the constitutive relations in a bianisotropic medium, there is an additional coupling between the electric and magnetic fields, which has given very attractive new fundamental problems and applications [9], [10]. Some particular cases of bianisotropic media are the dielectric and magnetic media, ferrites [11], and omega pseudochiral media [12]. Ferrites have magnetic anisotropy and dielectric isotropy. They have nonreciprocal properties and their applications include electromagnetic interference (EMI) suppression, antennas, microwave circulators, and phase shifters. Electromagnetic properties of the pseudochiral or media have generated considerable attention in the literature. They produce a phenomenon of field displacement. This property makes them very interesting for applications in novel integrated microwave and millimeter-wave devices. Manuscript received June 20, 2006; revised September 20, 2006. This work was supported by the Spanish MEC under Project TEC2006-13268C03-03/TCM J. Pitarch, J. M. Catalá-Civera, and F. L. Peñaranda-Foix are with the Departamento de Comunicaciones, Universidad Politécnica de Valencia, 46022 Valencia, Spain (e-mail: [email protected]; [email protected]; [email protected]). M. A. Solano is with the Departamento de Ingeniería de Comunicaciones, Universidad de Cantabria, 39005 Santander, Cantabria, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.888576

For the development of microwave devices, optimal design parameters must be found, which imply, most times, a minimization procedure. Accurate, efficient, and fast computer-aided design (CAD) tools are then of the utmost importance. In the analysis of inhomogeneously filled waveguides, the coupled mode method [13]–[31] is a well-known technique to solve propagation characteristics and scattering properties. This method is based on expressing the characteristic modes of the inhomogeneously filled waveguide (proper modes) in terms of the corresponding empty waveguide (basis modes). The main advantage of the coupled mode method is that all the proper modes are found in a single calculation, without the need to solve any time-consuming transcendent equation and, therefore, it is not necessary to make any search of zeroes in the complex plane. The paper by Schelkunoff [13], formulated for a hypothetical general media characterized by a permittivity tensor and permeability tensor , can be considered as the foundation of the coupled mode method. This original formulation was refined later in subsequent papers [14]–[22]. For instance, in [19], a new compact operator, called the transverse operator method, is developed and applied in [21] and [22] for the analysis of multidielectrics and anisotropic waveguides, respectively. The main drawback of Schelkunoff’s original approach is that it requires the inversion of two matrices, which was a numerical problem in those years. Another important work derived from Schelkunoff’s scheme was developed by Ogusu [23]. This study deals with isotropic dielectrics and it is applied to open guides, but the essential difference of Ogusu’s approach with respect to Schelkunoff’s formulation of the coupled mode method is the direct way of expressing the longitudinal field components as a function of the transverse ones, which avoids the problem of matrix inversion. Due to the simplicity of obtaining the longitudinal field components, this formulation has been widely used in the analysis of dielectric guides [24], magnetized ferrites [25], or general bianisotropic media [26]. Henceforth we will refer to Ogusu’s approach as “direct,” whereas Schelkunoff’s approach will be referred as “indirect” formulation. Starting from Schelkunoff’s original indirect formulation, Chaloupka analyzed the scattering by dielectric and ferromagnetic obstacles in rectangular waveguide [27], but with three important novelties: first, instead of a differential operator, an inverse integral operator is introduced, second, the scattering problem is reduced to a transmission-line model by directly using the generalized telegraphist’s equations, and third, a new term with no dependence on the transverse coordinates is added component. It has been confirmed in in the expansion of the

0018-9480/$25.00 © 2006 IEEE

PITARCH et al.: EFFICIENT MODAL ANALYSIS OF BIANISOTROPIC WAVEGUIDES BY COUPLED MODE METHOD

Fig. 1. Waveguide filled with a bianisotropic material with constitutive tensors ", ,  , and  , which have a spacial dependence on the transverse coordinates (u; v ).

subsequent papers dealing with more complicated gyrotropic structures [28], [29] or chiral media [30] that the inclusion of this new term is essential in the development of any formulation of the coupled mode method using the indirect strategy in order to obtain accurate results for both propagation constants and field distributions. However, in the direct strategy, the inclusion -field profile. of this term only affects the Several results comparing the direct formulation to the indirect formulation, measurements, and commercial software are reported in [30] and [31]. In these papers, it was concluded that after introducing the novelties of [27], the indirect formulation led to more accurate results and a faster convergence for the same number of basis modes than the direct formulation and, therefore, the final computational efficiency was enhanced. More recently, these improvements of the coupled mode method have been applied for the electromagnetic characterization of isotropic samples using a transmission-line waveguide method [32]. In this paper, this enhanced indirect formulation is extended to not only solve isotropic, biisotropic, or anisotropic waveguides, but also general bianisotropic waveguides, which include the cited materials as particular cases. Due to the novelty, the formulation is exposed in detail and complete expressions are given.

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are the transversal and axial magnetic basis functions, and are the transversal and axial electric amplitudes, and and are the transversal and axial magnetic amplitudes, respectively. Subscripts and also refer to the transversal and axial components, respectively, for all variables in this , which has no dependence on the paper. Note that the term in (4). This transversal coordinates [27], has been added to term takes into account the trivial solution for the modes (i.e., the “mode” as a part of the in the expansion for set of basis functions), which must be taken into account when a magnetic medium is considered, irrespective of whether it is isotropic or not. Here it is necessary to make a clarification. In (1)–(4), and , which are the fields of the material-filled waveguide, are expanded in terms of the basis modes. The boundary conditions on the perfect electric conductor walls are the same for the modes of the empty waveguide and also for the modes of the partially filled waveguide. In particular, this means that the normal in the partially filled waveguide do not fulcomponents of fill the appropriate boundary condition on the perfect conductor boundary in contact with a bianisotropic medium. The implications of this can be seen in [33]. The final goal is to obtain the generalized telegraphist’s equations of this waveguide, which relates the transversal modal amplitudes and and their derivatives in an eigenvalue problem

(5) The operator is the partial derivative with respect to , and are the modal vectors with elements and , and is a coupling matrix to be found for each physical configuration of waveguide and material. By substituting (1)–(4) in the Maxwell curl equations, apand the Galerkin procedure, the folplying the operator lowing expressions are obtained:

(6)

II. THEORY Let us consider a waveguide filled with an arbitrary material, as depicted in Fig. 1. The transversal and axial electric and magnetic fields are expanded as a function of the modes of the corresponding empty waveguide (1) (2)

is the cutoff wavenumber, is the propaThe variable is the transversal impedance of the th gation constant, and basis mode. The domain of integration is the whole transverse section of the waveguide. is the unitary vector on the -axis towards the positive direction. Analogously, substituting (1)–(4) in the Maxwell curl equations and applying the operator , the following expressions are obtained:

(3) (4)

(7)

In the following equations, spatial dependences will be omitted in order to simplify the notation. The functions and are the transversal and axial electric basis functions, and

Some information about the material filling the waveguide is needed in order to continue the analysis. The constitutive rela-

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tionships of a bianisotropic media are

and following the same procedure described in [30] for an can be expressed now as isotropic chiral medium, (8) (13)

where , , , and are 3 3 matrices. Decomposing the fields and constitutive parameters in transverse and axial components, e.g.,

Following the indirect formulation, multiplying (2) by and integrating, yields

(9)

(14)

(10)

By substituting the terms of (11) by (1), (3), (4), (7), and (13), and applying the Galerkin procedure, leads to solving for the following matrix equation:

and analogously for the rest of parameters and fields, we can write (8) in the form

(15) (11) Two different strategies can be established from this point for and in terms of the resolution of the axial amplitudes and , one “direct” and another “indirect,” as mentioned in Section I. and are derived in terms of In the direct approach, the rest of the amplitudes of the transverse electromagnetic components. This implies that there is no need for carrying out any matrix inversion. It is important to point out that, in the direct in (4) has no influence in the final eigenstrategy, the term value system from which the propagation constant and the transverse components of the electromagnetic field are derived. Due , the direct strategy, is well implemented in to this reason, [26]. However, the absence of this term would lead to incorrect -component. For more information about the results for the direct formulation, the reader is referred to [26] and [31]. Indirect and direct strategies provide similar results in some cases, and highly different results in other situations, for a reasonable number of modes. and are related to the rest In the indirect approach, of the amplitudes of the transverse electromagnetic field components through a set of linear equations. In matrix form, this implies to perform some matrix inversion. The implications of both formulations in the final results can be seen in [31] for a dielectric media and for anisotropic magnetic media in [29]. It is stated that, in general, the indirect formulation, , gives better accuracy in both with the inclusion of the term electromagnetic field and propagation constants than the direct formulation, especially for high values of the constitutive parameters, and this behavior is also justified in terms of the continuity or not of some components of the electromagnetic field. Thus, the same statements are expected for bianisotropic media. In this paper, we describe the details of the indirect formulation for bianisotropic media. As will be shown, the term contributes to both the component and the propagation constants in the indirect formulation. must be calculated. The series expanFirst, the value of sion of in (7) is integrated over , thus we have (12)

, and manipIn a similar manner, multiplying (4) by ulating (11) with the aid of (1)–(3), (7), and (13), the following expression is obtained:

(16) For the sake of clarity, matrix expressions are given in (A.1)–(A.7). Substituting (13) into (15) and (16), the axial amplitudes can be written in terms of the transversal amplitudes as (17) The matrices involved in (17) have been defined in (A.8)–(A.16). Note that the matrix inversion, which is inherent to the indirect strategy, appears in (A.16). can now be expressed as a function of the transversal amplitudes by defining the matrices (A.17) and (A.18) as follows:

(18) is not equal to zero, The reader can investigate when which depends on the geometry and constitutive parameters, and its influence in the final eigenvalue system. in (6) can be expressed as a function of At this point, known values

(19)

PITARCH et al.: EFFICIENT MODAL ANALYSIS OF BIANISOTROPIC WAVEGUIDES BY COUPLED MODE METHOD

111

In (19), substituting by (18), the axial amplitudes by (17), and writing the equation in matrix form, the following expression is derived: (20) , , and their components are given in where (A.19)–(A.28). in (6) is In a similar manner, the electric displacement substituted by

(21) and with the same procedure employed in (19), the following matricial expression is found:

Fig. 2. Rectangular piece of bianisotropic of thickness c and transverse dimensions a b inserted in a rectangular waveguide of transverse dimensions a b.

2

2

(22)

Fig. 3. Convergence of the normalized propagation constant with respect to the number of basis modes. Waveguide dimensions a = 15:8 mm, b = 7:9 mm, " = 1,  = 10, a = a=2, b = b, x = a=2, and y = 0. Frequency f = 10 GHz.

, , and their components are also where the matrices given in (A.29)–(A.35). The eigenvalue problem of (5) is now defined as follows:

and in the - and -coordinates, respectively. This sample has a constitutive materials , , , and following the notation of (8). A. Isotropic Material

(23) Solving (5), the eigenvalues of (23) are the propagation constants of the proper modes, and the eigenvectors are the coefficients of the basis modes that yield to the proper modes. Therefore, with the resolution of this matrix expression, an inhomogeneously filled waveguide can be fully characterized. When the inhomogeneously filled waveguide is part of a more complex structure, it can be interpreted as an obstacle or discontinuity in a cascade of microwave networks, and with the aid of the generalized circuital analysis, the generalized scattering matrix of the whole structure can be calculated. III. RESULTS AND DISCUSSION In this section, some results concerning some particular cases of different materials and waveguides are shown, both to validate and assess the performance of the method. The general formulation described in Section II is applied to a rectangular waveguide inhomogeneously filled with a rectangular sample in an arbitrary position, as is shown in Fig. 2. This describes a filled with a matewaveguide with transverse dimensions rial sample with a cross section and thickness , placed at

An isotropic material is a very simple particular case of bianisotropic material with constitutive parameters (24)

where is the identity matrix. Let us first consider a WR-75 rectangular waveguide ( mm, mm), partially filled with a magnetic sample, characterized by , (half width), (complete height), placed at , . The frequency is GHz. Although this is a well-known material, from set to the study of the convergence, valid statements can be derived for this and the rest of the filling materials in Sections III-B and C. of the Fig. 3 shows the normalized propagation constant as a funcfundamental mode of the structure tion of the number of basis modes. The propagation constant calmode) is also represented for comculated analytically ( parative purposes. The differences in the convergence of both formulations are significant. In order to reach the convergence, many more modes

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Fig. 4. Normalized axial magnetic field (H ) of the first mode for the case of Fig. 3, for five modes, 40 modes, and exact solution.

are needed in the direct formulation. This behavior is also corroborated in [31] for isotropic, in [33] for chiral, and in [28] and [29] for magnetic anisotropic media. Additionally, both formulations do not seem to converge exactly to the same value as the number of modes increases. The main reason for these differences in both formulations is the different way of solving the axial field components. As mentioned in Section II, the direct and by performing an analytical inverstrategy solves sion of the constitutive relationships (see [26, eq. (7)]), whereas the indirect approach reaches this solution through a weak formulation (Galerkin procedure) (14)–(17). This last approach has been demonstrated to be numerically more suitable for the inhomogeneities in the waveguides [29], [31]. To better explain these effects, Fig. 4 shows the normalized of the fundamental mode axial magnetic field component along the -axis in the partially filled waveguide for five and 40 basis modes. The same waveguide has been solved analytically component of the magnetic field of the and the normalized is also represented for comparative fundamental mode purposes. As can be appreciated, the indirect strategy is able to reproduce the analytical solution with 40 modes, whereas the direct approach presents noticeable differences even when the number of modes has reached convergence. Consequently, in this case, the indirect formulation is much more suitable for analyzing such structures in terms of accuracy and stability than the direct formulation, which means that fewer modes and CPU time and storage is required for a given precision. B. Ferrite Material Ferrites are a well-known case of magnetic anisotropy and their behavior has been widely reported [11]. The permeability . matrix can be controlled externally by a dc-bias field When this static field is parallel to the -axis, the permeability matrix takes the form (25)

Fig. 5. Convergence of the normalized propagation constant with respect to the number of basis modes in a CG-550-GA ferrite. a = 86:36 mm, b = 25 mm, x = 41:68 mm, y = 0, a = 9 mm, and b = b. Frequency f = 2:45 GHz.

These parameters are also related to the physical parameters of the ferrite [11]. To verify the simulation tool for this media, a waveguide with mm and mm is partially filled cross section mm, , and mm, with a ferrite slab of located at the center in the dimension of the waveguide ( mm, ). The ferrite is the CG-550 GA from Counties Laboratories, Grass Valley, CA, whose parameters are Lande , magnetization of saturation G, factor Oe, permittivity , and dielectric linewidth loss tangent . The ferrite is biased with an Oe. internal dc field mode, there is no If the structure is excited with the . The dependence with the dimension, due to mode set can then fully describe the fields in the ferrite. There is not a general rule to determine how many modes must be used in order to ensure acceptable results because this depends on all the geometric and electromagnetic parameters, and can be very variable for each case. Instead, the prior convergence analysis at the given frequency permits to choice an appropriate mode set. Fig. 5 shows the normalized propagation constant of the ferrite-filled waveguide as a function of the number of basis modes. From this figure, one sees that 20 basis modes are necessary to reach reasonable convergence. Again, the direct and indirect approaches converge to a different value, but not very significant for this case due to the small thickness of the ferrite slab. It must be noted that these deviations are increased for thicker materials due to the mentioned differences in both formulations. Figs. 6 and 7 show the magnitude and phase of the -parameters of this structure between 2–3 GHz computed with both the direct and indirect formulations. For validation purposes, finite-difference time-domain (FDTD) simulations obtained with commercial software (CONCERTO, Vector Fields, Oxford, U.K. [35]) and laboratory measurements of the described ferrite-filled structure are also provided. Measurements were performed with a ZVRE vectorial network analyzer from

PITARCH et al.: EFFICIENT MODAL ANALYSIS OF BIANISOTROPIC WAVEGUIDES BY COUPLED MODE METHOD

Fig. 6. a b

S

-parameter of a waveguide partially filled with a ferrite.

= 86:36 mm, b = 25 mm, x = 41:68 mm, y = 0, a = 9 mm, = b, and c = 20 mm. The results are referred to the WR340 port, placed

at 116.5 mm to the ferrite.

Fig. 7.

-parameters of a waveguide partially filled with a ferrite. = 41:68 mm, y = 0, a = 9 mm, a = 86:36 mm, b = 25 mm, x = b, and c = 20 mm. The results are referred to the WR340 port, placed b at 116.5 mm to the ferrite. S

Rhode & Schwarz, Munich, Germany. These measurements are referred to standard WR340 ports, at a distance of 116.5 mm to the ferrite in both ports. In order to make the comparison, these waveguide transitions were also taken into account in the simulations. As can be seen, simulations with the coupled mode method show good agreement with actual measurements and commercial software. In order to verify the behavior of the method with inhomogeneities in both dimensions, the previous ferrite-filled waveguide is analyzed as a function of the sample’s height . Figs. 8 and 9 show the scattering of such a configuration at mm and 2.45 GHz. These results are referred now to mm waveguide ports, placed at a distance of 10 mm to Oe. the ferrite. The external magnetic bias is set to , the incidence with the fundamental mode Given that excites the full set of TE and TM modes. 180 basis modes were taken into account for a good convergence in the solution. Simulations of the same structure with a commercial FDTD simulator

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Fig. 8. S -parameter of a waveguide partially filled with a ferrite as a function of the ferrite height b . a = 86:36 mm, b = 25 mm, x = 38:68 mm, y = 0, a = 9 mm, and c = 20 mm.

Fig. 9. S -parameter of a waveguide partially filled with a ferrite as a function of the ferrite height b . a = 86:36 mm, b = 25 mm, x = 38:68 mm, y = 0, a = 9 mm, and c = 20 mm.

are also included for comparison. Again, from the figures, simulations with the coupled mode method agree well with FDTD, which confirms the validity of the method for such structures. C. Pseudochiral Material A pseudochiral medium can produce field displacement in every direction depending on the orientation of the -shaped particles. For the case of a displacement in the -axis, the constitutive parameters take the form (26) (27)

(28)

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correspond to the indirect strategy because the direct strategy was unable to provide reasonable results. It is important to remark that, although the sample reaches the modes are an incomplete full height of the waveguide, the set to represent the fields. It is due to the field displacement component produced by the material, which couple the and modes with the other ones. The full set of must then be used in order to provide accurate results. This can be appreciated in Fig. 11, where the full set of modes better matches the results of [12]. However, given that the sample is centred in the axis, the modes which have odd symmetry ( odd) can be neglected. IV. CONCLUSION

Fig. 10. Convergence of the normalized propagation constant with respect to the number of basis modes, for an material. a = 20 mm, b = 10 mm, a = 1 mm, b = b, x = (a a )=2, y = 0, " = 9:8,  = 1, and

 = 4.

0

Fig. 11. Dispersion diagram of the fundamental mode for the case of Fig. 10 150 basis modes.

The generalized telegraphist’s equations for a waveguide partially filled with a general bianisotropic material have been derived following the indirect formulation. This procedure has proved to be more suitable for the analysis of inhomogeneously filled waveguides than other kind of formulations of the coupled mode method. It has been concluded that the indirect formulation leads to more accurate results and a faster convergence for the same number of basis modes. Although the indirect formulation is slightly slower for a given number of modes, fewer modes are then needed for a given precision, which makes the formulation faster and, therefore, the computational efficiency is enhanced. The method can be particularized for any known class of materials, and any spatial variation of the constitutive parameters in the cross section of the waveguide is allowed, such as ferrite or materials inside the same waveguide. With the aid of the generalized circuital theory, the scattering parameters of a discontinuity in a waveguide can also be computed, and different discontinuities can be connected in order to simulate more complex devices. Some results have been presented and compared with the literature in order to validate the method and to show its capability. A dielectric material, a ferrite, and an material have been analyzed with this method and a good performance has been proven in all cases. APPENDIX

and then the constitutive relationships take the form

Some coupling matrices were defined in Section II. These are shown in (A.1)–(A.35) as follows: (29) (A.1) Let us assume now a thin slab of an material placed on a rectangular waveguide, with the parameters mm, mm, mm, (complete height), (centred), , with the parameters , and . Fig. 10 shows the convergence analysis of the propagation constant of the first mode for the frequency of GHz. Due to the fact that the sample is very thin (5% of the width ), the convergence is reached at the value of approximately 150 modes. Taking into account this result, Fig. 11 shows the dispersion diagram for this mode, and this result is compared to the mode-matching results of [12]. Represented results uniquely

(A.2) (A.3) (A.4) (A.5) (A.6) (A.7)

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(A.8) (A.9)

(A.28)

(A.10)

(A.29)

(A.11)

(A.30) (A.31) (A.32)

(A.12)

(A.33)

(A.13) (A.14) (A.15)

(A.34) (A.16)

(A.35)

(A.17) REFERENCES (A.18) (A.19) (A.20) (A.21) (A.22) (A.23) (A.24) (A.25) (A.26)

[1] D. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE, vol. 56, no. 3, pp. 248–251, Mar. 1968. [2] J. A. Arnaud and A. A. M. Salen, “Theorems for bianisotropic media,” Proc. IEEE, vol. 60, no. 5, pp. 639–640, May 1972. [3] R. D. Graglia, P. L. E. Uslenghi, and R. E. Zich, “Dispersion relation for bianisotropic materials and its symmetry properties,” IEEE Trans. Antennas Propag., vol. 39, no. 1, pp. 83–90, Jan. 1991. [4] E. O. Kamenetskii, “Nonreciprocal microwave bianisotropic materials: Reciprocity theorem and network reciprocity,” IEEE Trans. Antennas Propag., vol. 49, no. 3, pp. 361–366, Mar. 2001. [5] D. L. Jaggard, A. R. Mickelson, and C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., vol. 18, pp. 211–216, 1979. [6] I. V. Lindell and A. J. Vitanen, “Duality transformations for general bi-isotropic (nonreciprocal chiral) media,” IEEE Trans. Antennas Propag., vol. 40, no. 1, pp. 91–95, Jan. 1992. [7] A. H. Sihvola, “Bi-isotropic mixtures,” IEEE Trans. Antennas Propag., vol. 40, no. 2, pp. 188–197, Feb. 1992. [8] A. J. Bahr and K. R. Clausing, “An approximate model for artificial chiral material,” IEEE Trans. Antennas Propag., vol. 42, no. 12, pp. 1592–1599, Dec. 1994. [9] D. Pissoort and F. Olyslager, “Study of eigenmodes in periodic waveguides using the Lorentz reciprocity theorem,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 542–553, Feb. 2004. [10] A. Alu, F. Bilotti, and L. Vegni, “Extended method of line procedure for the analysis of microwave components with bianisotropic inhomogeneous media,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1582–1589, Jul. 2003. [11] J. D. Adam, L. E. Davis, G. F. Dionne, E. F. Schloemann, and S. N. Stitzer, “Ferrite devices and materials,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 721–737, Mar. 2002. [12] J. Mazur and D. Pietrzak, “Field displacement phenomenon in a rectangular waveguide containing a thin plate of medium,” IEEE Microw. Guided Lett., vol. 6, no. 1, pp. 34–36, Jan. 2006. [13] S. A. Schelkunoff, “Generalized telegraphist’s equations for waveguides,” Bell Syst. Tech. J., vol. 31, pp. 784–801, Jul. 1952.

W

(A.27)

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[14] W. E. Hord and F. J. Rosenbaum, “Approximation technique for dielectric loaded waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-16, no. 4, pp. 228–233, Apr. 1968. [15] R. M. Arnold and F. J. Rosenbaum, “Nonreciprocal wave propagation in semiconductor loaded waveguides in the presence of a transverse magnetic field,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, no. 1, pp. 57–65, Jan. 1971. [16] ——, “An approximate analysis of dielectric-ridge loaded waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 10, pp. 699–701, Oct. 1972. [17] J. B. Ness and M. W. Gunn, “Microwave propagation in rectangular waveguide containing a semiconductor subject to a transverse magnetic field,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, no. 9, pp. 767–772, Sep. 1975. [18] F. Gauthier, M. Besse, and Y. Garault, “Analysis of an inhomogeneously loaded rectangular waveguide with dielectric and metallic losses,” IEEE Trans. Microw. Theory Tech., vol. MTT-25, no. 11, pp. 904–907, Nov. 1977. [19] A. D. Bresler, G. H. Joshi, and N. Marcuvitz, “Orthogonality properties of modes in passive and active uniform waveguides,” J. Appl. Phys., vol. 29, pp. 794–799, May 1958. [20] J. L. Amalric, H. Baudrand, and M. Hollinger, “Various aspects of coupled-mode theory for anisotropic partially-filled waveguides. Application to a semi-conductor loaded wave-guide with perpendicular induction,” in Proc. 7th Eur. Microw. Conf., Copenhagen, Denmark, 1977, pp. 146–150. [21] J.-W. Tao, J. Atechian, R. Ratovondrahanta, and H. Baudrand, “Transverse operator study of a large class of multidielectric waveguides,” Proc. Inst. Elect. Eng., vol. 137, pt. H, pp. 311–317, Oct. 1990. [22] J.-W. Tao, R. Andriamanjato, and H. Baudrand, “General waveguide problems studies by transverse operator method,” IEEE Trans. Magn., vol. 47, no. 12, pp. 2503–2511, Dec. 1999. [23] K. Ogusu, “Numerical analysis of the rectangular dielectric waveguide and its modifications,” IEEE Trans. Microw. Theory Tech., vol. MTT-25, no. 11, pp. 874–885, Nov. 1977. [24] J. Rodríguez, M. A. Solano, and A. Prieto, “Characterization of discontinuities in dielectric waveguides using Schelkunoff’s method: Application to tapers and transitions,” Int. J. Electron., vol. 66, pp. 807–820, May 1989. [25] Y. Xu and G. Zhang, “A rigorous method for computation of ferrite toroidal phase shifters,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 6, pp. 929–933, Jun. 1988. [26] Y. Xu and R. G. Bosisio, “An efficient method for study of general bi-anisotropic waveguides,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 4, pp. 873–879, Apr. 1995. [27] H. Chaloupka, “A coupled-line model for the scattering by dielectric and ferromagnetic obstacles in waveguides,” (AEU) Arch. Elektr. Ubertragung, vol. 34, pp. 145–151, Apr. 1980. [28] A. Vegas, A. Prieto, and M. A. Solano, “Rigorous analysis of scattering by partial height magnetized ferrite posts in rectangular waveguide,” Electron. Lett., vol. 28, pp. 913–915, May 1992. [29] M. A. Solano, A. Vegas, and A. Prieto, “Modelling multiple discontinuities in rectangular waveguide partially filled with non-reciprocal ferrites,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 5, pp. 797–802, May 1993. [30] A. Gómez, M. A. Solano, and A. Vegas, “New formulation of the coupled mode method for the analysis of chirowaveguides,” in Proc. SPIE—Int. Soc. Opt. Eng. Complex Mediums III: Beyond Linear Isotropic Dielectrics., Seattle, WA, 2002, pp. 290–301. [31] A. Vegas, A. Prieto, and M. A. Solano, “Optimisation of the coupled-mode method for the analysis of waveguides partially filled with dielectrics of high permittivity: Application to the study of discontinuities,” Proc. Inst. Elect. Eng., vol. 140, pt. H, pp. 401–406, Oct. 1993. [32] J. Pitarch, M. Contelles-Cervera, F. L. Peñaranda-Foix, and J. M. Catalá-Civera, “Determination of the permittivity and permeability for waveguides partially loaded with isotropic samples,” Meas. Sci. Technol., vol. 17, pp. 145–152, 2006. [33] M. A. Solano, A. Vegas, and A. G. c. on, “A comprehensive study of discontinuities in chirowaveguides,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1297–1298, Mar. 2006. [34] A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 5, pp. 686–696, May 1987. [35] Concerto User Manual. Oxford, U.K.: Vector Fields Ltd., 2003.

Jaime Pitarch was born on Valencia, Spain, on January 1979. He received the Dipl. Ing. degree in telecommunications engineering from the Universidad Politécnica de Valencia, Valencia, Spain, in 2002, and is currently working toward the Ph.D. degree at the Universidad Politécnica de Valencia. Since 2001, he has been a Research Assistant with the Grupo de Electromagnetismo Aplicado (GEA), Universidad Politécnica de Valencia. His current research areas are numerical methods for electromagnetics and CAD of waveguide circuits.

José M. Catalá-Civera (M’03) received the Dipl. Ing. and Ph.D. degrees in telecommunications engineering from the Universidad Politécnica de Valencia, Valencia, Spain, in 1993 and 2000, respectively. From 1993 to 1996, he was a Research Assistant with the Microwave Heating Group, Universidad Politécnica de Valencia, where he was involved with microwave equipment design for industrial applications. Since 1996, he has been with the Communications Department, Universidad Politécnica de Valencia, where he was appointed a Readership in 2000. He is currently Head of the Microwave Applications Research Group, Institute ITACA, Universidad Politécnica de Valencia. His research interests encompass the design and application of microwave theory and applications, the use of microwaves for electromagnetic heating, microwave cavities and resonators, and measurement of dielectric and magnetic properties of materials and development of microwave sensors for nondestructive testing. He has coauthored approximately 60 papers in refereed journals and conference proceedings and over 50 engineering reports for various companies. He holds five patents. Dr. Catalá-Civera is a member of the International Microwave Power Institute (IMPI). He is a reviewer of several international journals. He is currently a Board member of the Association of Microwave Power in Europe for Research and Education (AMPERE), a European-based organization devoted to the promotion of RF and microwave energy.

Felipe L. Peñaranda-Foix (M’92–SM’00) was born in Benicarló, Spain, in 1967. He received the M.S. degree in electrical engineering from the Universidad Politécnica de Madrid, Madrid, Spain, in 1992, and the Ph.D. degree in electrical engineering from the Universidad Politécnica de Valencia (UPV), Valencia, Spain, in 2001. In 1992, he joined the Departamento de Comunicaciones, UPV, where he is currently a Senior Lecturer. His current research interests include electromagnetic scattering, microwave circuits and cavities, sensors, and microwave heating applications. He has coauthored approximately 40 papers in referred journals and conference proceedings and over 40 engineering reports for various companies. Dr. Peñaranda-Foix is a member of the Association of Microwave Power in Europe for Research and Education (AMPERE). He is a reviewer of several international journals.

Miguel A. Solano (M’95) received the Licenciado en Física and Ph.D. degrees from Universidad de Cantabria, Santander, Spain, in 1984 and 1991, respectively. Since 1984, he has been with the Departamento de Electrónica, and is currently with the Departamento de Ingeniería de Comunicaciones, Universidad de Cantabria. In 1992, he was a Visitor Scholar with the University of Bristol, Bristol, U.K. Since 1995 he has been a Professor Titular with the Universidad de Cantabria. His current research activities include electromagnetic propagation in complex media, numerical methods in electromagnetics, biological effects of the electromagnetic field, and e-learning.

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HTS Quasi-Elliptic Filter Using Capacitive-Loaded Cross-Shape Resonators With Low Sensitivity to Substrate Thickness Alonso Corona-Chavez, Member, IEEE, Michael J. Lancaster, Senior Member, IEEE, and Hieng Tiong Su, Member, IEEE

Abstract—The design and implementation of a four-pole quasielliptic filter for -band radar applications using high-temperature superconductors is presented. The resonators used consist of cross-shape ring resonators with additional interdigital capacitive loads that allow miniaturization with reduction of sensitivity to substrate thickness. The experimental and simulated results are shown together with the power dependence of the filter. Index Terms—Bandpass filter, high-temperature superconductor (HTS), intermodulation distortion (IMD), quasi-elliptic response, radar systems, superconductor, -band.

I. INTRODUCTION

F

OR -BAND radar applications, superconducting materials offer the advantages of miniaturization of the circuits by using high-permittivity substrates without deterioration of power loss performance. At 10 GHz, the surface resistance of thin film YBa Cu O (YBCO) is in the order of 10 at 77 K, while copper at the same temperature has an of approx[1]. This low surface resistance in high-temperimately 10 ature superconductors (HTSs) translates into a longer range in radar receivers because the receiver sensitivity increases due to their lower noise figure compared to conventional technologies. Common radar systems for electronic warfare incorporate delay lines of several nanoseconds, filters, and low-noise amplifiers (LNAs) in their front-ends, all of which can use HTS materials [2]. It is, therefore, practical, by integrating all the components using one cryo-cooler, to produce compact and high-performance systems. Ryan [3] describes several HTS filters for improved warfare systems at -band. In [4], a delay line is presented having a delay of 35 ns at around 10 GHz with a loss of approximately 1 dB. In [5], an HTS diplexer at -band using two branch-line couplers and two-pole filters with half-wavelength-long resonators is shown. In [6], a narrow band (0.5%) HTS filter is also shown at -band and, in [7], a front-end receiver module for a radar at -band is demonstrated integrating an LNA and an eight-pole 2% bandwidth filter. For mass production of planar HTS filters, it is desirable to have trimmingless circuits [8]. The main causes of performance Manuscript received February 3, 2006; revised September 17, 2006. A. Corona-Chavez is with the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Gran Telescopio Milimetrico (GTM)/INAOE, Puebla, Mexico 72840 (e-mail: [email protected]). M. J. Lancaster and H. T. Su are with the Electronic and Electrical Engineering Department, University of Birmingham, Edgbaston B15 2TT, U.K. (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.888577

variation for the designed filters are manufacturing tolerances and nonuniformity of substrate thickness, the latter being the most significant one [9]. Numerous efforts have been carried out to reduce the sensitivity to tolerances in substrate thickness. In [10] and [11], novel single-mode microstrip resonators, which need little tuning, are shown. One other way of making resonators with little sensitivity to substrate thickness is using coplanar structures; however, for larger circuits, it is necessary to balance the ground planes with extra bond wires, incrementing crosstalk and manufacturing difficulties [12]–[14]. In this paper, the design of a novel quasi-elliptic four-pole -band radar applications is presented using filter for dual-mode cross-shaped ring resonators with capacitive loading. The advantage of these resonators is that they are considerably smaller than conventional ring resonators [15]–[19] and they present little sensitivity to substrate thickness. In [20], copper versions of such resonators were verified experimentally and filters with Chebyshev responses were demonstrated. However, it is possible to implement quasi-elliptic configurations. The principle behind this type of resonator lies in the fact that an interdigital capacitor in a microstrip structure would concentrate the electric field near the surface, thus making it less sensitive to variations in the thickness of the substrate, as less field would cross down to the ground plane [21]. One of the main drawbacks of HTS technology is their potentially poor power-handling capability caused by their power-dependent nonlinear behavior of surface resistance. This nonlinear behavior generates odd harmonics (third, fifth, seventh, etc.) and two-tone intermodulation in the filters. The power-handling capabilities of the filter will be mentioned later with experimental data. II. RESONATOR CHARACTERISTICS A ring resonator is a 360 closed-loop transmission line. Two orthogonal modes can be excited if one or several notches are inserted in one or more corners of the resonator [17]. The addition of the interdigital capacitor to the resonator has the effect of concentrating the electric fields near the surface, hence reducing its sensitivity to substrate thickness. Simulations, assuming a lossless metal on the MgO substrate and mm) at a center frequency of 9.4 GHz, ( were carried out to prove this concept. Using the cross-shape ring resonator, interdigital capacitors were added with different number of fingers (from 2 to 11). All the resonators were weakly

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Fig. 2. Sensitivity to substrate thickness of cross-shape resonators with different interdigital fingers. Simulations were carried out assuming MgO substrate (" = 9:7).

Fig. 1. Size comparison of conventional ring resonator and capacitive loaded cross resonator on MgO (" = 9:7, h = 0:5 mm) at 9.4 GHz.

coupled to prevent any effect from the feed lines and then were simulated in an electromagnetic simulator [22] at different substrate thicknesses. The structure dimensions are shown in Fig. 1. Fig. 2 shows the sensitivity of the resonator in terms of the deviation of center frequency per millimeter of substrate variation versus the number of fingers in the interdigital capacitor. is given by (1) where is the variation of the resonant frequency, difference in substrate thickness, and is defined as

is the

(2) where and are the resonant frequencies of the resonator with different substrate thicknesses. As can be seen in Fig. 2, /mm the sensitivity decreases from approximately /mm (with (with zero fingers) to approximately eight fingers). However, as the number of fingers increases, the parasitic capacitance to ground does as well. This effect becomes evident in this structure for nine, ten, and 11 fingers when the sensitivity to substrate thickness starts rising (4.8%/mm, 5.2%/mm, and 5.6%/mm, respectively). By adding the interdigital capacitance, reduction in size is also achieved by capacitive loading. Fig. 1 shows the conventional ring resonator, cross-shape ring resonator [23], and cross resonator with capacitive loading all to scale and operating at the same frequency. As can be seen, the introduction of interdigital fingers also reduces approximately 1.8 times the size of the resonator. III. QUASI-ELLIPTIC FILTER DESIGN A novel four-pole quasi-elliptic 2%BW filter has been designed using these resonators following the design procedure

Fig. 3. Final layout of quasi-elliptic four-pole filter at X -band.

presented in [24]. The desired coupling coefficients for a maxare , imum return loss of 20 dB and , , and , where is the frequency variable that determines the position of the transis the external coupling, and is the coupling mission zeros, coefficient between different resonant modes. To obtain the quasi-elliptic response, it is necessary to cross couple nonadjacent resonators or resonant modes. It is important to note that, for clarity, the two resonant modes of the first resonator are called 1 and 2, whereas for the second resonator, they are called 3 and 4, as shown in Fig. 3. For a four-pole Chebyshev filter, resonant mode 1 couples to 2 by the addition of the notch, resonant mode 2 is directly coupled to resonant mode 3, and finally, resonant mode 3 is coupled to 4 by the notch. However, if a quasi-elliptic response is to be achieved, an extra cross coupling between resonant modes 1 and 4 has to be added, as shown in Fig. 3. IV. SIMULATION AND EXPERIMENTS The filter was fabricated on an MgO substrate with permitand thickness mm. The superconductor tivity is YBCO. The circuit is packaged in a metal box with a titanium carrier coated with gold. This assembly was placed in a Grifford–MacMahon closed-cycle cryostat at 30 K. Fig. 4 shows the simulated lossless response using [22] and the tuned measured one. The center frequency is 9.25 GHz and the bandwidth is 2%. The insertion loss is less than 1 dB throughout the band, and the maximum ripple is 0.25 dB. The return loss of the tuned response is lower than 13 dB throughout the band. The two transmission zeros are placed at

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Fig. 4. Experimental and simulated data for four-pole quasi-elliptic filter. Fig. 6. Input power dependence of third-order IMD power and fundamental modes.

V. CONCLUSION

Fig. 5. Experimental setup for IMD measurement.

9.12 and 9.41 GHz. In this figure, the simulated response is overlapped for comparative purposes. There is good agreement between simulated and experimental data. A frequency shift of approximately 1.6% from 9.4 to 9.25 GHz is observed, which is thought to be due to substrate permittivity variation. Moreover, patterning tolerances and nonuniformities in the substrate thickness can cause variances in the different coupling coefficients throughout the structure. This is believed to be the main cause of the asymmetric lower than predicted rejection below the passband and higher rejection above the stopband. The nonlinear characteristics of the filter were examined by performing an intermodulation distortion (IMD) experiment. The experimental setup is shown in Fig. 5, and it consists GHz and of two power sources with frequencies kHz, which are combined by a power combiner connected to the filter placed inside the cryostat at a temperature K. The power levels entering the filter varied from 35 to 15 dBm. The values of the fundamental and third harmonic products were recorded and plotted in Fig. 6. As can be seen, the third-order intercept point (the value at which the magnitude of the third harmonic crosses the fundamental) is approximately 80 dBm. The value of the slope for the third harmonics is approximately 1.9 : 1. This is consistent with data from several authors [25]–[28] where the slopes vary from 1.6 : 1 to 3 : 1. Although the causes that induce this non 3 : 1 variation are not yet fully understood, some explanations have been presented [29]–[31].

A novel superconducting quasi-elliptic filter has been presented using cross-shape resonators with capacitive loading for -band radar applications. It has been proven that these resonators present lower sensitivity to substrate thickness and they offer size miniaturization. Good agreement was achieved from the electromagnetic simulation and the measured response. The loss was not greater than 1 dB throughout the band and the return loss better than 13 dB. Additionally, an IMD experiment was realized and the shown data coincides with the available literature. ACKNOWLEDGMENT The authors would like to thank the Emerging Device Technology Research Center, University of Birmingham, Edgbaston, U.K., especially Dr. Y. Wang, Dr. J. Zhou, Dr. G. Zhang, and I. Koutsonas, for their support during the experimental measurements. The authors are also grateful to Dr. A. Velichko, Department of Engineering, University of Cambridge, Cambridge, U.K., for his useful discussions regarding the IMD experiments. Finally, the authors are also thankful to Dr. T. Jackson and D. Holman, both with the Emerging Device Technology Research Center, University of Birmingham, for the manufacturing of the HTS device and to Ing. G. Diaz and Ing. G. Diaz-Burgos, both with Manufacturas Especiales Toluca S.A (METSA), Toluca, Mexico, for the fabrication of the packaging box. REFERENCES [1] S. Zhi-Yuan, High-Temperature Superconducting Microwave Circuits. Norwood, MA: Artech House, 1994. [2] C. Jackson and L. Cantfio, “High temperature superconductors for radar applications,” in Proc. IEEE Nat. Radar Conf., Los Angeles, CA, 1991, pp. 122–126. [3] P. A. Ryan, “High temperature superconducting filter technology for improved electronic warfare system performance,” in Proc. IEEE Aerosp. Electron. Conf., Columbus, OH, 1997, pp. 392–395. [4] H. T. Su, Y. Wang, F. Huang, and M. J. Lanchester, “Wide band superconducting microstrip delay line,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 12, pp. 2482–2487, Dec. 2004.

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[5] S. K. Han, Y. Ha, J. Kim, and K. Kang, “Characteristics of high power capability for high temperature superconducting microwave multiplexer,” in Asia–Pacific Microw. Conf., 1997, pp. 109–112. [6] A. Fathy, D. Kalikitis, and E. Belohoubek, “Critical design issues in implementing a YBCO superconductor X -band narrow bandpass filter at 77 K,” in IEEE MTT-S Int. Microw. Symp. Dig., 1991, pp. 1329–1332. [7] D. Niu, T. Huang, H. Lee, and C. Yang, “An X -band front end module using HTS technique for a commercial dual mode radar,” IEEE Trans. Appl. Supercond., vol. 15, no. 2, pp. 1008–1011, Jun. 2005. [8] I. Vendik et al., “Design of trimmingless narrowband planar HTS filters,” J. Supercond. Incorporating Magn., vol. 14, no. 1, pp. 21–28, 2001. [9] J. S. Hong, E. P. McErlean, and B. Karyamapudi, “Eighteen pole superconducting CQ filter for future wireless applications,” Proc. Inst. Elect. Eng.—Antennas Propag., vol. 153, no. 2, pp. 205–211, Apr. 2006. [10] H. R. Yi, S. K. Remillard, and A. Abdlelmonen, “A novel ultra compact resonator for superconducting thin film filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2290–2296, Dec. 2003. [11] M. Barra, C. Collado, and J. O’Callaghan, “Miniaturizaion of superconducting filters using Hilbert fractal curves,” IEEE Trans. Appl. Supercond., vol. 15, no. 3, pp. 3841–3846, Sep. 2005. [12] J. Zhou, M. J. Lancaster, and F. Huang, “HTS coplanar meander line resonator filters with suppressed slot line mode,” IEEE Trans. Appl. Supercond., vol. 14, no. 1, pp. 28–32, Mar. 2004. [13] Y. Song and C. C. Lee, “RF modeling and design of flip ship configurations of microwave devices in PCBs,” in Proc. IEEE 5th Electron. Compon., Technol., 2004, vol. 7, pp. 1837–1841. [14] N. Yang, Z. N. Chen, X. M. Qing, and Y. X. Guo, “Serially-connectedseries-stub resonators and their applications in coplanar stripline bandpass filter design,” in IEEE MTT-S Int. Microw. Symp. Dig, 2005, pp. 695–698. [15] A. Abramowicz, “Investigation of the HTS microstrip filters based on dual-mode ring resonators,” in 12th Int. Microw. Radar Conf., 1998, pp. 8–12. [16] L. Hwa, “Compact dual mode elliptic function bandpass filter using a single ring resonator with one coupling,” Electron. Lett., vol. 36, pp. 1626–1627, 2000. [17] A. Görür, “Description of coupling between degenerate modes of a dual mode microstrip loop resonator using a novel perturbation arrangement and its dual mode bandpass filter applications,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 671–677, Feb. 2004. [18] A. Hennings, “Design optimization of microstrip square ring bandpass filter with quasi-elliptic function,” in 3rd Eur. Microw. Conf., Munich, Germany, 2003, pp. 175–178. [19] J. S. Hong and M. J. Lancaster, “Realization of quasi-elliptic function filter using dual mode microstrip square loop resonators,” Electron. Lett., vol. 31, pp. 2085–2086, 1995. [20] A. Corona-Chavez, C. Gutierrez, M. J. Lancaster, and A. Torres, “Novel dual-mode ring-resonators with very low sensitivity to substrate thickness,” Microw. Opt. Technol. Lett., vol. 47, no. 4, pp. 381–384, 2005. [21] J. Zhou and M. J. Lancaster, “Superconducting microstrip filters using compact resonators with double spiral inductors and interdigital capacitors,” in IEEE MTT-S Int. Microw. Symp. Dig, 2003, pp. 1889–1892. [22] SONNET 7.0b. Sonnet Software Inc., Pittsburgh, PA, 2001. [23] F. Rouchard, “New classes of microstrip resonators for HTS microwave filter applications,” in IEEE MTT-S Int. Microw. Symp. Dig, 1998, pp. 1023–1026. [24] J. Hong and M. Lancaster, Microstrip Filters for RF/Microwave Applications, ser. Microw. Opt. Eng.. New York: Wiley, 2001. [25] M. A. Hein, R. G. Humphreys, P. J. Hirst, S. H. Park, and D. E. Oates, “Nonlinear microwave response of epitaxial YBaCuo films of varying oxygen content on MgO substrates,” J. Supercond., vol. 16, no. 5, pp. 895–904, 2004. [26] A. V. Velichko, M. J. Lancaster, and A. Porch, “Nonlinear microwave properties of high T c thin films,” Supercond. Sci. Technol., vol. 18, pp. R24–R49, 2005. [27] M. A. Bakar, A. Velichko, V, M. Lancaster, X. Xiong, and A. Porch, “Temperature and magnetic field effects on microwave intermodulation in YBCO films,” IEEE Trans. Appl. Supercond., vol. 13, no. 2, pp. 3581–3583, Jun. 2003. [28] D. Seron et al., “Linear and nonlinear microwave properties of Ca-doped YBa Cu O think films,” Phys. Rev. B, Condens. Matter, vol. 72, pp. 1–11, 2005, 104511.

[29] D. E. Oates, H. Park, D. Agassi, G. Koren, and K. Irgmaier, “Temperature dependence of intermodulation distortion in YBCO: Understanding nonlinearity,” IEEE Trans. Appl. Supercond., vol. 15, no. 2, pp. 3589–3595, Jun. 2005. [30] B. A. Willemsen, K. E. Kihlstrom, and T. Dahm, “Unusual power dependence of two tone intermodulation in high T c superconducting microwave resonators,” App. Phys. Lett, vol. 74, pp. 753–755, 2006. [31] A. V. Velichko, “Origin of the deviation of intermodulation distortion in high T c think films from the classical 3 : 1 scaling,” Supercond. Sci. Technol., vol. 17, pp. 1–7, 2004.

Alonso Corona-Chavez (M’02) received the B.Sc. degree from the Instituto Tecnologico y de Estudios Superiores de Monterrey (ITESM), Monterrey, Mexico, in 1997, and the Ph.D. degree from the University of Birmingham, Edgbaston, U.K., in 2001. His doctoral research concerned microwave applications of superconductivity. From 2001 to 2004, he was a Microwave Engineer with Cryosystems Ltd., Luton, U.K., where he was involved with the development of microwave circuits using superconducting materials. During this time, he was also an Honorary Research Fellow with the School of Electrical Engineering, University of Birmingham. In 2004, he joined the National Institute for Astrophysics, Optics and Electronics (INAOE), Puebla, Mexico, as an Associate Professor. His current interests include microwave applications of HTS, RF, and microwave devices for communications and radio astronomy.

Michael J. Lancaster (M’91–SM’04) received the Physics and Ph.D. degrees from Bath University, Bath, U.K., in 1980 and 1984, respectively. His doctoral research concerned nonlinear underwater acoustics. Upon leaving Bath University, he joined the Surface Acoustic Wave Group, Department of Engineering Science, Oxford University, as a Research Fellow, where his research concerned the design of novel surface acoustic wave (SAW) devices including filters and filter banks. These devices worked in the 10-MHz–1-GHz frequency range. In 1987, he became a Lecturer with the School of Electronic and Electrical Engineering, University of Birmingham, Edgbaston, U.K. Shortly after, he began the study of the science and applications of HTS, involved mainly with microwave frequencies. He currently heads the Emerging Device Technology Research Center, University of Birmingham. His current research interests include microwave filters and antennas, as well as the high-frequency properties and applications of a number of novel and diverse materials.

Hieng Tiong Su (S’98–M’01) was born in Sarawak, Malaysia, in 1970. He received the B.Eng degree in electrical and electronic engineering from the University of Liverpool, Liverpool, U.K., in 1994, and the Ph.D. degree from the University of Birmingham, Edgbaston, U.K., in 2001. His doctoral research concerned superconducting quasi-lumped element filters. From 1994 to 1997, he was a Communications Engineer with Telecom Malaysia, Jarawak, Malaysia, where he was involved with the operation and maintenance of various telecommunication equipments Since 2001, he has been a Research Fellow with the Electronic and Electrical Engineering Department, University of Birmingham, where he has been involved with the design of novel superconducting delay lines and filters. His more recent interests include superconducting coils for magnetic resonance imaging (MRI) and micromachining devices.

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Design of Compact Low-Pass Elliptic Filters Using Double-Sided MIC Technology Mariá del Castillo Velázquez-Ahumada, Jesús Martel, and Francisco Medina, Senior Member, IEEE

Abstract—A novel implementation of stepped-impedance low-pass elliptic filters is presented in this paper. The filters are based on the well-known technique of cascading high- and low-impedance sections to simulate the ladder LC lumped-circuit prototype. We propose in this study a new approach to build up the constitutive circuit elements by taking advantage of the use of both sides of the substrate. The use of double-sided technology yields both design flexibility and good circuit performance. High-impedance sections are achieved by using slots in the backside of the substrate, whereas low-impedance sections are obtained with parallel-plate capacitors. In order to achieve the transmission poles corresponding to the elliptic design, these capacitors are series connected to the ground plane by means of high-impedance coplanar-waveguide lines, which mainly act as inductors. As a final step, meandering techniques have been applied to the high-impedance sections of the filter to reduce the overall circuit size. The measurement of several fabricated filters shows fairly good agreement between theory and experiment. Index Terms—Double-sided microwave integrated circuit (MIC), elliptic response, low-pass filters (LPFs), steppedimpedance (SI) filters.

I. INTRODUCTION

L

OW-PASS filters (LPFs) are very important components used to eliminate unwanted harmonics, as well as spurious bands in microwave and millimeter-wave systems. The conventional stepped-impedance low-pass filter (SI-LPF) consists of a cascading of electrically short high- and low-impedance sections to approximate the corresponding ladder LC lumpedcircuit prototype. SI-LPFs fabricated using conventional microstrip lines are among the most commonly used LPFs owing to simple design and low fabrication cost [1]. However, this type of filter inherently presents two important practical problems. The first one is the degradation of the out-of-band rejection level because of the frequency-distributed behavior of the finite section lines beyond the cutoff frequency. The second problem comes from the limits imposed by the microstrip technology to achieve very narrow strip widths and, consequently, high-impedance section lines. Manuscript received February, 15 2006; revised September 26, 2006. This work was supported by the Spanish Ministry of Education and Science/European Union under FEDER Funds Project TEC2004-03214 and by the Spanish Junta de Andalucía under Project TIC-253. M. del Castillo Velázquez-Ahumada and F. Medina are with the Facultad de Física, Grupo de Microondas, Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla, 41012 Seville, Spain (e-mail: [email protected]; [email protected]). J. Martel is with the Departamento de Física Aplicada II, Grupo de Microondas, Escuela Técnica Superior de Arquitectura, Universidad de Sevilla, 41012 Seville, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.888578

The rejection level in the out-of-band region of the SI-LPF, when conventional Chebyshev or Butterworth implementations are used, can be obviously improved by increasing the number of filter sections. However, it is also evident that this solution implies increasing the circuit size, as well as the bandpass losses. Recently, the use of defected ground structures (DGSs) has been proposed [2], [3] as an efficient method to attain a good out-ofband response of SI-LPFs. Thus, the influence of using periodic DGSs of various shapes on the response of conventional SI-LPF designs is analyzed in [2]. An equivalent circuit for the combination of the line section and DGS is introduced in [3] to design LPFs with different DGSs in each filter section. Alternatively, complementary split-ring resonators (CSRRs) can be etched on the conductor strips of the low-impedance sections to suppress spurious bands in conventional SI-LPF [4]. However, if a sharper cutoff is required for the LPF, generalized Chebyshev [5] or elliptic [6]–[8] designs have to be chosen in order to introduce transmission zeros in the vicinity of the bandpass. A broadband LPF based on a generalized Chebyshev lumped-circuit prototype is designed in [5] making use of open-circuit stubs in suspended substrate stripline technology. Microstrip rectangular elements are cascaded in [6] to obtain a broadband elliptic filter response. Multiple cascaded stepped-impedance hairpin resonators have also been proposed in [7] and [8] in order to obtain a compact elliptic LPF with sharp cutoff frequency response. In this paper, the design of an elliptic response LPF that retains the simplicity of the stepped-impedance structure is described. The method is based on the implementation of the constituent filter components by properly patterning the metallization at both sides of the substrate. Circuits fabricated using this technology are usually called a double-sided microwave integrated circuit (MIC) [9]. They take advantage of using different transmission media (microstrip, slot line, coplanar waveguide (CPW), microstrip-slot lines) on both sides of the substrate, therefore adding more design flexibility. In Fig. 1, we show an elementary layout of the proposed SI-LPF concept (in this case, a three-pole filter with a single transmission zero) together with its corresponding lumped equivalent circuit. Note that the LC shunt section is implemented in a single stage involving both have been realsides of the substrate. The series inductors ized, as usual, by means of electrically short high-impedance transmission-line sections. The slot in the ground plane under the strip line contributes to achieve values of the characteristic impedance higher than those obtained with conventional microstrip lines [10], [11]. As a first-order approximation, lowimpedance sections are assimilated to parallel-plate capacitors . In order to approximate the parallel connected capacitances

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TABLE I SPECIFICATIONS AND VALUES OF THE COMPONENTS OF THE LUMPED-CIRCUIT PROTOTYPE OF FILTER A

TABLE II SPECIFICATIONS AND VALUES OF THE COMPONENTS OF THE LUMPED-CIRCUIT PROTOTYPE OF FILTERB

Fig. 1. (a) Lumped equivalent circuit of a three-pole LPF with a single transmission zero. (b) Layout of the planar circuit implementation of the corresponding SI-LPF proposed in this paper.

to synthesize the transmission poles corresponding to the elliptic design, those capacitors are then series connected to the ground plane through high-impedance CPW line sections, which act as . Please note the inductors present in the parallel branches that high-impedance CPW lines are located at opposite borders of the backside rectangular patch of the capacitors; therefore, they play the additional role of bridges joining the two sides of the ground plane at the bottom side of the filter layout. These bridges are essential for a good filter performance because they cancel out undesired ground-plane slot modes [12], [13]. It is worth mentioning here that, owing to their small electrical size, slots in the ground plane do not meaningfully behave as radiant elements in the frequency range of interest, although radiation might be important at higher frequencies. The geometrical parameters of the layout have been obtained starting from the standard tabulated lumped equivalent-circuit components. We have made use of the formula for the equivalent circuit of electrically short high characteristic impedance transmission-line sections [1] to calculate the length of those sections. The characteristic impedance values have been obtained from the fast quasi-TEM algorithm developed in [14] for the analysis of hybrid microstrip-CPW structures. The lengths of the capacitors have been roughly calculated from the trivial expression for the capacitance of an ideal parallel-plate capacitor without edge effects. Finally, we have made use of a commercial electromagnetic (EM) simulator to take into account these edge effects and to correct the capacitor lengths obtained from the crude parallel-plate capacitance expression. Although the use of slots in the backside of the high-impedance sections leads itself to a size reduction with respect to conventional microstrip implementation [11], we have also used a meander line to achieve more compact filters [18]. In this way, the high-impedance section lines have been substituted by series-connected open loops. The geometry of these open loops has been extracted from EM

simulation in such a way that they reproduce the frequency response of the original straight line sections fed by two 50transmission lines. Two selected examples are presented to illustrate the design procedure. Experimental confirmation of the theory is also provided. II. FILTER DESIGN In order to illustrate the use of the proposed structure and the design procedure, we have designed a couple of elliptic LPFs of and . The filter specifications and the correorders sponding lumped parameter values are shown in Tables I and II. These values can be easily computed from the tabulated element values for elliptic function low-pass prototype filters [1] after applying impedance (50 ) and frequency scaling. Our purpose is to design and fabricate the filters A and B using a commercial and thicksubstrate of relative dielectric permittivity ness mm. For this substrate, we have generated the design graphics shown in Figs. 2 and 3. We have made use of the well-known formula relating the length of a line section of with the value of the lumped high characteristic impedance inductance to be approximated (see [1, p. 97]) as follows: (1) where is the phase velocity in the transmission is the effecline, is the speed of light in vacuum, and tive dielectric constant of the transmission line. In Fig. 2, the of a transmission line consisting of a microstrip impedance with a centered slot in the ground plane has been represented as for different values of the strip a function of the slot width width . From the value of the characteristic impedance and the have been calcueffective dielectric constant, the ratios

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TABLE III FINAL GEOMETRICAL VALUES (IN MILLIMETERS) OF THE DESIGNED AND FABRICATED FILTERS (A AND B )

Fig. 2. Characteristic impedances Z and normalized values of line section lengths l =L of a microstrip with slotted ground plane as a function of the slot width s for several strip width values (w ).

Fig. 3. Characteristic impedances Z and normalized values of line section lengths l =L of a CPW as a function of the slot width s for several values of the center strip width (w ).

lated using (1). As expected, for each strip-width value, the characteristic impedance is a monotonously increasing function of the slot width. Since the use of higher characteristic impedance values yields smaller values of the line lengths, some circuit size reduction is achieved using the slot (when compared with , i.e., conventional microstrip lines). In Fig. 3, the case we have plotted the variation of the characteristic impedance versus the slot width for different values of a CPW and the corresponding of the center strip width . From , we have obtained the normalized length value of by applying again (1), but taking into account that the inducare approximated by using the parallel connection of tors two high-impedance CPWs shorted to ground (see Fig. 1). In has to be twice the rethis way, the correct value for sult provided by (1). As can be seen from Fig. 3, if we retain the same value of the center strip width , we can get higher and shorter line lengths characteristic impedance values by increasing the slot width . All the results for the characteristic impedances and effective dielectric constants required to generate Figs. 2 and 3 have been computed using a fast and

Fig. 4. (a) Electrical response of fabricated filter A obtained from EM simulation (Ensemble), measurements, and lumped-circuit modeling. The simulated response of a Chebyshev A (see Table III) filter is also included. (b) Detailed view of the insertion losses in the bandpass. Measurements and simulations including and excluding ohmic losses are shown.

accurate quasi-TEM code developed in [14]. This code can handle hybrid CPW-microstrip transmission lines. The analysis in [14] uses the free surface charge distribution as unknown in the strip-like interface, whereas the electric field is used in the slot-like region. The Galerkin method in the spectral domain is then employed to solve the resulting hybrid integral equation. This suitable choice of unknown functions is one of the key points for the high numerical efficiency of the method. The other point is that drastic exponential convergence of the

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TABLE IV GEOMETRICAL PARAMETERS (IN MILLIMETERS) OF STANDARD CHEBYSHEV SI-LPF IN MICROSTRIP TECHNOLOGY WITH THE SAME f AND L THAN FILTERS A AND B PROPOSED IN THIS PAPER

Fig. 5. (a) Electrical responses of fabricated filter B obtained from EM simulation (Ensemble), measurements, and lumped-circuit modeling. The simulated response of a Chebyshev B (see Table IV) filter is also included. (b) Detailed view of the insertion losses in the bandpass. Measurements and simulations including and excluding ohmic losses are shown.

spectral series is achieved by using the methods described in [15]–[17]. In conclusion, the algorithm in [14] allows us to generate the curves in Figs. 2 and 3 within a few seconds (Pentium IV PC platform). In order to clarify the design method, below we have enumerated the steps followed to obtain the filter geometry (as an example, we use filter A, but the geometry of filter B can be extracted in a similar way). Step 1) Determination of the strip width and the slot width (see Fig. 1). Many different values are reasonably possible (the design is flexible at this point) provided the choice leads to a high value of the characteristic (the approximate equation (1) should impedance yield good results). For filter A, we have chosen the mm and mm (we have values introduced a marker corresponding to those values and in Fig. 2). This corresponds to mm/nH. From the values of , mm. shown in Table I, we obtain Step 2) A similar procedure has been followed when and (again, the choice of these choosing values is arbitrary to some extent, provided high-impedance values are obtained). For filter A,

mm and mm. we have selected This point has been marked in Fig. 3 to obtain and mm/nH. From in Table I, we get mm. the value of Step 3) The capacitor widths have been determined in such is slightly smaller than (specifa way that mm). Therefore, ically, for filter A , the strips connecting the backside capacitor patches with the ground plane can be approximated as CPWs because only a very short section of these strips do not have lateral ground planes. Note that the value of has to be chosen in such a way that transverse resonances within the frequency range of interest (i.e., the bandpass and its vicinity) are precluded [1]. This condition also limits the maximum value of . The length has been calculated in a first-order approximation by using the expression for the capacitance of an ideal parallel-plate capacitor. From the value in Table I, we obtain the value mm. of This method has been preferred to use an equation similar to (1) for the length of a low-impedance line section owing to the particular etching in the backside of the substrate. However, because of edge effects, the length obtained from the parallel-plate capacitor expression must be conveniently shortened. In this step, we have chosen to use a commercial EM simulator (in our case, Ensemble) so as to obtain this geometrical parameter of the filter. The optimization criterion has been to determine the lengths ( even) so as to make coincident the frequencies of the transmission zeros of the lumped-circuit response and those of the simulated (full-wave) filter response. Using modern EM simulators instead of approximate formulas is preferred because existing parasitics and any possible full-wave effects are automatically taken into account. In case of our filter mm. A, the EM correction leads to The final geometrical parameters of filters A and B are shown in Table III. The total lengths of the filters are mm and mm. The electrical responses of

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Fig. 6. (a) Straight high-impedance line section with slot in the ground plane. (b) Its equivalent meander line with series-connected open loops.

Fig. 7. Extraction of the meander geometry of Section III of the reduced size filter. Narrow lines correspond to return losses of meandered lines when a is varied. The thick line is the return loss of the straight high characteristic impedance section to be substituted.

the designed filters A and B are shown in Figs. 4 and 5, respectively. In these figures, we compare the insertion and return losses obtained through EM simulation, measurements, and the lumped-circuit model prediction. Since the level of losses in the bandpass is very low, more detailed curves for the insertion losses are included in Figs. 4(b) and 5(b). A reasonably good agreement can be found between EM simulation and measurements (particularly good agreement is found in and around the passband). Discrepancies could be attributed to the limited accuracy of the available photoetching fabrication process. Moreover, these results agree reasonably well with those provided by the lumped-circuit model in the passband and its vicinity. Obviously, spurious bands appear at higher frequencies (above the frequencies of the transmission zeros) owing to the resonances of the transmission lines involved in the filters. To illustrate this fact, we have also included the EM responses of standard Chebyshev SI-LPFs [see Figs. 4(a) and 5(a)] with the same cutoff frequency and passband ripple designed in conventional microstrip technology. The geometrical parameters of these Chebyshev filters are shown in Table IV. Note that, in addition to the expected sharper bandpass, the section lengths of

Fig. 8. (a) Layout of the miniaturized SI-LPF with the meandered lines (dimensions in the text). (b) Electrical response of filter in (a) obtained from EM simulation (Ensemble) and measurements. (c) Detailed view of the insertion losses in the bandpass. Measurements and simulations including and excluding ohmic losses are shown.

the elliptic filters are physically shorter than those of Chebyshev designs, thus achieving a significant circuit size reduction mm and (the total length of Chebyshev filters are mm). Furthermore, it can be proved that the transmission lines involved in the elliptic filters are also electrically shorter than those of Chebyshev filters and, therefore, undesirable bands are moved to higher frequencies [11]. For instance,

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the spurious band of the Chebyshev response appearing in Fig. 5 at 5.2 GHz (owing to the half-wave resonance of Section III) has been shifted to 7.6 GHz in the new design. III. SIZE REDUCTION OF ELLIPTIC SI-LPF DESIGNS We have shown in Section II that a meaningful reduction of size (around 30%) can be achieved with the new elliptic designs based on double-sided technology (see Fig. 1) with respect to Chebyshev designs implemented in conventional microstrip technology. However, additional reduction can be achieved with a little bit more effort. For instance, the total length of filter B is four times its own width (the width of the filter is defined as the slot width plus the length of the longest CPW, i.e., in the case of filter B). The goal here is to propose a shorter version of filter B having the same electrical specifications. The filter length can be significantly reduced if, for instance, we substitute the straight high-impedance line sections by meandered lines built using series open loops such as those shown in Fig. 6. The geometry of the loops can be easily obtained using the EM simulator. The procedure basically consists of varying one of the geometrical parameters of the open loops until the electrical response of the single straight line (fed by 50- lines) to be substituted and the new meandered line become similar. As an example, the process can be visualized in Fig. 7. The thick solid line corresponds to the return loss of section 3 of filter B. The narrow lines correspond to the responses of meandered mm, mm, and lines in which we have fixed mm, while the value of varies from 0.3 to 0.6 mm. The final value used is mm. The same procedure has been employed in the design of sections 1 and 5 (final meander mm, mm, mm, line values are mm; mm, mm, mm, mm). Sections 2 and 4 remain unchanged. The final mm, i.e., a 50% smaller than design has a total length of filter B designed in the previous section. It must be pointed out that a coupling capacitive effect appears between the patches of sections 2 and 4 because of the reduction of and, therefore, the filter response slightly differs from that of filter B. These discrepancies become more important when the length decreases. Fig. 8 shows the EM simulation response and measurements of the miniaturized SI-LPF filter (a detailed view of the insertion losses in the bandpass is included again). Reasonably good agreement is found between both results, particularly at the low-frequency portion of the measured spectrum. As mentioned before, a small displacement of the transmission zeros can be observed when compared with the response of filter B shown in Fig. 5. IV. CONCLUSIONS In this paper, we have presented a new elliptic response SI-LPF filter with a sharp bandpass and reduced size. The design method is based on the use of the double-sided MIC technology to build the different components of the filter. The impedance of the high-impedance sections has been increased by using slots in the ground plane. The low-impedance sections have been approximated as parallel-plate capacitors. The new idea is that the patch of the capacitor in the backside of the substrate is connected to the ground plane by means

of high-impedance CPW lines, which act as inductors and yield the transmission zeros of the elliptic design. The lengths of the capacitors have been tuned using the EM simulator Ensemble in order to match the transmission zeros to those provided by the lumped equivalent-circuit model. Although the size of the new filters is meaningfully smaller than the size of the corresponding standard Chebyshev filters in microstrip technology, an additional improvement has been introduced in order to reduce the filter size. In this way, the straight high characteristic-impedance series-connected line sections have been substituted by meandered lines with series-connected open loops. The EM simulator has been used to calculate the dimensions of the meandered lines and, finally, a miniaturized modified version of the elliptic SI-LPF has been obtained. An additional reduction above 50% in the length of the filters has been then finally carried out. REFERENCES [1] J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [2] A. D. Abdel-Rahman, A. K. Verma, A. Boutejdar, and A. S. Omar, “Control of bandstop response of hi–lo microstrip low-pass filter using slot in ground plane,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 1008–1013, Mar. 2004. [3] J. S. Lim, C. S. Kim, D. Ahn, Y. C. Jeong, and S. Nam, “Design of low-pass filters using defected ground-plane structure,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 8, pp. 2539–2545, Aug. 2005. [4] J. Carcía-Carcía, J. Bonache, F. Falcone, J. D. Baena, F. Martìn, I. Gil, T. Lopetegui, M. A. G. Laso, A. Marcotegui, R. Marqués, and M. Sorolla, “Stepped-impedance lowpass filters with spurious passband suppression,” Electron. Lett., vol. 40, pp. 881–882, Jul. 2004. [5] S. A. Alseyab, “A novel class of generalized Chebyshev low-pass prototype for suspended substrate stripline filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 9, pp. 1341–1347, Sep. 1982. [6] F. Giannini, M. Salerno, and R. Sorrentino, “Design of low-pass elliptic filters by means of cascaded microstrip rectangular elements,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 9, pp. 1348–1353, Sep. 1982. [7] L. H. Hsieh and K. Chang, “Compact elliptic-function low-pass filters using microstrip stepped-impedance hairpin resonators,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 193–199, Jan. 2001. [8] W. Tu and K. Chang, “Compact microstrip low-pass filter with sharp rejection,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 6, pp. 404–406, Jun. 2005. [9] M. Aikawa and H. Ogawa, “Double-sided MIC’s and their applications,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 2, pp. 406–413, Feb. 1989. [10] L. Zhu, H. Bu, K. Wu, and M. Stubbs, “Unified CAD model of microstrip line with backside aperture for multilayered integrated circuit,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, vol. 2, pp. 981–984. [11] S. Sun and L. Zhu, “Stopband-enhanced and size-miniaturized lowpass filters using high-impedance property of offset finite-ground microstrip line,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2844–2850, Sep. 2005. [12] M. C. Velázquez, J. Martel, and F. Medina, “Parallel coupled microstrip filters with ground-plane aperture for spurious band suppression and enhanced coupling,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 1082–1086, Mar. 2004. [13] ——, “Parallel coupled microstrip filters with floating ground-plane conductor for spurious-band suppression,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1823–1828, May 2005. [14] J. Martel and F. Medina, “A suitable integral equation for the quasi-TEM analysis of hybrid strip/slot-like structures,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 224–228, Jan. 2001. [15] F. Medina and M. Horno, “Quasi-analytical static solution of the boxed microstrip line embedded in a layered medium,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 9, pp. 1748–1756, Sep. 1992. [16] E. Drake, F. Medina, and M. Horno, “Improved quasi-TEM spectral domain analysis of boxed coplanar multiconductor microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 2, pp. 260–267, Feb. 1993.

DEL CASTILLO VELÁZQUEZ-AHUMADA et al.: DESIGN OF COMPACT LOW-PASS ELLIPTIC FILTERS USING DOUBLE-SIDED MIC TECHNOLOGY

[17] ——, “Quick computation of [C ] and [L] matrices of generalized multiconductor coplanar waveguide transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2328–2335, Dec. 1994. [18] H. Peddibhotia and R. K. Settagori, “Compact folded line bandstop and low pass filters,” Microw. Opt. Technol. Lett., vol. 42, pp. 44–46, Jul. 2004. María del Castillo Velázquez-Ahumada was born in Lebrija, Sevilla, Spain, in 1976. She received the Licenciado degree in physics from the Universidad de Sevilla, Seville, Spain, in 2001, and is currently working toward the Ph.D. degree in electronics and electromagnetism at the Universidad de Sevilla. Her research focus is on printed passive microwave filters and couplers.

Jesús Martel was born in Seville, Spain, in 1966. He received the Licenciado and Doctor degrees in physics from the Universidad de Sevilla, Seville, Spain, in 1989 and 1996, respectively. Since 1992, he has been with the Departamento de Física Aplicada II, Universidad de Sevilla, where in 2000, he became an Associate Professor. His current research interest is focused on the numerical analysis of planar transmission lines, modeling of planar microstrip discontinuities, design of passive microwave circuits, microwave measurements, and artificial media.

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Francisco Medina (M’90–SM’01) was born in Puerto Real, Cádiz, Spain, in 1960. He received the Licenciado (with honors) and Doctor degrees in physics from the Universidad de Sevilla, Seville, Spain, in 1983 and 1987 respectively. From 1985 to 1989, he was an Assistant Professor with the Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla, where since 1990, he has been an Associate Professor. Since 1998, he has been the Head of the Grupo de Microondas, Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla. His research interest includes analytical and numerical methods for guiding, resonant, and radiating planar structures, passive planar circuits, periodic structures, and the influence of anisotropic materials (including microwave ferrites) on such systems. He is also interested in artificial media modeling and design. He is a member of the Editorial Board of the International Journal of RF and Microwave Computer-Aided Engineering. Dr. Medina has been a member of the Technical Programme Committees (TPCs) of several national and international conferences. He has been a reviewer for a number of IEEE, Institution of Electrical Engineers (IEE), U.K., and American Physics Association journals. He is a member of the Massachusetts Institute of Technology (MIT) Electromagnetics Academy. He was the recipient of two research scholarships presented by the Spanish Ministerio de Educación y Ciencia (MEC) (1983) and the Spanish MEC/French Ministère de la Recherche et la Technologie (ENSEEIHT, Toulouse, France) (1986).

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Novel Multifold Finite-Ground-Width CPW Quarter-Wavelength Filters With Attenuation Poles Chin-Hsuing Chen, Ching-Ku Liao, and Chi-Yang Chang, Member, IEEE

Abstract—This paper proposes a novel multifold finite-ground4 resonator filter that has width coplanar waveguide (CPW) the capability of realizing attenuation poles by cross coupling of nonadjacent resonators. The newly proposed multifold structures not only greatly shrink the length of a resonator, but also provide a convenient way to implement cross coupling in a filter. The proposed multifold finite-ground-width CPW resonators can have much stronger coupling than that of spiraling and meandering layouts, and any number of folds is possible. Two combline filters are designed and measured with twofold and threefold 4 finite-ground-width CPW resonators. A trisection and a quadruplet generalized Chebyshev filter formed by cross couplings between nonadjacent resonators have been implemented by modifying the layout of the resonator. These filters are smaller in size and have the capability of controlling attenuation poles. Index Terms—Attenuation poles, bandpass filter, cross coupling, finite-ground-width coplanar waveguide (CPW), generalized Chebyshev filter, multifold resonator.

I. INTRODUCTION

M

ODERN microwave filters need to be both highly selective and compact in size. Recently, the coplanar waveguide (CPW) has largely been applied in microwave circuits because of its uniplanar structure, performance at high frequency, and ease of connection to other CPW circuits [1]. It is less common for filter applications because the CPW value than the microstrip, resonator has a lower unloaded limiting the narrowband application of a CPW filter unless it is implemented by a superconducting material. Moreover, the CPW usually needs bonding wires at each discontinuity junction to maintain the proper mode of propagation, which often causes fabrication costs to rise. Nevertheless, the CPW possesses some benefits that microstrip do not have. First, it is resonator with a CPW due to much easier to implement a its lack of via-holes [2]. Second, some special structure, e.g., a series stub, can be easily implemented by the CPW [3], but are extremely difficult to implement using microstrip. Due to the differences between the CPW and microstrip, the resonator structures proposed in this paper are applicable only to the CPW. Manuscript received May 7, 2006; revised September 22, 2006. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 95-2752-E-009-003-PAE and Grant NSC 94-2213-E-009-072. The authors are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (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/TMTT.2006.886924

Fig. 1. Possible layouts of folded =4 CPW resonators. (a) Spiraling. (b) Meandering. (c) Proposed threefold finite-ground-width CPW.

A microstrip resonator can be reduced in size by different layouts, such as spiral, meander, and fractal [4]–[6]. These resonator. layouts are asymmetric as the resonator is a resonators The coupling between these shrunken-layout is orientation dependent in that some orientations are coupled very weakly. Due to weak coupling, only filters with narrow bandwidth can be realized with these shrunken-layout resonators. The coupling weakening effect is much more severe in the CPW than in a microstrip. Especially when capacitive, the coupling is weakened by the metal strips used as ground strips between two neighboring CPW resonators. As a result, the spiral layout, as shown in Fig. 1(a), and the meander layout, as shown in Fig. 1(b), are both unsuitable for CPW filter applications. Therefore, different resonator layouts should be developed specifically for a CPW filter with moderate bandwidth. For achieving high selectivity, a generalized Chebyshev filter realized by cross coupling [7], [8] is one of the most commonly resused methods. However, spiral, meander, and fractal onators are difficult to adequately implement the cross coupling between nonadjacent resonators. Using an extra line in the spiral structure to couple the nonadjacent resonators can create attenuation poles [4]. Nevertheless, the design of this kind of cross-coupled filter is usually trial-and-error, and rigorous synthesis methods have not yet been well developed. Regardless of meander resonator was weak coupling, a CPW filter with a meander resonator’s asymmetric reported in [9]. Due to this structure, only a few cross-coupling structures can be realized. In this paper, we propose a novel multifold finite-groundresonator, whose basic layout is shown in width CPW Fig. 1(c). This newly proposed resonator solves most of the

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above-mentioned problems. The weak coupling problem is solved because the coupling of two resonators in Fig. 1(c) is mainly inductive. The coupling is much stronger and experiences much less of the ground strip blocking effect than capacitive coupling. The open end of the multifold finite-ground-width CPW can be moved to the exterior of the resonator, creating strong capacitive coupling. This will be further discussed in Section II. By applying these resonators, various generalized Chebyshev filters can be adequately implemented because the layout of the proposed resonators are so flexible that both capacitive and inductive cross coupling can be easily realized. To show the feasibility of the proposed structures, two four-pole Chebyshev filters with twofold and threefold resonators are designed and realized. In addition, for better filter selectivity, a trisection and a quadruplet generalized Chebyshev filter are realized with various attenuation poles. II. THEORY A. Resonator The finite-ground-width CPW transmission line is similar to a center longitudinal slice of a coaxial line in that many fancy structures in coaxial lines can be emulated by a finite-groundwidth CPW, such as the multifold structure shown in Fig. 1(c), [10]. In a coaxial line, the structure is formed of many concentric cylinders, as is the case shown in Fig. 1(c). The even-numbered cylinders are shorted at one end, the odd-numbered cylinders are shorted at the other end, and the outmost cylinder is ground. We call it a multifold structure because it can have possible number of folds. There are three folds of center conductors in the structure shown in Fig. 1(c), so it is called a threefold resonator. The design procedures of a multifold resonator are described as follows. 1) General Description of Twofold Resonator: Since the twofold resonator will be the simplest one of the proposed resonators, we will describe the twofold resonator first. The layout of a twofold resonator is shown in Fig. 2(a), where the innermost strip is shorted at one end, the two medium strips are shorted at the other end, and the outmost frame is ground. In Fig. 2(a), there is a bonding wire at the open-end of strips – , which is essential to keep the finite-ground-width CPW wave mode propagating. It might be easier to understand if we think of Fig. 2(a) as a slice of coaxial line with two folds – represents the innermost cylinder, of cylinders where – represents a medium cylinder, and A-A’ represents the outermost cylinder. Here, because we have only a slice of coaxial line, the conductors becomes strips. 2) Characteristic Impedance of Each Fold: After we understand the twofold resonator, a equivalent circuit can be developed, as shown in Fig. 2(b). In Fig. 2(a), nodes – , as well as ground planes A and A’ correspond to the same numbered nodes and ground planes in the equivalent circuit shown corresponds to in Fig. 2(b). The characteristic impedance the finite-ground-width CPW with the innermost strip to be the signal conductor and two medium fold strips to be the ground corresponds to the conductor. The characteristic impedance finite-ground-width CPW with two medium fold strips to be the

Fig. 2. Proposed twofold =4 finite-ground-width CPW resonator. (a) Layout. (b) Equivalent circuit.

signal conductor and the outmost frame to be the ground conductor. 3) Input Admittance and Resonant Condition: According to the equivalent circuit shown in Fig. 2(b), the input admittance at open-end – may be calculated where it should be zero at resonant frequency. From the resonant condition, the electrical length of each fold finite-ground-width CPW can be obtained. , the circuit in Fig. 2(b) is equivalent to a quarterIf [11]. In our case, we choose wave resonator when and the total length of the resonator becomes half of the original quarter-wave resonator. This planar twofold finite-ground-width CPW resonator will be identical to a twofold coaxial resonator only if the field of innermost strip – is completely shielded by medium fold strips – . 4) -Fold Resonator: Following the above procedures, an -fold resonator can be developed, as shown in Fig. 3(a). The even-numbered strips are shorted at one end, and the odd-numbered strips are shorted at the other end, and the outmost frame is ground in Fig. 3(a). Theoretically, the resonator can have an arbitrary numbers of folds and, thus, Fig. 3(a) depicts the structure of an -fold resonator. According to Fig. 3(a), the -fold strips and bonding wires. resonator should have Fig. 3(b) shows the equivalent circuit of Fig. 3(a). and so the Let total length of the resonator should be of a quarter-wavelength. Based on the above analysis, we know that the length of the proposed resonators can be possibly shrunk by this multifold structure with a penalty of resonator width increase. 5) Resonator’s Physical Layout: In this paper, the proposed circuits are fabricated on an Al O substrate with a substrate thickness of 15 mil and dielectric constant of 9.8. Since the finite-ground-width CPW is a uniplanar transmission line, only

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TABLE I LENGTH AND WIDTH OF A TWOFOLD TO FIVEFOLD FINITE-GROUND-WIDTH CPW RESONATORS WITH A RESONANT FREQUENCY OF 2.4 GHZ

Fig. 3. Proposed N -fold =4 finite-ground-width CPW resonator. (a) Layout. (b) Equivalent circuit.

the circuit side of the substrate is deposited with 3- m-thick gold film, and the other side has no metal film. The substrate is suspended with a height of 120 mil. The filters are measured with a probe station where the ceiling does not exist. Electromagnetic (EM) simulation shows that the ceiling and bottom conductors have little influence on filter’s performance. As the distance of the ceiling and bottom conductor are larger than approximately 120 mils the influence is negligible. In most of our filters, the ratio of gap width to effective strip width (the width is called the effective strip width in an outer fold finite-ground-width CPW, as will be explained later) of each fold finite-ground-width CPW line kept at 1 : 2. This is done to maintain the characteristic impedance of each fold finite-ground-width CPW line to be approximately 50 . For a twofold resonator, the innermost finite-ground-width CPW has a strip width of 4 mil and a gap width of 2 mil, and the second fold finite-ground-width CPW has an effective strip width of 16 mil (three 4-mil strips and two 2-mil gaps is a total of 16 mil) and a gap width of 8 mil. Therefore, the width of the twofold resonator is 32 mil in total. The open end of each strip has a 5-mil gap to the ground plane. A Sonnet simulator [12] is used here to perform EM simulation. The totally simulated length of the twofold resonator is 313 mil at a center frequency of 2.4 GHz. A convenCPW resonator with a strip width of 16 mil and a gap tional width of 8 mil (similar resonator width with a twofold resonator) is simulated for comparison. The total length of the conventional CPW resonator is 545 mil at the same frequency. The length of the twofold resonator is approximately 15% more than half of CPW resonator. This occurs because the the conventional finite-ground-width CPW used in this case differs from a coaxial line, where the EM fields of the inner finite-ground-width CPW are not completely shielded by medium fold strips. Table I shows the length and width of twofold to fivefold resonators with a resonant frequency of 2.4 GHz. All strips are with a strip width of 4 mil for the entire multifold finite-groundwidth CPW resonator. The effective strip width of th-fold fiequals mil for nite-ground-width CPW , where mil and the gap width equals half of the effective strip width. It can also be seen that the length is a

Fig. 4. Structure and current distribution of twofold resonator. (a) Structure consists of resonator, feeding lines, and bonding wires. (b) Current distribution at 2.4 GHz of (a) (darker color represents higher RF current density).

of the conventional CPW resonator. little larger than The gap width to effective strip width ratio can be different from 1 : 2 of each fold finite-ground-width CPW line, and a different ratio causes the resonator length to change. The proposed resonator has a clear equivalent circuit and its length and width can be approximately obtained by a circuit model. However, the spiral and meander layout in Fig. 1(a) and (b) has no explicit equivalent-circuit model such that resonator length and width are not easy to estimate. value The CPW resonator usually has a lower unloaded than that of a microstrip resonator, and smaller resonator size causes a lower value. Since the proposed resonators are physvalue is necessary. Unically small, a study on resonator loaded value includes three parts, namely, for metal for dielectric loss, and for radiation loss, loss. The simulated results show that the highest loss term in proposed resonators come from the metal loss. For a twofold resonator at 2.4 GHz, the simufinite-ground-width CPW lated unloaded values are and respectively. The resonator is with 3- m thickness of gold metal film, Al O substrate of 15-mil thickness, and 0.0002 loss tangent. Since the simulated resonator value is not high, only a filter with moderate bandwidth is applicable, and narrowband application of a superconducting film is suggested.

CHEN et al.: NOVEL MULTIFOLD FINITE-GROUND-WIDTH CPW QUARTER-WAVELENGTH FILTERS WITH ATTENUATION POLES

Fig. 5. Main couplings of =4 finite-ground-width CPW resonators. (a) Two of threefold resonators. (b) Coupling coefficients versus ground-plane distance s (in mils) between two of threefold resonators, two of meander resonators, and two of spiral resonators.

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Fig. 6. Modified finite-ground-width CPW resonators to implement electric coupling. (a) Layout. (b) Coupling coefficient versus ground strip length d (in mils).

B. Coupling Structures A standard twofold resonator including feeding lines and bonding wires are shown in Fig. 4(a). The bonding wire at a taped point is for better maintaining of the CPW mode. Two extra bonding wires on a discontinuous ground plane are for the same purpose. Fig. 4(b) shows the current distribution of Fig. 4(a) at the resonant frequency of 2.4 GHz. Since the RF current is strong along the medium fold strips, the coupling between two of these threefold resonators, as shown in Fig. 5(a), will be magnetically coupled. As explained in Section II-A, the magnetic coupling is relatively strong. To demonstrate the relatively strong coupling, the coupling between two of the proposed threefold finite-ground-width CPW resonators, two of the spiral resonators in Fig. 1(a), and two of the meander resonators in Fig. 1(b) are obtained by the method in [13]. The coupling strength versus ground-plane distance between two resonators are depicted in Fig. 5(b). In Fig. 5(b), the resonator widths and gaps between the outmost signal strips and ground plane of all three types of resonators are chosen to be the same. According to Fig. 5(b), the coupling strength of the proposed resonator is more than three times stronger than the other two types of resonators.

Fig. 7. Layout of cross coupling of twofold finite-ground-width CPW resonators. (a) Magnetic coupling. (b) Electric coupling.

Moreover, a modified structure is shown in Fig. 6(a), where the open end of the resonator goes to the exterior region of the resonator to make a capacitive (electric) coupling. In Fig. 6(a), there is a ground strip inserted between open ends of two resonators to control the coupling strength. Fig. 6(b) depicts the electric coupling strength versus ground strip length of Fig. 6(a). Coupling can be stronger if and the gap between two open-ended strips becomes smaller.

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Fig. 8. Four-pole Chebyshev filter with twofold finite-ground-width CPW resonators. (a) Layout of top view with dimension (in mils) W = 150; L = 313; L1 = 301; D = 117; S 1 = 5; and S 2 = 12. (b) 3-D view. (c) Simulated and measured narrowband responses. (d) Wideband responses. (e) Photograph.

In the case of cross coupling, the layout of the resonator should be modified when coupling is nonadjacent. The layouts of cross coupling are somewhat similar to Fig. 6(a). Fig. 7(a)

and (b) shows the magnetic and the electric cross coupling of two nonadjacent resonators. The cross coupling strength could be found by a similar method as the main coupling.

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III. FILTER DESIGN A. Chebyshev Filters 1) Twofold Finite-Ground-Width CPW Filter: A four-pole Chebyshev filter with twofold finite-ground-width CPW resonators is designed at a center frequency of 2.4 GHz with a passband ripple of 0.05 dB and a fractional bandwidth of 10%. The corresponding normalized coupling coefficients are and , [13]–[16]. The coupling coefficients are obtained by adjusting the distances between adjacent resonators, and and are implemented by fine tuning the position of the tapped point. Fig. 8(a) and (b) depicts the layout of top view and 3-D view of this filter, respectively, and the area of the filter excluding the feeding lines is 150 mil 313 mil in Fig. 8(a). Fig. 8(c) depicts the simulated and measured narrowband responses of this filter. The metal losses are included in the simulation. The measured responses without tuning are described as follows: the insertion loss is approximately 2.8 dB, and the return loss is approximately 14.3 dB in the passband. The wideband responses are shown in Fig. 8(d). The measured center frequency shifts approximately 2.8% upward because the substrate dielectric constant, which is used to fabricate this filter, could be lower than the substrates, which built other filters. This filter is fabricated with a different lot of substrates with respect to other filters, and the dielectric constant might be different from lot to lot. The spurious passband occurs at a frequency of approximately 6.15 GHz, as shown in Fig. 8(d), and a photograph of this filter is shown in Fig. 8(e). 2) Threefold Finite-Ground-Width CPW Filter: To show the feasibility of resonators with higher numbers of folds, a threefold resonator Chebyshev filter with the same specifications as the previous twofold resonator filter is designed and fabricated for comparison. Fig. 9(a) shows the layout of the filter; the area of the filter excluding two feeding lines is 364 mil 243 mil. Fig. 9(b) depicts the simulated and measured responses of the threefold resonator filter. According to the measured responses, the insertion loss is approximately 1.8 dB and the return loss is approximately 16.8 dB in the passband. A higher number of folds is also feasible, but resonator widths will be larger. B. Generalized Chebyshev Filters 1) Trisection Finite-Ground-Width CPW Filter: A trisection generalized Chebyshev filter with a twofold finite-ground-width CPW resonator is proposed and realized with a center frequency of 2.4 GHz. The filter is designed with a passband ripple of 0.05 dB, a fractional bandwidth of 5%, and an attenuation pole of in the low-pass prototype. The main couplings are all magnetic and the cross coupling between resonators 1–3 is electric. The layout of the filter is shown in Fig. 10(a), and the area of the filter, excluding feeding lines, is 132 mil 381 mil. A trisection filter of this coupling configuration should generate an attenuation pole in the lower stopband. The corresponding normalized coupling matrix elements are and [15], [16]. Fig. 10(b) gives the simulated and measured results of this filter. The measured return loss is less than 20 dB, the

Fig. 9. Four-pole Chebyshev filter with threefold finite-ground-width CPW resonators. (a) Layout of top view with dimension (in mils) W = 364; L = 243; L1 = 231; D = 150; s1 = 10; and S 2 = 24. (b) Simulated and measured responses.

insertion loss is approximately 2.85 dB in the passband, and the designed attenuation pole position is approximately 2.19 GHz. An extra lower stopband attenuation pole at 2.08 GHz might come from parasitic couplings. 2) Quadruplet Finite-Ground-Width CPW Filter: A quadruplet generalized Chebyshev filter with resonators 2–3 coupled electrically and resonators 1–4 coupled magnetically is designed and fabricated. The filter is with the center frequency of 2.4 GHz, a passband ripple of 0.05 dB, the fractional bandin the low-pass width of 8%, and attenuation poles of prototype. The corresponding normalized coupling coefficient are and [15]–[17]. The layout is shown in Fig. 11(a); the coupling between resonators 1–2, 3–4, and 1–4 are all magnetic, and coupling between resonators 2–3 is electric. The size of the filter excluding feeding lines is 172 mil 285 mil. Fig. 11(b) gives the simulated and measured results of this filter. The measured return loss is approximately 22 dB and the insertion loss is approximately 3 dB in the passband. Fig. 11(c) shows a photograph of this filter. In Fig. 11(a), the magnetic coupling fades out slowly with respect

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Fig. 10. Trisection generalized Chebyshev filter with twofold finite-groundwidth CPW resonators. (a) Layout of top view with dimension (in mils) W = 132; L = 381; L1 = 309; L2 = 39; D = 283; S 1 = 13; and S 2 = 58. (b) Simulated and measured responses.

to resonator distance; therefore, couplings between resonators 1–3 and 2–4 still exist. Fig. 12 shows the coupling diagram of this filter, where represents magnetic couplings and represents electric coupling. The diagonal couplings in Fig. 12 cause shift of the attenand are magnetic, atuation poles [17], [18]. If both tenuation poles should shift upward. In contrast, the shift should and are both electric. and be downward if are magnetic in the proposed filter, thus the attenuation poles shift upward, as shown in Fig. 11(b). and We analyze the effect of unwanted couplings through the following procedures in order to understand attenuation poles shift. The reflection and transmission response of the low-pass prototype filter are (1)

Fig. 11. Quadruplet generalized Chebyshev filter with twofold =4 finite-ground-width CPW resonators. (a) Layout of top view with dimension (in mils) W = 172; L = 285; L1 = 128; L2 = 54; L3 = 37; L4 = 128; L5 = 71; D = 115; S 1 = 5; S 2 = 4; and S 3 = 20. (b) Simulated and measured responses. (c) Photograph.

and (2)

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(5)

(6) ferent from the original designed values because the filter is fine tuned during EM simulation of the whole filter.

IV. CONCLUSION

Fig. 12. Coupling and routing scheme of quadruplet filter.

where (3) Subscripts and in (2) represent the sixth row and first column matrix element, respectively. In (3), is a 6 6 matrix , and is whose only nonzero entries are . similar to a 6 unity matrix, except that is the normalized 6 6 coupling matrix. Due to circuit symmetry, and is equal zero to find the frequencies of attenuproposed. Let of the low-pass prototype filter so that the charation poles acteristic determinant is zero as follows: (4) Ignoring the negligible frequency shift of and in (4), is (5), shown at the top of this page. Using the diagnosis method described in [17], the symmetric -matrix in (6), shown at the top of this page, is extracted from the simulated results of Fig. 11(b) with the circuit loss excluded. are The extracted diagonal couplings and and cross coupling smaller than main couplings . The resonant frequency drift terms and are so small that they can be neglected. Substituting corresponding matrix parameters into are obtained to be (5), two frequencies of attenuation poles 1.616 and 2.244, respectively, so they both shift upward. The extracted values of coupling matrix elements are slightly dif-

resonator filNovel multifold finite-ground-width CPW ters have been demonstrated to be feasible in this paper. The proposed resonators are specifically designed for the CPW and have shown the capability of realizing a filter with narrow to moderate bandwidths. When we increase the number of folds, the resonator length can be reduced with a penalty of resonator width increase. The proposed resonators not only reduce the size of a filter, but also easily and accurately realize the demanded cross couplings. The filters with Chebyshev and generalized Chebyshev responses have been implemented. Two generalized Chebyshev filters have shown the possibility of implementation of various desired cross couplings. Due to the extreme compactness of the filter, some unwanted couplings are inevitable. However, we have demonstrated that filter performance has not been seriously affected by the unwanted couplings. REFERENCES [1] R. N. Simons, Coplanar Waveguide Circuits, Components, and Systems. New York: Wiley, 2001. [2] C. Y. Chang and D. C. Niu, “A novel CPW interdigital filter,” in Asia–Pacific Microw. Conf. Dig., Dec. 2001, pp. 621–624. [3] G. E. Ponchak and L. P. B. Katehi, “Open- and short-circuit terminated series stubs in finite-width coplanar waveguide on silicon,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 6, pp. 970–976, Jun. 1997. [4] G. Zhang, M. J. Lancaster, and F. Huang, “A high-temperature superconducting bandpass filter with microstrip quarter-wavelength spiral resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 559–563, Feb. 2006. [5] J. S. Hong and M. J. Lancaster, “Compact microwave elliptic function filter using novel microstrip meander open-loop resonators,” Electron. Lett., vol. 32, no. 6, pp. 563–564, Mar. 1996. [6] M. Barra, C. Collado, J. Mateu, and J. M. O’Callaghan, “Miniaturization of superconducting filters using Hilbert fractal curves,” IEEE Trans. Appl. Supercond., vol. 15, no. 3, pp. 3841–3846, Sep. 2005. [7] R. Levy, “Filters with single transmission zeros at real or imaginary frequencies,” IEEE Trans. Microw. Theory Tech., vol. MTT-24, no. 4, pp. 172–181, Apr. 1976. [8] J. S. Hong and M. J. Lancaster, “Design of highly selective microstrip bandpass filters with a single pair of attenuation poles at finite frequencies,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 7, pp. 1098–1107, Jul. 2000. [9] J. Zhou, M. J. Lancaster, and F. Huang, “HTS coplanar meander-line resonator filters with a suppressed slot-line mode,” IEEE Trans. Appl. Supercond., vol. 14, no. 1, pp. 28–32, Mar. 2004. [10] J. A. G. Malherbe, Microwave Transmission Line Filters. Dedham, MA: Artech House, 1979.

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[11] A. K. Rayit and N. J. McEwan, “Coplanar waveguide filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1993, pp. 1317–1320. [12] “EM User’s Manual,” Sonnet Software, Liverpool, NY, 2005. [13] J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [14] G. L. Matthaei, L. Young, and E. M. T. Johnes, Microwave Filters, Impedance-Matching Network, and Coupling Structure. Norwood, MA: Artech House, 1980. [15] R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 1–10, Jan. 2003. [16] S. Amari, U. Rosenberg, and J. Bornemann, “Adaptive synthesis and design of resonator filters with source/load-multiresonator coupling,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 8, pp. 1969–1978, Aug. 2002. [17] R. Levy, “Direct synthesis of cascaded quadruplet (CQ) filters,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 12, pp. 2940–2945, Dec. 1995. [18] C. K. Liao and C. Y. Chang, “Design of microstrip quadruplet filters with source-load coupling,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 7, pp. 2302–2308, Jul. 2005. Chin-Hsuing Chen was born in Taiwan, R.O.C., on May 21, 1970. He received the B.S. degree in industrial education and technology from National Changhua University of Education, Changhua, Taiwan, R.O.C., in 1994, M.S. degree in engineering and system science from the National Tsing-Hua University, Hsinchu, Taiwan, R.O.C., in 2001, respectively, and is currently working toward Ph.D. degree in communication engineering at National Chiao-Tung University, Hsinchu, Taiwan, R.O.C. His research interests include the analysis and design of microwave and millimeter-wave circuits.

Ching-Ku Liao was born in Taiwan, R.O.C., on October 16, 1978. He received the B.S. degree in electrophysics and M.S. degree in communication engineering from National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 2001 and 2003, respectively, and is currently working toward the Ph.D. degree in communication engineering at National Chiao-Tung University. His research interests include the analysis and design of microwave and millimeter-wave circuits.

Chi-Yang Chang (M’95) was born in Taipei, Taiwan, R.O.C., on December 20, 1954. He received the B.S. degree in physics and M.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1977 and 1982, respectively, and the Ph.D. degree in electrical engineering from The University of Texas at Austin, in 1990. From 1990 to 1995, he was an Associate Researcher with the Chung-Shan Institute of Science and Technology (CSIST), where he was in charge of development of uniplanar circuits, ultra-broadband circuits, and millimeter-wave planar circuits. In 1995, he joined the faculty of the Department of Communication, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., as an Associate Professor, and became a Professor in 2002. His research interests include microwave and millimeter-wave passive and active circuit design, planar miniaturized filter design, and monolithic-microwave integrated-circuit (MMIC) design.

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Novel Patch-Via-Spiral Resonators for the Development of Miniaturized Bandpass Filters With Transmission Zeros Shih-Cheng Lin, Chi-Hsueh Wang, and Chun Hsiung Chen, Fellow, IEEE

Abstract—A novel patch-via-spiral resonator based on the dual-metal-plane configuration is proposed and examined. With the microstrip patch on the top plane serving as a capacitor and linking to the quasi-lumped spiral inductor on the bottom plane through a connecting via, the proposed dual-plane resonator structure located on the opposite sides of the single substrate may form a miniaturized one in the printed-circuit board fabrication. By suitably combining the proposed patch-via-spiral resonators, useful coupled-resonator pairs may be constructed to simultaneously provide electric and magnetic couplings. Based on these coupled-resonator pairs, a second-order bandpass filter with multiple transmission zeros is realized without requiring either the cross-coupled path or the source–load coupling. For design purpose, the equivalent-circuit model is also derived and verified. In this study, a fourth-order patch-via-spiral bandpass filter with both good passband selectivity and miniaturized size 5.08 mm (i.e., of only 22.14 mm 0 0 ) is implemented, where 0 denotes the guided wavelength of the 50- microstrip line at center frequency.

0 188

0 043



Index Terms—Coplanar waveguide (CPW), coupled-resonator bandpass filter, dual-metal-plane structure, microstrip, miniaturization, transmission zero.

I. INTRODUCTION

D

UAL-PLANE configuration has increasingly attracted attention in the field of microwave design, especially for the development of mobile communication. Since most mobile devices leave limited space for the placement of filters, it is of importance to miniaturize the required filter size. In the dual-plane approach, which makes good use of two metal planes on the opposite sides of the single substrate, the arrangement of components is no longer limited on the single plane and, thus, the design becomes flexible. In addition, the fabricated components will end up with the three-dimensional structure, thus it may reduce their occupied planar size. Except for the size reduction, transmission zeros are usually required to attenuate the stopband level and also to sharpen the passband response. Several topologies, which are capable of creating transmission zeros around the passband, were presented in the literature [1]–[8]. The topologies such as canonical form, cascaded quadruplet [1]–[5], extracted pole [6], and

Manuscript received July 24, 2006; revised September 19, 2006. This work was supported by the National Science Council of Taiwan under Grant NSC 95-2752-E-002-001-PAE, Grant NSC 95-2219-E-002-008, and Grant NSC 95-2221-E-002-196. The authors are with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.888579

source–load coupling [7], [8] have been successfully realized using the coplanar waveguide (CPW) or microstrip structure. The above-mentioned planar filters are all fabricated on one plane of the single substrate and, thus, may occupy larger space. Over the past few years, several research studies have been reported to implement the filter topologies in the dual-plane configuration so as to economize the occupied size of filters. In a general sense, the dual-plane configuration may be classified into two principal catalogs. One catalog takes advantage of metallization patterns among two dielectric substrates with a common ground. Quarter-wavelength resonators are allocated on the outer planes of two substrates and the nonadjacent cross coupling is realized through the slots on the common ground [9]. Furthermore, microstrip open-loop resonators that lie on outer sides of two substrates with a common ground in between are coupled by the apertures on the ground so as to provide both electric and magnetic couplings for specified filtering characteristics [10], [11]. The filters belonging to this catalog need to tightly combine two dielectric substrates together, thus they require a complicated fabrication process to achieve high precision. The other catalog utilizes two metal planes on the opposite sides of a single substrate. Suspended-stripline filters were presented in [12] by arranging the quasi-lumped resonators in either an antipodal or parallel way. The main drawback of using a suspended stripline is the requirement of critical housing technology. Double-surface CPW filters [13] were constructed by introducing an electrode pattern on a double-sided metallized substrate with a high dielectric constant. A dual-plane combline filter having plural attenuation poles was investigated in [14]. However, the input and output ports are connected to CPW lines by tapped wires, which are hard to control accurately, and the analysis method is complicated when applied to higher order filters. An ultra-wideband bandpass filter based on the microstrip-to-CPW transitions and CPW/microstrip shorted stubs connected to ground was introduced [15]. However, its transmission zeros on the upper stopband are generated through complicated resonant mechanisms, therefore, it is difficult to establish the guidelines for designing such a filter. Another ultra-wideband filter using the hybrid microstrip and CPW structure in a dual-plane configuration was also presented in [16]. The CPW bandpass filters using both loaded air-bridge enhanced capacitors and broadside-coupled transition structures for wideband spurious suppression was recently proposed in [17] by also making use of dual-plane configuration without introducing any transmission zeros around the passband. In this study, novel patch-via-spiral resonators fabricated in the dual-metal-plane configuration are proposed and utilized

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Fig. 1. Three-dimensional geometry of the proposed second-order filter using patch-via-spiral resonators in the dual-plane configuration.

to achieve both filter-size miniaturization and transmission zeros creation originated from a completely different coupling scheme. The microstrip patches of the coupled-resonator pair on the top plane provide the electric coupling, while the spiral inductors on the bottom plane offer the mutual magnetic coupling. The simultaneously existed electric and magnetic couplings in a coupled-resonator pair facilitate the design of direct-coupled filters with transmission zeros, but without introducing either cross-coupling between nonadjacent resonators or source/load coupling between input/output ports. Based on the single patch-via-spiral resonator, several types of coupled-resonator pairs are carefully investigated and characterized. These coupled-resonator pairs are then applied to design miniaturized filters with prescribed frequency responses. For instance, the second-order filter (Fig. 1) may be realized for an elliptic-like or Chebyshev response by using different types of coupled-resonator pairs. By appropriately integrating the different types of coupled-resonator pairs, one may implement a higher order filter with improved performance. Specifically, the fabricated fourth-order filter provides sharp selectivity and possesses a miniaturized size of only 22.14 mm 5.08 mm (i.e., ). The good performance of the proposed filter using patch-via-spiral resonators makes it a useful component for the wireless system. II. PATCH-VIA-SPIRAL RESONATORS IN DUAL-PLANE CONFIGURATION Shown in Fig. 2(a) is the three-dimensional geometry of the proposed patch-via-spiral resonator. The top- and bottom-plane layouts viewed from the bottom side are also depicted in Fig. 2(b). The newly proposed resonator is mainly composed of three elements, i.e., the microstrip patch on the top plane, the connecting via through the substrate, and the grounded spiral inductor on the bottom plane. The microstrip patch has . The via-hole is drilled rectangle shape with area by the 1-mm-diameter bit and filled with the conductive liquid silver to provide good connecting capability. The spiral on is covered by the patch the bottom-plane of length on the top plane. Note that the spiral is short circuited to the CPW ground. This novel structure may roughly be regarded as a combination of quasi-lumped capacitors and inductors, as

Fig. 2. Proposed patch-via-spiral resonator structure. (a) Three-dimensional geometry. (b) Top- and bottom-plane layouts. (c) Equivalent-circuit model. With the structure looking from the bottom and shown in (b), the gray region denotes the spiral inductor and ground on the bottom plane, and the dashed line represents the rectangle microstrip patch on the top plane.

shown in Fig. 2(c). Since the elements of the patch-via-spiral resonator are located on the opposite sides of single substrate, its occupied size may be miniaturized. The dimensions of major resonator parameters, such as the patch area, spiral linewidth, and spiral line length, to provide a given response, are basically affected by the substrate. To be specific, hereafter, all circuits , in this study will be fabricated on an FR4 board ( mm, ). The approximate lumped-element equivalent circuit of the indicates proposed resonator is presented in Fig. 2(c). Here, the portion of the spiral inductance in the vicinity of the CPW denotes the rest of the spiral inductance ground, while measured from the via-hole location. The portion of the patch overlapping the spiral is represented by the capacitance shunt with the inductance ; while the rim of patch forms the to the ground. Note that the spiral parallel-plate capacitance on the bottom-plane contributes towards the total inductance . Although the models of shorted resonators with of grounding vias [18] were proposed and the spiral inductors for filter application [19] were also reported, the proposed patch-via-spiral resonator has a complicated composite structure, which is hardly analyzed using closed-form formulas. The microstrip patch and spiral inductor are strongly coupled to each other, thereby becoming inseparable. The corresponding lumped-element circuit model may be extracted using the quasi-static method-of-moment simulator (Ansoft Q3D v6.0).

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Fig. 3. Resonant frequencies with respect to different sets of physical dimensions (in millimeters) for the single patch-via-spiral resonator presented in Fig. 2, where W and D are given in parentheses, respectively.

To characterize this structure and to get its corresponding frequency characteristics, the full-wave electromagnetic simulator (Ansoft Designer v2.0) is also used. The patch-via-spiral resonator is carefully designed and adjusted to provide the resonant frequency at 1.43 GHz. To this end, the physical dimensions of the resonator are given as mm, mm, mm, and mm. Therefore, the extracted lumped-element values are nH, pF, and pF, respectively. With the favor of a dual-plane configuration, the novel patch-via-spiral resonator is compact and has a size of ). only 5.08 mm 5.08 mm (i.e., Using the weakly capacitive-gap-coupled excitation and the extraction technique provided in [20], the unloaded quality associated with the proposed resonator is found factor about 60, a value comparable to the conventional microstrip distributed-element resonators fabricated on the same FR4 is also examined by substrate. The influence of the via on varying the via diameter from 0.15 to 7.62 mm and, even so, values are found nearly constant. the corresponding To characterize the proposed resonator, the fundamental resonant frequency is simulated, using different sets of physical dimensions of the resonator. The simulated transmission coefficients are illustrated in Fig. 3. Intuitively, the larger the area of the microstrip patch, the larger the loaded capacitance for the resonator and, thus, the lower the resonant frequency. III. COUPLED-RESONATOR PAIRS Two patch-via-spiral resonators may properly be arranged to form the required coupled-resonator pairs. Shown in Fig. 4 are two possible types of coupled-resonator pairs with the tappedline-fed structures included for realizing the second-order filters discussed below. Fig. 5(a) shows the conventional coupling and routing diagram for these coupled-resonator pairs (Fig. 4). The coupling between two resonators is simply modeled by the overall . However, two coupled paths procoupling coefficient and magnetic coupling viding electric coupling may be realized by the proposed coupled-resonator pairs in Fig. 4(a) and (b). Fig. 5(b) shows a decomposed coupling divided into and . By utilizing diagram with

Fig. 4. Top- and bottom-plane layouts of the proposed coupled-resonator pairs. = 1:93, g = 0:7, (a) Symmetric type1 [S1]. (b) Flipped type1 [F1]. (W d = 3:3, g = 0:38, w = 0:38, W = 5:08, L = 5:08, s = 0:53, = 1:04. Unit: millimeters). w

Fig. 5. (a) Overall and (b) decomposed coupling and routing diagrams of the coupled-resonator pair.

the decomposed coupling diagram exhibited in Fig. 5(b), a filter may be designed to introduce a pair of transmission zeros without adopting the conventional cross-coupling or source–load coupling. In this diagram, two out-of-phase couplings, which simultaneously exist between the two coupled resonators, are essential in achieving the specified filtering response, as mentioned in [12]. Once the out-of-phase condition is met, the direct-coupled filter with an elliptic-like response may be realized. In contrast, a Chebyshev response may alternatively be achieved by creating the in-phase couplings. Since the proposed resonator is composed of a microstrip patch on the top plane and a spiral inductor on the bottom plane, one can easily arrange a coupled-resonator pair to provide an electric coupling path along the top plane and another magnetic coupling path along the bottom plane. The principal difference between Fig. 4(a) and (b) is reflected by the spiral-wound direction. Among the symmetric-type1 coupled-resonator pair shown in Fig. 4(a) (hereafter simply denoted as the S1 pair), the spiral inductors of the coupled resonators are wound in the opposite directions. Conversely, the spiral inductors in the flipped-type1 coupled-resonator pair shown in Fig. 4(b) (simply denoted as F1 pair) are wound in the same direction. In order to design the filters using different types of coupled-resonator pairs, the overall coupling

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Fig. 8. Layouts of another two coupled-resonator pairs. (a) Symmetric-type2 [S2]. (b) Flipped-type2 [F2].

plane may also be expressed as Fig. 6. Simulated overall coupling coefficients for the S1 and F1 pairs in Fig. 4(a) and (b), respectively.

Fig. 7. Field distributions and current flows along the spiral inductors belonging to coupled-resonator pairs. (a) S1. (b) F1.

coefficients as functions of gap spacing for the two coupled-resonator pairs [see Fig. 4(a) and (b)] are illustrated in Fig. 6 according to the information of resonant-mode splitting [2]. One can easily observe that the overall coupling for the F1 pair is larger than that for the S1 pair. It implies that the electric and magnetic couplings tend to add in the F1 pair, while tending to cancel in the S1 pair. Physically, the signs of electric and magnetic couplings may be distinguished by an approach extended from Zhang et al. [21]. Fig. 7 illustrates the field distributions and current flows along the spiral inductors belonging to two distinct coupled-resonator pairs. In our proposed coupled-resonator pair, the overall coupling may be divided into two different parts, i.e., the coupling between the microstrip patches and that between the spiral inductors . between the microstrip patches on the top The coupling plane may roughly be written as

(1)

The coupling

between the spiral inductors on the bottom

(2)

and denotes the vector electric fields belonging Here, to the microstrip patches of resonators 1 and 2, while and are the vector magnetic fields resulting from the spirals. The volumes of the air, substrate, and whole infinity space are , , and , respectively. Note that subdenoted by scripts and denote the patch and spiral, while the and represent electric and magnetic, respectively. On the top plane, the electric coupling dominates the interaction between two patches, thus the term in (1) corresponding to magnetic . Moreover, fields may approximately be neglected the two spirals on the bottom plane of the two patch-via-spiral resonators are so distant that their magnetic fields are dominant . in evaluating ; equivalently it implies that For the S1 pair [see Fig. 4(a)] with the spirals wound in opposite directions, the dot product of electric fields will always be positive, as observed from Fig. 7(a), resulting in a negative . When the two spirals are wound in the value for opposite directions, the currents in the spiral windings will be in the opposite directions as well. This makes the dot product of and positive . Thus, the coumagnetic fields plings between patches and spirals will be out-of-phase. Conversely, for the F1 pair Fig. 4(b) with the two spirals wound in the same direction, the couplings between patches and spi. The in-phase couplings rals will be in-phase among the F1 pair brings about a larger overall coupling coefficient than that of the S1 pair, as depicted by the curves in Fig. 6. According to the above-mentioned argument based on the field expressions in (1) and (2), one may relate the coupled resonator structure exhibited in Fig. 4(a) to the decomposed coubetween pling diagram depicted in Fig. 5(b). The coupling , while the microstrip patches realizes the electric coupling coupling between spiral inductors achieves the required magnetic coupling . Furthermore, another two coupled-resonator pairs are also depicted in Fig. 8(a) and (b) to facilitate the design of the higher order filter. Fig. 8(a) exhibits the so-called symmetric-type2 (S2) coupled-resonator pair, which has the two spirals wound in the opposite directions. In contrast, the two spirals among the flipped-type2 (F2) shown in Fig. 8(b) are wound in the same direction. The same argument may be applied to verify the out-ofphase couplings among the S2 pair and the in-phase couplings among the F2 pair. The simulated overall coupling coefficients

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Fig. 9. Simulated overall coupling coefficients for the S2 and F2 pairs in Fig. 8(a) and (b), respectively.

TABLE I SIGNS OF THE ELECTRIC AND MAGNETIC COUPLINGS FOR DIFFERENT TYPES OF COUPLED-RESONATOR PAIRS

for both coupled pairs as functions of spacing are illustrated in Fig. 9. In summary, Table I lists the signs of the electric and magnetic couplings, which are concluded from (1) and (2) for the four coupled-resonator pairs exhibited in Figs. 4 and 8. IV. ANALYSIS OF SECOND-ORDER FILTERS

Fig. 10. Equivalent-circuit models for the second-order filter using the proposed patch-via-spiral coupled-resonator pairs exhibited in Fig. 4(a). (a) Complete model. (b) Even-mode model. (c) Odd-mode model.

The parallel resonances for the two modes occur as and . Therefore, the even- and odd-mode resonant frequencies are given by

A. Equivalent Circuit The equivalent circuit required to design the proposed filter and to determine the locations of two transmission zeros will be derived here. Fig. 10(a) shows the approximate lumped-element equivalent-circuit model for the proposed second-order filter using the patch-via-spiral coupled-resonator pairs exhibited in Fig. 4. The inductive coupling through the bottom plane between the two spirals of the coupled-resonator pair is denoted . Furthermore, the capacitive couby the mutual inductance pling between the two patches on the top plane of the coupled. resonator pair is represented by the mutual capacitance Due to the symmetry of this configuration, even- and odd-mode analysis may be adopted to discuss this filter. By placing a virtual open or virtual short in the center reference plane, the even- and odd-mode equivalent-circuit models are illustrated in Fig. 10(b) and (c). The even- and odd-mode input admittances may be formulated as

(3)

(4)

(5)

(6) In addition, the transmission zeros would be created when condition holds. The lower and higher transmissionzero frequencies and are given by (7) and (8), shown at the bottom of the following page. For the filter requiring out-of-phase couplings to achieve transmission zeros, a positive mutual inductance is essential, and this condition may be realized through the value brings symmetric-type coupled pairs. The positive about two real solutions for (7) and (8), which are the two transmission zeros associated with the filter. , which ensures the On the contrary, condition in-phase couplings between coupled resonators, may be achieved through the flipped-type pairs and, thus, brings about a filter response without transmission zeros. Note that the value results in all imaginary solutions when negative

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Fig. 11. Circuit-model simulated results of the equivalent circuit in Fig. 10(a) for different C values (in picofarads) to adjust the transmission-zero frequencies (L = 3:374 nH, L = 9:1 nH, C = 0:524 pF, L = 0:727 nH, = 0:74 pF). C

calculating the transmission-zero frequencies through (7) and (8), implying the disappearance of transmission zeros. It is worth mentioning that the frequencies of these two transmission zeros may simply be altered by varying the values of or . A demonstrated example is exhibited in Fig. 11, with element values given in the caption, by varying the value , but keeping other elements unaltered. Apparently, the of is, the closer the transmission zeros approach the larger the passband edges. However, the bandwidth is almost not affected . due to the minor value of B. Design Procedure Based on the even- and odd-mode resonant frequencies de, rived from (5) and (6), the overall coupling coefficient which is the combination of electric and magnetic couplings between the two coupled resonators, can be calculated by the formula [22] (9) The required value is simply proportional to the predetermined filter bandwidth. In addition, one may also identify the signs of overall couplings by using (9). Note that the relative locations of even- and odd-mode resonances ( and ) are changed when the sign of mutual inis altered. By applying these two resonant freductance and quencies into (9) under two different conditions ( ), one may infer that will be positive as , will be negative as . Therefore, the sign of while the overall coupling of the coupled-resonator pair is principally

Fig. 12. External quality factor Q of the utilized tapped-line-fed structure (W = 1:93 mm, W = D = with respect to the tapped position T 5:08 mm, g = 0:38 mm, w = 0:38 mm).

between spidominated by that of the mutual inductance rals. Inspecting the full-wave simulated characteristics of the four coupled-resonator pairs exhibited in Figs. 4 and 8 and oband , one may lead to the taining the relative locations of conclusion that the symmetric-type coupled-resonator pairs (S1 value , while and S2) always give the positive the flipped-type coupled-resonator pairs bring about the negavalue . tive Consequently, the bandwidth and transmission-zero frequencies may be obtained using (5)–(8) derived thus far in this secmay also be evaltion. Note that the external quality factor uated [23] from the single tapped-line-fed patch-via-spiral resonator by using the full-wave simulator. The tapped-line CPW feed is used at input and output ports for this proposed filter, as shown in Fig. 4. Note that the 50- CPW possesses the center-conductor width of 5.08 mm, for a given slot width of 0.38 mm, which is too wide to serve as the tapped-feed line for the miniaturized patch-via-spiral resonator. To avoid this difficulty, a via-connected microstrip-to-CPW transition [24] is designed to connect the 50- microstrip line to the tapped-fed CPW with center-conductor width of 0.38 mm. The adoption of this transition would not affect the filter response. The evaluof the proposed resonator is plotted with respect to the ated in Fig. 12 in order to facilitate the practapped position tical design procedure. ’s, external Until now, the required coupling coefficients quality factor , and transmission-zero frequencies can all be obtained from either the full-wave simulator or the equivalent-

(7) (8)

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circuit model. The design procedure for the proposed filter may be summarized as follows. 1) Specify the filter passband response, center frequency, fractional-bandwidth, and transmission-zero frequencies. ’s and Accordingly, the related coupling coefficients may be decided. external quality factor 2) Obtain the corresponding element values , , , , , and of the equivalent-circuit model in accordance with the filter specification. Note that the center frequency , and for the is principally determined by , , single resonator. The mutual inductance and capacitance and collaterally control the fractional bandwidth, while the transmission-zero frequencies are dominated by . 3) Adjust the dimensions of single patch-via-spiral resonator to provide the required lumped-element values. In addition, the utilized coupled-resonator pairs should provide the reand to give the specified transmisquired mutual ’s. sion-zero frequencies and overall coupling 4) Finally, select appropriate tapped-line-fed position (as depicted in Fig. 12) so as to satisfy the required input and for the proposed filter. output quality factors V. IMPLEMENTATION OF SECOND-ORDER FILTERS To demonstrate the feasibility of realizing the filter with the coupling diagram shown in Fig. 5(b), a second-order filter using S1 coupled-resonator pair Fig. 4(a) is designed with the center frequency at 1.4 GHz, a 3 dB-fractional bandwidth of 11%. and ) are assigned The two required transmission zeros ( at 1.05 and 2.01 GHz, respectively. The corresponding overall and the external quality coupling coefficient is factors associated with the input/output resonators are . To satisfy this specification, the coupling gap is chosen as 0.53 mm and as 1.04 mm. For the dimensions given in Fig. 4(a), the corresponding element values of nH, nH, the equivalent-circuit model are pF, nH, pF, and pF, respectively. The full-wave simulated frequency response of the S1 filter in Fig. 4(a) with dielectric loss excluded is exhibited in Fig. 13 together with the equivalent-circuit simulated result for comparison. Very good agreement is observed between the fullwave and equivalent-circuit simulated results, implying that the equivalent circuit is useful in predicting the filter response. The narrowband measured result of the fabricated filter is illustrated in Fig. 14. The measured center frequency is at 1.408 GHz, the measured 3-dB fractional bandwidth is approximately 10.9%, and the minimum insertion loss is 2.25 dB. Two transmission zeros appear at 1.047 and 2.083 GHz around the passband, as predicted. As a result, a filter with the elliptic-like response using the proposed patch-via-spiral resonators is realized. The locations of the two transmission zeros appearing in the S1 filter response may be adjusted by changing the mutual capacitance between the microstrip patches on the top plane. In this connection, one can simply bring the two microstrip patches closer to enhance the mutual capacitive coupling between them. As indicated in Fig. 11, the nearer the two patches

Fig. 13. Comparison of the results, for the proposed second-order filter in Fig. 4(a), based on the equivalent-circuit model [see Fig. 10(a)] and full-wave simulation with dielectric loss excluded. (a) Amplitude. (b) Phase.

Fig. 14. Measured frequency responses of the second-order filters using S1 and F1 coupled-resonator pairs in Fig. 4.

Fig. 15. Coupling and routing diagram of the proposed fourth-order filter.

approach, the closer the two transmission zeros move toward the passband.

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Fig. 16. Top- and bottom-plane layouts of the fourth-order S1–S2–S1 filter using patch-via-spiral resonators on 1-mm-thick FR4 substrate. (W W = 5:08, L = 5:08, w = 0:38, g = 0:38, s = 0:53, w = 1:04, s = 0:76, w = 1:27, d = 1:6. Unit: millimeters).

= 1:93,

With reference to the filter in Fig. 4(a), the resonator on the right-hand side is flipped horizontally to form the F1 filter in Fig. 4(b) with all other structure dimensions kept almost unaltered. As illustrated in the curves of the coupling coefficients shown Fig. 6, the coupling between the F1 coupled-resonator when compair will increase under the same coupling gap pared with that of the S1 pair. The increment in coupling may be verified from the bandwidth expansion of the response for the F1 filter. The measured frequency response of this F1 filter is also presented in Fig. 14. The measured center frequency is at 1.403 GHz, the measured 3-dB fractional bandwidth is approximately 12.9% [apparently wider than that of the S1 filter in Fig. 4(a)], and the minimum insertion loss is 2.26 dB. Most important of all, the transmission zeros completely disappear, thus a filter response without transmission zeros is achieved. VI. FOURTH-ORDER FILTERS The filter performance may be improved by cascading several path-via-spiral resonators to form a higher order filter. To this end, the generalized coupling and routing diagram of the fourth-order filter is also extended from that of the second-order filter, as shown in Fig. 15. There are three pairs of coupled resonators that may be arbitrarily determined in Fig. 15 for different filter responses. As long as one or more coupled-resonator pairs provide out-of-phase electric and magnetic couplings, the filters adopted these coupled-resonator pairs would possess the transmission zeros, as expected. To realize a filter with transmission zeros, any one of the S1 and S2 coupled-resonator pairs is required to develop the filter. For convenience in the tapped feed, only the type1 pair is used as the first pair of coupled resonators. Consequently, the type2 pair is inserted as the middle coupled resonators. According to the permutation and combination, six sequences of the fourth-order filter are possible, which are S1–S2–S1, S1–S2–F1, S1–F2–S1, S1–F2–F1, F1–S2–F1, and F1–F2–F1, respectively. Among these six possible sequences, only the filter of the specific sequence F1–F2–F1 possesses no transmission zeros near the passband since no symmetric-type coupled-resonator pair is utilized. An experimental fourth-order filter of the sequence S1–S2–S1 based on the proposed patch-via-spiral resonators is designed and fabricated on the 1-mm-thick FR4 substrate. The three adopted coupled-resonator pairs are all of the symmetric type. The filter is preliminarily designed at a center frequency of 1.43 GHz with 10% 3-dB fractional bandwidth. The corresponding overall coupling coefficients and external

Fig. 17. (a) Narrowband and (b) wideband frequency responses (measured and simulated) of the fourth-order S1–S2–S1 filter exhibited in Fig. 16.

quality factors are

, , and . The required resonator spacings may be obtained according to the overall coupling coefficients illustrated in Figs. 6 and 9. Moreover, the tapped position may be determined from Fig. 12. The via-connected microstrip-to-CPW transition is also utilized here for the tapped feed. Fig. 16 shows the top- and bottom-plane layouts of this proposed filter. The measured and simulated results are both illustrated in Fig. 17. The measured center frequency is at 1.424 GHz, the measured 3-dB fractional bandwidth is approximately 8.99%, and the minimum insertion loss is 5.811 dB. The two transmission and are observed around 1.2 and 2.01 GHz, zeros respectively. The first spurious passband occurs at . The insertion loss is mainly contributed 5.32 GHz by the high dielectric loss of the adopted FR4 substrate. Note that the four reflection zeros of an ideal fourth-order elliptic

LIN et al.: NOVEL PATCH-VIA-SPIRAL RESONATORS FOR DEVELOPMENT OF MINIATURIZED BANDPASS FILTERS WITH TRANSMISSION ZEROS

Fig. 18. Top- and bottom-plane photographs of the fabricated fourth-order S1–S2–S1 filter using patch-via-spiral resonators.

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Another filter of the particular sequence F1–F2–F1 is also implemented to demonstrate the design of the Chebyshev response without transmission zeros. It is also designed at 1.43 GHz with 10% 3-dB factional bandwidth. Therefore, its design parameters are the same with those of the previous S1–S2–S1 filter. Its measured and simulated responses are depicted in Fig. 19. The measured center frequency is at 1.403 GHz, the measured 3-dB fractional bandwidth is approximately 10.7%, and the minimum insertion loss is 4.652 dB. Since no coupled-resonator pair of symmetric type is adopted, the two transmission zeros disappear, as predicted. Note that the four reflection zeros of ideal fourth-order Chebyshev response degenerate into two as well due to the dielectric loss. It is also worth mentioning that the two measured 3-dB fractional bandwidths of the S1–S2–S1 and F1–F2–F1 filters are 8.99% and 10.7%, respectively, varying from the designate fractional bandwidth (i.e., 10%) due to the deviation and nonuniformity in the dielectric constant. In order to examine the effects of radiation losses, the simulated results with dielectric loss excluded are also carried out for the two fourth-order filters. It is found that the simulated insertion losses are significantly improved in the lossless cases, which are both nearly 0 dB, implying that the radiation losses may be neglected in our proposed patch-via-spiral filters. The poor performances of measured insertion losses are mainly atof the emtributed to the high dielectric loss ployed FR4 substrate. VII. CONCLUSIONS

Fig. 19. (a) Narrowband and (b) wideband frequency responses (measured and simulated) of the fourth-order F1–F2–F1 filter using patch-via-spiral resonators with Chebyshev response.

In this study, novel miniaturized filters using patch-via-spiral resonators based on different coupled-resonator pairs have been proposed and constructed. Originated from the proposed coupled-resonator pairs, novel filters without either cross-coupling or source–load coupling have been analyzed and implemented in the dual-plane configuration. With adjustable electric and magnetic couplings simultaneously existing between coupled-resonator pair, the transmission-zero frequencies may easily be shifted or even be removed by changing the couplings or even by removing. The proposed filters were realized in a dual-plane configuration, hence possessing a miniaturized occupied size. Specifically, the fourth-order filter using patch-via-spiral resonators has been fabricated with either an elliptic-like or a Chebyshev response and possesses a very . Due to the merits of the compact size of proposed filters, they are good candidates for wireless communication for which miniaturized size and sharp selectivity are required. REFERENCES

response degenerate into two due to the dielectric loss. The topand bottom-plane photographs of the fabricated filter are presented in Fig. 18. The fabricated filter possesses a compact size ), which is of 22.14 mm 5.08 mm (i.e., much smaller than the conventional uniplanar or planar filters. The miniaturized size of this filter reveals the advantage of using the proposed patch-via-spiral resonators for filter design.

[1] J. Zhou, M. J. Lancaster, and F. Huang, “Coplanar quarter-wavelength quasi-elliptic filters without bond-wire bridges,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1150–1156, Apr. 2004. [2] J.-S. Hong and M. J. Lancaster, “Couplings of microstrip square openloop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 11, pp. 2099–2109, Nov. 1996. [3] ——, “Design of highly selective microstrip bandpass filters with a single pair of attenuation poles at finite frequencies,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 7, pp. 1098–1107, Jul. 2000.

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[4] S.-Y. Lee and C.-M. Tsai, “New cross-coupled filter design using improved hairpin resonators,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2482–2490, Dec. 2000. [5] J. S. Hong and M. J. Lancaster, Microstrip Filter for RF/Microwave Applications. New York: Wiley, 2001, pp. 56–63. [6] K. S. K. Yeo, M. J. Lancaster, and J.-S. Hong, “The design of microstrip six-pole quasi-elliptic filter with linear phase response using extractedpole technique,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 321–327, Feb. 2001. [7] C.-K. Liao and C. Y. Chang, “Design of microstrip quadruplet filters with source–load coupling,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 7, pp. 2302–2308, Jul. 2005. [8] P.-H. Deng, Y.-S. Lin, C.-H. Wang, and C. H. Chen, “Compact microstrip bandpass filters with good selectivity and stopband rejection,” IEEE Trans. Microw. Theory Tech., vol. 54, no. .2, pp. 533–539, Feb. 2006. [9] S.-J. Yao, R. R. Bonetti, and A. E. Williams, “Generalized dual-plane multicoupled line filters,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 12, pp. 2182–2189, Dec. 1993. [10] J.-S. Hong and M. J. Lancaster, “Aperture-coupled microstrip openloop resonators and their applications to the design of novel microstrip bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1848–1855, Sep. 1993. [11] A. Djaiz and T. A. Denidni, “A new two-layer bandpass filter using stepped impedance hairpin resonators for wireless applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 1487–1490. [12] W. Menzel and A. Balalem, “Quasi-lumped suspended stripline filters and diplexers,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 10, pp. 3230–3237, Oct. 2005. [13] T. Tsujiguchi, H. Matsumoto, and T. Nishikawa, “A miniaturized double-surface CPW bandpass filter improved spurious responses,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 5, pp. 879–885, May 2001. [14] T. Kitamura, Y. Horii, M. Geshiro, and S. Sawa, “A dual-plane combline filter having plural attenuation poles,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1216–1219, Apr. 2002. [15] T.-N. Kuo, S.-C. Lin, and C. H. Chen, “Compact ultra-wideband bandpass filters using composite microstrip–coplanar waveguide structure,” IEEE Trans. Microw. Theory Tech., to be published. [16] H. Wang, L. Zhu, and W. Menzel, “Ultra-wideband bandpass filter with hybrid microstrip/CPW structure,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 12, pp. 844–846, Dec. 2005. [17] S.-C. Lin, T.-N. Kuo, Y.-S. Lin, and C. H. Chen, “Novel coplanarwaveguide bandpass filters using loaded air-bridge enhanced capacitors and broadside-coupled transition structures for wideband spurious suppression,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, pp. 3359–3369, Aug. 2006. [18] G. A. Kouzaev, M. J. Deen, N. K. Nikolova, and A. H. Rahal, “Cavity models of planar components grounded by via-holes and their experimental verification,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1033–1042, Mar. 2006. [19] G.-A. Lee, M. Megahed, and F. D. Flaviis, “Design of multilayer spiral inductor resonator filter and diplexer for system-in-a-package,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 527–530. [20] C. A. Tavernier, R. M. Henderson, and J. Papapolymerou, “A reducedsize silicon micromachined high- resonator at 5.7 GHz,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2305–2314, Oct. 2002. [21] G. Zhang, F. Huang, and M. J. Lancaster, “Superconducting spiral filters with quasi-elliptic characteristic for radio astronomy,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 947–951, Mar. 2005.

Q

[22] K. A. Zaki and C. Chen, “Coupling of non-axially symmetric hybrid modes in dielectric resonators,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 12, pp. 1136–1142, Dec. 1987. [23] J. S. Hong and M. J. Lancaster, Microstrip Filter for RF/Microwave Applications. New York: Wiley, 2001, pp. 258–271. [24] R. N. Simons, Coplanar Waveguide Circuits, Components, and Systems. New York: Wiley, 2001, pp. 289–297. Shih-Cheng Lin was born in Taitung, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Sun Yet-Sen University, Kaohsiung, Taiwan, R.O.C., in 2003, and is currently working toward the Ph.D. degree in communication engineering at National Taiwan University, Taipei, Taiwan, R.O.C. His research interests include the design and analysis of microwave filter circuits and passive components.

Chi-Hsueh Wang was born in Kaohsiung, Taiwan, R.O.C., in 1976. He received the B.S. degree in electrical engineering from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1997, and the Ph.D. degree from National Taiwan University, Taipei, Taiwan, R.O.C. in 2003. He is currently a Post-Doctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University. His research interests include the design and analysis of microwave and millimeter-wave circuits and computational electromagnetics.

Chun Hsiung Chen (SM’88–F’96) was born in Taipei, Taiwan, R.O.C., on March 7, 1937. He received the B.S.E.E. and Ph.D. degrees in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1960 and 1972, respectively, and the M.S.E.E. degree from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1962. In 1963, he joined the Faculty of the Department of Electrical Engineering, National Taiwan University, where he is currently a Professor. From August 1982 to July 1985, he was Chairman of the Department of Electrical Engineering with National Taiwan University. From August 1992 to July 1996, he was the Director of the University Computer Center, National Taiwan University. In 1974, he was a Visiting Scholar with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley. From August 1986 to July 1987, he was a Visiting Professor with the Department of Electrical Engineering, University of Houston, Houston, TX. In 1989, 1990, and 1994, he visited the Microwave Department, Technical University of Munich, Munich, Germany, the Laboratoire d’Optique Electromagnetique, Faculte des Sciences et Techniques de Saint-Jerome, Universite d’Aix-Marseille III, Marseille, France, and the Department of Electrical Engineering, Michigan State University, East Lansing, respectively. His areas of interest include microwave circuits and computational electromagnetics.

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Microstrip Realization of Generalized Chebyshev Filters With Box-Like Coupling Schemes Ching-Ku Liao, Pei-Ling Chi, and Chi-Yang Chang, Member, IEEE

Abstract—This paper presents generalized Chebyshev microstrip filters with box-like coupling schemes. The box-like coupling schemes taken in this paper include a doublet, extended doublet, and fourth-order box section. The box-like portion of the coupling schemes is implemented by an E-shaped resonator. Synthesis and realization procedures are described in detail. The example filters show an excellent match to the theoretical responses. Index Terms—Bandpass filters, design, elliptic filters, resonator filters, transmission zero.

I. INTRODUCTION HE microstrip filters with a generalized Chebyshev response attract considerable attention due to its light weight, easy fabrication, and ability to generate finite transmission zeros for sharp skirts. In the literature, most of them are based on cross-coupled schemes such as a cascade trisection and cascade quadruplet. Some representative examples of cross-coupled microstrip filters are available in [1]. Recently, with the progress of the synthesis technique, new coupling schemes such as the “doublet,” “extended doublet,” and “box section” are introduced [2]–[4]. As shown in Fig. 1, these coupling schemes have a box-like center portion so we refer to them in this paper as box-like coupling schemes. These coupling schemes impact the filter design since they not only provide new design possibilities, but also exhibit some unique and attractive properties. They differ from the conventional cascade trisection and cascade quadruplet mainly on two aspects. First, there are two main paths for the signal from source to load, while there is only one main path in the case of cascade trisection and cascade quadruplet. Second, the configuration of the doublet and box section exhibit the zero-shifting property, which makes it possible to shift the transmission zero from one side of the passband to the other side simply by changing the resonant frequencies of the resonator while keeping other coupling coefficients unchanged. The zero-shifting property implies that the similar physical layout can implement a filter with a transmission zero at the lower stopband or at the upper stopband,

T

Manuscript received July 21, 2006; revised October 4, 2006. This work was supported in part by the National Science Council, R.O.C., under Grant NSC 95-2752-E-009-003-PAE and Grant NSC 95-2221-E-009-042-MY3. C.-K. Liao and C.-Y. Chang are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]). P.-L. Chi is with the Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA 90024 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/TMTT.2006.888580

Fig. 1. Basic box-like coupling schemes for generalized Chebyshev-response filters discussed in this paper. (a) Doublet. (b) Extended doublet. (c) Box section. (The gray area is realized by the proposed E-shaped resonator.)

which is not feasible on the conventional trisection configuration. Besides, the third-order extended-doublet configuration, as shown in Fig. 1(b), exhibits one pair of finite transmission zeros as that of a cross-coupled quadruplet filter [5]. Pairs of finite transmission zeros can be used to improve the selectivity of the filter or flatten the in-band group delay. However, to the authors’ knowledge, only a few studies in the literature are focused on realization of the coupling schemes shown in Fig. 1 with a microstrip line [6], [7]. An important property of the schemes in Fig. 1 is that one of the coupling coefficients on the two main paths must be negative while others are positive. The simplest way to obtain the required negative sign is to use higher order resonance [3], [7]. Unfortunately, higher order resonance leads to a spurious resonance in the lower stopband. Instead of using higher order resonance, loop resonators are arranged carefully to satisfy the required sign of coupling coefficients for the box-section configuration [5]. However, a similar method cannot apply to the doublet or extended doublet. To overcome these difficulties, an E-shaped resonator, as shown in Fig. 2(a), is proposed in this paper to implement the required coupling signs. The E-shaped resonator can achieve the required magnitude and sign of the coupling schemes shown in Fig. 1. As shown in Fig. 2(a), the E-shaped resonator comprises a hairpin resonator and an open stub on its center plane. This symmetric structure can support two modes, i.e., an even mode and odd mode. Thus, the source and load are coupled to both modes of the E-shaped resonator. That is, even though only one physical path exists between source and load, there are two electrical paths between them. Consequently, the layout in Fig. 2(a) can be

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There are some interesting properties of the doublet filter in Fig. 2(a). First, since the E-shaped resonator exhibits symand metry, the relationship holds. Second, is always true for this structure since the coupling strength between the odd mode and external feeding network is always larger than that of the even mode. To get more insight of how to control a transmission zero of a doublet filter in this configuration, an explicit expression relating the coupling elements and the transmission zero is provided in a low-pass prototype as follows: Fig. 2. Doublet filter. (a) Proposed layout (gray area indicate the E-shaped resonator). (b) Corresponding coupling scheme.

modeled by the coupling scheme, a doublet, shown in Fig. 2(b). The doublet filter illustrates how a E-shaped resonator directly couples to external feeding network. Furthermore, based on the proposed E-shaped resonator, filters with an extended-doublet and a box-section configuration can be realized as well. The E-shaped resonator can use either its even mode or odd mode to couple an extra resonator. Thus, the extended-doublet configuration in Fig. 1(b) is achievable. Besides, the E-shaped resonator can couple to external resonators with two of its modes simultaneously and forms the box-section configuration in Fig. 1(c). The feasibility of realization of the basic coupling schemes in Fig. 1 with the proposed E-shaped resonator makes it possible to realize a class of coupled microstrip filters in a unified approach.

(2) Note that the mapping between normalized frequency and actual frequency is , where and are center frequency and bandwidth of a filter, respectively. Based on the (2), observations are summarized as follows. 1) The transmission zero is always located at finite frequency . In other words, the structure exhibits since finite transmission zero inherently. 2) The transmission zero can be moved from the upper stopband to lower stopband, or vice versa, by changing the sign and simultaneously. This property makes it of possible to generate upper stopband or lower stopband finite transmission zero with similar structure. and , would be greater than zero. In 3) If and can be related to a more explicit expression, and even mode the resonant frequencies of odd mode , respectively by the following equations:

II. CIRCUIT MODELING (3)

A. Filters in the Doublet Configuration The E-shaped resonator filter in Fig. 2(a) was originally reported in [8]. In [8], the E-shaped resonator was not modeled as a two-mode resonator. Instead, the circuit was modeled as two quarter-wavelength resonators with a tapped open stub in the center plane. The open stub is considered as a K-inverter between two quarter-wavelength resonators to control the coupling strength, and as a quarter-wave open stub to generate a transmission zero at the desired frequency. However, the filter cannot be designed with a prescribed quasi-elliptic response since there is no suitable prototype corresponding to the circuit model in [8]. In this paper, a doublet, as shown in Fig. 2(b), is used to model the circuit in Fig. 2(a). In Fig. 2(b), resonator 1 represents the odd-mode resonance, where the center plane of the E-shaped resonator is an electric wall ( -plane). On the other hand, resonator 2 represents even-mode resonance, where the center plane of the E-shaped resonator is a magnetic wall ( -plane). With the notation shown in Fig. 2(b), the correcan be written down as [3] sponding coupling matrix

(1)

(4) where and are the center frequency and bandwidth and , of a filter, respectively; i.e., if the transmission zero would be on the upper stopband. and , would be smaller than zero; 4) If and , the transmission zero i.e., if would be on the lower stopband. To get the related electrical parameters indicated in Fig. 2(a), one can take the following procedures. First, synthesize a coucorresponding to the prescribed response by pling matrix methods provided in [7]. Then consider parameters concerning the odd mode only by removing the open stub on the center plane. Once the open stub is removed, the circuit becomes a first-order hairpin filter. The first-order hairpin filter can be synthesized by the conventional method [8] with the center frequency set to be the resonant frequency of odd mode, which . At this can be expressed as , , and and obtain step, one can specify the values of and by an analytical method [8]. Second, the values of put the open stub back. The two parameters of the open stub, and , can be adjusted to achieve the desired resoi.e., nant frequency and the needed external coupling strength of the

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Fig. 4. Layout of extended-doublet filter and its corresponding coupling scheme. The design is for flat group-delay response.

Fig. 3. Responses generated from the coupling matrix and from electrical network shown in Fig. 2(a) with synthesis parameters.

even mode. Here, the resonant frequency of the even mode is . To illustrate the procedure, an example is taken of a secondorder generalized Chebyshev filter with a passband return loss of 20 dB and a single transmission zero at a normalized frequency . The corresponding coupling coefficients are , , , and . GHz and fractional For filter with center frequency bandwidth , the ideal response is depicted in Fig. 3 could first as solid lines. After getting the coupling matrix, , , and be specified. Here, we set and obtain and for a uniform impedance resonator with characteristic impedance at frequency GHz. Next, put the open and by the optimization stub back and adjust the values of method to let the response of the circuit match with the ideal response calculated from the matrix. The optimized values of and are 62 and 86.8 , respectively, at frequency GHz. According to the obtained electrical parameters in Fig. 2(a), the corresponding response is shown in Fig. 3 as circled lines. The frequency response contributed only by the odd mode is also depicted in Fig. 3 as dashed lines to let us understand the procedures clearer. B. Extended-Doublet Filters Based on the doublet filters developed in Section II-A, the emphasis here is put on how to extend the design to extendeddoublet filters [4]. There are two possible arrangements suitable to form extended-doublet filters. One possible arrangement is indicated in Fig. 4, where the extended doublet filter consists of a doublet filter plus a grown resonator. The grown resonator is a half-wavelength resonator with both ends open. In this case, the grown resonator would mainly couple to the odd mode of the E-shaped resonator. For the even mode of the E-shaped resonator, it acts as a nonresonant element, which slightly perturbs the resonant frequency of the even mode. Another possible design is shown in Fig. 5 where both ends of the grown resonator are shorted to ground. In this case, the grown resonator mainly couples to the even mode of the E-shaped resonator and acts as a nonresonant element to the odd mode of the E-shaped resonator.

Fig. 5. Layout of extended-doublet filter and corresponding coupling scheme. The design is for skirt selectivity response.

To clarify the coupling relationship between each resonator, the coupling routes are accompanied with layouts in Figs. 4 and 5. The extended-doublet filter has a pair of finite transmission zeros [4]. For the design in Fig. 4, the pair of transmission zeros is on the imaginary-frequency axis. On the other hand, to generate a pair of real-frequency transmission zeros, the design in Fig. 5 must be adopted. The difference between the two designs can be understood from the governing equation of finite transmission zeros. Since the proposed extended doublet filters and are symmetric structures, the relations always hold. Thus, the governing equation of finite transmission zero can be expressed as (5) As discussed in the design of the doublet, the coupling coefficient of source to odd mode is stronger than that of source to even mode. Thus, for the design in Fig. 4, , . On the contrary, for the design in Fig. 5, which leads to , which results in . In conclusion, the design in Fig. 4 can be used to generate delay-flattening transmission zeros, while the design in Fig. 5 can be used to generate a pair of attenuation poles. To illustrate the procedure of design, a generalized Chebyshev filter with passband return loss of 20 dB and a pair of transis taken as an example. The design of mission zeros at an extended doublet starts from the synthesis of the coupling matrix, which can be done using the technique in [9]. The synthesized coupling matrix is shown in Fig. 6(a). Using the inforand , one can construct the doublet by the mation of and in method provided in Section II-A. Excluding the coupling matrix, one can calculate the response contributed

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Fig. 6. Extended-doublet filter with in-band return loss RL = 20 dB, normalized transmission zeros at = 2. (a) Its coupling matrix. (b) Responses of extended doublet filter and responses contributed by doublet only.

6

from the doublet only. For instance, if the center frequency and fractional bandwidth of the designed filter are 2.4 GHz and 5%, respectively, the responses of the doublet are shown as dotted lines in Fig. 6(b). After getting the initial design of the doublet, in this case, the layout add the grown resonator. Since in Fig. 5 must be adopted. Ideally, the response of the extended doublet would be the solid lines shown in Fig. 6. The physical implementation of this design will be presented in Section III to confirm the validity. C. Box-Section Filters The fourth-order filter in the “box-section” configuration was first proposed in [2] and realized by coaxial resonators. With the zero-shifting property, it is possible to use the similar filter structure to realize the finite transmission zero either on the upper or lower stopband. The box-section filter is suitable for the complementary filters of a transmit/receive duplexer [7] since it has an asymmetric response with high selectivity on one side of the passband. The microstrip box-section filter was first reported in [6] with open square loop resonators. Since the box-section cou, , pling diagram is symmetric where , and should be held in the coupling route shown in Fig. 7(a). Therefore, it is preferable to layout the filter symmetrically because a symmetrical-layout filter can inherently obtain symmetrical coupling coefficients. The asymmetrical layout causes the filters in [6] to be difficult to keep the coupling coefficients symmetric. Another microstrip box-section filter was proposed in [7]. Although the layout of the filters in [7] is symmetric, it suffers from a spurious response in the filter’s lower stopband due to one of the filter’s resonators being a higher order mode resonator. In this paper, the layout depicted in Fig. 7(b) solves the problems mentioned. The E-shaped resonator is symmetric and is free from lower stopband spurious

Fig. 7. Box-section filter. (a) Filter’s coupling scheme. (b) Proposed layout.

resonances. Due to the symmetry, only half of the electrical parameters are shown in Fig. 7(b). As explained in the doublet filter, the circuit layout in Fig. 7(b) satisfies the required sign of couplings. To illustrate how to obtain the corresponding electrical parameters in Fig. 7(b) from a prescribed response, examples are taken as follows. The first example is a fourth-order generalized Chebyshev filter with a passband return loss of 20 dB, , which gives a and a single transmission zero at lobe level of 48 dB on the lower side of the passband. The is shown in Fig. 8(a). After corresponding coupling matrix the low-pass-to-bandpass transformation, the ideal bandpass response of this filter with a center frequency of 2.4 GHz and fractional bandwidth of 5% is shown in Fig. 8(b). The design procedures are described as follows. First, remove the open stub in the E-shaped resonator in Fig. 7(b), which is equivalent to discarding the even mode [resonator 3 in Fig. 7(a)] of the E-shaped resonator. After removing the open stub, the circuit becomes a third-order hairpin-like filter. The coupling of this hairpin-like filter is identical to the coupling matrix in Fig. 8(a), except and are zero. The ideal matrix response of this hairpin-like filter can be calculated from the matrix, as denoted by circled lines in Fig. 8(b). To get the electrical parameters associated with the asynchronously tuned third-order hairpin-like filter, a synchronous tuned third-order hairpin filter provides the initial design and is synthesized first. The synchronous tuned hairpin filter has the coupling matrix , which is identical to , except . When synthe, sizing the synchronously tuned hairpin filter, we set , , and at , and the characteristic impedance of each resonator to 50 . With these settings, the electrical parameters of the synchronously tuned hairpin filter are calculated and shown in Table I(a), which provides the initial values for the asynchronous-tuned hairpin-like filter. An optimization routine is then involved. The goal of the

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Fig. 8. Fourth-order box-section filter. (a) Its coupling matrix. (b) Responses of the box-section filter and ideal responses of the asynchronous tuned third-order hairpin-like filter calculated by M matrix.

TABLE I ELECTRICAL PARAMETERS CORRESPONDING TO BOX-SECTION FILTERS SHOWN = 90 , # = 60 , Z = 50 , AND Z = 50 . IN FIG. 7(b). HERE, # ALL OF THE ELECTRICAL LENGTHS CORRESPOND TO THE CENTER FREQUENCY OF THE FILTER. DESIGN 1: IN-BAND RETURN LOSS RL = 20 dB, = 2:57, AND FBW = 5% DESIGN 2: IN-BAND RETURN LOSS RL = 20 dB,

= 2:57, AND FBW = 5%

0

optimization routine is to find a set of electrical parameters, which can make the response match with the response of the ideal asynchronously tuned hairpin-like filter calculated from matrix. The optimized parameters are shown in Table I the for comparison. Note that the optimized values of associated parameters are nearly identical to the initial values; therefore, the optimization routine can converge within a few times. Finally, put the open stub back and optimize the parameters and to make the response match with the response of the desired box-section filter’s response, as denoted by solid lines in and are given in Table I Fig. 8(b). The optimized values of as well. Instead of a low-pass prototype filter with a transmission zero in the first example, the second example locates at the transmission zero at a normalized frequency

Fig. 9. Responses of the box-section filters. (a) Responses obtained by electrical parameters of design #1 in Table I and its coupling matrix, respectively. (b) Responses obtained by electrical parameters of design #2 in Table I and its coupling matrix, respectively.

and keeps all other parameters unchanged. According to the synthesis procedures in [2], the inter-resonator couplings are unchanged, but self-couplings [principal diagonal matrix eleetc., of the coupling matrix in Fig. 8(a)] ments, must change sign. Following the same procedures in the previous design, one can get the electrical parameters given in Table I. In Table I, the column of design #1 corresponds to a and low-pass prototype filter transmission zero at the column of design #2 corresponds to a low-pass prototype . The responses obtained filter transmission zero at from the electrical parameters are listed in Table I and responses matrix in Fig. 8(a) are both plotted in calculated from the Fig. 9 for comparison. III. DESIGN EXAMPLES AND EXPERIMENTAL RESULTS The extended-doublet filter discussed in Section II-B with its ideal response. shown in Fig. 6, and the design #1 of box-section filter discussed in Section II-C with its ideal response, shown in Fig. 9(a), are fabricated to verify the designs. Although all of the electrical parameters obtained in Section II can be transformed to physical parameters, it does not include the junction effect. Therefore, a commercial electromagnetic (EM) simulator Sonnet [11] is adopted to take all the EM effects into consideration. To efficiently tune the physical dimensions of the filter to achieve the prescribed response, the diagnosis and tuning

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Fig. 10. Fabricated extended-doublet filter. (a) Layout (unit: mils). (b) Simulated and measured response.

methods proposed in [12] are taken. Fig. 10 shows the physical dimensions and the corresponding responses for the extended-doublet filter where an RO6010 substrate with a dielectric constant of 10.8 and thickness of 50 mil is used. Fig. 11 depicts the physical dimensions and corresponding responses for the box-section filter where an RO4003 substrate with a dielectric constant of 3.63 and a thickness of 20 mil is used. The measured in-band insertion loss of the filters in Figs. 10 and 11 are 1.4 and 2.7 dB, respectively. In Fig. 10(b), the experimental results show a larger passband than the simulated ones. The deviation mainly results from the fabrication error. In Fig. 11(b), the measured response is shifted about 30 MHz. Further investigation showed that the dielectric constant of the substrate is closer to 3.4 rather than 3.63. The wideband measurement results of the fabricated box-section filter are shown in Fig. 11(c).

IV. DISCUSSION In Section II, we have discussed how to get the electrical parameters of a filter network in a doublet, an extended-doublet, and a box-section configuration from the corresponding coupling matrices. With the understanding of the correspondence between the coupling matrix and physical structure, the layout is not limited to those provided in this paper. A filter can be

Fig. 11. Fabricated box-section filter. (a) Layout (unit: mil). (b) Simulated and measured response. (c) Measured wideband response.

Fig. 12. Possible filter layout that can be modeled as a doublet configuration.

modeled by the box-like coupling scheme, as long as it contains a two-mode resonator that is physically symmetric and supports two resonant modes. For instance, the filters in Fig. 12 can also be modeled as a doublet filter since it is symmetric and has two resonant modes. However, for the filter in Fig. 12, it is not easy to get the initial physical dimensions. On the contrary, the initial dimensions of the layouts proposed in this paper can easily

LIAO et al.: MICROSTRIP REALIZATION OF GENERALIZED CHEBYSHEV FILTERS WITH BOX-LIKE COUPLING SCHEMES

be obtained. Besides, using the E-shaped resonator and the design procedures provided in this paper, all electrical parameters of a filter with box-like coupling schemes can be easily obtained. These parameters can be applied to filters with the same low-pass prototype and fractional bandwidth, but a different center frequency and a different substrate. Having clear initial dimensions of a filter can save quite a lot of time in the design when comparing to the conventional design procedures of cross-coupled filters, e.g., the filters in [5]. In the design of a conventional cross-coupled filter, once the substrate, shape of the resonator, or center frequency of a filter is changed, one must redo the design from the very beginning of the procedures. The sensitivity analysis of the box-like coupling routes can be performed by the method proposed in [13]. The most sensitive part of the proposed structures is the coupling section between the E-shaped resonator and the source/load or other resonators because the coupling section controls the coupling strengths of two modes of the E-shaped resonator to an external circuit simultaneously. V. CONCLUSION The three box-like coupling schemes, namely, the doublet, extended doublet, and box section, have been illustrated. How an E-shaped resonator constructs the box-like portion of the coupling schemes has been explained. The couplings between an E-shaped resonator, external feeding network, and other single-mode resonators have also been modeled. The correspondence between an electrical network and coupling matrix has been established, which make it possible to get the initial dimensions of a filter from the information of a coupling matrix. Besides, with the aid of a proper coupling matrix served as a surrogate model, a systematic way of tuning a filter has been applied, which saves a lot of time for optimizing the response of a filter. The proposed filters have provided an effective way to design a class of filters exhibiting a generalized Chebyshev response to meet the stringent specification in modern communication systems. REFERENCES [1] J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [2] R. J. Cameron, A. R. Harish, and C. J. Radcliffe, “Synthesis of advanced microwave filters without diagonal cross-couplings,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2862–2872, Dec. 2002. [3] U. Rosenberg and S. Amari, “Novel coupling schemes for microwave resonator filters,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2896–2902, Dec. 2002. [4] S. Amari and U. Rosenberg, “New building blocks for modular design of elliptic and self-equalized filters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 721–736, Feb. 2004. [5] J. S. Hong and M. J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2099–2108, Dec. 1996. [6] S. Amari, G. Tadeson, J. Cihlar, R. Wu, and U. Rosenberg, “Pseudo-elliptic microstrip line filters with zero-shifting properties,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 7, pp. 346–348, Jul. 2004.

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[7] A. G. Lamperez and M. S. Palma, “High selectivity X -band planar diplexer with symmetrical box section filters,” in Eur. Microw. Conf., Paris, France, Oct. 2005, vol. 1, pp. 105–108. [8] J. R. Lee, J. H. Cho, and S. W. Yun, “New compact bandpass filter using microstrip =4 resonator with open stub inverter,” IEEE Microw. Guided Wave Lett., vol. 12, no. 12, pp. 526–527, Dec. 2000. [9] R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 1–10, Jan. 2003. [10] M. Makimoto and S. Yamashita, Microwave Resonators and Filters for Wireless Communication. New York: Springer, 2001. [11] Em User’s Manual. Liverpool, NY: Sonnet Software Inc., 2004. [12] A. G. Lamperez, S. L. Romano, M. S. Palma, and T. K. Sarka, “Fast direct electromagnetic optimization of a microwave filter without diagonal cross-couplings through model extraction,” in Eur. Microw. Conf., Munich, Germany, 2003, vol. 2, pp. 1361–1364. [13] S. Amari and U. Rosenberg, “On the sensitivity of coupled resonator filters without some direct couplings,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 6, pp. 1767–1773, Jun. 2003. Ching-Ku Liao was born in Taiwan, R.O.C., on October 16, 1978. He received the B.S. degree in electrophysics and M.S. degree in communication engineering from National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 2001 and 2003, respectively, and is currently working toward the Ph.D. degree in communication engineering at National Chiao-Tung University. Since March 2006, he has been a Visiting Researcher with University of Florida, Gainesville, under a sponsorship of the National Science Council Graduate Student Study Abroad Program (GSSAP). His research interests include the analysis and design of microwave and millimeter-wave circuits.

Pei-Ling Chi was born in Taiwan, R.O.C., on March 25, 1982. She received the B.S. and M.S. degrees in communication engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 2004 and 2006, respectively, and is currently working toward the Ph.D. degree in electrical engineering at the University of California at Los Angeles. Her research interests include the analysis and design of microwave and millimeter-wave circuits.

Chi-Yang Chang (S’88–M’95) was born in Taipei, Taiwan, R.O.C., on December 20, 1954. He received the B.S. degree in physics and M.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1977 and 1982, respectively, and the Ph.D. degree in electrical engineering from the University of Texas at Austin, in 1990. From 1979 to 1980, he was with the Department of Physics, National Taiwan University, as a Teaching Assistant. From 1982 to 1988, he was with the Chung-Shan Institute of Science and Technology (CSIST), as an Assistant Researcher, where he was in charge of development of microwave integrated circuits (MICs), microwave subsystems, and millimeter-wave waveguide E -plane circuits. From 1990 to 1995, he returned to CSIST as an Associate Researcher in charge of development of uniplanar circuits, ultra-broadband circuits, and millimeter-wave planar circuits. In 1995, he joined the faculty of the Department of Communication, National Chiao Tung University, Hsinchu, Taiwan, R.O.C., as an Associate Professor and became a Professor in 2002. His research interests include microwave and millimeter-wave passive and active circuit design, planar miniaturized filter design, and monolithic microwave integrated circuit (MMIC) design.

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Tunable Dielectric Resonator Bandpass Filter With Embedded MEMS Tuning Elements Winter Dong Yan, Student Member, IEEE, and Raafat R. Mansour, Fellow, IEEE

Abstract—This paper presents a novel approach for constructing a tunable dielectric resonator bandpass filter by using the microelectromechanical system (MEMS) technology. The tunability is achieved by unique MEMS tuning elements to perturb the electrical and magnetic fields surrounding the dielectric resonators. The use of such elements as a tuning mechanism results in a wide tuning range at a relatively low tuning voltage and fast tuning speed. A three-pole tunable dielectric resonator bandpass filter is designed, fabricated, and tested. The experimental filter has a center frequency of 15.6 GHz, a 1% relative bandwidth, and an of 1300. A tuning range of 400 MHz is obtained by unloaded using MEMS tuning elements with 2 mm 2 mm tuning disks. The measured results demonstrate the feasibility of the proposed concept. Index Terms—Actuators, dielectric resonators, microelectromechanical devices, tunable filters.

I. INTRODUCTION

T

UNABLE filters with a fast tuning speed, high quality factor, and broad tuning range are key elements in reconfigurable systems. Tunable filters effectively utilize the frequency bandwidth, suppress interfering signals, and ease the requirements for the oscillator phase noise and dynamic range. In terms of the tuning mechanisms, most tunable filters reported in the literature are categorized into three different types, which are: 1) magnetic tuning; 2) electronic tuning; and 3) mechanical tuning. Each tuning approach has a different impact on the performance of tunable filters, including insertion loss, tuning speed, and tuning range. Yttrium–iron–garnet (YIG) magnetically tunable filters have a wider tuning range factor; however, they are quite bulky and and a higher consume a considerable amount of dc power (typically from 0.75 to 3 W) [1], [2]. Electronically tunable filters, realized by integrating solid-state or microelectromechanical systems (MEMS) devices with microstrip or coplanar filters, offer a compact size, fast tuning speed, and wide tuning range [2]–[5]. However, this type of tunable filters results in high insertion loss due to the low factor of planar filters that are in the range of 200–400. The limited power-handling capability of electronically tunable filters is another drawback. Often mechanically Manuscript received August 10, 2006; revised October 17, 2006. This work was supported in part by the Natural Science and Engineering Research Council of Canada and by COM DEV Ltd. The authors are with the Center for Integrated RF Engineering, Electrical and Computer Engineering Department, University of Waterloo, Waterloo, ON, Canada N2L 3G1 (e-mail: [email protected]; raafat.Mansour@ece. uwaterloo.ca). 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/TMTT.2006.888582

tunable filters are realized in a waveguide cavity or dielectric resonator format, but are bulky and suitable for only low-speed applications. than miDielectric resonator filters have a much higher crostrip or coplanar filters. The ability to integrate MEMS devices with dielectric resonators will open the door for the realization of MEMS-based high tunable filters. Panaitov et al. [8] proposed the integration of MEMS loaded slots with dielectric resonators to achieve tunability. However, only a limited tuning range has been achieved. In this paper, a novel approach for constructing a tunable dielectric resonator bandpass filter by using MEMS technology is proposed. A high factor tunable bandpass filter with a wide tuning range and moderate tuning speed can then be achieved. Tuning disks, powered by MEMS thermal actuators, are embedded into the dielectric resonator filter to serve as tuning elements. The tunability is attained by perturbing the electrical fields with the nearby tuning elements. A wide tuning range of mode is the dielectric resonator filter that operates at the accomplished with the assistance of the large deflection generated by MEMS thermal actuators. Moreover, since the MEMS tuning elements are not in the main electrical path, the powerhandling capability of this proposed design can be maintained at a relatively high level. Additionally, the tuning speed of the new tunable filter is also much higher than that of any traditional mechanically tunable filter due to the low mass and the fast response to the electrical control signal of the MEMS tuning elements. In order to prove the proposed concept, a three-pole tunable dielectric resonator bandpass filter is designed and constructed. Three MEMS tuning elements with large deflections are embedded into the experimental dielectric resonator filter. II. DESIGN AND ANALYSIS OF THE PROPOSED TUNABLE FILTER The proposed tuning mechanism of the tunable dielectric resonator filter includes a dielectric resonator with an operating , a tuning disk, and MEMS thermal actuators that mode of connect the tuning disk to the cavity wall. All the components are contained in a metallic cavity. A cross section of the proposed tuning mechanism is exhibited in Fig. 1. The concept of the newly developed tuning mechanism is to use MEMS thermal actuators and tuning disks to imitate the tuning screws employed in conventional mechanically tunable filters [6]. The tunability of the proposed structure is achieved by moving the tuning disk along the -axis with the use of MEMS thermal actuators. Initially, MEMS thermal actuators are at the relaxation state, and the tuning disk is at the closest position to the dielectric resonator, i.e., the value of tuning gap is at a minimum. At this state, the MEMS tuning elements have the

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Q

Fig. 3. Frequency shift and factor as a function of the tuning gap when the tuning disk diameter is fixed.

Fig. 1. Schematic of the proposed tuning structure.

Fig. 2. Cross section of the electrical field of the dielectric resonator: (a) without the tuning element and (b) with the tuning element.

most impact on the resonant frequency of the dielectric resonator. When a dc voltage is applied to the MEMS actuators, the tuning disks are pulled away from the dielectric resonator, and the resonant frequency begins to decrease. A comparison of the magnitude of the electrical field of the dielectric resonator with and without the tuning disk is depicted in Fig. 2. The substantial difference indicates the impact of the tuning disk on the resonant frequency of the dielectric resonator.

There are several constraints in developing such MEMS tuning elements. In order to select the optimal electrical and mechanical configurations for the proposed design, High Frequency Structure Simulator (HFSS) software is employed to determine the optimum tuning disk size for maximum tuning factor. The most important design range and maximum parameters of this novel structure are tuning gap and diameter of tuning disk since they have the most effect on the tuning factor of the dielectric resonator. Two sets of range and simulations are performed to study the effects of these two parameters. Fig. 3 illustrates the variation in the frequency shift and factor as a function of the tuning gap when the diameter of the tuning disk is fixed. A 2-mm-diameter circular tuning disk of 2- m gold and 3.5- m polysilicon is employed to construct the simulation model. For simplicity, the zigzag configuration of MEMS thermal actuators are replaced by a gold rod configuration. With the decrease in the tuning gap, the tuning range is increased, whereas the factor of the dielectric resonator is decreased. In view of Fig. 3, it can be seen that, at a tuning gap of 0.5 mm, the tuning structure can achieve a tuning range of only 200 MHz. Based on the cavity size and the maximum deflection of the MEMS thermal actuators, the theoretical minimum tuning gap that can be achieved is approximately 0.1 mm. When the tuning disk is at this location, the proposed structure exhibits a maximum tuning range of approximately 700 MHz. However, factor is decreased to 2000, approximately 45% lower the than that of the dielectric resonator without the presence of the tuning elements. The loss is mainly attributed to the lossy polysilicon. The novel structure exhibits a good tuning linearity, one of the most important criteria for tunable filters [1]. By balancing the tuning range and factor, for a 2-mm-diameter tuning disk, the optimal operating range of the proposed design in terms of tuning gap should be higher than 0.2 mm, which corresponds to a tuning range of around 500 MHz. The focus of the second set of simulations is to investigate the effect of the tuning disk size on the tuning range and factor of the proposed structure. In terms of the tuning disk size, two groups of simulation results are compared: one group is generated with a 0.1-mm tuning gap, and the other one, with a 0.2-mm tuning gap. As indicated in Fig. 4, the larger the tuning

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Fig. 4. Frequency shift and factor as a function of the tuning disk diameter when the tuning gap is 0.1 and 0.2 mm.

Fig. 6. Proposed MEMS thermal actuator. (a) Top view. (b) Isotropic view.

Fig. 5. HFSS simulation results of a three-pole dielectric resonator filter tuned at different frequencies (without loss).

disk, the wider the tuning range when the tuning disk is moved in the same distance. The factor of the proposed structure starts to decrease when the diameter of the tuning disk is increased. However, the factor is constant at 1800, when the diameter of the tuning disk is larger than 3 mm. A tuning disk with a 4-mm-diameter size seems to perform better than smaller tuning disks since the 4-mm disk has more than a 400% increase in the tuning range with a only 50% decrease in the factor. However, such large tuning disks are subject to other design issues such as mechanical stability and warpage. After careful consideration of the overall performance of the proposed MEMS tuning elements (the mechanical and RF properties), tuning disks with a 2- and 3-mm diameter are chosen for this study. Fig. 5 shows the HFSS simulation results of a three-pole tunable dielectric resonator filter. Circular tuning disks of a 3-mm diameter are employed in the simulation model. With a deflection of 0.7 mm provided by MEMS tuning elements, the filter can be continuously tuned from 15.65 GHz to as high as 16.45 GHz. III. MEMS TUNING ELEMENT DESIGN In the proposed tunable dielectric resonator filter, the MEMS actuator is one of the key elements. In order to achieve a wide

tuning range, it is essential to design a MEMS actuator that can generate enough force to lift a relatively large tuning disk, as well as generate a large deflection. In addition, a low actuation voltage and fast tuning speed are desirable. MEMS thermal actuators are good candidates to meet such requirements. An out-of-plane rotation of a 400 m 2500 m 2 m plate, produced by MEMS thermal actuators, is successfully demonstrated [9]. The actuation voltage is less than 10 V and a 90 rotation is achieved within 1 ms. The basic unit for constructing the MEMS thermal actuator in [9] is a bimorph of gold and polysilicon. After the thermal plastic deformation assembly treatment [10], the bimorph unit deflects due to the difference of the thermal expansion coefficient between gold and polysilicon. In order to achieve a large out-of-plane rotation, several units are connected in series, requiring a large footprint for one single actuator. This, though, is not desirable for the proposed design since each MEMS tuning disk requires at least two MEMS thermal actuators. In this study, the concept of thermal plastic deformation assembly is adopted to design the MEMS tuning element with a vertical displacement. Since a linear motion that is perpendicular to the substrate is required for the newly devised tunable dielectric resonator filter, bimorphs of gold and polysilicon are connected by a passive polysilicon beam in a zigzag manner instead of in series (Fig. 6). After the thermal plastic deformation assembly treatment, each bimorph unit has a small deflection in the -direction; meanwhile, the passive polysilicon beam keeps straight since it is made of one single material. The straight passive polysilicon beam can transfer the small deflection of

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Fig. 8. Side view of the proposed MEMS thermal actuator. Fig. 7. MEMS tuning element with a 3-mm square tuning disk lifted by four proposed MEMS actuators.

one bimorph unit to an adjacent unit. Large deflection can be achieved by repeating this configuration several times in the actuator design. The zigzag configuration ensures the deflection is accumulated in only the -direction. As shown in Fig. 6(a), the new MEMS thermal actuator demonstrates a negligible deflection in the - and -direction and maximum deflection in the -direction [see Fig. 6(b)]. The payload of the MEMS thermal actuators in Fig. 6 is 1500 m 300 m 4.75 m. It is demonstrated that four such MEMS thermal actuators can lift a 3000 m 3000 m 2 m plate more than 1 mm in the -direction (Fig. 7). Each MEMS thermal actuator in this design contains four bimorph units with a dimension of 100 m 700 m. The maximum deflection of the proposed MEMS thermal actuator depends on the length of each bimorph unit and the number of the bimorph units used in the design. Obviously the length of the bimorph unit and the number of the bimorph units is proportional to the total deflection of the design. However, the increase of the bimorph unit length and the number of the bimorph units decreases the stiffness of the entire structure, affecting the actuator’s capability to lift a large tuning disk. Thus, these two parameters must be carefully selected based on the size of the tuning disk and the maximum deflection required. Fig. 8 shows a side view of one of the proposed MEMS thermal actuators that can achieve a deflection of 1200 m by using six bimorph units with a dimension of 700 m 100 m. The inset of Fig. 8 is an isotropic view of the thermal actuator. A MEMS tuning element with four of the novel MEMS thermal actuators and one 3-mm-diameter circular tuning disk is first constructed. However, after thermal plastic deformation assembly treatment, the tuning disk is subjected to large warpage due to the mismatch between the thermal expansion coefficients of the gold and polysilicon [see Fig. 9(a)]. This warpage decreases the effective deflection of the thermal actuators, which results in a narrower tuning range. Instead, a solid circular disk and a hexagonal plate is employed to solve the warpage problem. As demonstrated in Fig. 9(b), the large hexagonal plate consists of many small hexagonal plates. Each

Fig. 9. MEMS tuning elements. (a) Solid circular tuning disk with warpage. (b) Hexagonal tuning disk without warpage.

plate consists of six equilateral triangles that are connected by short beams. The small air gap between the triangles can significantly reduce the warpage of the tuning disk. The dimension of the small air gap between the triangles is only a small fraction of the wavelength so it will not affect the size of the tuning disk. The size of this hexagonal tuning disk is approximately 2000 m 2000 m 4.75 m. The resistance between two dc pads is 3.35 k . For a single MEMS tuning element, it requires less than 0.1-W dc power to achieve maximum deflection. Therefore, the total dc power consumption of the tunable filter is less than 0.3 W over the entire tuning range. Latching mechanisms can be embedded into the MEMS tuning elements to further reduce the power consumption and increase the resistance to the vibration and temperature variation. IV. EXPERIMENTAL RESULTS In order to facilitate the assembly of MEMS tuning elements, the filter cavity of the proposed design is manufactured into

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Fig. 12. Measured results of a single dielectric resonator tuned at different resonance frequencies.

Fig. 10. Proposed three-pole tunable dielectric resonator filter. (a) Top cover with MEMS tuning elements. (b) Filter body. (c) Assembled filter.

Fig. 11. Comparison of the simulated and measured results of the proposed dielectric tunable filter.

two detached parts: the top cover in Fig. 10(a) and the body in Fig. 10(b). These two parts are gold plated to decrease the metallic loss from the metal cavity. Dielectric resonators are assembled with the body and the MEMS tuning elements are integrated on the top cover. Each MEMS tuning element is connected to two dc feed-through pins for applying the control voltage. Synchronously tuning method is employed to tune the proposed filter. All tuning elements are connected to one control signal to ensure they have the same deflection, i.e., the same loading effect on dielectric resonators. This configuration can also allow tuning elements to be independently controlled. Therefore, nonsynchronous tuning of each resonator can then be applied to achieve a wider tuning range, a better return loss, and a constant absolute bandwidth within the tuning range of the tunable dielectric resonator filter. The gold coating of the MEMS actuators is instrumental to reduce the loss from the polysilicon structure layers and the silicon substrate. A few mechanical tuning screws are also included in the tunable filter design to compensate for the machining tolerance and variation of material properties. A comparison of the simulated and measured results of the proposed dielectric resonator filter is illustrated in Fig. 11. From

the simulated results, the proposed filter is predicted to have a center frequency of 15.6 GHz with a relative bandwidth of 1% and the insertion loss at the midband is around 0.5 dB. The measured response of the fabricated filter has a slightly lower center frequency. This minor difference could be attributed to the machining and assembly error, and the approximation of the material properties and the dimensional tolerances of the dielectric resonators. The insertion loss of the measured results at the midband is approximately 1.5 dB, which corresponds to an unloaded of approximately 1300. The discrepancy between the insertion loss of the simulated results and that of measured results is partially due to the discontinuity on the top cover since there are small gaps between the MEMS tuning elements and the metal body, and the small holes required for the dc feed-through pins. Moreover, lossy assembly materials such as the dielectric resonator support structures also contribute to the higher insertion loss. Fig. 12 illustrates the measured unloaded values of a single dielectric resonator as it is tuned by the newly devised approach. As shown in Fig. 12, the dielectric resonator can be tuned from 15.62 to 15.86 GHz. The extracted unloaded varies from 1638 to 421 over this tuning range. The method used to extract the unloaded can be found in [11]. The measured results for the insertion loss and return loss responses of the proposed three-pole tunable dielectric resonator bandpass filter are presented in Fig. 13. The filter’s center frequency is synchronously tuned from 15.6 to 16.0 GHz. There is a 15-MHz variation of the bandwidth observed during the entire tuning range. Less variation in the bandwidth can be achieved by utilizing nonsynchronously tuning where resonators are tuned at a different rate [12], [13]. However, we believe that the main cause for this variation is the input/output coupling of the proposed filter, which is fixed during the entire tuning range. In order to achieve a constant bandwidth for a wide tuning range, tuning mechanisms must be implemented for the input and output coupling as well. When the filter is tuned from 15.6 to 16.0 GHz, the midband insertion loss of the filter increases from 1.5 to 4.5 dB. The simulation results reveal that this type filters can exhibit an unloaded as high as 1500 over a 400-MHz tuning range when losses from dielectric resonators, metal cavity, and MEMS tuning disks are only considered.

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tuning range. The measured results have proven the feasibility of the proposed concept. With the superior performance of the MEMS tuning element, the proposed approach can be used to -, and -band tunable filters with relatively construct -, wide tuning. ACKNOWLEDGMENT The authors wish to thank B. Jolley, Center of Integrated RF Engineering (CIRFE), Electrical and Computer Engineering Department, University of Waterloo, Waterloo, ON, Canada, for his technical support, and N. Sarkar, Zyvex Corporation, Richardson, TX, for his valuable discussions. The authors also appreciate the input from S. Chang, M. Daneshmand, and T. Oogarah, all with the CIRFE, Electrical and Computer Engineering Department, University of Waterloo. REFERENCES

Fig. 13. Comparison of the measured results of the proposed tunable dielectric resonator filter at different tuning states.

This high insertion loss is attributed to several factors such as the assembly procedure and lossy materials (epoxy, stainless tuning screws, etc.) used to construct this prototype filter. As mentioned earlier, the MEMS actuators are partially made of polysilicon, which is very lossy at high frequency. This also contributes to the high insertion loss of the prototype filter. To further improve the insertion loss, amorphous silicon can be employed to replace polysilicon since it has almost the same mechanical properties as polysilicon and much better electrical performance than polysilicon [14]. V. CONCLUSIONS In this paper, a novel approach to construct tunable dielectric resonator filters has been presented and discussed in detail. To the best of the authors’ knowledge, this is the first time that MEMS tuning elements have been applied to demonstrate a high tunable dielectric resonator filter with a wide tuning range. By using MEMS tuning elements, the size of the tunable dielectric resonator filter has been significantly reduced; in addition, the tuning speed has been dramatically increased. The integration of MEMS with dielectric resonators can lead to the realization of miniature tunable filters with a reasonably high factor. A three-pole tunable dielectric resonator bandpass filter has been -band filter with a reldesigned, fabricated, and tested. This ative bandwidth of 1% has achieved an approximately 400-MHz

[1] J. Uher and W. J. R. Hoefer, “Tunable microwave and millimeter-wave bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 4, pp. 643–653, Apr. 1991. [2] K. Entesari and G. M. Reibeiz, “A differential 4-bit 6.5–10-GHz RF MEMS tunable filter,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 1103–1110, Mar. 2005. [3] E. Fourn, A. Pothier, C. Champeaux, P. Tristant, A. Catherinot, P. Blondy, G. Tanne, E. Rius, C. Person, and F. Huret, “MEMS switchable interdigital coplanar filter,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 320–324, Jan. 2003. [4] I. C. Hunter and J. D. Rhodes, “Electronically tunable microwave bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 9, pp. 1354–1360, Sep. 1982. [5] M. Makimoto and M. Sagawa, “Varactor tuned bandpass filters using microstrip line ring resonators,” in IEEE MTT-S Microw. Symp. Dig., May 2001, pp. 411–414. [6] S.-W. Chen and K. A. Zaki, “Tunable, temperature-compensated dielectric resonators and filters,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 8, pp. 1046–1052, Aug. 1990. [7] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures. New York: McGraw-Hill, 1964, ch. 17. [8] G. Panaitov, R. Ott, and N. Klein, “Discrete tunable dielectric resonator for microwave applications,” in IEEE MTT-S Microw. Symp. Dig., Jun. 2005, pp. 265–268. [9] M. Daneshmand, R. R. Mansour, and N. Sarkar, “RF MEMS waveguide switch,” in IEEE MTT-S Microw. Symp. Dig., Jun. 2004, pp. 589–592. [10] A. Geisberger, N. Sarkar, M. Ellis, and G. Skidomre, “Modeling electrothermal plastic deformation self-assembly,” Nanotech, vol. 1, pp. 482–485, Feb. 2003. [11] D. Kajfez, Factor. Oxford, MS: Vector Forum, 1994. [12] A. E. Atia and A. E. Williams, “Narrow-bandpass waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 4, pp. 258–265, Apr. 1972. [13] P. Laforge, “Tunable microwave bandpass filters,” Masters degree, Dept. Elect. Comput. Eng., Univ. Waterloo, Waterloo, ON, Canada, 2003. [14] S. Chang and S. Sivoththaman, “Development of a low temperature MEMS process with a PECVD amorphous silicon structural layer,” J. Micromech. Microeng., vol. 16, pp. 1307–1313, Jul. 2006.

Q

Winter Dong Yan (S’04) received the B.Sc. degree in automatic control from the Beijing Institute of Technology, Beijing, China, in 2000, the M.Sc. degree in mechanical engineering from the University of Waterloo, Waterloo, ON, Canada, in 2002, and is currently working toward the Ph.D. degree in electrical engineering at the University of Waterloo. He is currently a Research Assistant with the Center for Integrated RF Engineering (CIRFE), University of Waterloo, where he is involved in the area of MEMS actuator development for RF applications.

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Raafat R. Mansour (S’84–M’86–SM’90–F’01) received the B.Sc. (with honors) and M.Sc. degrees from Ain Shams University, Cairo, Egypt, in 1977 and 1981, respectively, and the Ph.D. degree from the University of Waterloo, Waterloo, ON, Canada, in 1986, all in electrical engineering. In 1981, he was a Research Fellow with the Laboratoire d’Electromagnetisme, Institut National Polytechnique, Grenoble, France. From 1983 to 1986, he was a Research and Teaching Assistant with the Department of Electrical Engineering, University of Waterloo. In 1986, he joined COM DEV Ltd., Cambridge, ON, Canada,

where he held several technical and management positions with the Corporate Research and Development Department. In 1998, he became a Scientist. In January 2000, he joined the University of Waterloo, as a Professor with the Electrical and Computer Engineering Department. He holds an Natural Science and Engineering Research Council of Canada (NSERC) Industrial Research Chair in RF engineering with the University of Waterloo. He has authored or coauthored numerous publications in the areas of filters and multiplexers and high-temperature superconductivity. He holds several patents related to microwave filter designs for satellite applications. His current research interests include superconductive technology, MEMS technology, and computer-aided design (CAD) of RF circuits for wireless and satellite applications.

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A Broadband Compact Microstrip Rat-Race Hybrid Using a Novel CPW Inverter Ting Ting Mo, Student Member, IEEE, Quan Xue, Senior Member, IEEE, and Chi Hou Chan, Fellow, IEEE

Abstract—A new compact microstrip rat-race hybrid with an octave bandwidth employing a novel frequency-independent coplanar waveguide (CPW) phase inverter is reported in this paper. The 270 branch of a conventional rat-race is replaced by a 90 branch, which is realized by a 90 microstrip line and the CPW phase inverter. A new microstrip-to-CPW transition is introduced for which a lumped-element model is devised to facilitate parameter optimization. The designed transition has an insertion loss less than 0.33 dB across the designed frequency band from 1.5 to 3.5 GHz. The footprint of the proposed design is reduced by 75% and shows almost 60% and 80% enhancements in the 0.5-dB mismatch bandwidth of amplitude and 10 mismatch bandwidth of phase, respectively, when compared with the conventional implementation. The proposed hybrid can be fabricated using a conventional printed-circuit and plated thru-hole technologies. Index Terms—Broadband, coplanar waveguide (CPW) phase inverter, microstrip-to-CPW transition, rat-race hybrid.

I. INTRODUCTION

A

PPLICATIONS of rat-race hybrids can be seen in mixers, phase shifters, and beam-forming networks or antenna arrays. Much effort has been pursued to design the “perfect” ratrace with a minimum circuit footprint and maximum bandwidth. In [1], a broadband microstrip antenna fed by an L-shaped probe having an impedance bandwidth of 36% has been reported. Due to the limitation of the rat-race/power divider, the bandwidth of the array antenna using the same antenna element was reduced to around 24%, whether directional couplers [2] or T-junctions [3] were employed as its feeding network. and Conventionally, a rat-race is composed of three sections where is the guided wavelength. As one shown in Fig. 1(a), the total circumference of the conventional . Based on a scattering matrix analysis, a design rat-race is method for and rat-race using and sections, respectively, was introduced in [4]. The footprint of the rat-race can be reduced by 70% if it is terminated by arbitrary impedances [5]. Other miniaturization strategies are to adopt folded lines [6], artificial lines [7], and synthetic meander lines [8], taking advantage of the state-of-the-art fabrication technology. However, designers have to deal with parasitic coupling of these folders and meanders. Lumped elements are Manuscript received March 14, 2006; revised September 14, 2006. This work was supported by the Hong Kong Research Grant Council under Grant CityU 121905. The authors are with the Wireless Communications Research Centre, City University of Hong Kong, Kowloon, Hong Kong (e-mail: 50003643@student. cityu.edu.hk; [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/TMTT.2006.888938

Fig. 1. Diagram of conventional rat-race circuit and proposed rat-race circuit. (a) Conventional rat-race. (b) Diagram of the proposed rat-race. “T” denotes the transition between microstrip and CPW. (c) Bird’s eye view of the proposed rat-race.

also employed, as their sizes are always much smaller than their distributed counterparts, e.g., [9] adopted a partially lumped circuit and [10] an entirely lumped one. Their disadvantage lies in the parasitic effect of lumped elements beyond a certain frequency range. Recently, we also see the application of a left-handed transmission line in replacing the 270 arm with that of 90 [11]. Due to its use of lumped elements, difficulty similar to that in [9] and [10] may not be overlooked. Size reduction and harmonic suppression techniques using electromagnetic bandgap (EBG) and defected ground structure have also been reported [12]–[14]. Accompanying size reduction is bandwidth enhancement. However, this bandwidth improvement is limited in extent, as all the techniques used in [4]–[14] are frequency dependant. To achieve a wide bandwidth, in [15] Ho et al. introduced an

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ideal 180 reverse-phase section independent of frequency. Obviously, this kind of 180 phase inverter is easy to realize on a uniplanar circuit, such as CPW and slot lines, by simply swapping the signal line and the ground line. Unfortunately, the microstrip line is not authentically uniplanar and reports on utilizing the 180 phase-reversal technique in the microstrip rat-race have yet to be seen. In this paper, we propose a new compact broadband microstrip rat-race hybrid employing a CPW 180 phase inverter, as shown in Fig. 1(b), in which its circumference is equal . The total footprint of the hybrid can be reduced by to 50% when compared with the conventional implementation. The phase inverter and the two transition regions will collectively provide the needed 270 arm of the rat-race. Fig. 1(c) shows an entire image of the proposed rat-race. As the phase inverter is frequency independent, a broadband performance can be achieved in addition to reducing the circuit footprint. A new transition between the microstrip line and CPW on opposite sides of a thin dielectric substrate is introduced in Section II along with a lumped-element model to reveal the physical interpretation of the design parameters. In Section III, a frequency-independent CPW phase inverter is presented. By combining the microstrip-to-CPW transition, the CPW phase inverter and the CPW-to-microstrip transition, the proposed compact rat-race is realized in Section IV, achieving a broadband performance of 80% bandwidth with 0.5-dB mismatch for 1.5–3.5 GHz. A conclusion is presented in Section V.

Fig. 2. Geometry of the proposed microstrip-to-CPW transition. For substrate: h = 0:254 mm, " = 3:5. 50- microstrip: w = 0:57 mm. 50- CPW: w = 2:13 mm, s = 0:1 mm. Other dimension (unit: millimeters): L1 = 2:7, L2 = 2:2, L3 = 2:7, d1 = 1, d2 = 1,W 1 = 2:13, W 2 = 1:6, R = 0:4.

Fig. 3. Proposed LC equivalent circuit for the transition.

TABLE I KEY GEOMETRY PARAMETERS OF THE FOUR STUDY CASES

II. DESIGN OF MICROSTRIP-TO-CPW TRANSITION There are two major methods to construct a transition between the microstrip line and CPW; one is by electric continuity, as suggested in [16], and the other is electromagnetic coupling, as proposed in [17]. Electrical contact is suitable for wideband application, but it takes risk of energy loss if mode matching of the propagating wave is overlooked between the two different types of transmission lines. On the other hand, electromagnetic coupling could be perfectly matched at the design frequency and easy to fabricate as it is free of a via or wire-bond, but the bandwidth will be much confined compared with that of electrical contact. Combining the merits of these two methods, the proposed transition is depicted in Fig. 2. The microstrip line, on top of the substrate, shares the ground plane with the CPW, which is on the bottom side of the substrate. A via connects the microstrip and the center line of the CPW to secure electric continuity. In the transition region, the width of the microstrip line and the split ground under the microstrip vary gradually to reduce wave reflection. It is well known that the dominant field of the microstrip line is in the vertical direction, while that of the CPW is horizontal. The function of the split ground under the microstrip is to adjust its field gradually to that of the CPW. Since the impedance of the microstrip line will increase due to the slot at the ground, its width should be increased accordingly to keep the line impedance constant as wider line yields lower impedance. As to the CPW, its characteristic impedance is determined by the ratio of the width of the center line to that of the slot so only a slight change in the slot width can help to balance the impedance of the CPW to a uniform line. After considering

field matching as well as impedance matching, an excellent electromagnetic coupling is achieved. Despite its complexity, we are able to find an equivalent circuit for this microstrip-to-CPW transition, which is shown in Fig. 3, knowing that the via behaves as an inductor, the overlapping of the microstrip line and CPW line introduces capacitance and, meanwhile, the slope of the split ground affects the impedance and phase velocity of the microstrip line by generating another capacitance. Here we use a “ ” model to simplify the microstrip line with the split ground and another “ ” model to represent the via. Considering the effect of the overlapping is introduced in pararea between the microstrip and CPW, in the “ ” model of the via. is the sum of allel with shunt capacitance of the two models. We observe that the following factors have the most influence on the level of impedance matching, namely, the diameter of the via, the overlapping area between the microstrip line and the CPW, and the slope, i.e., , of the split ground under the microstrip. Here we list four cases. The parameters of the geometry of the four cases are summarized in Table I, and Case 1 is set as a reference. In

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Fig. 4. Different reflection levels of the four cases of transitions.

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Fig. 6. Experiment results of the proposed transition compared with simulation results.

TABLE II PARAMETER VALUES OF THE FOUR STUDY CASES

Fig. 7. CPW swap and its phase performance. (a) CPW swap with unconnected ground. (b) Its phase performance.

Fig. 5. Comparison of the performance of the microstrip-to-CPW transition predicted by the proposed LC model and electromagnetic (EM) simulator.

Case 2, the diameter of the via is reduced from 0.4 to 0.1 mm, resulting in large increase in inductance ( ) and capacitance ( ) since the effective overlapping area is also increased. In Case 3, the extended portion of the microstrip beyond the via overlapping the CPW is reduced slightly, leading to the reduction of capacitance between them ( ). Lastly, in Case 4, the slope of the split ground under the microstrip is reduced slightly

from that of Case 1 and it changes the values of its transmission ). Since reflects the matching level line model ( and representing each case are displayed directly, four lines of in Fig. 4. Accordingly, the component values of the equivalent circuits for each case are listed in Table II. To prove the effectiveness of the equivalent model, a comparison of the two simulation results, one from electromagnetic full-wave simulator IE3D and the other from the proposed model, is presented in Fig. 5. It can be seen that they are quite close to each other. By recognizing how each parameter affects the return loss and with the help of the equivalent circuit, it is easy for us to obtain the optimized dimensions.

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Fig. 9. Simplification of the proposed transition and CPW swap geometry. (a) Two vias with wire bonds. (b) Five vias with no wire bond. (c) Three vias with no wire bond.

the transition can be adjusted to compensate for the additional electrical length caused by the via. Following the same procedure, the CPW-to-microstrip transition can be designed. The designed microstrip-to-CPW transition with the dimensions showed in the caption of Fig. 2 was fabricated and measured. Its insertion loss over the frequency band of 1.5 to 3.5 GHz is less than 0.33 dB, as shown in Fig. 6. III. DESIGN OF THE 180 PHASE INVERTER

Fig. 8. CPW swap with connected ground. (a) Diagram of CPW swap with connected ground. (b) Its amplitude performance. (c) Its phase performance.

Ideally, the transition should add as little extra physical length as possible to the circuit. In fact, the proposed transition causes very little additional electrical length. The difference in the phase delay of a uniform microstrip line and that of a microstrip-to-CPW transition with equal horizontal length is approximately 1 from 0.5 to 5 GHz. The slight discrepancy at higher frequencies is due to the vertical length of the short via. For applications at higher frequencies, the geometry of

As mentioned before, here we employ a CPW polarity swap as a phase inverter. The function of the polarity swap using bond wires is to force the center (signal) line of the CPW on the righthand side of Fig. 7(a) to become the return path (ground) of the CPW on the left and, hence, creating a 180 phase change, which is shown in Fig. 7(b). The deviation from 180 phase difference is negligible throughout the whole frequency band from dc to 5 GHz. However, this phase inversion occurs only under the condition of disconnected grounds. For a continued ground shown in Fig. 8(a), a slot is cut through the ground plane whose length and width are chosen to balance their phase difference between the two CPWs and prevent energy radiation. Fig. 8(b) shows the frequency response of this phase inverter with connected ground. By controlling the slot width and length, we can choose the operating frequency band for the designed rat-race Here, the

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Fig. 10. Dimensions of the proposed rat-race circuit (unit: millimeters): L0 = 18:5, L1 = L6 = 3:85, L2 = L5 = 2:5; L3 = 1:4, L4 = 2:9; L7 = 0:5, L8 = 1:5; L9 = 1:2, L10 = 2:7; L11 = 1:7, W 1 = 1:47, W 2 = W 3 = 1:94, W 4 = 0:5, R = 0:4, Ls = 15:25.

10-dB bandwidth is 120% at the center frequency of 2.5 GHz. The maximum deviation of the 180 phase difference is 25 , as shown in Fig. 8(c). Although the bandwidth is reduced, its performance is still superior to most 180 phase shifters. IV. PROPOSED RAT-RACE HYBRID The microstrip-to-CPW transition, CPW phase inverter, and CPW-to-microstrip transition are now incorporated in the proposed rat-race. It has two vias for transition and two wire bonds for the polarity swap, as depicted in Fig. 9(a). These wire bonds can be eliminated using combinations of vias and short microstrip lines, as shown in Fig. 9(b). We further simplify this circuit by eliminating the two vias in the CPW-to-microstrip transition altogether and bring the center line of the CPW up to the level of the microstrip line. It is noted that the transition between an elevated CPW and microstrip is quite smooth, no transition method other than the split ground under the microstrip line mentioned before is needed. Finally, a rat-race circuit with three vias and no wire bonds is shown in Fig. 9(c). This final

Fig. 11. Experimental and simulated results of the proposed rat-race circuit. (a) Amplitudes of Ports 2 and 3 when input is at Port 1. (b) Phase difference of Ports 2 and 3 at the same condition of: (a) and (c) amplitudes of Ports 2 and 3 when input is at Port 4. (d) Phase difference of Ports 2 and 3 at the same condition of (c).

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TABLE III PERFORMANCE COMPARISON BETWEEN THE CONVENTIONAL RAT-RACE AND THE PROPOSED ONE

the microstrip line and CPW has been developed, and we have demonstrated that it has low mismatch over a large frequency band. The study of the CPW polarity swap has demonstrated its potential as a 180 phase inverter for wideband application. Combining the transition and polarity swap, we arrived at a new microstrip rat-race circuit with 80% bandwidth at a center uniform microstrip section frequency of 2.5 GHz. As the is substituted by the proposed structure with a total length , the total circumference of the proposed rat-race is of and the overall footprint is reduced by 75% reduced to when compared to the conventional implementation. On the other hand, for low-frequency application below approximately 1 GHz, the lumped-element approach may result in a more compact design.

REFERENCES

version is much easier to fabricate and can avoid a parasitic effect introduced by the wire bonding. All the dimensions of the proposed structure are given in Fig. 10. The proposed rat-race circuit was fabricated on Taconic and thickness RF-35 with a dielectric constant mm. Fig. 11 displays all the measurement results with the comparison of the simulated results by Ansoft’s High Frequency Structure Simulator (HFSS). Fig. 11(a) and (b) shows the outputs from Ports 2 and 3 when input at Port 1, i.e., the in-phase case. They are equal in amplitude, as well as phase. The amplitude bandwidth defined by 0.5-dB imbalance between Ports 2 and 3 is 93% from 1.2 to 3.5 GHz, and their phase bandwidth, defined by 10 imbalance, is over 150%; meanwhile, the isolation is kept around or below 20 dB throughout the operating frequency band. Fig. 11(c) and (d) shows the case of input at Port 4. The amplitudes of Ports 2 and 3 are equal, but their phase difference is around 180 . The amplitude bandwidth defined by 0.5-dB imbalance is 80% from 1.5 to 3.5 GHz, and their phase bandwidth defined by 10 imbalance is again over 150%. The isolation is kept below 20 dB almost throughout the operating frequency band. All the experimental results are consistent with the simulated ones to some extent. Table III gives a summary of the features of the proposed rat-race circuit. The values for the conventional rat-race, as seen in Fig. 1(a), are obtained from simulation, while values for the proposed rat-race are taken from measurement. It is obvious that the proposed rat-race outperforms the conventional one in all aspects of the comparison, especially in the operating bandwidth. V. CONCLUSION In this paper, a microstrip rat-race circuit employing a CPW polarity swap has been proposed. A new transition between

[1] C. L. Mak, K. M. Luk, K. F. Lee, and Y. L. Chow, “Experimental study of a microstrip patch antenna with an L-shaped probe,” IEEE Trans. Antennas Propag., vol. 48, no. 5, pp. 777–783, May 2000. [2] H. Wong, K. L. Lau, and K. M. Luk, “Design of dual-polarized L-probe patch antenna arrays with high isolation,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 45–52, Jan. 2004. [3] K. L. Lau, K. M. Luk, and D. Lin, “A wideband dual polarization patch antenna with directional coupler,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 186–189, 2002. [4] D. I. Kim and G.-S. Yang, “Design of new hybrid-ring directional coupler using =8 or =6 sections,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 10, pp. 1779–1784, Oct. 1991. [5] H.-R. Ahn, I. S. Chang, and S.-W. Yun, “Miniaturized 3-dB ring hybrid terminated by arbitrary impedances,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2216–2221, Dec. 1994. [6] R. K. Settaluri, G. Sundberg, A. Weisshaar, and V. K. Tripathi, “Compact folded line rat-race hybrid couplers,” IEEE Microw. Guided Wave Lett., vol. 10, no. 2, pp. 61–63, Feb. 2000. [7] K. W. Eccleston and S. H. M. Ong, “Compact planar microstripline branch-line and rat-race couplers,” IEEE Trans. Microw. Theory Tech., vol. 51, pp. 2119–2125, Oct. 2003. [8] C.-C. Chen and C.-K. C. Tzuang, “Synthetic quasi-TEM meandered transmission lines for compacted microwave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 6, pp. 1637–1647, Jun. 2004. [9] T. Hirota, A. Minakawa, and M. Muraguchi, “Reduced-size branchline and rat-race hybrids for uniplanar MMIC’s,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 3, pp. 270–275, Mar. 1990. [10] I. Sakagami, M. Tahara, Y. Hao, and Y. Iwata, “Simplified lumpedelement rat-race for a mobile receiver,” in Proc. 14th IEEE Int. Pers., Indoor, Mobile Radio Commun. Symp., 2003, pp. 2465–2469. [11] H. Okabe, C. Caloz, and T. Itoh, “A compact enhanced-bandwidth hybrid ring using an artificial lumped-element left-handed transmission-line section,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 798–803, Mar. 2004. [12] K. M. Shum, Q. Xue, and C. H. Chan, “A 180 ring hybrid incorporating a 1-D photonic bandgap structure,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 6, pp. 258–260, Jun. 2001. [13] Y. J. Sung, C. S. Ahn, and Y. S. Kim, “Size reduction and harmonic suppression of rat-race hybrid coupler using defected ground,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 1, pp. 7–9, Jan. 2004. [14] B.-L. Ooi, “Compact EBG in-phase hybrid-ring equal power divider,” IEEE Microw. Theory Tech., vol. 53, no. 7, pp. 2329–2334, Jul. 2005. [15] C. H. Ho, L. Fan, and K. Chang, “New uniplanar coplanar waveguide hybrid-ring couplers and magic-T’s,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2440–2448, Dec. 1994. [16] B. Golja, H. B. Sequeira, S. Duncan, G. Mendenilla, and N. E. Byer, “A coplanar-to-microstrip transition for W -band circuit fabrication with 100-m-thick GaAs wafers,” IEEE Microw. Guided Wave Lett., vol. 3, pp. 29–31, Feb. 1993. [17] L. Zhu and W. Menzel, “Broadband microstrip-to-CPW transition via frequency-dependant electromagnetic coupling,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5, pp. 1517–1522, May 2004.

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Ting Ting Mo (S’04) received the B.S. degree in electronic engineering from Shanghai Jiao Tong University, Shanghai, China, in 2001, and the Ph.D. degree in electronic engineering from the City University of Hong Kong, Hong Kong, in 2006. She is currently with the Wireless Communication Research Centre, City University of Hong Kong. She is scheduled to join the School of Microelectronics, Shanghai Jiao Tong University soon. Her research interest is focused on the applications of CPWs in microwave and RF circuits.

Quan Xue (M’02–SM’04) received the B.S., M.S., and Ph.D. degrees in electronic engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1988, 1990, and 1993, respectively. In 1993, he joined the institute of Applied Physics, University of Electronic Science and Technology of China, as a Lecturer. In 1995, he became an Associate Professor 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. Since June 1999, he has been with the Wireless Communications Research Centre, City University of Hong Kong, Hong Kong, where he is currently an Associate Professor, the Director of the Applied Electromagnetism Laboratory, and the Director of the Industrial Technology Center. His research interests include microwave circuits and antennas. Dr. Xue is serving on the Technical Program Committee (TPC) of IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) in 2007.

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Chi Hou Chan (S’86–M’86–SM’00–F’02) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1987. In 1996, he joined the Department of Electronic Engineering, City University of Hong Kong, and became a Professor (Chair) of electronic engineering in 1998. Prior to his current appointment as Dean of Faculty of Science and Engineering, he was the Faculty’s Associate Dean for six years. He is also an Adjunct Professor with the University of Electronic Science and Technology of China, Peking University, and the Electromagnetic (EM) Academy, Zhejiang University. His research interests include computational electromagnetics, antennas, and microwave and millimeter-wave components and systems. Dr. Chan was the recipient of the 1991 U.S. National Science Foundation (NSF) Presidential Young Investigator Award and the 2004 Joint Research Fund for Hong Kong and Macau Young Scholars, National Science Fund for Distinguished Young Scholars, China. His undergraduate and graduate students have been the recipient of numerous awards including the First Prize in the 2004 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) Student Paper Contest and the IEEE MTT-S Graduate Fellowship and Undergraduate/Pre-Graduate Scholarship for 2004–2005 and 2006–2007, respectively.

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On the Fast and Rigorous Analysis of Compensated Waveguide Junctions Using Off-Centered Partial-Height Metallic Posts Ángel A. San Blas, Fermín Mira, Vicente E. Boria, Senior Member, IEEE, Benito Gimeno, Member, IEEE, Marco Bressan, Member, IEEE, and Paolo Arcioni, Senior Member, IEEE

Abstract—In this paper, we present an efficient and rigorous method, based on the 3-D boundary integral-resonant-mode expansion technique, for the analysis of multiport rectangular waveguide junctions compensated with partial-height cylindrical metallic posts. The electrical performance of a great variety of commonly used wideband microwave circuits has been improved drastically thanks to the introduction of a new design parameter, i.e., the relative position of the metallic post in the structure. To the authors’ knowledge, this parameter has not been taken into account in previous studies concerning compensated junctions using partial-height metallic posts. The developed tool has been successfully used to design compensated - and -plane right-angled bends and power dividers, as well as optimal magic-T junctions. This novel tool has been fully verified through comparisons between our results and those provided by a well-known commercial finite-element method software. Index Terms—Compensation, integral equations, method of moments, waveguide components, waveguide junctions.

I. INTRODUCTION

M

ULTIPORT rectangular waveguide junctions, such as right-angled bends, T-junctions, crossings, magic-Ts, and turnstile and six-port cross junctions, are key basic building blocks of most common passive waveguide devices as orthomode transducers, diplexers and multiplexers, couplers, and power dividers, etc., widely used in microwave and millimeter-wave applications [1], [2]. Over the past years, several efficient modal methods, formulated either in terms of generalized admittance matrices or generalized scattering matrices, have been continuously proposed for solving - and -plane T-junctions [3], [4], magic-Ts [5], [6], turnstile junctions [7], Manuscript received May 30, 2006; revised September 10, 2006. This work was supported by the European Commission under the Research and Training Networks Program Contract HPRN-CT-2000-00043 and by the Ministerio de Educación y Ciencia, Spanish Government, under the Coordinated Research Project TEC 2004/04313-C02. Á. A. San Blas is with the Área de Teoría de la Señal y Comunicaciones, Universidad Miguel Hernández de Elche, 03202 Elche, Alicante, Spain (e-mail: [email protected]). F. Mira is with the Centro Tecnológico de Telecomunicaciones de Catalúna, 08860 Castelldefels, Barcelona, Spain (e-mail: [email protected]). V. E. Boria is with the Departamento de Comunicaciones, Universidad Politécnica de Valencia, 46022 Valencia, Spain (e-mail: [email protected]). B. Gimeno is with the Departamento de Física Aplicada, Instituto de Ciencia de los Materiales, Universidad de Valencia, 46100 Burjasot, Valencia, Spain (e-mail: [email protected]). M. Bressan and P. Arcioni are with the Department of Electronics, University of Pavia, 27100 Pavia, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.886928

and six-port cross junctions [8], [9]. More recently, different versions of the well-known boundary integral-resonant mode expansion method [10] have been extended for the efficient computation of the generalized admittance matrix of arbitrarily shaped 3-D cavities [11] and cubic junctions with arbitrarily shaped access ports [12]. Over the last years, modern communications systems are increasingly requiring broader operation bandwidths, which has led the attention of researchers to extend the usable frequency range of previously cited waveguide junctions employing compensation techniques. For instance, in 1991, Hirokawa et al. [13] first solved a T-junction perturbed with an inductive cylindrical post, by means of a Green’s function-based formulation, where the radius and position of the post were the design parameters. Afterwards, Alessandri et al. [14] studied compensated -plane junctions using a segmentation technique (in such work, different structures were compensated by the introduction of triangular wedges), and Wang and Zaki [15] investigated double-ridge waveguide T-junctions using the three-plane mode-matching method. More recently, waveguide junctions have been compensated using partial-height cylindrical posts, thus adding a new degree of freedom with respect to the work developed by Hirokawa et al. Following this method of compensation, Ritter and Arndt [16] performed a full-wave analysis of a compensated magic-T through a finite-difference time-domain technique, and Wu and Wang [17] designed optimized -plane T-junctions using a rigorous modal analysis technique. In this paper, the authors propose to use a new design parameter for compensating the electrical response of any multiport waveguide junction, i.e., the relative position of a partial-height cylindrical metallic post placed within the structure. The authors have just verified this idea in a recent publication [18], but in such a case, it was only applied to -plane T-junctions. In this study, we generalize the previous analysis method for using this concept with most common multiport waveguide junctions (up to five ports), and we demonstrate the effectiveness of such a compensation technique with - and -plane right-angled bends, -plane T-junctions, and magic-Ts (see Fig. 1). II. THEORY The 3-D boundary integral-resonant mode expansion method is a rigorous and efficient technique for the multimodal analysis of lossless microwave components with arbitrary 3-D geometry [19]. It provides a wideband electromagnetic characterization of

0018-9480/$25.00 © 2006 IEEE

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Fig. 2. Key building block of the developed tool: five-port rectangular waveguide junction (a b c) perturbed with a partial-height cylindrical metallic post of radius r and height h, placed on the base of the cavity and centered at the generic x ; 0; z position.

f

Fig. 1. Multiport waveguide junctions perturbed with a partial-height cylindrical metallic post placed at an arbitrary position within the structure. (a) Right-angled H -plane bend. (b) Right-angled E -plane bend with a matching iris. (c) E -plane T-junction with a matching iris. (d) Magic-tee junction with a matching iris.

these structures by giving an approximation of its generalized , where is the admittance matrix in the band requested maximum frequency of interest. The method is based on the solution of an electric-type integral equation, where the unknown is the current density lying on the boundary of the structure, and the electric field distribution over the ports constitutes the impressed term. The knowledge of the current density distribution permits us to calculate, by a Green’s integral, the magnetic field over the ports and to associated with the impressed compute the modal currents modal voltages . Following the original formulation of the method (see [19]), it is possible to determine an approximation of the required generalized admittance matrix, which is valid for , in the form (1) where is the wavenumber, matrices and are real, symrepresent metric, and frequency independent, the first resonant wavenumbers of the short-circuited juncare related to the eigenvectors assotion, and vectors ciated to . , the following In order to determine the values of and generalized eigenvalue problem must be solved:

2 2 g

of some coefficients used in the representation of the surface current density with respect to a suitable set of basis functions, and of the amplitudes of the resonant modes considered in the representation of the fields. More details about the definition of variables and matrices appearing in (1) and (2) can be found in [19]. Once the 3-D boundary integral-resonant mode expansion method has been introduced, its application to the rigorous analysis of a compensated multiport (up to five) waveguide junction (see Fig. 2) is then thoroughly discussed. This junction constitutes the key basic building block of all waveguide structures under analysis (see Fig. 1). Up to now, this method has been successfully applied to perturbed junctions with only two parallel waveguide access ports [21]. In this study, we extend the method for solving the more general case (five ports), thus allowing the existence of orthogonal access waveguides. In order to implement the 3-D boundary integral-resonant mode expansion method, we must select an external cavity of canonical shape where the investigated structure is embedded. In our particular problem (Fig. 2), the best option is to choose a as the external rectangular box cavity of dimensions resonator, whose Greens’s functions are numerically computed by means of the Ewald technique in a very fast and accurate way [22]. Furthermore, proceeding in this way, the unknown current density must be defined only in the contour of the metal post placed within the resonator, thus widely reducing the required computational effort. According to (1), for computing the generalized admittance matrix of the key building block under analysis, we also need , and that depend on the to calculate the matrices ports. Following the notation of [19, Table II], these matrices can be easily obtained from the following:

(3) (4)

(2) where matrices and , which are symmetric and positive definite [20], depend on the shape of the structure and on the representation of the unknown current density over its boundary; and vector includes all the state variables of the system, consisting

(5) (6)

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with denoting a waveguide access port, whereas the meaning of the remaining symbols can be easily inferred from [19]. For the efficient evaluation of integrals (3)–(6), we start from and the following expressions of the Green’s functions , valid for any cylindrical cavity of axis and cross section , and deduced from their eigenfunction expansion [10] by expressing in closed form their dependence on the axial co: ordinates and

(13)

(7)

(14)

(8) where and are the normalized magnetic-type vector mode functions for the th TE-to- and TM-to- mode, , are their corresponding cutoff wavenumbers, , and functions , , and are defined as

is the normalized electric-type vector mode function, where . In previous expressions (12)–(14), and the first line corresponds to the case in which is a TE mode and the second line corresponds to the TM case. The integrals in (12) and (13) can be evaluated analytically for all port pairs, and their calculation requires only the explicit relationship between the and coordinate systems. The expressions of matrices and are collected in the Appendix. Integrals in (14) involve the basis functions used to represent the current density on the post surface and are evaluated by a numerical integration. To compute the elements of the matrix , we can express any as the sum of the TE-to- and TM-toresonant mode modes with the same modal indices in the same directions as follows:

(9) (10)

(15) With this decomposition, it is found

(16) (11) , , In the above equations, and is the dimension of the cavity in the axial direction. Let represent a pair of transverse coordinates related and be the corresponding axial to the rectangular port coordinate (see, as an example, Fig. 2 for the case of the port ). If we use the previous expressions (3)–(6) referred to this particular reference system, we finally obtain

(12)

where are the modal indices of the th cavity mode with respect to the , , and directions, respectively, are the corresponding modal indices of the th mode represents the Kronecker’s delta of the port, if otherwise), is the cavity mode wavenumber in the direction , with being the dimension of the cavity in the direction ), is the is the Neuresonant wavenumber of the cavity mode, and if otherwise). The coeffimann symbol ( cients and can be deduced from Table I, where the decomposition of a TE-to- or TM-to- cavity mode with respect to the corresponding TE-to- and TM-to- modes is considered. Table I only reports the absolute values of the and coefficients; in fact, since in (15) all the modal fields have

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TABLE I VALUES OF THE COEFFICIENTS jj AND j j

Fig. 3. Return losses of an H -plane right-angled bend implemented in WR-90 rectangular waveguide perturbed with a partial-height cylindrical post (r = 0:5 mm, h = 9:0 mm).

a sign uncertainty, we must be careful to choose their correct sign for the actual definitions. III. NUMERICAL RESULTS Here, we present results related to the design of several multiport rectangular waveguide junctions compensated with offcentered partial-height cylindrical posts. All simulation results, which have been obtained using a PC Pentium [email protected] GHz with 512-MB RAM, have been successfully compared with numerical data provided by Ansoft HFSS,1 a commercial finite-element method tool. In order to get convergent results for all the examples considered here, we have required to use 20 modes in each waveguide access port, 300 modes in the rectangular boxed cavity, and 250 basis functions for expanding the unknown current density. The authors have verified that convergence properties of the method are basically not dependent on the post and rectangular cavity dimensions. For design purposes, the authors have followed the practical procedures described in [13] and [17]. The aspect ratio of the posts has been determined in order to recover a minimum value of reflection [13] and flat -parameters in the operation bandwidth [17]. On the other hand, the position of the post has helped to center the desired frequency response [13]. A. Design of Compensated Right-Angled Bends In order to validate our proposed method, we first discuss the design of compensated two-port waveguide junctions such as - and -plane right-angled bends [see Fig. 1(a) and (b)]. First of all, we present in Fig. 3 the return losses of an -plane right-angled bend implemented in WR-90 rectangular wavemm, mm) perturbed with a parguide ( tial-height cylindrical post placed in an off-centered position in the structure (coordinates of the center of the post basis: mm and mm, referred to in the axis 1Ansoft

HFSS, ver. 9.0, Ansoft Corporation, Pittsburgh, PA, 2005.

system of Fig. 2). We have also compared this case with the one in which the post is located at a centered position. In the same figure, the results obtained for an uncompensated bend are also included. As observed, the position of the post plays a crucial role when optimizing the electrical performance of the -plane bend. With respect to the computational efficiency of the proposed method, our response in the whole band of interest was computed in 4.6 s, whereas HFSS simulation took approximately 20 min to compute the considered 51 discrete frequency points in this example. This comparison fully validates our new tool for computer-aided design purposes. Next, we present the design of an -plane right-angled bend implemented in WR-90 rectangular waveguide compensated using a partial-height metallic post. In this case, in order to match the structure, it has been necessary to resort to an admm. In order to ditional iris whose thickness is analyze the planar discontinuity represented by the presence of this matching iris, the rigorous integral equation technique proposed in [23] has been implemented. With respect to the designed -plane right-angled bend, we show in Fig. 4 the return losses of the structure for the following three different cases. Case 1) Uncompensated bend. Case 2) Bend perturbed with a compensating post placed in a centered position and matched with an iris. Case 3) Bend compensated with a post placed in an off-centered position and matched with an iris. mm, (Coordinates of the center of the post basis: mm, referred to the axis system of Fig. 2). Observe that Case 3) is the best one in terms of electrical performance. The CPU effort was only 8.1 s in the whole operation band of the WR-90 waveguide. B. Design of Compensated

-Plane T-Junctions

One of the most important applications of a three-port waveguide junction is the design of a two-way power divider by means of a T-junction configuration, as the one shown in Fig. 1(c). As it is well known, in this case it is desirable that and the magnitude of the transmission coefficients

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Fig. 4. Return losses of an E -plane right-angled bend in WR-90 waveguide perturbed with a partial-height cylindrical post (r = 3:75 mm, h = 1:5 mm). = 7:0 mm, l = Port (2) of the bend has been matched using an iris (h 20:46 mm, t = 1:5 mm).

Fig. 5. S -parameters of an E -plane T-junction implemented in WR-75 waveguide compensated with a partial-height metallic post placed in a centered position (r = 2:9 mm, h = 4:65 mm). Comparison with the case in which an = 1:5 mm, l = 15:3 mm, additional matching iris is placed at port (1) (h = 1:25 mm). t

will approximately be 3 dB in the whole band of interest. With the aim of optimizing the electrical performance of an -plane rectangular waveguide power divider, a relevant investigation was performed in [18], where it was demonstrated that a compensating partial-height cylindrical post placed at an off-centered position within the structure improves the electrical response of a power divider. Here, our aim is to present new design examples concerning -plane power dividers compensated using partial-height cylindrical posts. Thus, we present in Fig. 5 the design of an -plane T-juncmm, tion implemented in WR-75 waveguide ( mm) perturbed with a partial-height metallic post placed in a centered position in the structure [see Fig. 1(c)]. In this case, the authors have observed that the position of the post does not significantly improve the electrical performance of the T-juncmm is tion. However, if a matching iris of thickness added at port (1) of the structure, the scattering parameters get highly improved.

Fig. 6. S -parameters of a magic-T implemented in WR-90 waveguide perturbed with a partial-height metallic post placed in a centered position (r = 0:65 mm, h = 9:5 mm). An additional matching iris has been placed in port (4) = 4:0 mm, l = 16:11 mm, t = 1:5 mm). (a) Reflection parame(h ters. (b) Transmission parameters.

C. Design of Compensated Magic Tees Here, we present the design of a compensated magic-T junction, which plays an important role in many modern microwave circuits such as directional couplers or balanced mixers. We will also prove in this design the benefits of perturbing the magic-T with a partial-height metallic post placed in an off-centered position in the structure. Let us consider a magic-T such as the one shown in Fig. 1(d) implemented in a WR-90 rectangular waveguide. Firstly, we perturb the magic-T with a partial-height metallic post placed in a centered position. Moreover, for matching purmm. poses, we place at port (4) an iris of thickness The obtained scattering parameters have been represented in Fig. 6. Next, we demonstrate that if we put the post off center, the electrical performance of the magic-T improves drastically. In Fig. 7, we present the -parameters for a magic-T in which the mm, basis post has been situated in the coordinates mm (see Fig. 2). We observe that all the reflecGHz, i.e., tion parameters have been optimized around the central frequency of the operation band. On the other hand,

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results for other common multiport waveguide junctions, such as crossings and turnstile junctions.

APPENDIX ELEMENTS OF THE MATRICES

AND

The

matrix elements relating parallel ports, i.e., the elements with and the elements with are (17) (18)

The matrix elements relating orthogonal ports, i.e., the elements with and or with and , are (19) if where auxiliary functions

, and and if if if

Fig. 7. S -parameters of a magic-T implemented in WR-90 waveguide perturbed with a partial-height metallic post (r = 0:65 mm, h = 9:5 mm) placed in an off-centered position. An additional matching iris has been placed in port (4) (h = 4:0 mm, l = 16:11 mm, t = 1:5 mm). (a) Reflection parameters. (b) Transmission parameters.

we appreciate that the magnitude of the transmission parameters and are equal to 3 dB in a wide frequency band. Moreis very low around the central over, the isolation parameter frequency. Therefore, the position of the post becomes a crucial design parameter. Making reference to the computational efficiency of our tool, the complete CPU effort was approximately 16.5 s.

if

and

if

and

if

and

if are defined as and and and

, and the

(20)

(21)

denoting as the well-known Neumann symbol. The matrix elements relating parallel ports, i.e., the elements with and the elements , are with

(22)

IV. CONCLUSION In this paper, we have extended the 3-D boundary integral-resonant mode expansion method for analyzing multiport waveguide cavities (up to five ports) with a partial-height cylindrical metallic post. We have shown that the relative position of the post improves drastically the electrical response of most common compensated junctions. Successful results for right-angled bends and T-junctions, as well as magic-Ts, have been presented. Finally, the authors foresee to present similar

(23) (24) (25) (26)

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The and

matrix elements relating orthogonal ports, i.e., the elements with and or with are

(27) (28)

(29)

(30)

(31) , where the auxiliary functions , and are defined as follows:

,

,

if

and

if

and

if

and

,

(32) if if

and and if if

and and

if if

and and and

if if

and if if if

and and and

(33) (34)

(35) (36) (37)

REFERENCES [1] J. Uher, J. Bornemann, and U. Rosenberg, Waveguide Components for Antenna Feed Systems: Theory and CAD. Norwood, MA: Artech House, 1993. [2] Y. Rong, H. Yao, K. A. Zaki, and T. G. Dolan, “Millimeter-wave Ka-band H -plane diplexers and multiplexers,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2325–2330, Dec. 1999.

[3] X. Liang, K. A. Zaki, and A. E. Atia, “A rigorous three plane modematching technique for characterizing waveguide T-junctions, and its application in multiplexer design,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 12, pp. 2138–2147, Dec. 1991. [4] F. Alessandri, M. Mongiardo, and R. Sorrentino, “A technique for the fullwave automatic synthesis of waveguide components: Application to fixed phase shifters,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 7, pp. 1484–1495, Jul. 1992. [5] T. Sieverding and F. Arndt, “Modal analysis of the magic-T,” IEEE Microw. Guided Wave Lett., vol. 3, no. 5, pp. 150–152, May 1993. [6] V. E. Boria and M. Guglielmi, “Efficient admittance matrix representation of a cubic junction of rectangular waveguides,” in IEEE MTT-S Int. Microw. Symp. Dig., Baltimore, MD, Jun. 1998, pp. 1751–1754. [7] W. Wessel, T. Sieverding, and F. Arndt, “Mode-matching analysis of general waveguide multiport junctions,” in IEEE MTT-S Int. Microw. Symp. Dig., Anaheim, CA, Jun. 1999, pp. 1273–1276. [8] T. Sieverding and F. Arndt, “Rigorous analysis of the rectangular waveguide six-port cross junction,” IEEE Microw. Guided Wave Lett., vol. 3, no. 7, pp. 224–226, Jul. 1993. [9] V. E. Boria, S. Cogollos, H. Esteban, M. Guglielmi, and B. Gimeno, “Efficient analysis of a cubic junction of rectangular waveguides using the admittance-matrix representation,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 147, no. 6, pp. 111–119, Dec. 2000. [10] G. Conciauro, M. Guglielmi, and R. Sorrentino, Advanced Modal Analysis. New York: Wiley, 2000. [11] M. Bozzi, M. Bressan, and L. Perregrini, “Generalized Y-matrix of arbitrary 3-D waveguide junctions by the BI-RME method,” in IEEE MTT-S Int. Microw. Symp. Dig., Anaheim, CA, Jun. 1999, pp. 1269–1272. [12] S. Cogollos, V. E. Boria, P. Soto, A. A. San Blas, B. Gimeno, and M. Guglielmi, “Direct computation of the admittance parameters of a cubic junction with arbitrarily shaped access ports using the BI-RME method,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 150, no. 2, pp. 111–119, Apr. 2003. [13] J. Hirokawa, K. Sakurai, M. Ando, and N. Goto, “An analysis of a waveguide T-junction with an inductive post,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 563–566, Mar. 1991. [14] F. Alessandri, M. Dionigi, and R. Sorrentino, “Rigorous analysis of compensated E -plane junctions in rectangular waveguide,” in IEEE MTT-S Int. Microw. Symp. Dig., Orlando, FL, May 1995, pp. 987–990. [15] C. Wang and K. A. Zaki, “Full wave modeling of generalized double ridge waveguide T-junctions,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2536–2542, Dec. 1996. [16] J. Ritter and F. Arndt, “Efficient FDTD/matrix-pencil method for the full-wave scattering parameter analysis of waveguiding structures,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2450–2456, Dec. 1996. [17] K. Wu and H. Wang, “A rigorous modal analysis of H -plane waveguide T-junction loaded with a partial-height post for wideband applications,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 5, pp. 893–901, May 2001. [18] A. A. San Blas, F. Mira, V. E. Boria, B. Gimeno, M. Bressan, G. Conciauro, and P. Arcioni, “Efficient CAD of optimal multi-port junctions loaded with partial-height cylindrical posts using the 3-D BI-RME method,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 67–70. [19] P. Arcioni, M. Bozzi, M. Bressan, G. Conciauro, and L. Perregrini, “Frequency/time-domain modeling of 3-D waveguide structures by a BI-RME approach,” Int. J. Numer. Modeling, vol. 15, pp. 3–21, 2002. [20] P. Arcioni, M. Bressan, and L. Perregrini, “A new boundary integral approach to the determination of resonant modes of arbitrarily shaped cavities,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 8, pp. 1848–1856, Aug. 1995. [21] F. Mira, M. Bressan, G. Conciauro, B. Gimeno, and V. E. Boria, “Fast S -domain modeling of rectangular waveguides with radially symmetric metal insets,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1294–1303, Apr. 2005. [22] M. Bressan, L. Perregrini, and E. Regini, “BI-RME modeling of 3-D waveguide components enhanced by the Ewald technique,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, Jun. 2000, pp. 1097–1100. [23] G. Gerini, M. Guglielmi, and G. Lastoria, “Efficient integral formulations for admittance or impedance representation of planar waveguide junctions,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1998, vol. 3, pp. 1747–1750.

SAN BLAS et al.: FAST AND RIGOROUS ANALYSIS OF COMPENSATED WAVEGUIDE JUNCTIONS

Ángel A. San Blas was born in Fortaleny, Spain, on September 20, 1976. He received the Ingeniero de Telecomunicación degree from the Universidad Politécnica de Valencia, Valencia, Spain, in 2000, and is currently working toward the Ph.D. degree at the Universidad Politécnica de Valencia. In 2001, he became a Researcher with the Departamento de Comunicaciones, Universidad Politécnica de Valencia, where he was involved in the development of simulation tools for the analysis and design of waveguide devices. Since 2003, he has been an Assistant Professor with the Área de Teoría de la Señal y Comunicaciones, Universidad Miguel Hernández de Elche, Elche, Alicante, Spain. His current research interests include numerical methods for the analysis and design of waveguide components.

Fermín Mira was born in Elda, Spain, on April 19, 1976. He received the Ingeniero de Telecomunicación degree and Doctor Ingeniero de Telecomunicación degree from the Universidad Politécnica de Valencia, Valencia, Spain, in 2000 and 2005, respectively. In 2001, he joined the Department of Electronics, University of Pavia, Pavia, Italy, where he was a Pre-Doctoral Fellow (2001–2004) involved with the “Millimeter-Wave and Microwave Components Design Framework for Ground and Space Multimedia Network (MMCODEF)” research project funded by the European Commission (5th Framework Programme). In May 2004, he joined the Departamento de Comunicaciones, Universidad Politécnica de Valencia. He is currently with the Centro Tecnológico de Telecomunicaciones de Catalúna, Castelldefels, Barcelona, Spain. His current research interests are numerical methods for the analysis of microwave passive components.

Vicente E. Boria (S’91–A’99–SM’02) was born in Valencia, Spain, on May 18, 1970. He received the Ingeniero de Telecomunicación degree (with first-class honors) and Doctor Ingeniero de Telecomunicación degree from the Universidad Politécnica de Valencia, Valencia, Spain, in 1993 and 1997, respectively. In 1993, he joined the Departamento de Comunicaciones, Universidad Politécnica de Valencia, where since 2003 he has been a Full Professor. In 1995 and 1996, he was held a Spanish Trainee position with the European Space Research and Technology Centre (ESTEC)–European Space Agency (ESA), Noordwijk, The Netherlands, where he was involved in the area of electromagnetic (EM) analysis and design of passive waveguide devices. He has authored or coauthored five chapters in technical textbooks, 40 papers in refereed international technical journals, and over 100 papers in international conference proceedings. His current research interests include numerical methods for the analysis of waveguide and scattering structures, automated design of waveguide components, radiating systems, measurement techniques, and power effects (multipactor and corona) in waveguide systems. Dr. Boria is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and the IEEE Antennas and Propagation Society (IEEE AP-S) since 1992. He serves on the Editorial Boards of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He is also a member of the Technical Committees of the IEEE MTT-S International Microwave Symposium (IMS) and of the European Microwave Conference. He was the recipient of the 2001 Social Council of Universidad Politécnica de Valencia First Research Prize for his outstanding activity during 1995–2000.

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Benito Gimeno (M’01) was born in Valencia, Spain, on January 29, 1964. He received the Licenciado degree in physics and Ph.D. degree from the Universidad de Valencia, Valencia, Spain, in 1987 and 1992, respectively. From 1987 to 1990, he was a Fellow with the Universidad de Valencia. Since 1990, he has been an Assistant Professor with the Departmento de Física Aplicada y Electromagnetismo, Universidad de Valencia, where in 1997 he became an Associate Professor. During 1994 and 1995, he was with the European Space Research and Technology Centre (ESTEC) of the European Space Agency (ESA) as a Research Fellow. In 2003, he was with the Università degli Studi di Pavia, Pavia, Italy, as a Visiting Scientist for a three-month period. His current research interests include computer-aided techniques for analysis of passive components for space applications, waveguides, and cavities including dielectric objects, electromagnetic-bandgap structures, frequency-selective surfaces, and nonlinear phenomena appearing in power microwave subsystems (multipactor and corona effects). Dr. Gimeno was the recipient of a Spanish Government Fellowship.

Marco Bressan (M’94) was born in Venice, Italy, on February 13, 1949. He received the Laurea degree in electronic engineering from the University of Pavia, Pavia, Italy, in 1972. Since 1973, he has been a Researcher of electromagnetics with the Department of Electronics, University of Pavia. In 1987, he joined the Faculty of Engineering, University of Pavia, where he is currently an Associate Professor lecturing on antennas and propagation and radio communications. His research interests include electromagnetic modeling of microwave and millimeter-wave components and efficient algorithms for computer-aided design of electromagnetic devices. Dr. Bressan is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and the IEEE Antennas and Propagation Society (IEEE AP-S).

Paolo Arcioni (M’90–SM’03) was born in Busto Arsizio, Italy, in 1949. He received the Laurea degree in electronic engineering from the University of Pavia, Pavia, Italy, in 1973. In 1974, he joined the Department of Electronics, University of Pavia, where he currently teaches a course in microwave theory as a Full Professor. In 1991, he was a Visiting Scientist with the Stanford Linear Accelerator Center (SLAC), Stanford, CA, where he worked in cooperation with the RF Group to design optimized cavities for the PEP II Project. From 1992 to 1993, he collaborated with the Istituto Nazionale de Fisica Nucleare (INFN), Frascati, Italy, on the design of accelerating cavities for the DAHNE storage ring. His main research interests are in the area of microwave theory, modeling, and design of interaction structures for particle accelerators and development of numerical methods for the electromagnetic computer-aided design (CAD) of passive components. His current research activities concern the modeling of planar components on semiconductor substrates and integrated structures for millimeter-wave circuits. Prof. Arcioni is a member of the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.

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Nonlinear Distortion Due to Cross-Phase Modulation in Microwave Fiber-Optic Links With Optical Single-Sideband or Electrooptical Upconversion Linghao Cheng, Student Member, IEEE, Sheel Aditya, Senior Member, IEEE, Zhaohui Li, Student Member, IEEE, Ampalavanapillai Nirmalathas, Senior Member, IEEE, Arokiaswami Alphones, Senior Member, IEEE, and Ling Chuen Ong, Senior Member, IEEE

Abstract—A new analytical model is presented to study nonlinear distortion due to cross-phase modulation in dispersive and nonlinear wavelength-division-multiplexing microwave fiber-optic links. The model is not based on the pump-probe approach. Hence, it can be used to analyze a larger variety of links, including, in particular, electrooptical upconversion links. Our simulations and experiments show that the model predicts the nonlinear distortion due to cross-phase modulation quite accurately, even when the modulating microwave frequency is in tens of gigahertz and the fiber length is in tens of kilometers. Detailed analyses of the distortion due to cross-phase modulation for links with optical single-sideband modulation are presented. Measured results for optical single-sideband are shown to match our theoretical predictions very well. Our results show that the nonlinear distortion due to cross-phase modulation can be a limiting factor for optical launch power in wavelength-division-multiplexing microwave fiber-optic links; moreover, the maximum possible nonlinear distortion level for a higher frequency may be lower than that for a lower frequency. Also presented are some simple approximations for a quick estimate of the level of the nonlinear distortion. Index Terms—Cross-phase modulation, electrooptical upconversion, fiber dispersion, fiber nonlinearity, microwave fiber-optic link, wavelength division multiplexing.

I. INTRODUCTION N RECENT years, there has been increasing interest in the exploitation of the microwave region of the electromagnetic spectrum with the help of fiber-optic techniques[1]–[3]. The extremely low loss and wide bandwidth of the fiber enable distribution of microwave signals from a central control station to a number of base stations, resulting in simple base-station architectures with low cost and good system flexibility. Microwave fiber-optic links have been widely proposed for many applications such as broadband wireless access and antenna remoting. Moreover, it is desirable to incorporate wavelength di-

I

Manuscript received May 17, 2006; revised August 18, 2006. L. Cheng, S. Aditya, Z. Li, and A. Alphones are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). A. Nirmalathas is with the Department of Electrical and Electronic Engineering, University of Melbourne, VTC 3010 Melbourne, Australia (e-mail: [email protected]). L. C. Ong is with the Institute for Infocomm Research, Singapore 117674 (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.886931

vision multiplexing into microwave fiber-optic links to improve the bandwidth efficiency [4], [5]. It is well known that the performance of a microwave fiberoptic link using optical double-sideband modulation is severely limited by the chromatic dispersion of the fiber. which leads to power fading of the RF signal. To mitigate this limitation, optical single-sideband modulation has been proposed[6]. Alternatively, some electrooptical upconversion systems, such as those operating a dual-drive Mach–Zehnder modulator at the minimum transmission bias point [7], [8] and those using a heterodyning approach are also immune to the limitation imposed by chromatic dispersion. Moreover, electrooptical upconversion helps to overcome the limited bandwidth of electronic devices such as the Mach–Zehnder modulator. Therefore, systems using electrooptical upconversion have been widely investigated. Generally, microwave fiber-optic links impose stringent limits on nonlinear distortion. Therefore, nonlinear distortion in these links needs to be studied carefully. Both the modulator and fiber can introduce significant nonlinear distortion in the links. For single wavelength microwave fiber-optic links, Mach–Zehnder modulator is the dominant source of nonlinear distortion. The nonlinear distortion due to the combined effect of Mach–Zehnder modulator and fiber dispersion has been analyzed in the past through exact general analytical models [7]–[9]. For the nonlinear distortion due to fiber, stimulated Brillouin scattering must be considered in the system design[10]. Due to the very low stimulated Brillouin scattering threshold of fibers, the launched optical power of a microwave fiber-optic link has to be very low, typically less than 6 dBm, when no special measures are taken to suppress stimulated Brillouin scattering. Such a low launched optical power makes other fiber nonlinearities, such as third-order intermodulation due to self-phase modulation, rather insignificant in a single wavelength link. However, the RF power level needs careful consideration. The scenario is different in a wavelength-division-multiplexing microwave fiber-optic link where the nonlinear distortion due to cross-phase modulation between adjacent optical channels can be significant. The nonlinear crosstalk due to cross-phase modulation in wavelength-division-multiplexing systems has been studied by many researchers [11]–[15]. Their studies show that the nonlinear crosstalk level due to cross-phase modulation increases rapidly as the modulation frequency increases, which suggests that the nonlinear

0018-9480/$25.00 © 2006 IEEE

CHENG et al.: NONLINEAR DISTORTION DUE TO CROSS-PHASE MODULATION IN MICROWAVE FIBER-OPTIC LINKS

Fig. 1. Typical electrooptical upconversion system configuration with wavelength division multiplexing. Dual-drive Mach–Zehnder modulator: DD-MZM. Laser diode: LD. Photodetector: PD. Frequency of IF oscillator: f . Frequency of master oscillator: f .

Fig. 2. Spectrum of two of the optical channels in Fig. 1, as well as the generation of the nonlinear distortion at f f due to cross-phase modulation between the two channels in an electrooptical upconversion system. Optical carrier: O.C.

+

crosstalk level for wavelength division multiplexing microwave fiber-optic links may be significant even when the launched optical power is well below stimulated Brillouin scattering threshold. Further, optical double-sideband is the primary modulation format studied in most of these papers. Since optical single-sideband is preferable in microwave fiber-optic links, a thorough study of the nonlinear crosstalk due to cross-phase modulation in a system using optical single-sideband is very desirable. Turning our attention to electrooptical upconversion systems, the pump-probe approach used in most of the earlier studies [11]–[15] is inadequate for the analysis of the nonlinear distortion. Consider a typical electrooptical upconversion system configuration with wavelength division multiplexing, as shown in . Fig. 1. The final desired signal in such a system is at Fig. 2 shows the spectrum of two of the optical channels, as well as the interactions leading to the generation of nonlinear distordue to cross-phase modulation between the tion at two channels. In each channel, there are two types of contribu. The first type of contion to nonlinear distortion at tribution, called the nonlinear crosstalk, is shown by the dashed

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lines at the bottom in this figure. This is induced by the interacand can tion of the optical carrier and the subcarrier at be analyzed through the pump-probe approach. The second type , denoted by of contribution to nonlinear distortion at the solid lines in the lower part of Fig. 2, cannot be analyzed by the pump-probe approach. This part of nonlinear distortion in is induced by the interaction between the subcarrier at one channel and the subcarrier at in the other channel. In this paper, we call the second type of contribution to nonlinear distortion as nonlinear intermodulation products because these involve the interaction between two subcarriers at different optical channels through cross-phase modulation. There are two objectives in this paper. First, we present a new analytical model, not based on the pump-probe approach, to study the nonlinear distortion due to cross-phase modulation in dispersive wavelength-division-multiplexing microwave fiber-optic links. The model predicts quite well the nonlinear distortion due to cross-phase modulation, including both nonlinear crosstalk and nonlinear intermodulation products, even when the modulating microwave frequencies are in the range of tens of gigahertz and the fiber length is tens of kilometers. Moreover, the study shows that in an electrooptical upconversion system, the nonlinear intermodulation products can be several times greater than the nonlinear crosstalk. We also show that after reasonable approximation, the nonlinear distortion can be expressed in a very simple form, clearly showing the influence of various parameters. As the second objective, detailed analysis of the nonlinear distortion for optical single-sideband is presented since this is the preferred modulation format in this context. Numerical simulations, as well as experiments confirm the accuracy of our analytical model. The remainder of this paper is organized as follows. Section II presents our analytical model for the nonlinear distortion due to cross-phase modulation. Sections III and IV present the results of the simulations and experiments for nonlinear intermodulation product and nonlinear crosstalk, respectively. After a brief discussion in Section V, Section VI summarizes and concludes this paper. II. NONLINEAR DISTORTION DUE TO CROSS-PHASE MODULATION IN MICROWAVE FIBER-OPTIC LINKS A. General Analytical Model We begin with the derivation for a single optical channel and the nonlinear distortion due to self-phase modulation and then generalize the model to cross-phase modulation. The optical field along the fiber length can be written as (1) where is the modal distribution, is the angular is the wavenumber, and frequency of the optical field, denotes the complex conjugate. is the slowly varying pulse envelope of the optical field with , and is the group velocity in the fiber. We assume that the nonlinear distortion due to self-phase modulation is a small perturbation of the case of transmission with fiber dispersion only. Therefore, according to the perturbation theory, the optical field

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envelope of the optical channel in a nonlinear dispersive fiber can be written as

Therefore, the nonlinear distortion that comes from the optical in (7a) is intensity of

(2) where only the first-order perturbation is considered for simplicity, and where is the fiber nonlinearity parameter. is the optical field envelope when there is no self-phase is the first-order nonlinear dismodulation and tortion due to self-phase modulation with coefficient . It is well known that the nonlinear propagation of optical field envelope in single-mode fibers is governed by nonlinear Schrödinger equation [16] (3) where is the second-order dispersion parameter and is the fiber loss parameter. Substituting (2) into (3) and using the perturbation theory, we get the following equations:

(4a) (4b) Equation (4b) is obtained by equating the terms with coefficient . (4a) gives the evolution of the optical field envelope with fiber dispersion only. Generally, the expression for the correis known. For example, when sponding intensity evolution of using a Mach–Zehnder modulator, the intensity evolution has been expressed in terms of exact analytical expressions in [9]. This makes (4b) ready for solution. Assuming , the Fourier transform of (4) is

(5a) (5b) where denotes the Fourier transform and volution. The solution of (5) is

denotes the con-

(6a)

(6b) . From the conjugate equation of (5), we can where and where denotes the complex get the solutions of conjugate. Since the system is intensity modulation/direct detection, the intensity is our concern. The spectrum of the optical intensity at the photodetector is then (7a) (7b)

(8) The photodetection converts the optical intensity to a current, which consists of the current of the signal converted from the optical intensity of , and the current of the nonlinear distortion . Since the amplitude converted from the optical intensity of of the current is directly proportional to the optical intensity during the photodetection, we use the optical intensity of and afterwards to denote the amplitude of the signal and the nonlinear distortion that resulted from the photodetection, respectively. , and consist For microwave fiber-optic links, of Dirac delta functions, and the convolutions in (8) can be performed easily through frequency shift and multiplication. Therefore, (8) enables us to find the corresponding nonlinear distortion due to fiber at different frequencies. The model derived above for a single optical channel can be extended to wavelength division multiplexing and cross-phase modulation easily by modifying (4b) as follows: (9) where is the ininducing cross-phase tensity of another optical channel at is the anmodulation to the channel at , gular frequency difference between the two optical channels, denotes the optical field envelope time delay beand tween the two optical channels due to the walk-off effect. When we do not consider the walk-off effect, the Fourier transform of is generally of the form (10) is the intensity evolution without fiber loss. where When the walk-off effect is included, we find that

(11) where . The introduction of the new complex with the fiber loss parameter results in the same form of walk-off effect as that without the walk-off effect. Therefore, (8) can also be used for cross-phase modulation after replacing by , by , and multiplying by 2 since cross-phase modulation is two times greater than self-phase modulation.

CHENG et al.: NONLINEAR DISTORTION DUE TO CROSS-PHASE MODULATION IN MICROWAVE FIBER-OPTIC LINKS

Fig. 3. Setup for simulations and experiments. Dual-drive Mach–Zehnder modulator: DD-MZM. Wavelength demultiplexer: Demux. Photodetector: PD.

B. Optical Double-Sideband and Optical Single-Sideband Next, we present a solution of (8) for optical double-sideband and optical single-sideband, the two most commonly used modulation methods in microwave fiber-optic links. For optical double-sideband and optical single-sideband, the evolution of under the influence of dispersion alone optical field envelope has been well studied and generally can be expressed in closed form, e.g., [9]. We study the nonlinear distortion due to cross-phase modulation in dispersive wavelength-division-multiplexing microwave fiber-optic links by using the setup shown in Fig. 3, which is a simplified version of Fig. 1 so that we can observe the two types of nonlinear distortion separately. Each of the two tunable lasers is modulated by an RF tone at and , with modulation and , respectively, and the outputs are combined index and transmitted in the same fiber. The nonlinear distortion due to cross-phase modulation will show up as an RF tone at in both channels, which is easily observed since no upconverted is generated. For the derivation of the expressignal at sions here, we assume that the nonlinear distortion is observed in the channel with tunable laser 1. For optical double-sideband modulation, the general expres, and are sions for ,

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for microwave fiber-optic links that use a remote heterodyning method to generate a microwave signal optically by beating two correlated laser source outputs at the desired frequency spacing. Such a method can also be viewed as optical single-sideband. For the cases of optical double-sideband and optical single-sideband, (8) is generally integratable and results in and . many new frequencies such as those at Different from nonlinear crosstalk located at either or , these new frequencies are a result of the interaction between at least two subcarriers, and their magnitude is generally . As mentioned in Section I, these new proportional to frequencies are called nonlinear intermodulation products. Generated due to cross-phase modulation, these nonlinear intermodulation products will cause interference across optical channels. In an electrooptical upconversion system, these nonlinear intermodulation products will distort the upconverted , which is also proportional to . signal at Here we derive the analytical expressions for the nonlinear for different types of modintermodulation product at ulation on the two optical channels. . When both and are optical Let single-sideband modulated, we find that the nonlinear distortion at is given by

(14) is optical double-sideband modulated and When optical single-sideband modulated, then

is optical single-sideband modulated and When tical double-sideband modulated, then

is

(15) is op-

(12a) (12b) (12c) and . and are the where average optical powers of the two channels, respectively. For optical single-sideband, the general expressions are

(16) When both lated, then

and

are optical double-sideband modu-

(13a) (13b) (13c) Typically, optical single-sideband modulation can be generated by using a dual-drive Mach–Zehnder modulator biased at the quadrature point and shifting the phase of one RF drive with . However, (13) can also be used respect to the other by

(17)

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In (14)–(17), (18a) (18b) (18c) Note that (14)–(17) can be used to estimate both the nonlinear , ) and nonlinear intermodulation product (when ). crosstalk (when For most wavelength-division-multiplexing systems, the is much larger than the subcarrier frequenchannel spacing cies and . Hence, all in (14)–(17) can be approxi. When and are less than 0.25 , our mated as calculation shows that this approximation normally produces an error less than approximately 2 dB. Equations (14)–(17) can is optical single-sideband then be simplified further. When modulated, then

(19) When

is optical double-sideband modulated, then

(20) After the approximation, it is clearly seen that the nonlinear distortion experiences power fading. Besides, it is the modu, not that of , that decides whether the lation format of nonlinear distortion follows (19) or (20). Moreover, the power fading period of nonlinear distortion is basically governed by . The power fading period of nonlinear crosstalk is alone because . governed by For the maximum nonlinear distortion level, we can further approximate (19) and (20), resulting in a single relationship as follows:

(21)

III. NONLINEAR INTERMODULATION PRODUCTS In Fig. 3, at the output of the fiber, the two channels are filtered out by a wavelength demultiplexer, and the nonlinear intermodulation products due to cross-phase modulation are measured by

Fig. 4. Calculated power ratio of nonlinear intermodulation products to the upconverted subcarrier at 27 GHz versus fiber length for the two channels, respectively. Channels 1 and 2 are optical single-sideband modulated by a 22and 5-GHz subcarrier, respectively. The nonlinear intermodulation product is at 27 GHz for both channels. The continuous lines and dotted lines show numerical and analytical calculation results, respectively. The dashed lines show the sine approximation [see (21)] for the nonlinear intermodulation products at both channels. Nonlinear intermodulation product: NIMP.

an RF spectrum analyzer after photodetection. To estimate the , a fixed reference nonlinear intermodulation products at is used. This refvalue of the upconverted subcarrier at and erence value is calculated by using Fig. 1 with and is the magnitude at the Mach–Zehnder modulator output. At the photodetector, the magnitude of the ratio between after complete fiber loss compensation the RF tone at and the reference value of the upconverted subcarrier is a measure of the nonlinear distortion level between the two channels. Fig. 4 shows simulation results based on (14) for the level of the nonlinear intermodulation product for both channels in a GHz and two-wavelength-channel system with GHz. In practice, a small value of the modulation index is chosen for optical modulation to meet the requirement on linearity of the system [9]. Hence, a modulation index of 0.1 is used for our simulations. Both channels are optical single-sideband modulated. The optical channel spacing is 100 GHz and the launched optical power is 0 dBm for both channels. The dB/km, the fiber dispersion pafiber loss parameter ps/nm km , and the fiber rameter nonlinearity parameter W km. The numerical simulations shown as the continuous curves in this and the next figure are based on the split-step Fourier (SSF) method and verify our analytical results. Due to cross-phase modulation, 27-GHz nonlinear intermodulation products are generated in both channels. Fig. 4 shows that the results calculated using our analytical model very well match those obtained by numerical simulation for such high-frequency subcarriers, even when the fiber length is quite long. The power fading character of nonlinear intermodulation products is clearly visible. However, even though both nonlinear intermodulation products are at 27 GHz, the power fading periods are different. Moreover, their periods are not the period of the power fading of a 27-GHz optical double-sideband subcarrier. Our analytical model, as well as the approximate expressions, accurately predict these periods. As mentioned earlier, the previous analyses based on the pump-probe

CHENG et al.: NONLINEAR DISTORTION DUE TO CROSS-PHASE MODULATION IN MICROWAVE FIBER-OPTIC LINKS

Fig. 5. Calculated power ratio of crosstalk to subcarrier versus fiber length for optical single-sideband case. The continuous lines and dotted lines show numerical and analytical calculation results, respectively. The dashed line shows the sine approximation for the 10-GHz subcarrier.

approach cannot yield information on nonlinear intermodulation products, which may be significant in applications such as electrooptical upconversion. IV. NONLINEAR CROSSTALK For an estimate of nonlinear crosstalk, tunable laser 1 in as the probe Fig. 3 operates in continuous wave mode channel. Thus, with , (14)–(21) reduce to expressions for the nonlinear crosstalk. A. Simulations Following the same method and the same values of the optical parameters as in Section III, the simulated nonlinear crosstalk evolution for the case of optical single-sideband is shown in Fig. 5. Both 10- and 20-GHz subcarrier frequencies, with a modulation index of 0.1, are investigated. The slight difference between numerical results and the results based on our analytical method arises from higher order nonlinear distortion, which is not included in the first-order perturbation model used by our analytical method, as shown in (2). Fig. 5 show that for wavelength-division-multiplexing microwave fiber-optic links, the crosstalk level increases sharply at the beginning of the fiber and reaches its maximum very soon. Hence, the crosstalk may cause a problem even for short fiber links. An interesting feature is that the maximum value of the crosstalk for the 20-GHz case is smaller than that for the 10-GHz case. It has been reported [13] that a higher subcarrier frequency induces a higher crosstalk among channels through cross-phase modulation. Indeed, this is true at the beginning of the fiber, which shows that the crosstalk in the 20-GHz case is higher and increases faster than that in the 10-GHz case. However, it is not true for the maximum crosstalk level. This interesting phenomenon means less nonlinear crosstalk problem for some systems operating at higher microwave frequency. Equation (21) shows that the maximum nonlinear distortion level decreases as the optical channel spacing and modulating subcarrier frequency increase. Given that the nonlinear distortion evolution can be approximated by sine function (19), the

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Fig. 6. Relationship between the power of the maximum nonlinear crosstalk and optical channel spacing. The results are normalized to the nonlinear distortion power at 50-GHz channel spacing. The continuous line shows the result of our analytical model; the short dashed line is the result based on theoretical approximation; the square markers are numerical results. The subcarrier frequency f = 0; f = 10 GHz and fiber length is 37 km.

nonlinear distortion due to cross-phase modulation can now be estimated quickly by using these simple expressions. The continuous curve in Fig. 6 shows the calculated results according to our analytical model for a 37-km fiber link, for which case the nonlinear crosstalk of the 10-GHz system reaches its maximum value (see Fig. 5). A curve based on the approximation in (21) is also shown as a short dash line. The approximate results match the analytical results very well, the typical error being 5%. Including all higher order nonlinear distortion, the numerical calculation produces slightly worse results than those based on our analytical method, which considers the first-order nonlinear distortion only. This figure shows that the nonlinear distortion level drops gradually as the optical channel spacing increases, indicating that cross-phase modulation typically affects the closest channels. The rate of drop also gradually slows down. Fig. 7 shows the maximum nonlinear crosstalk level versus the modulating subcarrier frequency for a 100-GHz optical channel spacing. Numerical results are obtained by finding the . maximum crosstalk level, which occurs at The continuous curve is calculated from (21). The theoretical approximation matches the numerical results very well. This figure also shows that the maximum nonlinear distortion level decreases as the modulating subcarrier frequency increases. This trend has not been reported earlier. B. Experiments For experiments, the fiber length is taken as 40 km. The fiber parameters are the same as in Section III. One of the tunable lasers is operated in continuous wave mode with nm. The other tunable lasers is operated at nm and is optical single-sideband modulated by an RF tone, which can be tuned from 1 to 18 GHz. The is 1.7 dBm and that of the launched optical power at continuous-wave channel is 5.75 dBm. No significant stimulated Brillouin scattering is observed for both channels. Since the power of the nonlinear crosstalk is quite small, a fairly large launched optical power at the continuous-wave channel

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Fig. 7. Relationship between the power of the maximum nonlinear crosstalk and the modulating subcarrier frequency f . The results are normalized to the power of the nonlinear crosstalk at 5 GHz. The continuous line shows the result of our theoretical approximation; the square markers are the numerical simulation results. The optical channel spacing is 100 GHz and f = 0.

Fig. 8. Analytical and experiment results of the nonlinear crosstalk due to cross-phase modulation for a 40-km fiber. The continuous line shows the results based on analytical model; the square (15-dBm RF source) and triangle markers (12-dBm RF source) are experiment results.

is chosen so that the nonlinear crosstalk can also be observed at high frequencies. The crosstalk-to-subcarrier power ratio is defined as the power ratio between the detected RF tone at the and that in the optical fiber output in the optical channel . The adjacent channel isolation of wavelength channel division multiplexing is seen to be greater than 40 dB, resulting in a linear crosstalk well below the noise floor. In Fig. 8, two RF power levels, i.e., 15 and 12 dBm, are used to study the nonlinear crosstalk response with respect to RF frequency in the continuous-wave optical channel. This figure shows that the results of our analytical model match the experiment results very well. The figure also confirms the “periodic” nature of crosstalk level and that for a higher frequency, the maximum possible crosstalk level may not be higher than that at a lower frequency. Fig. 8 also shows that the crosstalk-to-subcarrier ratio is independent of a modulation index since both RF power levels produce the same crosstalk-to-subcarrier ratio; this

Fig. 9. Measured and linear fitted power ratio of the crosstalk-to-subcarrier versus launched optical power at the continuous wave channel for 5- and 10-GHz subcarrier.

can also be easily deduced from our expressions. Since the frequency response of the optical modulator is not flat, the same input RF power produces different values of a modulation index at different frequencies. In our experiment, the 15- and 12-dBm RF power results in a modulation index value of approximately 0.1 at frequencies around 12 and 8 GHz, respectively. Note that due to the degraded response of the experiment link at high frequency, for 12-dBm RF source power, the crosstalk level beyond 11 GHz is very close to or below noise level (which is roughly from 80 to 75 dBm), making the measurement very difficult. Although the crosstalk-to-subcarrier ratio is independent of the modulation index, the launched optical power does have an impact on this ratio. From the expressions [e.g., (21)], every 1-dB increase of the launched optical power at either channel leads to an increase of 2 dB in crosstalk. Fig. 9 graphically shows this tendency by varying the launched optical power at while keeping the launched optical power at constant at 1.7 dBm. The slopes of the linear fit curves for 5- and 10-GHz subcarrier are 1.97 and 2.22, respectively. V. DISCUSSION Given that the crosstalk-to-subcarrier ratio is optical power dependent, but RF power independent, the nonlinear crosstalk due to cross-phase modulation may impose stringent limits on the launched optical power if one wants to keep the nonlinear crosstalk well below the noise level across the entire spuriousfree dynamic range (SFDR). Assuming both channels have the , with (20) and same average launched optical power (21), the maximum crosstalk-to-subcarrier ratio can be approximated as follows: (22) W /km and For a fiber nonlinearity parameter a launched optical power 0 dBm, (22) predicts that the crosstalk-to-subcarrier ratio can be much greater than 60 dB for a practical system. For example, a 10-GHz microwave fiber-optic link with 100-GHz optical channel spacing results in a value of this ratio as high as 46.6 dB, while typically

CHENG et al.: NONLINEAR DISTORTION DUE TO CROSS-PHASE MODULATION IN MICROWAVE FIBER-OPTIC LINKS

the SFDR requirement for a microwave fiber-optic link is , which means a third-order-inapproximately 100 dB Hz termodulation-to-subcarrier power ratio less than 65 dB for a system with 200-kHz bandwidth. The maximum nonlinear intermodulation product-to-subcarrier power ratio also follows (22) since both nonlinear intermodulation products and upconverted subcarriers are proportional to . Moreover, for nonlinear intermodulation products, be, which is much lower than the upconverted cause can be , the maximum nonlinear subcarrier frequency at intermodulation product-to-subcarrier power ratio is approxitimes the maximum crosstalk-to-subcarmately rier ratio. Therefore, the nonlinear distortion due to cross-phase modulation is more detrimental in electrooptical upconversion systems.

VI. CONCLUSION A new analytical model has been presented to study the nonlinear distortion due to cross-phase modulation in dispersive wavelength-division-multiplexing microwave fiber-optic links. The model is not based on the pump-probe approach. Hence, it can be used to analyze a larger variety of links, including, in particular, electrooptical upconversion links. Verified by simulations and experiments, our analytical model can predict the nonlinear distortion quite accurately even for long fiber lengths and high subcarrier frequencies. In an electrooptical upconversion link, the relatively low leads to high nonlinear distortion and the high results in a sharp increase of the nonlinear distortion to its maximum value within a short distance of transmission in the fiber. Hence, the nonlinear distortion due to cross-phase modulation is detrimental in electrooptical upconversion links. Next, based on our model, optical single-sideband has been studied in detail since it is the preferred modulation format in microwave fiber-optic links. The studies show that closer optical channel spacing results in a higher nonlinear distortion level. Besides, the nonlinear distortion experiences a variation similar to power fading even when the original subcarrier does not experience power fading. Moreover, even though higher subcarrier frequencies lead to a sharper increase of nonlinear distortion level at the beginning of the fiber, the maximum nonlinear distortion level gradually drops as the subcarrier frequency increases. This behavior has been confirmed by experiments, as well as numerical simulations. Thus, a microwave fiber-optic link can benefit by using a high subcarrier frequency. Further, very simple approximate expressions have been presented to estimate the nonlinear distortion quite accurately. These expressions make the physical mechanism of nonlinear crosstalk generation clearer. It is seen that the impact of a fiber loss parameter on the nonlinear distortion level is negligible in dispersive wavelength-division-multiplexing microwave fiber-optic links. The experiment results confirm this aspect. In summary, the nonlinear distortion due to cross-phase modulation in microwave fiber-optic links, especially in electrooptical upconversion systems, can be significant even when the

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launched optical power is very low and, therefore, should be considered carefully. REFERENCES [1] A. J. Seeds, “Microwave photonics,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 877–887, Mar. 2002. [2] X. Chen, Z. Deng, and J. Yao, “Photonic generation of microwave signal using a dual-wavelength single-longitudinal-mode fiber ring laser,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 804–809, Feb. 2006. [3] A. J. C. Vieira, P. R. Herczfeld, A. Rosen, M. Ermold, E. E. Funk, W. D. Jemison, and K. J. Williams, “A mode-locked microchip laser optical transmitter for fiber radio,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 10, pp. 1882–1887, Oct. 2001. [4] X. Zhang, B. Liu, J. Yao, K. Wu, and R. Kashyap, “A novel millimeter-wave-band radio-over-fiber system with dense wavelength-division multiplexing bus architecture,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 929–937, Feb. 2006. [5] T. Kuri and K. Kitayama, “Optical heterodyne detection technique for densely multiplexed millimeter-wave-band radio-on-fiber systems,” J. Lightw. Technol., vol. 21, no. 12, pp. 3167–3179, Dec. 2003. [6] G. H. Smith, D. Novak, and Z. Ahmed, “Overcoming chromatic dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 8, pp. 1410–1415, Oct. 1997. [7] J. L. Corral, J. Martí, and J. M. Fuster, “General expressions for IM/DD dispersive analog optical links with external modulation or optical up-conversion in a Mach–Zehnder electrooptical modulator,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 10, pp. 1968–1976, Oct. 2001. [8] K. Kojucharow, M. Sauer, and C. Schaffer, “Millimeter-wave signal properties resulting from electrooptical upconversion,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 10, pp. 1977–1985, Oct. 2001. [9] L. Cheng, S. Aditya, Z. Li, and A. Nirmalathas, “Generalized analysis of subcarrier multiplexing in dispersive fiber-optic links using Mach–Zehnder external modulator,” J. Lightw. Technol., vol. 24, no. 6, pp. 2296–2304, Jun. 2006. [10] S. L. Zhang and J. J. O’Reilly, “Effect of dynamic stimulated Brillouin scattering on millimeter-wave fiber radio communication systems,” IEEE Photon. Technol. Lett., vol. 9, no. 3, pp. 395–397, Mar. 1997. [11] M. Shtaif, “Analytical description of cross-phase modulation in dispersive optical fibers,” Opt. Lett., vol. 23, no. 15, pp. 1191–1193, Aug. 1998. [12] R. Hui, K. R. Demarest, and C. T. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightw. Technol., vol. 17, no. 6, pp. 1018–1026, Jun. 1999. [13] M. R. Phillips and D. M. Ott, “Crosstalk due to optical fiber nonlinearities in WDM CATV lightwave systems,” J. Lightw. Technol., vol. 17, no. 10, pp. 1782–1792, Oct. 1999. [14] Z. Jiang and C. Fan, “A comprehensive study on XPM- and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightw. Technol., vol. 21, no. 4, pp. 953–960, Apr. 2003. [15] W. H. Chen and W. I. Way, “Multichannel single-sideband SCM/DWDM transmission systems,” J. Lightw. Technol., vol. 22, no. 7, pp. 1679–1693, Jul. 2004. [16] G. P. Agrawal, Nonlinear Fiber Optics. New York: Academic, 2001.

Linghao Cheng (S’04) was born in Sichuan, China, in 1977. He received the B.Eng. degree in opto-electronic technology from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2000, and is currently working toward the Ph.D. degree at Nanyang Technological University, Singapore. For two years, he was a Research and Development Engineer with the Wuhan Research Institute of Post and Telecommunication (WRI), Hubei, China. His research interests are subcarrier multiplexing optical communications, radio-over-fiber, and microwave/millimeter-wave fiberoptic links.

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Sheel Aditya (S’74–M’79–SM’94) received the B.Tech. and Ph.D. degrees in electrical engineering from the Indian Institute of Technology (IIT) Delhi, Delhi, India, in 1974 and 1979, respectively. From 1979 to 2001, he was a faculty member with the Electrical Engineering Department, IIT Delhi, where held academic positions ranging from Lecturer to Professor. He is currently an Associate Professor with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He has held visiting assignments with Chalmers University of Technology, Göteborg, Sweden, the Indian Institute of Science, Bangalore, India, Florida State University, Tallahassee, and Nanyang Technological University, Singapore. His teaching and research interests are microwave integrated circuits and antennas, microwave photonics, integrated optics, and optical-fiber communication. He has been involved in numerous research and development projects. He co-holds three Indian patents. Dr. Aditya is a Fellow of the Institution of Electronics and Telecommunication Engineers (IETE), India.

Zhaohui Li (S’04) was born in Inner Mongolia, China. He received the B.S. degree in physics and M.S. degree from the Institute of Modern Optics, Nankai University, Tian Jin, China, in 1999 and 2002, respectively, and is currently working toward the Ph.D. degree at Nanyang Technological University, Singapore. His research interests are Raman amplifiers, transmission systems, and optical access networks.

Ampalavanapillai Nirmalathas (S’96–M’97– SM’03) received the B.E. degree (Hons.) in electrical and electronic engineering and Ph.D. degree in electrical and electronic engineering from the University of Melbourne, Melbourne, Australia, in 1993 and 1997, respectively. He is currently an Associate Professor and Reader with the Department of Electrical and Electronic Engineering, University of Melbourne. He is also the Program Leader for the Network Technologies Research Program with the Victoria Research Laboratory, National Institute of Information and Communications Technology Australia (NICTA). From 1997 to 2003, he was a Research Fellow, Senior Research Fellow, and Senior Lecturer with the University of Melbourne, prior to his current position. From 2001 to 2005, he was also the Director of Photonics Research Laboratory, University of Melbourne, and the Program Manager of the Telecommunications Technologies Research Program in the Australian Photonics CRC. In 2004, he was a Guest Researcher with the Ultra-fast Photonic Network Group, National Institute of Information and Communications Technology (NICT), Tokyo, Japan, and a Visiting Scientist with the Lightwave Department, I R, Singapore. His current research interests include microwave and terahertz photonics, optical access networks, optical performance monitoring, photonic packet switching technologies, and ultrafast optical communications systems.

Arokiaswami Alphones (M’92–SM’98) received the B.Tech. degree from the Madras Institute of Technology, Madras, India, in 1982, the M.Tech. degree from the Indian Institute of Technology Kharagpur, Kharagpur, India, in 1984, and the Ph.D. degree in optically controlled millimeter wave circuits from the Kyoto Institute of Technology, Kyoto, Japan, in 1992. From 1996 to 1997, he was a Japan Society for the Promotion of Science (JSPS) Visiting Fellow in Japan. From 1997 to 2001, he was with Centre for Wireless Communications, National University of Singapore, Singapore, as a Senior Member of Technical Staff, where he was involved in research on optically controlled passive/active devices. He is currently an Associate Professor with the School of Electrical and Electronic Engineering, Nanyang Technological University. He possesses 22 years of research experience. He has authored/coauthored and presented over 130 technical papers in international journals/conferences. His current interests are electromagnetic analysis on planar RF circuits and integrated optics, microwave photonics, and hybrid fiber-radio systems. His research has been cited in Millimeter Wave and Optical Integrated Guides and Circuits (Wiley, 1997). He has delivered tutorials and short courses in international conferences. He authored the chapter “Microwave Measurements and Instrumentation” in Encyclopedia of Electrical and Electronic Engineering (Wiley, 2002). Dr. Alphones is on the Editorial Review Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He was involved with the organization of the APMC99, ICCS 2000, ICICS 2003, PIERS 2003, IWAT 2005, ISAP 2006, and ICICS 2007 conferences.

Ling Chuen Ong (M’96–SM’02) received the Ph.D. degree from the University of Birmingham, Edgbaston, U.K., in 1996. From 1992 to 1994, he was a Research Associate with the University of Birmingham. From 1996 to 1999, he was a Network Planner and Project Manager with Singapore Telecom, where he was involved with its first digital trunked radio system. He is currently an Associate Lead Scientist with the Institute for Infocomm Research, Agency for Science, Technology and Research (A*STAR), Singapore. His research interests include radio-over-fiber technology for intelligent transport systems and future wireless communications, low-temperature co-fired ceramics, and ultra-wideband technology. He is also an Adjunct Assistant Professor with the National University of Singapore and the Nanyang Technological University. Dr. Ong was the recipient of a Science and Engineering Research Council grant and an Institute of Electrical Engineers postgraduate scholarship.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

185

Letters Comments on “Extension of the Leeson Formula to Phase Noise Calculation in Transistor Oscillators With Complex Tanks”

Authors’ Reply Jean-Christophe Nallatamby, Michel Prigent, Marc Camiade, and Juan Obregon

Takashi Ohira The above paper [1] attempts to extend Leeson’s oscillator noise formula, with the result being [1, eq. (21)]. It is expected to be useful for any kind of feedback oscillator since the phase slope against frequency is employed in place of Leeson’s Q factor. Although one can agree that the extended formula is more persuasive than using the conventional energy-based Q factor, this is not always true for actual circuits [2]. Generally speaking, a passive circuit consisting of multiple elements such as R; L; and C has a frequency-dependent response not only in its phase, but also in its magnitude. This takes place even at the circuit’s resonant frequency. For example, the circuit shown in [1, Fig. 10(b)] exhibits an immittance whose slope in the real part vanishes at resonance only if the two resonators Lp Cp and Ls Cs have exactly the same resonant frequency. To strictly extend Leeson’s formula, the phase slope d=d! in [1, eq. (21)] should be replaced by the modulus of the logarithmic derivative of complex immittance as

1 dY 1 dZ or : Y d! Z d!

(1)

Consequently, [1, eq. (31)] should also be replaced by

!0 !0 1 dY =  2 Y d! ! P

1 4

dP d!

2

+ !0

2 d ("m 0 "e ) : d!

(2) This equation clarifies the physical relationship among the stored energy, dissipated power, and Leeson’s coefficient, which should involve not only the group delay, but also the slope of the entire complex immittance. The mathematical proof of this can be found in [3] along with the deduced spectrum- and energy-based Q factor formulas for typical oscillator circuits.

In Dr. Ohira’s comments on the above paper [1], he notices that [1, eq. (21)], in which we only take into account the variation of the imaginary part of the transfer function of the feedback circuit, can be improved by also taking into account the variation of the real part. We agree with his comments, as well as on the modification resulting from [1, eq. (31)]. These comments can be extended to the more general case in which the two-port feedback circuit is described by a chain matrix, as in [2]

V1 = V2

A B C D

V2

:

0I2

(1)

The interest of the calculations by a chain matrix is due to the fact that, contrary to the two port matrices [Z ] or [Y ] (see [3]), this matrix is always defined. Here, we recall for convenience [2, eq. (13)],

C d' =0 I d! ! Co 0

where Co is real and negative and constitutes the third term of the feedback circuit chain matrix calculated at the oscillation frequency !o , and CI0 is the slope factor of the imaginary part of C as follows: CI0 = (dCI =d!)j! . Now, by taking into account [2, eqs. (12) and (13)], we obtain the improved formula

~ 1 dG 1 d jG j d' = +j ~ d! j G j d! d! G

(2)

~ is the open-loop gain of the oscillator circuit. where G Reference [2, eq. (14)] now becomes

! QLoscill = o 2

1 jGj2!

d jG j d!

2

d' d!

+

! QLoscill = 0 o (CR )2 + (CI )2 : 2C o 0

Q

Manuscript received July 21, 2006. The author is with ATR Wave Engineering Laboratories, Kyoto 619-0288, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.886910

(3)

and, finally,

REFERENCES [1] J.-C. Nallatamby, M. Prigent, M. Camiade, and J. J. Obregon, “Extension of the Leeson formula to phase noise calculation in transistor oscillators with complex tanks,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 690–696, Mar. 2003. [2] B. Razavi, “A study of phase noise in CMOS oscillators,” IEEE J. Solid-State Circuits, vol. 31, no. 3, pp. 331–343, Mar. 1996. [3] T. Ohira, “Rigorous factor formulation for one- and two-port passive linear networks from an oscillator noise spectrum viewpoint,” IEEE Trans. Circuits Syst. II, Reg Papers, vol. 52, no. 12, pp. 846–850, Dec. 2005.

2

0

(4)

CR now takes into account the slope factor of the magnitude jGj of the 0

open loop gain. 0 = 0. we found [2, eq. (14)]. If CR We calculated the Q factor of several circuits according to whether CR0 is or is not taken into account. Practically, at the resonant fre0 because for real-world circuits, quency, Q is affected very little by CR 0 (CR =CI0 )  1 for high-Q, as well as for low-Q oscillators. Manuscript received August 22, 2006. J.-C. Nallatamby, M. Prigent, and J. Obregon are with the Institut de Recherche en Communications Optiques et Microondes–Centre National de la Recherche Scientifique, 19100 Brive, France (e-mail: [email protected]). M. Camiade is with the UMS Domaine de Corbeville, 91404 Orsay, France. Digital Object Identifier 10.1109/TMTT.2006.886911

0018-9480/$25.00 © 2006 IEEE

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

185

Letters Comments on “Extension of the Leeson Formula to Phase Noise Calculation in Transistor Oscillators With Complex Tanks”

Authors’ Reply Jean-Christophe Nallatamby, Michel Prigent, Marc Camiade, and Juan Obregon

Takashi Ohira The above paper [1] attempts to extend Leeson’s oscillator noise formula, with the result being [1, eq. (21)]. It is expected to be useful for any kind of feedback oscillator since the phase slope against frequency is employed in place of Leeson’s Q factor. Although one can agree that the extended formula is more persuasive than using the conventional energy-based Q factor, this is not always true for actual circuits [2]. Generally speaking, a passive circuit consisting of multiple elements such as R; L; and C has a frequency-dependent response not only in its phase, but also in its magnitude. This takes place even at the circuit’s resonant frequency. For example, the circuit shown in [1, Fig. 10(b)] exhibits an immittance whose slope in the real part vanishes at resonance only if the two resonators Lp Cp and Ls Cs have exactly the same resonant frequency. To strictly extend Leeson’s formula, the phase slope d=d! in [1, eq. (21)] should be replaced by the modulus of the logarithmic derivative of complex immittance as

1 dY 1 dZ or : Y d! Z d!

(1)

Consequently, [1, eq. (31)] should also be replaced by

!0 !0 1 dY =  2 Y d! ! P

1 4

dP d!

2

+ !0

2 d ("m 0 "e ) : d!

(2) This equation clarifies the physical relationship among the stored energy, dissipated power, and Leeson’s coefficient, which should involve not only the group delay, but also the slope of the entire complex immittance. The mathematical proof of this can be found in [3] along with the deduced spectrum- and energy-based Q factor formulas for typical oscillator circuits.

In Dr. Ohira’s comments on the above paper [1], he notices that [1, eq. (21)], in which we only take into account the variation of the imaginary part of the transfer function of the feedback circuit, can be improved by also taking into account the variation of the real part. We agree with his comments, as well as on the modification resulting from [1, eq. (31)]. These comments can be extended to the more general case in which the two-port feedback circuit is described by a chain matrix, as in [2]

V1 = V2

A B C D

V2

:

0I2

(1)

The interest of the calculations by a chain matrix is due to the fact that, contrary to the two port matrices [Z ] or [Y ] (see [3]), this matrix is always defined. Here, we recall for convenience [2, eq. (13)],

C d' =0 I d! ! Co 0

where Co is real and negative and constitutes the third term of the feedback circuit chain matrix calculated at the oscillation frequency !o , and CI0 is the slope factor of the imaginary part of C as follows: CI0 = (dCI =d!)j! . Now, by taking into account [2, eqs. (12) and (13)], we obtain the improved formula

~ 1 dG 1 d jG j d' = +j ~ d! j G j d! d! G

(2)

~ is the open-loop gain of the oscillator circuit. where G Reference [2, eq. (14)] now becomes

! QLoscill = o 2

1 jGj2!

d jG j d!

2

d' d!

+

! QLoscill = 0 o (CR )2 + (CI )2 : 2C o 0

Q

Manuscript received July 21, 2006. The author is with ATR Wave Engineering Laboratories, Kyoto 619-0288, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.886910

(3)

and, finally,

REFERENCES [1] J.-C. Nallatamby, M. Prigent, M. Camiade, and J. J. Obregon, “Extension of the Leeson formula to phase noise calculation in transistor oscillators with complex tanks,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 690–696, Mar. 2003. [2] B. Razavi, “A study of phase noise in CMOS oscillators,” IEEE J. Solid-State Circuits, vol. 31, no. 3, pp. 331–343, Mar. 1996. [3] T. Ohira, “Rigorous factor formulation for one- and two-port passive linear networks from an oscillator noise spectrum viewpoint,” IEEE Trans. Circuits Syst. II, Reg Papers, vol. 52, no. 12, pp. 846–850, Dec. 2005.

2

0

(4)

CR now takes into account the slope factor of the magnitude jGj of the 0

open loop gain. 0 = 0. we found [2, eq. (14)]. If CR We calculated the Q factor of several circuits according to whether CR0 is or is not taken into account. Practically, at the resonant fre0 because for real-world circuits, quency, Q is affected very little by CR 0 (CR =CI0 )  1 for high-Q, as well as for low-Q oscillators. Manuscript received August 22, 2006. J.-C. Nallatamby, M. Prigent, and J. Obregon are with the Institut de Recherche en Communications Optiques et Microondes–Centre National de la Recherche Scientifique, 19100 Brive, France (e-mail: [email protected]). M. Camiade is with the UMS Domaine de Corbeville, 91404 Orsay, France. Digital Object Identifier 10.1109/TMTT.2006.886911

0018-9480/$25.00 © 2006 IEEE

186

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 1, JANUARY 2007

Idealized oscillator circuits, such as the ideal two-integrator oscillator, must be handled with care before any steady-state calculation in the real frequency domain, Nyquist plot, or poles in the complex frequency domain must first be computed. Because without care, idealized circuit models can lead to impasse points [4]. Finally, at all events, as suggested by Dr. Ohira, it is always preferable, if only for theoretical reasons, to use improved expressions in analytical calculations.

REFERENCES [1] J. C. Nallatamby, M. Prigent, M. Camiade, and J. Obregon, “Extension of the Leeson formula to phase noise calculation in transistor oscillators with complex tanks,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 690–696, Mar. 2003.

[2] ——, “Phase noise in oscillators—Leeson formula revisited,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1386–1394, Apr. 2003. [3] T. Ohira, “Rigorous factor formulation for one- and two-port passive linear networks from an oscillator noise spectrum viewpoint,” IEEE Trans. Circuits Syst. II, Reg. Papers, vol. 52, no. 12, pp. 846–850, Dec. 2005. [4] L. O. Chua, “Dynamic nonlinear networks: State-of-the art,” IEEE Trans. Circuits Syst., vol. CAS-27, no. 11, pp. 1059–1087, Nov. 1980.

Q

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Digital Object Identifier 10.1109/TMTT.2007.890650

EDITORIAL BOARD Editors: D. WILLIAMS AND A. MORTAZAWI Associate Editors: D. DE ZUTTER, K. ITOH, J. LIN, Y. NIKAWA, J. PEDRO, Z. POPOVIC, S. RAMAN, R. SNYDER, K.-L. WU, R. WU, A. YAKOVLEV P. Aaen M. Abe R. Abhari A. A. Abidi A. Abramowicz M. Acar L. Accatino R. Achar D. Adam E. Adler M. Adlerstein K. Agawa S. Aggarwal A. Agrawal D. Ahn H. R. Ahn M. Aikawa J. S. Aine C. Aitchison M. Akaike J. Akhtar E. Akmansoy S. Aksoy C. Albert M. Ali R. Allam D. Allstot J. I. Alonso B. Alpert A. Alphones A. Alu S. Amari H. An J. P. Anderson Y. Ando A. Andrenko W. Andress C. L. Andrew M. Andrés K. S. Ang I. Angelov G. Antonini C. Antonopoulos H. Aoki R. Araneo J. Archer F. Ares J. Armstrong F. Arndt F. Aryanfar M. Asai Y. Asano P. Asbeck H. Ashoka A. Atalar A. Atia S. Auster P. Awai A. Aydiner M. S. Ayza R. Azadegan M. T. Azar V. I. B A. Babakhani P. Baccarelli M. Baginski I. Bahl D. Bajon W. Bakalski S. Bakhtiari B. Bakkaloglu M. Bakr S. Balasubramaniam J. V. Balbastre J. Ball K. G. Balmain S. Banba J. Bandler R. Bansal D. Barataud A. Barbosa M. F. Barciela Z. Bardai F. Bardati I. Bardi S. Barker F. Barnes R. Bashirullah D. Becker J. P. Becker H. C. Bell P. Bell J. P. Berenger P. V. Berg M. Berroth H. Bertoni E. Bertran A. Bessemoulin A. Bevilacqua A. Beyer A. V. Bezooijen F. Bi M. Bialkowski E. Biebl P. Bienstman R. Biernacki S. Bila A. L. Billabert H. Bilzer A. Biswas D. Blackham M. Blank Y. Bliokh P. Blondy D. Boccoli G. Boeck L. Boglione R. Boix P. H. Bolivar C. R. Bolognesi G. Bonaguide G. Bonmassar F. Boone V. Boria O. Boric-Lubecke A. Borji J. Bornemann W. Bosch R. Bosisio M. V. Bossche S. Boumaiza K. Boutros M. Bozzi T. Brabetz J. E. Bracken P. Bradley R. Bradley J. Brannan J. R. Bray T. Brazil J. Breitbarth M. Bressan P. Bretchko K. Breuer B. Bridges J. Brinkhoff A. Brown F. Broyde S. Brozovich S. Bruce E. Bryerton

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A. Deutsch V. Devabhaktuni Y. Deval T. Dhaene N. Dib C. Dietlein L. Ding A. Djordjevi J. Dobrowolski W. B. Dou P. Draxler R. Drayton A. Dreher J. Drewniak L. Dunleavy J. Dunsmore L. Dussopt M. Dvorak J. East K. W. Ecclestone M. L. Edwards R. Egri R. Ehlers N. Ehsan G. Eleftheriades F. Ellinger T. Ellis B. Elsharawy A. Elsherbeni N. Engheta T. Enoki M. Enqvist K. Entesari H. Eom K. Erickson N. Erickson C. Ernst D. Erricolo I. Eshrah M. Essaaidi H. Esteban M. C. Fabres C. Fager D. G. Fang S. J. Fang A. Faraone M. Farina W. Fathelbab A. E. Fathy A. P. Feresidis A. Fernandez A. Ferrero T. Fickenscher S. J. Fiedziuszko G. Fikioris I. Filanovsky F. Filicori D. Filipovic F. D. Flaviis B. Floyd P. Focardi N. H. Fong S. S. Fort K. Foster P. Foster C. C. Franco M. C. Francos J. C. Freire F. Frezza I. Frigyes J. Fu R. Fujimoto O. Fujiwara C. Fumeaux V. Fusco D. Gabbay T. Gaier J. D. Gallego B. Galwas A. Gameiro O. Gandhi S. Gao J. R. Garai H. Garbe J. A. Garcia K. Gard P. Gardner R. Garg J. L. Gautier B. Geelen F. Gekat B. Geller J. Gering F. German M. Geshiro S. Gevorgian H. Ghali M. Ghanevati F. Ghannouchi K. Gharaibeh G. Ghione K. Ghorbani O. Giacomo E. D. Giampaolo F. Giannini P. Gilabert A. Goacher M. Goano E. Godshalk M. Goldfarb P. Goldsmith M. Golosovsky R. Gonzalo S. Gopalsami D. Gope A. Gopinath R. Gordon A. Gorur M. Gottfried G. Goussetis W. Grabherr J. Graffeuil R. Graglia L. Gragnani J. Grahn G. Grau A. Grebennikov I. Gresham J. Grimm A. Griol E. Grossman Y. Guan J. L. Guiraud S. E. Gunnarsson L. Guo Y. X. Guo C. Gupta K. C. Gupta M. Gupta W. Gwarek J. Hacker S. Hadjiloucas S. H. Hagh S. Hagness A. Hajimiri D. Halchin P. Hale P. Hall D. Ham K. Hanamoto T. Hancock

A. Hanke E. Hankui G. Hanson Z. Hao H. Happy A. R. Harish L. Harle L. D. Haro F. J. Harris H. Harris M. Harris P. Harrison R. G. Harrison O. Hartin H. Hashemi K. Hashimoto O. Hashimoto J. Haslett S. Hay J. Hayashi L. Hayden T. Heath J. Heaton M. P. Heijden G. Heiter J. Helszajn R. Henderson D. Heo P. Herczfeld H. Hernandez J. J. Herren K. Herrick F. Herzel J. S. Hesthaven K. Hettak P. Heydari T. S. Hie M. Hieda A. Higgins A. Hirata J. Hirokawa T. Hirvonen J. P. Hof K. Hoffmann R. Hoffmann M. Hoft E. Holzman J. S. Hong S. Hong W. Hong A. Hoorfar K. Horiguchi Y. Horii T. S. Horng J. Horton J. Hoversten H. Howe H. M. Hsu H. T. Hsu J. P. Hsu P. Hsu C. W. Hsue M. Z. Hualiang C. W. Huang F. Huang G. W. Huang J. Huang T. W. Huang W. Huei M. Huemer H. T. Hui J. A. Huisman A. Hung C. M. Hung J. J. Hung I. Hunter M. Hussein E. Hutchcraft B. Huyart J. C. Hwang J. N. Hwang R. B. Hwang M. Hélier Y. Iida S. Iitaka P. Ikonen K. Ikossi M. M. Ilic A. Inoue T. Ishikawa T. Ishizaki S. Islam Y. Isota M. Ito N. Itoh T. Itoh Y. Itoh T. Ivanov D. Iverson M. Iwamoto Y. Iyama D. Jablonski D. Jachowski R. Jackson R. W. Jackson A. Jacob M. Jacob S. Jacobsen D. Jaeger B. Jagannathan V. Jamnejad V. Jandhyala M. Janezic M. Jankovic R. A. Jaoude B. Jarry P. Jarry J. B. Jarvis A. Jastrzebski B. Jemison W. Jemison S. K. Jeng A. Jenkins Y. H. Jeong A. Jerng T. Jerse P. Jia X. Jiang B. Jim J. G. Jiménez J. M. Jin J. Joe R. Johnk L. Jonathan J. Joubert E. J. Jr N. C. Jr R. Judaschke J. Juntunen D. Junxiong T. Kaho M. Kahrs T. Kaiser S. Kalenitchenko V. Kalinin T. Kalkur Y. Kamimura H. Kanai S. Kanamaluru H. Kanaya K. Kanaya

Digital Object Identifier 10.1109/TMTT.2007.890648

S. Kang P. Kangaslahtii V. S. Kaper B. Karasik N. Karmakar A. Karwowski T. Kashiwa L. Katehi H. Kato K. Katoh A. Katz R. Kaul R. Kaunisto T. Kawai K. Kawakami A. Kawalec T. Kawanishi S. Kawasaki H. Kayano M. Kazimierczuk R. Keam S. Kee L. C. Kempel P. Kenington A. Kerr A. Khalil A. Khanifar A. Khanna F. Kharabi R. Khazaka J. Kiang J. F. Kiang Y. W. Kiang B. Kim C. S. Kim D. I. Kim H. Kim H. T. Kim I. Kim J. H. Kim J. P. Kim M. Kim W. Kim S. Kimura N. Kinayman A. Kirilenko V. Kisel M. Kishihara A. Kishk T. Kitamura K. I. Kitayama T. Kitazawa T. Kitoh M. Kivikoski G. Kiziltas D. M. Klymyshyn R. Knochel L. Knockaert Y. Kogami T. Kolding B. Kolundzija J. Komiak G. Kompa A. Konczykowska H. Kondoh Y. Konishi B. Kopp K. Kornegay T. Kosmanis P. Kosmas Y. Kotsuka A. Kozyrev N. Kriplani K. Krishnamurthy V. Krishnamurthy C. Krowne V. Krozer J. Krupka W. Kruppa D. Kryger R. S. Kshetrimayum H. Ku H. Kubo A. Kucar A. Kucharski W. B. Kuhn T. Kuki A. Kumar M. Kumar C. Kuo J. T. Kuo H. Kurebayashi K. Kuroda D. Kuylenstierna M. Kuzuhara Y. Kwon G. Kyriacou P. Lampariello M. Lancaster L. Langley U. Langmann Z. Lao G. Lapin L. Larson J. Laskar M. Latrach C. L. Lau A. Lauer J. P. Laurent D. Lautru P. Lavrador G. Lazzi B. H. Lee C. H. Lee D. Y. Lee J. Lee J. F. Lee J. H. Lee J. W. Lee R. Lee S. Lee S. G. Lee S. T. Lee S. Y. Lee T. Lee T. C. Lee D. M. Leenaerts Z. Lei G. Leizerovich Y. C. Leong R. Leoni P. Leuchtmann G. Leuzzi A. Leven B. Levitas R. Levy G. I. Lewis H. J. Li L. W. Li X. Li Y. Li H. X. Lian C. K. Liao M. Liberti E. Lier L. Ligthart S. T. Lim E. Limiti C. Lin F. Lin H. H. Lin

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V. Volman A. Vorobiev A. V. Vorst B. Vowinkel L. D. Vreede M. A. Vérez B. Z. Wang K. Wagner K. Wakino P. Waldow M. Wale A. Walker D. Walker C. Walsh C. Wan S. Wane C. Wang C. F. Wang H. Wang N. Wang S. Wang T. Wang X. Wang Y. Wang J. Ward K. Warnick P. Warr S. Watanabe Y. Watanabe R. Waugh J. Webb K. Webb R. Webster C. J. Wei D. V. Weide R. Weigel G. Weihs B. Weikle R. M. Weikle C. Weil T. Weiland A. Weily S. Weinreb T. Weller S. Wentworth D. D. Wentzloff R. Wenzel J. Whelehan J. Whitaker D. A. White J. Wiart W. Wiesbeck J. Wight D. Willems D. Williams J. Wiltse D. Wittwer I. Wolff K. Wong W. Woo J. Wood C. Woods G. Woods R. C. Woods D. Woolard M. Wren C. Wu H. Wu K. Wu K. L. Wu Q. Wu T. L. Wu C. Wwang T. Wysocki S. Z. Xiang G. Xiao Y. Xiaopeng C. Xie Z. Xing H. Xu J. Xu S. Xu W. Xu X. B. Xu Y. Xu Y. P. Xu Q. Xue M. Yagoub T. Yakabe A. Yakovlev K. Yamamoto K. Yamauchi F. Yang G. M. Yang H. Y. Yang J. Yang K. Yang L. Yang X. Yang H. Yano F. Yanovsky H. W. Yao J. Yao B. Yarman A. G. Yarovoy Y. Yashchyshyn K. Yashiro K. Yasumoto Y. Yasuoka S. Ye K. S. Yeo S. P. Yeo L. K. Yeung H. R. Yi W. Y. Yin D. Yongsheng J. G. Yook Y. Yoon Y. J. Yoon R. York J. L. Young H. K. Yu M. Yu P. Yu W. Yu Z. Yuanjin P. Yue S. W. Yun A. I. Zaghloul A. G. Zajic K. Zaki J. Zamanillo P. J. Zampardi J. Zapata L. Zappelli J. Zehentner H. Zhang L. Zhang Q. J. Zhang R. Zhang X. Zhang A. P. Zhao J. Zhao Y. Zhao F. Zhenghe W. Zhiguo W. Zhou A. Zhu L. Zhu N. H. Zhu H. Zirath S. Zouhdi A. J. Zozaya T. Zwick