AUGUST 2010 IEEE MTT-V058-I08 (2010-08) [58, 8 ed.]

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AUGUST 2010

VOLUME 58

NUMBER 8

IETMAB

(ISSN 0018-9480)

Editorial .. ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .. D. Williams and A. Mortazawi

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PAPERS

Active Circuits, Semiconductor Devices, and ICs A Low-Power Full-Band Low-Noise Amplifier for Ultra-Wideband Receivers . .... R.-M. Weng, C.-Y. Liu, and P.-C. Lin CMOS Active Quasi-Circulator With Dual Transmission Gains Incorporating Feedforward Technique at -Band .... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... H.-S. Wu, C.-W. Wang, and C.-K. C. Tzuang Analysis and Design of a 1.6–28-GHz Compact Wideband LNA in 90-nm CMOS Using a -Match Input Network ... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ...... H.-K. Chen, Y.-S. Lin, and S.-S. Lu

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Signal Generation, Frequency Conversion, and Control Phase Noise of Distributed Oscillators ... ......... ........ ... ....... ......... ..... X. Li, O. O. Yildirim, W. Zhu, and D. Ham Analytical Modeling of Microwave Parametric Upconverters .... ......... ........ .. B. Gray, B. Melville, and J. S. Kenney Poly-Harmonic Modeling and Predistortion Linearization for Software-Defined Radio Upconverters .. ......... ......... .. .. ........ ......... ......... .... X. Yang, D. Chaillot, P. Roblin, W.-R. Liou, J. Lee, H.-D. Park, J. Strahler, and M. Ismail

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Wireless Communication Systems Design of Doherty Power Amplifiers for Handset Applications . ......... ........ .. D. Kang, J. Choi, D. Kim, and B. Kim A CMOS Wideband RF Front-End With Mismatch Calibrated Harmonic Rejection Mixer for Terrestrial Digital TV Tuner Applications ... ......... ........ ......... ......... ........ ......... ... H.-K. Cha, K. Kwon, J. Choi, H.-T. Kim, and K. Lee

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Field Analysis and Guided Waves Waves in a Semiconductor Periodic Layered Resonator . ......... ......... ........ .... A. A. Bulgakov and O. V. Shramkova Fields at a Finite Conducting Wedge and Applications in Interconnect Modeling ....... . T. Demeester and D. De Zutter

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover) CAD Algorithms and Numerical Techniques Robust Trust-Region Space-Mapping Algorithms for Microwave Design Optimization ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... S. Koziel, J. W. Bandler, and Q. S. Cheng Large Overlapping Subdomain Method of Moments for the Analysis of Frequency Selective Surfaces ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ A. Fallahi, A. Yahaghi, H. Abiri, M. Shahabadi, and C. Hafner Filters and Multiplexers Theory of Coupled Resonator Microwave Bandpass Filters of Arbitrary Bandwidth .... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... S. Amari, F. Seyfert, and M. Bekheit Dual-Band and Wide-Stopband Single-Band Balanced Bandpass Filters With High Selectivity and Common-Mode Suppression .... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... J. Shi and Q. Xue Varactor-Tuned Dual-Mode Bandpass Filters ..... ........ ......... ......... ........ ......... ......... W. Tang and J.-S. Hong Accurate Synthesis and Design of Wideband and Inhomogeneous Inductive Waveguide Filters ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ....... P. Soto, E. Tarín, V. E. Boria, C. Vicente, J. Gil, and B. Gimeno Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements Analysis and Measurement of a Time-Varying Matching Scheme for Pulse-Based Receivers With High- Sources ... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... X. Wang, L. P. B. Katehi, and D. Peroulis Design of Flip-Chip Interconnect Using Epoxy-Based Underfill Up to -Band Frequencies With Excellent Reliability .. .. ........ ......... ......... . L.-H. Hsu, W.-C. Wu, E. Y. Chang, H. Zirath, Y.-C. Hu, C.-T. Wang, Y.-C. Wu, and S.-P. Tsai An Intrinsic Circuit Model for Multiple Vias in an Irregular Plate Pair Through Rigorous Electromagnetic Analysis .. .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .. Y.-J. Zhang and J. Fan Miniaturized Coupled-Line Couplers Using Uniplanar Synthesized Coplanar Waveguides ...... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... C.-C. Wang, C.-H. Lai, and T.-G. Ma A Novel Miniaturized Forward-Wave Directional Coupler With Periodical Mushroom-Shaped Ground Plane . ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... S.-K. Hsu, C.-H. Tsai, and T.-L. Wu

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Instrumentation and Measurement Techniques Extraction of Intrinsic and Extrinsic Parameters in Electroabsorption Modulators ....... ... M. Yañez and J. C. Cartledge

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Microwave Photonics A Self-Started Laser Diode Pulsation Based Synthesizer-Free Optical Return-to-Zero On–Off-Keying Data Generator .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... Y.-C. Chi and G.-R. Lin

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Biological, Imaging, and Medical Applications Microwave Human Vocal Vibration Signal Detection Based on Doppler Radar Technology .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .... C.-S. Lin, S.-F. Chang, C.-C. Chang, and C.-C. Lin Exploring Joint Tissues With Microwave Imaging ....... ... S. M. Salvador, E. C. Fear, M. Okoniewski, and J. R. Matyas

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LETTERS

Corrections to “Analytical Extraction of Extrinsic and Intrinsic FET Parameters” ....... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... B. L. Ooi, Z. Zhong, and M.-S. Leong

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

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CALLS FOR PAPERS

Special Issue on RF Nanoelectronics ..... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .

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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 $17.00, plus an annual subscription fee of $23.00 per year for electronic media only or $46.00 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 S. M. EL-GHAZALY, President L. BOGLIONE W. CHAPPELL M. GUPTA

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

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Editorial UR EDITING team is key to the success of this TRANSACTIONS. The Associate Editors who commit to a term of service are dedicated individuals who work on a strictly voluntary basis to ensure the high quality of the manuscripts that are published; this is not only a serious commitment, but also ensures the continued success of the journal. The term of each Associate Editor varies because each handles a total of roughly 200 papers during their term of office. Our current team of Associate Editors includes Daniël De Zutter, Ian Gresham, Wolfgang Heinrich, Wei Hong, Robert W. Jackson, Jen-Tsai Kuo, Youngwoo Kwon, Jenshan Lin, Mauro Mongiardo, Jose Carlos Pedro, Zoya Popovic´, Dick Snyder, and Chi Wang. We would like to thank our indispensable team of Associate Editors for the time that they give freely and voluntarily to this TRANSACTIONS. Their work is important to us all.

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DYLAN WILLIAMS, Editor-in-Chief National Institute of Standards And Technology (NIST) Boulder, CO 80305 USA AMIR MORTAZAWI, Editor-in-Chief Department of Electrical and Computer Engineering The University of Michigan at Ann Arbor Ann Arbor, MI 48109-2122 USA

Daniël De Zutter (M’92–SM’96–F’01) is a Full Professor of Electromagnetics with the Engineering Faculty of Ghent University, Gent, Belgium. Over the past four years, he has served as Dean of the Engineering Faculty and is currently the Head of the Department of Information Technology, Ghent University. His research focuses on all aspects of circuit and electromagnetic modeling of high-speed and high-frequency interconnections and packaging, electromagnetic compatibility (EMC), and numerical solutions of Maxwell’s equations.

Ian Gresham (S’91–M’93–SM’03) joined the Boston Design Center, NXP Semiconductors, Boston, MA, as Director of Research and Development. His research interests have focused on the commercialization of millimeter-wave systems with a particular emphasis on automotive radar. His research has also focused on the deployment of Si-based integrated circuits as an enabling technology for millimeter-wave transceiver chipsets, initially at 24 GHz.

Digital Object Identifier 10.1109/TMTT.2010.2052663 0018-9480/$26.00 © 2010 IEEE

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Wolfgang Heinrich (M’84–SM’95–F’09) has been with the Ferdinand-Braun-Institut (FBH) Berlin, Germany, since 1993, as Head of the Microwave Department and Deputy Director. Since 2009, he has also been a Professor with the Technical University of Berlin. His current research activities focus on monolithic microwave integrated circuit (MMIC) design with emphasis on GaN power amplifiers, millimeter-wave packaging, and electromagnetic simulation.

Wei Hong (M’92–SM’07) has been with the State Key Laboratory of Millimeter Waves, Southeast University, Nanjing, China, since 1998 and has been the Director since 2003. He is currently a Professor and the Associate Dean of the School of Information Science and Engineering, Southeast University. His research interests include numerical methods for electromagnetic problems, millimeter-wave theory and technology, antennas, electromagnetic scattering, inverse scattering and propagation, RF front-ends for mobile communications, and interconnect parameter extraction.

Robert W. Jackson (M’82–SM’88–F’04) joined the faculty of the University of Massachusetts at Amherst, in 1982 and is currently a Professor of Electrical and Computer Engineering. His primary research and teaching interests center on microwave and millimeter-wave electronics, especially integrated circuits. He has specifically contributed in the areas of numerical modeling of microstrip and coplanar waveguide circuits, novel circuit structures, and the modeling of packages for microwave and millimeter-wave integrated circuits. His current interests include miniature low-cost devices for sensing applications, active antennas, and high-frequency CMOS.

Jen-Tsai Kuo (S’88–M’92–SM’03) has been with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, Japan, where he has been a Professor since 1984. His research interests include analysis and design of microwave integrated circuits and numerical techniques in electromagnetics.

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Youngwoo Kwon joined the faculty of the School of Electrical Engineering, Seoul National University, Seoul, Korea, in 1996, where he is currently a Professor. His research interests include microwave integrated circuits and systems, RF microelectromechanical systems (MEMS), power amplifier design, nonlinear device modeling, and biomedical application of microwaves.

Jenshan Lin (S’91–M’91–SM’00–F’10) joined the University of Florida, Gainesville, in 2003, as an Associate Professor, and became a Professor in August 2007. His research interests include sensors and biomedical applications of microwave and millimeter-wave technologies, wireless energy transmission, RF system-on-chip integration, and integrated antennas.

Mauro Mongiardo (M’91–SM’00) has been a Full Professor with the University of Perugia, Perugia, Italy, since 2001. His research interests include numerical methods to model electromagnetic (EM) fields, in particular for computer-aided design (CAD) of microwave and millimeter-wave passive components. He is also interested in the development of new designs for microwave components and filters.

Jose Carlos Pedro (S’90–M’95–SM’99–F’07) is currently a Full Professor with the University of Aveiro, Aveiro, Portugal. He is also a Senior Research Scientist with the Institute of Telecommunications, University of Aveiro. His main scientific interests include active device modeling and the analysis and design of various nonlinear microwave and opto-electronics circuits, in particular, the design of highly linear multicarrier power amplifiers and mixers.

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Zoya Popovic´ has been with the University of Colorado at Boulder, since 1990, where she is currently the Hudson Moore Jr. Chaired Professor with the Department of Electrical, Computer, and Energy Engineering. Her research interests include high-efficiency, low-noise, and broadband microwave and millimeter-wave circuits, quasi-optical millimeter-wave techniques for imaging, smart and multibeam antenna arrays, intelligent RF front ends, RF optics, and wireless powering for batteryless sensors.

Dick Snyder (S’58–M’63–SM’80–F’97–LF’05) is President of RS Microwave, Butler, NJ, where he combines the business of filters with the pleasures of the technical discipline. His interests include teaching the filter subject matter. He also enjoys authoring, reviewing, and editing manuscripts with filters and related structures as subject matter. Dr. Butler is currently vice president of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S).

Chi Wang (S’95–A’97–SM’98) has been with the Orbital Sciences Corporation, Sterling, VA, for eight years, where he is currently a Program Director. His current research interests include satellite and wireless communication systems, advanced microwave resonator and filter technology, numerical technology, and computer-aided design of advanced RF and microwave circuits for wireless communication systems.

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A Low-Power Full-Band Low-Noise Amplifier for Ultra-Wideband Receivers Ro-Min Weng, Member, IEEE, Chun-Yu Liu, Student Member, IEEE, and Po-Cheng Lin

Abstract—In this paper, a low-power full-band low-noise amplifier (FB-LNA) for ultra-wideband applications is presented. The proposed FB-LNA uses a stagger-tuning technique to extend the full bandwidth from 3.1 to 10.6 GHz. A current-reused architecture is employed to decrease the power consumption. By using an input common-gate stage, the input resistance of 50 can be obtained without an extra input-matching network. The output matching is achieved by cascading an output common-drain stage. FB-LNA was implemented with a TSMC 0.18- m CMOS process. On-wafer measurement shows an average power gain of 9.7 dB within the full operation band. The input reflection coefficient and the output reflection coefficient are both less than 10 dB over the entire band. The noise figure of the full band remained under 7 dB with a minimum value of 5.27 dB. The linearity of input third-order intercept point is 2.23 dBm. The power consumptions at 1.5-V supply voltage without an output buffer is 4.5 mW. The chip area occupies 1.17 0.88 mm2 .



Index Terms—Common gate, low-noise amplifier (LNA), ultrawideband (UWB).

I. INTRODUCTION LTRA-WIDEBAND (UWB) systems have been used on consumer electronic applications such as vehicular radars for high-data-rate wireless connections and high-accuracy locating ability. UWB standards have been defined by the U.S. Federal Communications Commission (FCC), Washington, DC, since 2002.1 UWB systems are approved for use in bandwidth from 3.1 to 10.6 GHz. An extremely low-power signal over a full-band radio spectrum of 7.5 GHz is allowed to transmit and/or receive. Direct-sequence architectures and orthogonal frequency-division multiplexing architectures are commonly seen in UWB systems [1]. Low-noise amplifiers (LNAs) in front-end receivers of both architectures are required to amplify all band signals from antennas. Low noise, full bandwidth, good gain flatness, large power gain, and low power consumption are noted issues in UWB LNA design. The feed-forward noise-cancelling technique was proposed to break the tradeoff between noise factor and input impedance matching [2]. However, high power consumption is inevitable to drive the circuit. The bandwidth, limited by

U

Manuscript received September 21, 2009; revised March 02, 2010; accepted April 23, 2010. Date of publication June 28, 2010; date of current version August 13, 2010. This work was supported by the National Science Council under Contract NSC95-2221-E-259-038 and Contract NSC97-2221-E-259-034. R.-M. Weng and C.-Y. Liu are with the Department of Electrical Engineering, National Dong Hwa University, Hualien 97401, Taiwan (e-mail: romin@mail. ndhu.edu.tw). P.-C. Lin was with the Department of Electrical Engineering, National Dong Hwa University, Hualien 97401, Taiwan. He is now with Mobile Internet Device, Tao Yuan 33383, Taiwan. Digital Object Identifier 10.1109/TMTT.2010.2052404 1FCC,

Jun. 2007. [Online]. Available: http://www.fcc.gov

a 3-dB gain, is also not wide enough for UWB systems. An LNA with resistive shunt feedback topology can achieve good input matching, high gain, and wideband performance [3]–[6]. A feedback resistor was utilized to extend the bandwidth. Unfortunately, the performance was degraded when the LNA was operated in the high frequency band; this is due to the effect of parasitic capacitances. Distributed amplifiers were presented to obtain wideband characteristics [7]–[9]. Since distributed amplifiers are composed by cascading several stages, it is difficult to meet the low-power requisition. Multistage LNAs were designed for full-band operation and broadband matching [10], [11]. However, multistage topologies need multiple dc biasing paths, which largely increase the total power consumption. L-degenerated LNAs with input low-pass filters can provide low power, wide input matching, and low noise [12], [13]. Such topologies require several high- inductors to achieve wideband input matching. Wideband LNAs using input low-pass filters are hardly realized if the chip area is the main design issue. The concept of the mutual coupling technique was implemented through a symmetric center-tap inductor to obtain low power and low noise [14]. The output reflection coefficient cannot, however, be kept lower than 10 dB during the entire bandwidth. A low-power full-band low-noise amplifier (FB-LNA) for UWB receivers was previously discussed with only simulation results [15]. In this paper, more theoretical analysis and measurement results, which emphasize the performances of the proposed FB-LNA, are provided. In Section II, the design concepts, such as bandwidth extension, input/out impedance matching, and noise analysis are analyzed. The implementation of FB-LNA is illustrated in Section III. Section IV presents both the simulated and measured results of the FB-LNA. Section V presents the conclusion of this design. II. PROPOSED LOW-POWER FB-LNA The proposed FB-LNA is shown in Fig. 1. There are two and stages including an input common-gate (CG) amplifier a common-source (CS) amplifier . The first stage provides an input impedance matching of 50 for RF signals from an antenna to LNA. The second stage acts as a gain stage, which assures that the weak RF signals will be greatly amplified. The first and second stages are constructed as cascode configurations beto lower power consumption. The input impedance of . An output buffer, recognized as comes an output load for a source follower , is cascaded with the second stage. is . a current source, which provides a biasing current for and are part of the input matching network. and are . is added to supply ac coupling paths for signals into a dc bias for . A high-frequency ac current into source of flows to ground by adding the bypass-capacitor , which

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Fig. 3. Illustration of full-band frequency response.

Fig. 1. Proposed low-power FB-LNA.

Fig. 4. First inter-stage: (a) schematic and (b) equivalent-circuit model.

Fig. 2. Two-stage CG-CS amplifiers of: (a) cascaded topology. (b) Cascoded as current-reusing topology.

avoids the signal interference coupling back to . and are part of the output matching network. For a conventional two-stage CG-CS amplifier design, as deand are connected as a cascaded picted in Fig. 2(a), provides a signal coupling path bestructure. The capacitor tween and . It is shown that two current biasing streams to match the input impedance and for to are required for amplify the received signal. Large power consumption of such a CG-CS cascaded topology is inevitable. With the same sizes of and , the simulated power consumption of the cascaded amplifier in Fig. 2(a) is 8.62 mW, while the current-reusing amplifier in Fig. 2(b) consumes only 4.05 mW. Considering the similar specifications of noise, bandwidth, and linearity for both circuits in Fig. 2, the power consumption of the cascaded amplifier is decreased to 5.02 mW; the currentreusing amplifier is operated at 4.05 mW. The simulation results demonstrate that the power-saving efficiency for the currentreusing amplifier is about 24%. To lower the power consumption, FB-LNA employs a cascoded structure known as a currentreusing topology, shown in Fig. 2(b). Cascoded structures are transformed from cascaded structures without changing the amand allow plifier types. The passive elements between

the RF signal to pass through and then through without an extra dc current path. A stagger tuning technique is adopted to extend ultra-wide bandwidth, as shown in Fig. 3. The inter-stage resonant circuits generate two different resonate frequencies and . The parasitic inductances and capacitances, embedded and , is designed to resin the first inter-stage between onate at in low band. The second inter-stage, between and , is resonated at in high band. These two resonant frequencies, provided by the stagger tuning technique, achieves the flat output gain for the first and second inter-stages bracing all UWB bandwidth. A. Low-Band Resonant Frequency of the First Inter-Stage The first inter-stage with parasitic impedances is shown in Fig. 4(a). The low-band resonant frequency of the FB-LNA is decided by the elements of the first inter-stage, which conand . sists of passive elements such as is the sum of the gate–source parasitic capacitance and the . represents the parasitic capaciMiller capacitance of . The small-signal equivalent-cirtance at the drain node of cuit model is simplified and illustrated in Fig. 4(b). The output can be calculated as impedance of (1) with

(2) (3)

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Fig. 5. Second inter-stage: (a) schematic and (b) equivalent-circuit model.

where is the transconductance of impedance is simplified as

. The output

(4)

Fig. 6. Simulated power gain with different LC elements.

The denominator can be derived as (5) (6) is a third-order frequency-dependent function, which is is relative to . calculated as (7). The denominator of , decided by , are located at the Since the poles of infinite of the frequency spectrum, it will not affect the low-band becomes resonant frequency. The voltage gain of

where and are the total parasitic resistance and capaci, respectively. intance obtained from the drain node of and the cludes both the drain–source parasitic resistance of resistance of the spiral inductor . There are also three parasitic . They are the gate–source capacicapacitances included in , gate–drain capacitance of , and drain–source tance of . is the denominator of the output capacitance of impedance, which can be derived as (13)

(7) (8) where is the source impedance and is the transconductance of . Hence, the low-band resonant frequency can be derived as

For bypasses ac signal to ground, the voltage gain of the CS stage becomes (14) The high-band resonant frequency of the second inter-stage is (15)

(9) It is noted that the low-band resonant frequency can be deterand according to the specificamined by adjusting tion of the lower bound in the FB-LNA design. B. High-Band Resonant Frequency of the Second Inter-Stage and the The high-band resonant frequency is provided by parasitic capacitances of the second inter-stage between and . The schematic of the second inter-stage is depicted in Fig. 5(a). The small-signal equivalent-circuit model is shown in can be calFig. 5(b). The output equivalent impedance of culated as (10) (11) (12)

It is seen that the high-band resonant frequency can be deterproperly. mined by choosing As derived in (9) and (15), the low-band resonant frequency and , whereas the high-band resocan be varied by nant frequency is tuned by . The theoretical analysis is proven by the simulation results given in Fig. 6. The adjusting paramand , which eters of an on-chip spiral inductor are represent the width of the metal line in micrometers, the radius in micrometers, and the number of turns. For a standard capacitor, the length and width of conductive plates, using the fifth and sixth metal layers, are 30 and 30 m, respectively. Hence, the element value of a capacitor is decided by choosing the number of standard capacitors . As shown in Fig. 6, the low-band resonant frequency moves to a lower frequency band and/or . The high-band resonant frequency by increasing also shifts to a higher frequency band by decreasing . However, the gain flatness is greatly affected while adjusting different passive elements.

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Fig. 7. Small-signal equivalent circuit of input common stage.

C. Input Impedance-Matching Design provides good performance in input The input CG stage impedance matching of 50 . Fig. 7 shows the small-signal . The spiral inductor is chosen equivalent-circuit model of from the standard elements provided by TSMC with a relatively is 8.9 at 5 GHz. high- factor. The maximum factor of During the full band operation, factors of are larger than and generate a resonance at the center of the entire 5. band. The input resistance is designed around 50 in the Smith chart. The input impedance of the FB-LNA can be given as

Fig. 8. Simulated NF under different power consumption.

TABLE I ELEMENT SIZES OF FB-LNA

(16) where is the drain–source resistance of . is the . is the load impedance at the output drain node of impedance obtained from source to gate of , including the and other parasitic elements. The gate–source capacitance and are relatively large in the modified impedances frequency and can be neglected in the input impedance. Therefore, the input impedance matching can be obtained as (17) is smaller than over a wide frequency range in the CG stage. Considering a 50- matching condition, is by selecting the width and the bias designed to be equal to . voltage of The CG topology also provides a better isolation and electrostatic discharge protection [16]. In a conventional cascode structure, a CS stage is used to achieve both large gain and input is between input matching. For the gate–drain capacitance, and output nodes, the isolation from input to output is degraded and , in in the high-frequency band. On the other hand, a CG topology, are both connected to ground and the isolation can therefore be improved. D. Noise Analysis The noise figure (NF) of the FB-LNA is mainly affected by . According to the noise analysis, the the input CG stage noise factor can be given by [11] (18)

where is the coefficient of the channel thermal noise and is is kept smaller the zero-bias drain conductance. Since than within the entire band, the noise factor becomes

(19) Ideally, the NF remains constant according to the above equation. However, the low-band resonant frequency depends slightly on . As resonates at 3 GHz, the power gain of the first stage can achieve the maximum value and the minimum NF. Nevertheless, the quality factor of an on-chip inductor becomes worse, which causes unwanted noises to inject into the system. When compared to a CS stage with a high-order bandpass filter for UWB input matching, only one inductor, , is required in the FB-LNA to conform the wideband matching. Since there are less inductors, the FB-LNA can resist the noise greatly. . is affected by The NF is inversely proportional to power consumption, which is mainly decided by the biasing cur. The relationship between the power consumption rent of and the NF is shown in Fig. 8. As obtained from Fig. 8, of a smaller NF can be achieved by increasing the power. This is

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Fig. 9. Simulated and measured results of input reflection coefficient.

Fig. 11. Simulated and measured results of power gain.

Fig. 10. Simulated and measured results of output reflection coefficient.

Fig. 12. Simulated and measured results of isolation.

commonly used in the design of LNAs to trade power consumption for the NF. In order to keep the NF below 6 dB within the entire UWB band, the power consumption of the FB-LNA is chosen to be 4.05 mW. The most common interference to UWB systems is the strong narrowband signals received from wireless local area network (WLAN) systems. It is necessary to filter out the WLAN signals at a frequency of 5–6 GHz in the allocated UWB spectrum. An active notch filter was added in an UWB LNA to eliminate the interference [17]. However, more power is required to operate the tunable notch filter. Typically, such interference can be avoided by implementing microwave passive filters, which are embedded in the receiving path of UWB receivers after antennas.

and ground at the capacitance between the bottom plate of . Additionally, of the second inter-stage the drain node of and is chosen to resonate at 11 GHz. The inductances of are 9.7 and 3.3 nH, respectively. , implemented by a symmetric inductor, is 1.6 nH. is selected to be 3.3 nH. acts is adjusted to 4 pF as a coupling capacitor, which is 2 pF. . is 5.5 k , in order to provide an ideal ac ground for for . The output which provides the bias voltage from impedance is matched to 50 by a buffer stage for RF measurement purposes. The output buffer is biased at 4.86 mA. The element sizes of the FB-LNA are listed in Table I.

III. IMPLEMENTATION OF FB-LNA In order to achieve low-power consumption, the bias current of the FB-LNA without a buffer is set to be 2.7 mA. The transconductance is designed to be approximately 20 mS. is calculated to be 144 m. The width Hence, the width of of is 80 m, to meet the linearity specification. is chosen to resonate at 3 GHz with the parasitic capacitances, as well as

IV. SIMULATED AND MEASURED RESULTS The proposed FB-LNA is simulated with a TSMC 0.18- m CMOS RF model. The performances of gain, input/output matching, power consumption, noise, and linearity are specified in order to regulate the values of the circuit components. The post-layout simulation results are provided at 1.5 V. The measurement of the FB-LNA is performed by an on-wafer method. The power gain is related to the reflection is decided by the input matching condition. coefficients. is adjusted by the source The output reflection coefficient

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TABLE II PERFORMANCE COMPARISON OF UWB LNAs WITH CMOS PROCESS

Fig. 13. Simulated and measured results of NF. Fig. 15. Measurement result of P

.

Fig. 14. Two-tone test measurement at 6 GHz.

follower. Figs. 9 and 10 are the measured results of the input reflection coefficient and the output reflection coefficient , respectively. These transmission zeros, which existed and , are caused by the series at 3 GHz for both resonance of the parasitic inductance and capacitance. The measured results of both and are kept less than 10 dB within full band, which allows the transferring of RF signals and conform the UWB standard. As shown in Fig. 11, the output matching load of 50 can achieve a maximum power gain of 12 dB from 3.1 to 10.6 GHz. , as measured in Fig. 12, is less than 43 dB; The isolation, this reduces the local oscillation leakage arising from the capacitive paths and the substrate coupling. The degradation of

Fig. 16. Die photograph of the proposed FB-LNA.

the measured isolation in the low-frequency band is due to the coupling effects, which are caused by the parasitic capacitances between interconnects and elements. A minimum NF of 5.27 dB is shown in Fig. 13. To observe the nonlinear behavior, a two-tone signal test at 6 and 6.01 GHz, with equal power levels, is applied. Fig. 14 indicates that FB-LNA has an input third-order intercept point (IIP3) of 2.23 dBm and 1-dB compression point. is 10 dBm, as shown in Fig. 15.

WENG et al.: LOW-POWER FB-LNA FOR UWB RECEIVERS

The die photograph is shown in Fig. 16. The chip area is 0.88 mm . The circuit floor plan is carefully arranged 1.17 to minimize the lengths of the interconnect metal lines in order to reduce the effects of the parasitic capacitance. The characteristic performance is summarized in Table II along with other UWB LNAs for comparison. V. CONCLUSION A low-power FB-LNA for receiver front-ends in UWB systems is demonstrated. The CG, cascoded to CS topology, provides a current-reuse technique to save the power consumption. FB-LNA not only meets the UWB standard, but also achieves high gain, low noise, full-band operation, small chip area, and low-power consumption. From 3.1 to 10.6 GHz, the maximum power gain is 12 dB. The minimum NF is 5.27 dB. The total power consumption, including the buffer, is 12 mW with 1.5-V supply voltage. It is shown that the designed FB-LNA is capable of both reducing the power dissipation and achieving good gain flatness. Since no off-chip components are required, it can be easily integrated into a single chip of UWB systems with high-data-rate transmission and high-accuracy locating ability. ACKNOWLEDGMENT The authors wish to thank the Chip Implementation Center (CIC), Hsinchu, Taiwan, and the National Nano Device Laboratory (NDL), Hsinchu, Taiwan, for supporting TSMC process parameters, chip fabrication, and measurement. REFERENCES [1] Y. Lu, K. S. Yeo, J. G. Ma, M. A. Do, and Z. Lu, “A novel CMOS low-noise amplifier design for 3.1–10.6 GHz ultra-wideband wireless receivers,” IEEE Trans. Circuit Syst. I, Fund. Theory Appl., vol. 53, no. 8, pp. 1683–1692, Aug. 2006. [2] F. Bruccoleri, E. A. M. Klumperink, and B. Nauta, “Wide-band CMOS low-noise amplifier exploiting thermal noise canceling,” IEEE J. SolidState Circuits, vol. 39, no. 2, pp. 275–282, Feb. 2004. [3] H. Knapp, D. Zoschg, T. Meister, K. Aufinger, S. Boguth, and L. Treitinger, “15 GHz wideband amplifier with 2.8 dB noise figure in SiGe bipolar technology,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., June 2003, pp. 287–290. [4] S. Andersson, C. Svensson, and O. Drugge, “Wideband LNA for a multistandard wireless receiver in 0.18 m process,” in Proc. Eur. SolidState Circuits Conf., Sept. 2003, pp. 655–658. [5] R. Gharpurey, “A broadband low-noise front-end amplifier for ultra wideband in 0.13 m CMOS,” in Proc. IEEE Custom Integr. Circuits Conf., Oct. 2004, pp. 605–608. [6] C.-W. Kim, M.-S. Kang, P. T. Anh, H.-T. Kim, and S.-G. Lee, “An ultra-wideband CMOS low noise amplifier for 3–5-GHz UWB system,” IEEE J. Solid-State Circuits, vol. 40, no. 2, pp. 544–547, Feb. 2005. [7] R.-C. Liu, C.-S. Lin, K.-L. Deng, and H. Wang, “A 0.5–14 GHz 10.6 dB CMOS cascode distributed amplifier,” in VLSI Circuits Symp. Tech. Dig., Jun. 2003, pp. 139–140. [8] F. Zhang and P. Kinget, “Low power programmable-gain CMOS distributed LNA for ultra-wideband applications,” in VLSI Circuits Symp. Tech. Dig., Jun. 2005, pp. 78–81. [9] M.-U. Nair, Y.-J. Zheng, and Y. Lian, “1 V 0.18 m-area and power efficient UWB LNA utilising active inductors,” IET Electron Lett., vol. 44, no. 19, pp. 1127–1129, Sep. 2008. [10] C. T. Fu and C. N. Kuo, “3–11 GHz CMOS UWB LNA using dual feedback for broadband matching,” in IEEE Radio Freq. Integr. Circuits Symp., Jun. 2006, pp. 53–56. [11] Y. Shim, C.-W. Kim, J. Lee, and S.-G. Lee, “Design of full band UWB common-gate LNA,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 10, pp. 721–723, Oct. 2007.

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[12] A. Bevilacqua and A. M. Niknejad, “An ultra-wideband CMOS LNA for 3.1 to 10.6 GHz wireless receivers,” IEEE Int. Solid-State Circuits Conf. Tech. Dig., pp. 382–383, Feb. 2004. [13] A. Bevilacqua, C. Sandner, A. Gerosa, and A. Neviani, “A fully integrated differential CMOS LNA for 3–5-GHz ultrawideband wireless receivers,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 3, pp. 134–136, Mar. 2006. [14] A. Meaamar, B. C. Chye, D. M. Anh, and Y. K. Seng, “A 3–8 GHz low-noise CMOS amplifier,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 4, pp. 245–247, Apr. 2009. [15] R.-M. Weng and P.-C. Lin, “A 1.5-V low-power common-gate low noise amplifier for ultra-wideband receivers,” in Proc. IEEE Int. Circuits Syst. Symp., May 27–30, 2007, pp. 2618–2621. [16] S. K. Tank, C. F. Chan, C. S. Choy, and K. P. Pun, “A 1.2 V, 1.8 GHz CMOS two-stage LNA with common-gate amplifier as an input stage,” in Proc. 5th Int. ASIC Conf., Oct. 2003, vol. 2, pp. 1042–1045. [17] B.-H. Park, S.-S. Choi, and S.-C. Hong, “A low-noise amplifier with tunable interference rejection for 3.1- to 10.6-GHz UWB systems,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 1, pp. 40–42, Jan. 2010. [18] C.-F. Liao and S.-I. Liu, “A broadband noise-canceling CMOS LNA for 3.1–10.6-GHz UWB receivers,” IEEE J. Solid-State Circuits, vol. 42, no. 2, pp. 329–339, Feb. 2007. [19] H.-Y. Yang, Y.-S. Lin, and C.-C. Chen, “2.5 dB NF 3.1–10.6 GHz CMOS UWB LNA with small group-delay variation,” IET Electron Lett., vol. 44, no. 8, pp. 528–529, Apr. 2008. [20] J.-F. Chang and Y.-S. Lin, “3–10 GHz low-power, low-noise CMOS distributed amplifier using splitting-load inductive peaking and noise-suppression techniques,” IET Electron. Lett., vol. 45, no. 20, pp. 1033–1035, Sep. 2009.

Ro-Min Weng (M’08) received the B.S. degree in electrical engineering from National Cheng Kung University, Tainan City, Taiwan, the M.S. degree in electronics engineering from National Chiao Tung University, Hsinchu, Taiwan, and the Ph.D. degree in electrical engineering from National Taiwan University, Taiwan. She is currently an Associate Professor and Chair with the Department of Electrical Engineering, National Dong Hwa University, Hualien, Taiwan. Her research is focused on the design of low-power analog and mixed-signal integrated circuits for wireless communication systems.

Chun-Yu Liu (S’08) was born in Tainan, Taiwan, in 1980. He received the B.S. and M.S. degrees in electrical engineering from National Dong Hwa University, Hualien, Taiwan, in 2004 and 2006, respectively, and is currently working toward the Ph.D. degree in electrical engineering from National Dong Hwa University. His research interests include RF front-end circuits and mixed-signal circuits.

Po-Cheng Lin was born in Kaohsiung, Taiwan, in 1982. He received the M.S. degrees in electrical engineering from National Dong Hwa University, Hualien, Taiwan, in 2007. He is currently a Design Engineer with Mobile Internet Device, Tao Yuan, Taiwan, where he specializes in RF design for GSM and WCDMA applications. His research has focused on CMOS RF integrated circuits.

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CMOS Active Quasi-Circulator With Dual Transmission Gains Incorporating Feedforward Technique at -Band

K

Hsien-Shun Wu, Member, IEEE, Chao-Wei Wang, Student Member, IEEE, and Ching-Kuang Clive Tzuang, Fellow, IEEE

Abstract—This paper presents an innovative architecture for the active quasi-circulator to break the structural limitation on the leakage suppression. A practical prototype is implemented at -band and fabricated by using standard 0.18- m 1P6M CMOS technology. The comparisons between simulations and on-wafer measurements are reported in detail to confirm the feasibility and capability of the proposed active quasi-circulator. At 24 GHz, the prototype has dual transmission gains of 22.4 and 12.3 dB in the transmitting and receiving paths, respectively. The measured isolation, which is defined by the ratio of the forward transmission to reverse transmission coefficient in the same path, is higher than 50.0 dB at 24.0 GHz. By comparing the leakage phenomena between the prototype with and without feedforward cancellations, the leakage suppression can be improved with a maximum value of 44.7 dB at 23.63 GHz. Index Terms—CMOS, quasi-circulator.

I. INTRODUCTION HE ferrite-based circulator had been recognized as the industry-promised solution to realize the RF signal processing in the first stage of an RF module [1]–[6]. Parallel to the circulator designs that rely on magnetic materials, the active circulator and quasi-circulator had also been reported [7]–[26]. The active circulator can be simply realized by only three transistors [7], [8] or the composite networks consisting of transistors and the power combiners [9]–[12]. The elements in these active circulators, which are connected in the ring-type topology, can realize complete circularity for power flow in the same direction of the three-port network [see Fig. 1(a)]. The transmission gains in the forward direction of these circulators are dependent on the forward transmission coefficients of the transistors and the losses of the combiner. The isolations in the backward direction of the circulators are relied on the reverse transmission coefficients of the transistors and the isolations of

T

Manuscript received November 24, 2009; revised April 06, 2010; accepted April 13, 2010. Date of publication June 28, 2010; date of current version August 13, 2010. This work was supported by the National Science Council of Taiwan under Grant NSC 97-2221-E-002-059-MY3 and by the Excellent Research Projects of National Taiwan University, 98R0062-03. The authors are with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan (e-mail: mickwu2@yahoo. com.tw; [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.2010.2052405

Fig. 1. Three-port circulators. (a) Circulator. (b) Quasi-circulator. (c) Active quasi-circulator with feedforward cancellation.

the power combiners. On the other hand, the three-port quasicirculator supports only two power flows in the same direction, showing an incomplete rotational symmetry, as illustrated in Fig. 1(b) [13]–[26]. The quasi-circulators can be realized in the following two types. First is the circulator constructed by the in-phase/out-of-phase power combiner, and the out-ofphase/in-phase power divider [13]–[19]. The second approach consists of the multiple hybrids and amplifiers [20]–[26]. Similar to those of active circulators, the forward transmission gains of the quasi-circulator are dependent upon the losses of the hybrids and the forward transmission coefficients of the amplifiers. These composite networks result in the quasi-circulators either having the losses on both two power flows [13]–[15], [19], [20] or achieving the transmission gain on one power flow [21], [22], [24]–[26]. Some prototypes, which are realized by the transistor-based power divider and combiner, can also establish the dual transmission gains for both power flows [16]–[18]. The isolations of the first kind of quasi-circulators are relied on the nonreciprocity of the transistors and the precisions on the phase and

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WU et al.: CMOS ACTIVE QUASI-CIRCULATOR WITH DUAL TRANSMISSION GAINS

amplitude matching between the in-phase/out-of-phase power combiner and out-of-phase/in-phase power divider [13]–[19]. In the second kind of quasi-circulator, the isolation is dominated by the isolation of the hybrids, and the attention must also be paid on the increasing of the leakage inside the circulator [24]. The increasing of the leakage is often proportional to the transmission gain. The hybrids need to have high isolation for alleviating the leakage phenomena occurring within [24]. This paper presents the feedforward technique to design active quasi-circulator, simultaneously achieving the transmission gains and the high isolations in two power flows. Fig. 1(c) shows the concept of the proposed architecture. Differing from the quasi-circulators with dual transmission gains in [16]–[18], the core of the quasi-circulator consists of two amplifiers (AMP1 and AMP2) and two directional couplers (C1 and C2). The feedforward path integrated inside the quasi-circulator is applied to resolve the leakages caused by the couplers with finite isolation. The operating principles of the architecture are reported in Section II. Followed by the designs of the building blocks presented in Section III, the measured results of a -band prototype in the standard CMOS 0.18- m 1P6M technology are compared with the simulated results in Section IV. Section V concludes this paper. II. ACTIVE QUASI-CIRCULATOR WITH FEEDFORWARD CANCELLATION As shown in Fig. 1(c), the active quasi-circulator defined two power flows. One is the transmitting path from transmitting (TX) terminal to the antenna (ANT) terminal, and the other is the receiving path from ANT terminal to receiving (RX) terminal. Any transmissions between TX and RX terminals are regarded as the leakages of the quasi-circulator and can be eliminated by the integrated feedforward cancellation. The transfer functions for the transmitting and receiving paths in Fig. 1(c) can be simply expressed by (1) and (2), respectively, as follows: (1) (2) , , and are the signals at the In (1) and (2), terminals of the active quasi-circulator. All building blocks of Fig. 1(c) have a common reference impedance of 50 . Consequently, the forward transmission of a given signal path is the product of the individual transmission coefficient in the signal , for example, is the product of , , path. , , and . is the complex forward transmission coefficient of the two-port -parameter of the active-bandpass filter (ABPF). and are the thru-port and coupled-port transmission coefficients of the coupler C1, and coupler C2. For the , and purpose of clarity, (1) abbreviates the product terms to represents the forward transmission coefficient and phase shifting of the receiving path. Since coupler C2 has finite isolation, the output signals from the thru-port of C2 consist of the signals from the ANT terminal and the transmitting path. Therefore, the second term of (2) represents the leakage content. The quasi-circulator in [24] used the directional coupler with an isolation over 30 dB to reduce the

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leakage. However, in the proposed quasi-circulator, the leakage can be eliminated from the output of the RX terminal by performing the feedforward cancellation. As shown in Fig. 1(c), the feedforward path, which is realized by the magnitude/phase control unit (MPCU), will reproduce the replica of the leakage , in the receiving path, except a phase difference of 180 . the third term in (2), represents forward transmission gain and phase shifting of the feedforward path. By combining the signals from the thru-port of coupler C2 with the output signals from the MPCU, the leakage contributed by the coupler C2 can be removed. Since the out-of-phase leakage cancellation is made by the open-loop fashion, the leakage cancellation is thus of feedforward type [27]. Additionally, (3) defines the leakage suppresis equal to sion of the proposed quasi-circulator. When , the signal in the RX terminal is equal to the product and . The leakage suppression is approached to of the infinite, revealing the perfect isolation between TX and RX terminals Leakage Suppression (3)

III. IMPLEMENTATION OF ACTIVE QUASI-CIRCULATOR: CMOS SYNTHETIC TRANSMISSION LINES (TLs) APPROACH A. CMOS Active Quasi-Circulator The fundamental principles of the proposed active quasi-circulator with feedforward cancellation [see (2)] are based on a major assumption that the leakage signals in the receiving path are only from the transmitting path. The assumption dictates that the additional coupling of the adjacent building blocks must be low enough to keep the assumption legitimate. The coupling-free assumption becomes the most challenge on the silicon integrated circuit (IC) integration of the active quasi-circulator. The CMOS synthetic quasi-TEM TLs are applied to construct and integrate all the building blocks shown in Fig. 1 for minimizing the adjacent coupling of components and building blocks in the proposed quasi-circulator at -band. In the presented CMOS realizations, the synthetic quasi-TEM TLs fall in two categories. First is the complementary conducting-strip transmission line (CCS TL); the second is the complementary conducting-strip coupled line (CCS CL). These two synthetic TLs have been proven for their good flexibility on the syntheses of guiding characteristics [28]–[30], and high efficiency in circuit miniaturization [31]–[35]. Fig. 2 shows the 24-GHz prototype of the active quasi-circulator based on standard 0.18- m 1P6M CMOS technology, consisting of four-stage common-source amplifiers (AMP1–AMP4), two directional couplers (C1 and C2), two phase shifters (in MPCU), one attenuator (in MPCU), and one second-order active bandpass filter (ABPF) in a chip area of 1940 m 1660 m including pads. All building blocks are marked one-to-one corresponding to those shown in Fig. 1(c) for clarity purpose. The quasi-circulator is fully shielded by the mesh ground plane provided by the synthetic TLs, thereby

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Fig. 2. Chip photograph of the active quasi-circulator: chip size 1940 m 1660 m including pads; standard 0.18-m 1P6M CMOS technology.

2

eliminating the coupling through the silicon substrate substantially. The connections between the adjacent components are realized by the CCS TL with a 50- characteristic impedance to support the well-defined electromagnetic (EM) propagation. Inside the building blocks, either TL elements or coupled lines are randomly arranged in the meandered form for the compactness. The dummy metals are also filled into the prototype to maintain the global metal density of 30.0% for each metal layer [29]. As shown in Fig. 2, the architectures of the components, which include the amplifiers (AMP1–AMP4) and ABPF, are identical to those of reported in the [33] and [34]. This amplifier had demonstrated the highest gain-area efficiency achievable for the multistage TL-based amplifier using 0.18- m CMOS technology at 24 GHz incorporated with a CCS TL. The proposed quasi-circulator integrates the ABPF as the first stage of the input in the transmitter path. This feature can assist in suppressing the harmonic signals generated from the transmitting terminal, resulting in the output spectrums with good quality at antenna terminal. The half-wavelength resonators of the ABPF are constructed by the CCS TL and the complementary current-reuse cross-coupled pairs. The current-reuse pairs generate the negative resistances to enhance the quality factor of the resonator. In this design, the order of the bandpass filter is two, and the coupling of the two resonators is defined by the metal–oxide–metal (MOM) capacitors with a capacitance of 50 fF for the 5.8% passband bandwidth at 24.0 GHz. The 24-GHz power combiner shown on the right-hand side of Fig. 2 is designed based on the conventional architecture of a 3.0-dB Wilkinson power combiner [36]. Two quarter-wavelength TLs are realized by the CCS TL. The extracted results, which are calculated by Ansoft’s High Frequency Structure Simulator (HFSS), show that the quality factor ( factor) and the slow-wave factor (SWF) are 7.82 and 2.24 at 24.0 GHz, respectively. On the other hand, the coupler design is based on the CCS CL [30], where the even- and odd-mode propagations are supported by the edge-coupled design. For achieving a 3.0-dB coupling coefficient, the characteristic impedances of

Fig. 3. MPCU of the presented active quasi-circulator shown in Fig. 1. (a) Block diagrams. (b) Schematic of attenuator. (c) Schematic of phase shifter.

the even and odd modes of the CCS CL are synthesized as 143.0 and 25.2 , respectively. Concurrently the propagation constants of the even and odd modes are obtained by optimizing the following five structural parameters of the CCS CL. In this design, the period of the unit cell is 30.0 m and the mesh area is 29.5 m 29.5 m. The linewidth and spacing are 2.0 and 1.2 m, respectively. The metal thickness of the mesh ground plane, which is thinner than those in [31], is only 0.53 m. The extracted results from scattering analyses of the CCS CL, following the procedures reported in [37], show that the SWFs of the even and odd modes are 1.84 and 2.42 at 24.0 GHz. factors of the even and odd modes are 6.2, and 2.7, The respectively. The full-wave EM simulations, also performed by Ansoft’s HFSS, show that the coupler has a coupling coefficient of 3.0 dB with 0.5-dB variation from 22.8 to 28.9 GHz. The isolation is 19.2 dB at 24.0 GHz. The phase difference between coupled and thru-ports is kept from 90 to 95 from 11.7 to 26.1 GHz. The amplitude imbalance between the two output ports is less than 1.0 dB from 18.0 to 35.0 GHz. B. MPCU As shown in Fig. 1(c), the implementation of feedforward cancellation is initiated from the coupled-port of the coupler C1. The MPCU can be realized by the composite networks of attenuator and phase shifters shown in Fig. 3. The attenuator, which is shown in Fig. 3(b), is the reflection-type topology [38]. This design, which consists of a 3-dB directional coupler, two NMOS transistors ( and ), and the necessary matching and ) are conducted by following the proceelements ( and , biased in the triode region, dure outlined in [39]. can be regarded as a voltage-control resistor. , which represents the impedance of the reflection load, can be controlled by . Varying will make the reflection load impedance changed accordingly, resulting in the adjustment of the attenuation at the output port. Doing the adjustments, if the magnitude is higher than , the phase of output signal lags beof is smaller than hind the input signal. If the magnitude of , the phase of the output signal has a constant lead over the

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TABLE I SIMULATED PERFORMANCES OF BUILDING BLOCKS FOR -BAND ACTIVE QUASI-CIRCULATOR IN STANDARD 0.18- m 1P6M CMOS TECHNOLOGY



K

Fig. 4. Simulated performances of the reflection-type attenuator and phase shifter using 0.18-m 1P6M CMOS technology at 24.0 GHz.

input signal. These two states mentioned above are regarded as the phase reversal, which can be applied to increase the range of the phase shift of the MPCU. Two identical phase shifters in cascade, which are shown on the right-hand side of Fig. 3(a), are also based on the reflection-type topology [38]. The variable capacitors, denoted by and , are realized by the transistor-based varactors. The , are realized by the CCS TLs inductive loads, denoted by are inserted to isolate the dc control [29]. The capacitors from the RF signals and the isolated ports of the voltage coupler. Fig. 4 superimposes the plot of the relative attenuation (in on decibels) of the attenuator against the control voltage another plot of the relative phase shifting (in degrees) of the phase shifter. The simulations are conducted by Agilent ADS 2008. The models of NMOS transistors, resistors, and varactors are foundry based. The characteristics of the directional couplers are identical to those of reported in Section III-A. The operation of the MPCU is as follows. When the phase shifting is set higher than 0.68 V, taking more than 180 is required, advantage of the phase reversal property of the reflection-type higher than 0.68 V, appropriate attenuaattenuator. For tion can be selected, followed by phase shifting contributed by two phase shifters, accounting for a maximum of 360 phase shifting. When the total phase shifting less than 180 is desired, is set lower than 0.68 V for the desired value of attenuation. Accurate phase control is, therefore, obtained by changing and . Additionally, the connections for the adjacent building blocks are also carefully designed with a 50- CCS TL to maintain the accuracies of the magnitude and phase control inside the quasi-circulator. These connections include, as shown in Fig. 2, the paths between C2, the ANT terminal, and the power combiner, the paths between AMP4, C1, and MPCU, and the path between the MPCU and the power combiner. The effects of the extra phase shifting and magnitude changing, which are generated by these connections, need to be included in the operations of the MPCU for precisely estimating the specific frequency for

the leakage suppression. Finally, Table I summarize the theoretical performances for all the building blocks reported in this section. In Section IV, the experimental characterizations for the CMOS prototype shown in Fig. 2 are reported in detail. IV. MEASUREMENTS The prototype is fully characterized through the on-wafer measurements. All the measured scattering parameters are collected with a 50.0- system after the short-open-load-thru (SOLT) calibration procedures that have been carried out to de-embedded the parasitics of the contacting pads. The measured results are also compared with those of the simulations, which are performed by using the commercial software Agilent ADS2008. In the following sections, the definitions of the ports and signal flows are identical to those reported in Section II and Fig. 1(c). The prototype consumes a total power of 144.8 mW , from the supplying voltages of 1.8 V. The voltages of , and are set as 0.0 V to initialize the magnitude/phase control unit. A. Transmission Gains Fig. 5 shows the forward transmission coefficients of the transmitting and receiving paths in the active quasi-circulator. The measured transmitting gain is 20.0 dB from 23.0 to 24.5 GHz and has a maximum value of 22.6 dB at 23.8 GHz. The measured receiving gain is 12.0 dB from 22.2 to 24.1 GHz and has a maximum value of 14.1 dB at 23.2 GHz. As shown in Fig. 2, the ABPF is the first stage in the transmitting path. Therefore, the curves with dots in Fig. 5 indicate the transmission gain with a bandpass response. The measured center frequency of the passband is at 23.6 GHz, and the rejection is higher than 20 dBc at 2.0-GHz offset from the center frequency. The measured center frequency of transmitting path is 0.15% lower than simulation. However, the comparisons of receiving

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Fig. 5. Simulated and measured forward transmission coefficients of the active quasi-circulator with the feedforward cancellation.

Fig. 7. Leakage phenomena of the active quasi-circulator with and without feedforward cancellation.

The isolation of transmitting path is higher than 70.0 dB with a bandwidth from 23.2 to 24.8 GHz. The isolation of the receiving path is 53.6 dB at 24.0 GHz and has a maximum value of 65.9 dB at 22.0 GHz. The isolation is higher than 50.0 dB from 20.7 to 25.0 GHz. The comparison between two curves in Fig. 6 shows about 9 dB of isolation in the transmitting path higher than that of in the receiving path. This difference is caused by the higher transmitting gain in the transmitting path shown in Fig. 5. Additionally, the presented isolation of the prototype based on 0.18- m CMOS technology is at least 15 dB higher than those of the ferrite-based circulator.1 2 3 C. Leakage Suppression: Improving TX-to-RX Isolation by Feedforward Cancellation Fig. 6. Measured isolations of the active quasi-circulator with the feedforward cancellation. The isolation is defined by the ratio of the forward transmission to the reverse transmission coefficient.

gain show a good agreement from 20.0 to 24.0 GHz. Two curves with squares in Fig. 5 have a smooth response of receiving gain, revealing the broadband characteristics of the directional coupler, power combiner, and amplifier in the receiving path. By referring the transfer functions defined by (2) and (3), the , and , confirming the dual curves in Fig. 5 represent transmission gains of the active quasi-circulator. B. Isolations The isolation of a circulator is defined as the power ratio of the forward transmission to the reverse transmission between each two of the three terminals of the circulator [40, p. 28]. As presented in the beginning of Section II, only two power flows are relevant in the active quasi-circulator. As shown in Fig. 6, the curve with dots and squares represent the isolation of transmitting and receiving paths, respectively. The isolation of the transmitting path is 75.8 dB at 24.0 GHz and has a maximum value of 78.3 dB at 24.3 GHz. The forward transmission of the transmitting path has a gain with a bandpass response and the reverse transmission coefficient is nearly kept a constant value.

Fig. 7 shows the characterizations on the leakage phenomena of the prototype. A dashed line and solid line with dots theoretically and experimentally show the transmission coefficients of the prototype without feedforward cancellation higher than 23.0 dB from 23.0 to 24.4 GHz. As represented by (2), the signal in the RX terminal includes those from the ANT terminal, TX terminal, and feedforward path. No signal is injected into the ANT terminal, and AMP4 is turned off for inactivating the feedforward path. The dashed and solid lines with dots in Fig. 7 also inside the prototype. However, when AMP4 represent the is activated, the magnitude/phase control unit, whose , , are set as 0.8, 0.6, and 1.8 V, can provide the relative atand tenuation and phase shifting of 11.5 dB and 355.5 at 24.0 GHz. The leakage from the TX to RX terminals is seriously reduced. The solid line with squares shows that the measured transmission coefficient is lower than 5.0 dB from 23.6 to 23.65 GHz and has a minimum value of 20.5 dB at 23.63 GHz. By referring the leakage suppression defined by (3), the difference between the solid lines with dots and squares experimentally 1H40.84, Aerocomm Company, Bangkok, Thailand, 1994. [Online]. Available: http://www.aerocommthailand.com/Div3/CoaxialCirculator/Products.htm 2D3C2030, DiTom Microwave, Fresno, CA, 1987. [Online]. Available: http:// www.ditom.com/ 3SR2123T01, Quest Microwave Inc., Morgan Hill, CA, 1996. [Online]. Available: http://www.questmw.com/

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Fig. 9. Noise figures in the receiving path in the prototype with feedforward cancellation. TABLE II MEASURED PERFORMANCES OF THE PROPOSED ACTIVE QUASI-CIRCULATOR IN STANDARD 0.18-m 1P6M TECHNOLOGY

Fig. 8. Input 1-dB compression points and IIP of the prototype with feedforward cancellation. (a) Transmitting path. (b) Receiving path.

indicates that the leakage suppression is 20.0 dB from 23.56 to 23.68 GHz and has a maximum value of 44.7 dB at 23.63 GHz. The observations summarized above successfully demonstrate the feasibility of the proposed feedforward technique for the narrowband leakage suppression. As shown in Table I, the isolation of the coupler in the prototype is only about 18.0 and 15.0 dB lower than those reported in [24]. The comparison mentioned above indicates that the proposed feedforward technique can break the structural limitation of the quasi-circulators reported in [24], and can effectively assist the active quasi-circulator in carrying out the leakage suppression without sacrificing the transmission gains. Additionally, as reported in Section II, the maximum leakage suppression can be made when is equal to . The accuracy of the leakage suppression is mainly relied on the precision of the MPCU. As shown in Fig. 3, the linearity and mismatch of the MOS varactors mainly dominate the performance of the leakage suppression. The layout of the varactors and inter-connections between the MPCU and the other building blocks need to be considered carefully for the narrowband leakage suppression. Fig. 8 shows the linearity of the prototype under large-signal operations. The measured input 1-dB compression point of the transmitting path is 19.8 dBm, showing

14.4 dB lower than that of the receiving path at 23.63 GHz. The characterizations for the input third-order intercept points ) of the transmitting and receiving paths are conducted by ( two-tone test. Doing the measurements, two coherent signals with identical power at 23.55 and 23.65 GHz are injected into the prototype with the feedforward cancellation. The measured of the transmitting and receiving pathes are 11.0 and 0.6 dB, respectively. The differences mentioned above are caused by a 9-dB difference on the transmitting gains between two paths, as shown in Fig. 5. Additionally, Fig. 9 shows the noise figure in the receiving path of the prototype with the feedforward cancellation. The measured noise figure is less than 17 dB from 20.0 to 24.7 GHz. The comparison between the two curves shows the difference less than 1.0 dB from 20.0 to 24.35 GHz, revealing a fair agreement between the measurements and simulations. As shown in Fig. 1, the first and second stages in the receiving path are all passive components. Therefore, the transmission losses of these components and the noise contributed by AMP3 are directly accumulated, resulting in the noise figure shown in Fig. 9. Finally,

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Table II summarizes the measured performances of the prototype reported in this section.

V. CONCLUSION The simulations and experiments prove that the presented quasi-circulator can simultaneously achieve transmission gains with high isolations for both two power flows, and effectively suppress the leakage from TX terminals to RX terminals of the quasi-circulator, confirming the validity of the concept of feedforward cancellation.



ACKNOWLEDGMENT The authors wish to extend their sincere thanks to Dr. M.-J. Chiang, HTC Corporation, Xindian, Taiwan, for providing the preliminary design of the directional coupler, Y.-H. Wu, CMSC Inc., Hsinchu, Taiwan, for providing the design of amplifiers, and K.-K. Huang, The University of Michigan at Ann Arbor, for providing the design of the ABPF.

REFERENCES [1] L.-R. Whicker, “Active phase array technology using coplanar packaging technology,” IEEE Trans. Antennas Propag., vol. 43, no. 9, pp. 949–952, Sep. 1995. [2] B.-A. Kopp, M. Borkowski, and G. Jerinic, “Transmit/receive modules,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 827–834, Mar. 2002. [3] R.-E. Blight and E. Schloemann, “A compact broadband microstrip circulator for phased array antenna modules,” in IEEE MTT-S Int. Microw. Symp. Dig., Albuquerque, NM, Jun. 1992, pp. 1389–1392. [4] V. Krishnamurthy, B. Whitmore, and K. Paik, “Integral microwave circulators for multi-chip module (MCM) applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Denver, CO, Jun. 1997, pp. 1829–1832. -band [5] M. Akaishi, T. Kaneko, N. Yakuwa, and K. Wada, “A fully integrated transceiver multi-chip module based on aluminum nitride multi-layer LCC package with the waveguide interface,” in IEEE MTT-S Int. Microw. Symp. Dig., Anaheim, CA, Jun. 1999, pp. 471–474. [6] A. Tessmann, S. Kudszus, T. Feltgen, M. Riessle, C. Sklarczyk, and -band FMCW radar modules W.-H. Haydl, “Compact single-chip for commercial high-resolution sensor applications,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2295–3001, Dec. 2002. [7] S. Tanaka, N. Shimomura, and K. Ohtake, “Active circulators—The realization of circulators using transistors,” Proc. IEEE, vol. 53, no. 3, pp. 260–267, Mar. 1967. [8] M.-A. Smith, “GaAs monolithic implementation of active circulators,” in IEEE MTT-S Int. Microw. Symp. Dig., New York, May 1988, pp. 1015–1016. [9] Y. Ayasli, “Field effect transistor circulators,” IEEE Trans. Magn., vol. 25, no. 5, pp. 3242–3247, Sep. 1989. [10] Y. Naito, M. Iwakuni, K. Araki, and A. Ikeda, “A new type of electronic circulator at 800 MHz band,” in Proc. 10th Eur. Microw. Conf., 1990, pp. 502–506. [11] I.-J. Bahl, “The design of a 6-port active circulator,” in IEEE MTT-S Int. Microw. Symp. Dig., 1988, pp. 1011–1014. [12] G. E. Stratakos, “CAD design of an active MMIC circulator at 21–26 GHz frequency band,” in Proc. 1st Eur. Gallium Arsenide and Other Compound Semicond. Appl. Symp., Oct. 1998, pp. 355–360. [13] S. Hara, T. Tokumitsu, and M. Aikawa, “Novel unilateral circuits for MMIC circulators,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 10, pp. 1399–1406, Oct. 1990. [14] D. Köther, B. Hopf, T. Sporkmann, and I. Wolff, “Active CPW MMIC circulator for the 40 GHz band,” in Proc. 2nd Eur. Microw. Conf., Oct. 1994, pp. 542–547.

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[15] A. Gasmi, B. Huyart, E. Bergeault, and L. Jallet, “Quasi-circulator module design using conventional MMIC components in the frequency range 0.45–7.2 GHz,” Electron. Lett., vol. 31, no. 15, pp. 1261–1262, Jun. 1995. [16] M. Berg, T. Hackbarth, B.-E. Maile, S. Koßlowski, and J. Dickmann, “Active circulator MMIC in CPW technology using quarter micron InAlAs/InGaAs/InP HEFTs,” in Proc. 8th Int. Indium Phosphide Rel. Mater. Conf., Apr. 1996, pp. 68–71. [17] A. Gasmi, B. Huyart, E. Bergeault, and L. Jallet, “MMIC quasi-circulator with low noise and medium power,” in IEEE MTT-S Int. Microw. Symp. Dig., 1996, pp. 1233–1236. [18] A. Gasmi, B. Huyart, E. Bergeault, and L. Jallet, “Narrow band quasicirculator module design used in a transmit/receive module,” presented at the Gallium Arsenide Appl. Symp. Dig., Jun. 1996, Paper 4C6. [19] A. Gasmi, B. Huyart, E. Beregeault, and L. Jallet, “Noise and power optimization of a MMIC quasi-circulator,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 9, pp. 1572–1577, Sep. 1997. [20] S.-C. Shin, J.-Y. Huang, K.-Y. Lin, and H. Wang, “A 1.5–9.6 GHz monolithic active quasi-circulator in 0.18- m CMOS technology,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 12, pp. 797–799, Dec. 2008. [21] S. W. Y. Mung and W. S. Chan, “Novel active quasi-circulator with phase compensation technique,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 12, pp. 800–802, Dec. 2008. [22] Y. Zheng and C.-E. Saavedra, “Active quasi-circulator MMIC using OTAs,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 4, pp. 218–220, Apr. 2009. [23] C.-Y. Kim, J.-G. Kim, and S. Hong, “A quadrature radar topology with tx leakage canceller for 24-GHz radar applications,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 7, pp. 1438–1444, Jul. 2007. [24] C.-E. Saavedra and Y. Zheng, “Active quasi-circulator realization with gain elements and slow-wave couplers,” IET Microw. Antennas Propag., vol. 1, no. 5, pp. 1020–1023, Oct. 2007. [25] S. Cheung, T. Halloran, W. Weedon, and C. Caldwell, “Active quasicirculators using quadrature hybrids for simultaneous transmit and receive,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, Jun. 2009, pp. 381–384. [26] S. K. Cheung, T. Halloran, W. Weedon, and C. Caldwell, “MMICbased quadrature hybrid quasi-circulators for simultaneous transmit and receive,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 3, pp. 489–497, Mar. 2010. [27] S.-L. Karode and V.-F. Fusco, “Feedforward embedding circulator enhancement in transmit/receive applications,” IEEE Microw. Guided Wave. Lett., vol. 8, no. 1, pp. 33–34, Jan. 1998. [28] 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. [29] M.-J. Chiang, H.-S. Wu, and C.-K. C. Tzuang, “Design of synthetic quasi-TEM transmission line for CMOS compact integrated circuit,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 12, pp. 2512–2520, Dec. 2007. [30] M.-J. Chiang, H.-S. Wu, and C.-K. C. Tzuang, “Artificial-synthesized edge-coupled transmission lines for compact CMOS directional coupler designs,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 12, pp. 3410–3417, Dec. 2009. [31] M.-J. Chiang, H.-S. Wu, and C.-K. C. Tzuang, “A CMOS 3-dB directional coupler using edge-coupled meandered synthetic transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, Jun. 2008, pp. 771–774. [32] M.-J. Chiang, H.-S. Wu, and C.-K. C. Tzuang, “A -band CMOS Wilkinson power divider using synthetic quasi-TEM transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 12, pp. 837–839, Dec. 2007. [33] S. Wang, K.-H. Tsai, K.-K. Huang, S.-X. Li, H.-S. Wu, and C.-K. C. Tzuang, “Design of -band RF CMOS transceiver for FMCW monopulse radar,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 1, pp. 61–70, Jan. 2009. [34] Y.-H. Wu, M.-J. Chiang, H.-S. Wu, and C.-K. C. Tzuang, “24-GHz 0.18- m CMOS four-stage transmission line-based amplifier with high gain-area efficiency,” in Proc. Asia–Pacific Microw. Conf., Dec. 16–20, 2008, pp. 1–5. [35] K.-K. Huang, M.-J. Chiang, and C.-K. C. Tzuang, “A 3.3 mW -band 0.18- m 1P6M CMOS active bandpass filter using complementary current-reuse pair,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 2, pp. 94–96, Feb. 2008.

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[36] D.-M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005. [37] K.-H. Tsai and C.-K. C. Tzuang, “Mode symmetry assessment of structurally non-uniform asymmetric coupled lines meandered for CMOS passive component design,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, Jun. 2009, pp. 285–288. [38] S. Lucyszyn and I.-D. Robertson, “Analog reflection topology building blocks for adaptive microwave signal processing applications,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 3, pp. 601–611, Mar. 1995. [39] C.-W. Wang, H.-S. Wu, M.-J. Chiang, and C.-K. C. Tzuang, “A 24 GHz CMOS miniaturized phase-invertible variable attenuator incorporating edge-coupled synthetic transmission line,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, Jun. 2009, pp. 841–844. [40] D.-K. Linkhart, Microwave Circulator Design, 1st ed. Norwood, MA: Artech House, 1989. Hsien-Shun Wu (S’97–M’05) received the B.S. degree in electronic engineering from the National Taipei University of Technology, Taipei, Taiwan, in 1999, and the M.S. and Ph.D. degrees in communication engineering from National Chiao Tung University, Hsinchu, Taiwan, in 2001, and 2005, respectively. He is currently a Post-Doctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University. His research interests include the design of wireless system modules and the development of synthetic waveguides for RF circuits.

Chao-Wei Wang (S’07) was born in Taichung, Taiwan, in 1982. He received the B.S. and M.S. degrees in electrical engineering from National Taiwan University of Science and Technology, Taipei, in 2005 and 2007, respectively, and is currently working toward the Ph.D. degree at the Graduate Institute of Communication Engineering, National Taiwan University. His research activities involve design and development of microwave/millimeter-wave ICs and the phased-array system.

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Ching-Kuang Clive Tzuang (S’80–M’80–SM’92– F’99) received the B.S. degree in electronic engineering from National Chiao Tung University, Hsinchu, Taiwan, in 1977, the M.S. degree from the University of California at Los Angeles (UCLA), in 1980, and the Ph.D. degree in electrical engineering from the University of Texas at Austin, in 1986. From 1981 to 1984, he was with TRW, Redondo Beach, CA, where he was involved in the design of the high-speed analog and digital data converter ICs. He became an Associate Professor with the Institute of Communication Engineering, National Chiao Tung University, in 1986, and a Full Professor in 1991. In February 2004, he joined the Graduate Institute of Communication Engineering, Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, where he conducted research on advanced guiding structures for research and development of RF sensor system-on-chip, integrating active and passive microwave/millimeter-wave RF signal-processing components into a single CMOS chip. His research activities also involved the field-theoretical analysis and design of waveguide structures and large-array antennas for integrating RF systems in a package. From 1992 to 1994, he was a team member who supervised the installation of the tracking radar system placed at the Center for Space and Remote Sensing Research, National Central University, Chung Li, Taiwan. He helped execute an eight-year Academic Excellent Program (2000–2008) funded by the Ministry of Education and National Science Council of Taiwan, which focused on the advanced microwave/millimeter-wave RF and communication technology development. The results enabled the investigation of scaled microwave RF system-on-chip (SOC) technology when he visited Stanford University and the contribution to the IEEE 802.15 TG3c 60-GHz WPAN standardization in collaboration with CoMPA, Japan. He has supervised 25 Ph.D. students and 71 M.S. students. Dr. Tzuang helped establish the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Taipei Chapter, and served as secretary, vice chairman, and chairman in 1988, 1989, and 1990, respectively. He was the recipient of the 2008 Excellent Project Award presented by the Ministry of Transportation and Communications for practically demonstrating the real-time multilane traffic sensor using a CMOS-based lightweight radar. He recently explored the feasibility of applying the CMOS foundry to the design and development of a terahertz imaging sensor. In January 2010, he became the editor-in-chief for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS.

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Analysis and Design of a 1.6–28-GHz Compact Wideband LNA in 90-nm CMOS Using a  -Match Input Network Hsien-Ku Chen, Yo-Sheng Lin, Senior Member, IEEE, and Shey-Shi Lu, Senior Member, IEEE

Abstract—This paper presents a wideband low-noise amplifier (LNA) based on the cascode configuration with resistive feedback. Wideband input-impedance matching was achieved using a shunt–shunt feedback resistor in conjunction with a preceding -match network, while the wideband gain response was obtained using a post-cascode inductor ( ), which was inserted between the output of the cascoding transistor and the input of the shunt–shunt resistive feedback network to enhance the gain and suppress noise. Theoretical analysis shows that the frequency response of the power gain, as well as the noise figure (NF), can be described by second-order functions with quality factors or damping ratios as parameters. Implemented in 90-nm CMOS technology, the die area of this wideband LNA is only 0.139 mm2 including testing pads. It dissipates 21.6-mW power and achieves 1.1 dB, 11 below 10 dB, 22 below 10 dB, flat 21 of 9.6 and flat NF of 3.68 0.72 dB over the 1.6–28-GHz band. Besides, excellent input third-order inter-modulation point of 4 dBm is also achieved. The analytical, simulated, and measured results are mutually consistent.

+

Index Terms—CMOS, feedback, flatness, low-noise amplifier (LNA), matching, noise figure (NF), series peaking, wideband.

I. INTRODUCTION

R

ECENTLY, wideband technology has attracted much attention because of its high data-rate transmission capability [1], [2]. In such a wideband system, a low-noise amplifier (LNA) with wideband input-impedance matching is a must. To reach this goal, various wideband-matching techniques for wideband LNAs have been proposed. For example, a cascode CMOS LNA with a bandpass response at the input for wideband impedance-matching has been reported [3]. Such topology incorporates the input impedance of the cascode amplifier as a part of the filter. However, the adoption of the filter at the input requires a number of reactive elements, increasing die-size and the noise figure (NF) due to the finite quality ( ) factor of the passive components when implemented on-chip. Alternatively, broadband matching with low power dissipation and Manuscript received December 24, 2009; revised April 02, 2010; accepted May 11, 2010. Date of publication June 28, 2010; date of current version August 13, 2010. This work was supported by the National Science Council of the R.O.C. under Contract NSC 98-2221-E-002-153-MY2. H.-K. Chen and S.-S. Lu are with the Graduate Institute of Electronics Engineering and Department of Electrical Engineering, National Taiwan University, 106 Taipei, Taiwan (e-mail: [email protected]; [email protected]). Y.-S. Lin is with the Department of Electrical Engineering, National Chi Nan University, 545 Puli, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2052406

small die size can be realized by common-gate input topology [4]. Nevertheless, a single-stage common-gate amplifier cannot provide sufficient power gain, and hence, extra stages are required to boost the gain, resulting in ripples in the passband due to nonbroadband inter-stage matching. In another work [5], a noise-cancelling technique was introduced to resolve the high NF issue arising from the common-gate amplifier. However, according to the experimental results, the achieved NF was still high, even higher than that of the LNAs without the use of noise cancelling [6], [7]. Although the distributed amplifier (DA) has the inherent advantages of wideband impedance matching and gain [7]–[9], it is notorious for its low gain, high power dissipation, and large chip area. In [10], an excellent 3–5-GHz cascode based wideband LNA has been proposed. Resistive feedback was adopted to reduce the factor of the input series circuit to achieve wideband impedance matching. However, the impedance/gain bandwidth of this design was still limited due to the nonoptimized choice of the factor of the input network and the large capacitive effect from the active devices. Notably, while the frequency responses of gain can be very flat, the frequency responses of the NF are not flat in most of the published wideband LNAs [11]–[13] because there is an intrinsic conflict between flat gain and flat NF responses. It is noteworthy that though low and flat NF (3.58 0.41 dB) was obtained in our previous work ([14]) over the 3.1–10.6-GHz has only one zero limits it band of interest, the fact that for wider bandwidth applications, such as 20 GHz. Besides, -branch wideband input the LNA topology with a dualmatching network proposed in our previous work ([15]) has the disadvantage of occupying a large chip size because four spiral inductors are needed. In this work, to demonstrate that wideband input impedance matching, flat and high gain, flat and low NF, and compact size can be achieved for a wideband CMOS LNA at the same time, we propose a wideband LNA architecture with a shunt–shunt feedback resistor in connetwork and a postjunction with a preceding -match cascode series-peaking inductor. In contrast to [14], a parallel input capacitor ( ) was added to enhance the input impedance matching bandwidth (by introducing a new zero) and to sup– ). With press the gate noise (by series resonance of this approach, compact size could be achieved because only two spiral inductors were needed. In addition, wideband input impedance matching could be achieved because the input network was equivalent to a third-order ladder-type low-pass filter has two zeros) [16], to be explained in more detail (i.e.,

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in Section II. The reason why a post-cascode series-peaking inductor ( ) is located between the output of the cascoding transistor and the input of the shunt–shunt resistive feedback network is for improving gain and suppressing noise at high frequencies. The frequency response of the NF is derived to identify the key parameters that control the shape of the NF. The frequency response of gain, as well as that of the NF, can be described by second-order functions with factors or damping ratios as parameters. Design equations and tradeoffs relevant to the input matching bandwidth, power gain, and NF are also presented. Implemented in 90-nm CMOS technology, the proposed circuit only occupied a die area of 0.139 mm and consumed 21.6 mW. Wideband input impedance matching from 1.6 to 30.4 GHz was achieved by properly selecting the shunt–shunt feedback resistor and its preceding -match network based on the design equations, whereas the wideband gain response was obtained by the post-cascode peaking inductor. Furthermore, a very flat frequency response of the NF was obtained by appropriately choosing the factor from the design equations for the NF. Although the feedback resistor inevitably degraded the NF, the achieved NFs were still better than the values of wideband CMOS LNAs (with about the same bandwidth) previously reported [1], [2]. This paper demonstrates the design principles and analytical equations of the proposed wideband LNA. The remainder of this paper is organized as follows. Section II elaborates on the input-impedance-matching/gain bandwidth along with the frequency responses of the NF. Section III describes the proposed wideband LNA. Section IV discusses the measurements of the proposed wideband LNAs and compares them with the results of previous works. Section V presents the conclusions.

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Fig. 1. (a) Schematic and (b) small-signal equivalent circuit of resistive shunt–shunt feedback amplifier. (c) Small-signal equivalent circuit of resistive shunt–shunt feedback amplifier with additional series input inductor L and parallel input capacitor C .

bandwidth is limited by the growth of as frequency inof the amplifier can be derived from (1) creases, the

II. PRINCIPLES OF CIRCUIT DESIGN A. Wideband Input Impedance Matching Resistive shunt–shunt feedback is a commonly used technique for widening the bandwidth of an amplifier [17]. Fig. 1(a) and (b) schematically depicts a typical resistive shunt–shunt feedback amplifier (biasing not shown) and its small-signal equivalent circuit, respectively. For simplicity, the is neglected. The input intrinsic gate–drain capacitance (i.e., the inverse of the input impedance ) of admittance the shunt–shunt resistive feedback amplifier is given by

(1) ( ) represents the equivwhere alent resistance looking into the feedback resistor [see and are the gate–source capacitance and Fig. 1(b)]. , respectively. is the the transconductance of transistor is dominated by . load resistance. At low frequencies, Traditionally, is chosen to be close to 20 mS to ensure that the input impedance of the amplifier is well matched before the ) becomes appreciable at high frequencapacitive effect ( cies. To gain more insight into how input impedance matching

(2) is the resistance of the voltage source, and is the normalized value of . Generally speaking, should be less than 10 dB over the in RF amplifier design, band of interest, i.e., where

dB (3) From (3), the corresponding input-matching bandwidth ( ) and the range of can be derived as (4a) or

(4b)

For a shunt–shunt resistive feedback amplifier, it is a common in order to achieve wideband practice to choose characmatching from dc. Fig. 2 shows the calculated teristic of the feedback amplifier on the Smith chart, where fF, mS, and

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The results of (6a) and (6b) show that there are two frequencies at which a perfect match can be obtained. Equation (6b) can be is chosen further simplified if the condition (6c)

Fig. 2. Loci of S of resistive shunt–shunt feedback amplifier both with and without series input inductor L and parallel input capacitor C .

are assumed. The dashed circle with a radius of 0.316 represents the input-matching boundary obtained from dB. Clearly, equals to 0, implying a perfect match at dc. Due to the capacitive effect of , follows the constant conductance ( ) circle of mS as frequency increases, and finally intersects with the boundary circle at point A when the frequency is 12.1 GHz. That is, the and the boundary circle limits the matching intersection of from interbandwidth. Consequently, if we can prevent secting with the boundary circle, the matching bandwidth can be extended. An easy solution suggested by the Smith chart and a parallel capacitor to is to add a series inductor the input of the amplifier to form a third-order ladder-type low-pass-filter- based input network to reduce the imaginary , delaying the intersection of with the boundary part of circle and increasing the input matching bandwidth. This effect can be mathematically proven as follows. Fig. 1(c) shows the small-signal equivalent circuit of a resistive shunt–shunt feedback amplifier with a series input inductor and a parallel input capacitor . The input impedance of the amplifier can be represented as follows:

(5) Assume and , which is usually the case, , i.e., perfect input the frequencies corresponding to impedance matching, can be derived as follows: (6a) (6b)

is to be explained The physical meaning of is not too far away from , then can be shortly. If less than 10 dB from up to (see Fig. 3). In fact, since is normally chosen, of the amplifier can be represented as (7), shown at the bottom of this page. Assuming (i.e., the input network is equivalent to a symmetrical third-order ladder-type low-pass filter), then from (7) it is contains two zeros at clear that the frequency response of and given by (6a) and (6b), as expected. That is, two are predicted. If we furdips in the frequency response of ther assume that (i.e., ), then : (7) can be simplified as follows for frequencies around (8) should be less than 10 dB over the band of interest, Since a constraint is imposed on (8) as follows: dB

(9)

From (9), the corresponding input-matching bandwidth can be derived as follows: dB

(10)

Note that is normally not too far away from the desired (see Figs. 2 and 3), the assumption that is around is reasonable. Fig. 3 plots calculated versus frequency, using (2) and (7). Clearly, the input impedance matching bandwidth is extended from 12.1 to 22.5 GHz by adding a series input inductor and a parallel input capacitor. The Smith chart in Fig. 2 and (1) reveals that the addition of moves point B (the intercept of and the mS curve at 18.2 GHz), to point B’ ( mS mS) of the Smith chart. Since the admittances of point B and point B’ are mutually complex conjugated, i.e., outside of the input-matching boundary to the same extent, it is reasonable that the input matching bandwidth only improves a little (from 12.1 to 14.8 GHz (see point A’ in Fig. 2) after is added. In contrast, the addition of both and cancels the imaginary part of point B, and moves point B to the center B” [20 mS (or 50 )] of the Smith chart, postponing the intersection of with the 10-dB

(7)

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Fig. 3. Calculated S versus frequency for resistive shunt–shunt feedback amplifier both with and without series input inductor L and parallel input capacitor C .

input-matching boundary circle to 22.5 GHz (see point A’’ in Fig. 2). This value (22.5 GHz) is very close to the result (22.1 GHz) estimated by (10), which justifies the proposed analysis. Note that, in practice, a dc decoupling capacitor ( in Fig. 3) is normally included in the input terminal and hence the dip ), as can be seen in Fig. 3, will move to a at dc (i.e., (1.4 GHz) in Fig. 3) due to the higher frequency (see degradation of input matching at low frequencies around dc. B. Frequency Response of The cascode architecture is one of the most popular LNA topologies due to its merits of low power consumption, high of gain, and high reverse isolation. The following analyses a resistive shunt–shunt feedback cascode amplifier with a series and a parallel input capacitor , as shown in input inductor Fig. 4(a) (biasing not shown), which is the cascode version of the circuit in Fig. 1(c). Fig. 4(b) shows the small-signal equivalent circuit. To simplify the analysis, assume , which is usually the case for frequencies around 0 and . of this amplifier, equal to twice of the voltage gain The ) in a 50- system ( ) [18], ( can thus be represented as follows:

Fig. 4. (a) Schematic and (b) small-signal equivalent circuit of cascode amplifier with resistive shunt–shunt feedback. Small-signal equivalent circuit of cascode amplifier with resistive shunt–shunt feedback and post-cascode inductor. (c) Feedback after inductor L . (d) Feedback before inductor L .

(11c) (11d) (11a) where

(11b)

and represent the voltage gains and , respectively, and and are the pole frefactor of the input network of the LNA, quency and pole respectively. represents the pole frequency of the core ) of the circuit of the LNA. Clearly, the 3-dB bandwidth ( , , and . Normally, the amplifier is determined by bandwidth of the amplifier is limited by the large capacitance

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drain of M2. Fig. 4(d) shows the corresponding small-signal equivalent circuit. The voltage gain of the amplifier is given by

(13a)

where (13b)

Fig. 5. Calculated f versus R of cascode amplifier with resistive shunt–shunt feedback with/without post-cascode inductor L .

(13c) . A possible means to increase of the amplifier is at the drain to add a post-cascode series peaking inductor terminal of . Fig. 4(c) shows the corresponding small-signal equivalent circuit. The voltage gain of the amplifier is given by

To achieve wider bandwidth, it is found that the sampling of the adopted shunt–shunt feedback should be applied at load instead of the drain of M2, as shown in Fig. 5. C. Frequency Response of NF

(12a)

where (12b) (12c) and represent the pole frequency and pole factor, respectively, of the core circuit of the LNA with . Since and a post-cascode series-peaking inductor are both standard second-order low-pass filtering of the amplifier is determined by , functions, , , and . Fig. 5 plots the calculated versus of the LNAs in Fig. 4(b) and (c) under the conmS, fF, fF, ditions nH, fF, and nH. Clearly, adding a series peaking inductor at the drain terminal of significantly improves the bandwidth of because the can be chosen to frequency response of the core circuit with of the circuit be critically damped or under damped, while without will start to decline once the frequency exceeds . Notably, the sampling node of the adopted shunt–shunt feedback can also be chosen at the

In wideband applications, a flat and low-NF frequency response is required in addition to wideband input impedance matching and a flat and high-gain frequency response. The traditional narrowband low-noise design, based on optimum NF at each frequency, cannot be usefully exploited in wideband design because the resultant NF frequency response of the amplifier is not flat. However, the conventional wideband resistive shunt–shunt feedback amplifier is also inappropriate because, while the frequency response of its gain can be very flat, the frequency response of NF is not [11]–[13]. An intrinsic conflict is shown herein between flat gain and flat NF responses. Therefore, to mitigate the increase in NF at high frequencies and to achieve a flat NF response, the factors that govern the shape of the NF frequency response must be identified at first. Notably, although many NF formulas have been derived for wideband amplifiers, most of these estimate NF only at dc [13]. This section analyzes the NF response of the circuit that is shown in Fig. 4(c). Fig. 6(a) shows the complete equivalent circuit of the LNA in Fig. 4(c) such that its NF can be calculated. Fig. 6(b) shows the simplified equivalent circuit of Fig. 6(a), where is modified as to take into account the effects of the cas) and the peaking inductor code transistor ( and The noise ) derived in (20) can be factor of the LNA ( rewritten as follows to yield more insight:

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Fig. 6. (a) Complete and (b) simplified equivalent circuit of Fig. 4(c) for noise calculation.

(14a)

where (14b)

(14c) (14d) (14e)

(14f) Since the NF [see (14a)] has been expressed as a second-order function of , its frequency response is well known. It is conand . Fig. 7(a) plots the trolled by the quality factors , , and (defined in the Appendix A) calculated versus frequency of the wideband amplifier with the following

Fig. 7. (a) Calculated F , F , and F versus frequency of the amplifier =F versus R for various g . Calculated in Fig. 4(a). (b) Calculated F (c) NF and (d) S versus frequency for various Q and Q .

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parameters:

mS,

fF, , nH, fF, and nH. Under these conditions, [if the effect of pH is considered (see (16), which will be explained shortly)], indicating that the corresponding gain response of the input network in (11a)] is close to the maximally flat response. The re[ . is sult shows that the noise factor was dominated by over the frequencies of interest, and is much lower than for frequencies lower than 22.8 GHz. The low lower than noise contribution of the gate-induced noise at high frequencies being (around 20 GHz) is partly attributed to the effect of since is equivalent to a low-pass netsuppressed by work [see Fig. 6(b) and (20)]. In other words, at high frequenin series with – is equivalent to in series cies, with . This, in turn, results in small at high frequen(i.e., series resonance frequency of cies around – , equal to 22 GHz in this work). can be larger than at low frequencies is Whether worthy of study. To simplify the analysis, suppose . From at low frequencies can be represented (20), the ratio as follows: fF,

mS,

fF,

(15) versus for various Fig. 7(b) plots calculated of the wideband amplifier, according to (15). As can be corresponds to a smaller ratio , seen, a smaller . The value that corresponds to the maxregardless of imum is exactly 50 for any value of . Since normally exceeds 50 , decreases the needed increases over the range of interest. In the case as mS), is smaller than 1 under study ( for any value of and equals 0.398 if . The (i.e., ) is exactly equal to 1 maximum of 165.6 mS is adopted. if a larger Fig. 7(c) shows the characteristics of calculated NF versus frequency of the wideband amplifier under various values and , which are correlated [see (16) and (14c)]. of versus freFig. 7(d) shows the corresponding calculated quency of the wideband amplifier under various values of and . Note that the values of and are changed and while keeping the same by varying the values of of 21.25 GHz and ( ) of 99.42 GHz. Some previously presented wideband LNAs [11]–[13] exhibited flat gain. However, the corresponding NF response was not sufficiently flat because was chosen for the input matching network. over-damped That is, their NF responses were similar to the two over-damped ; ) and ( ; curves [( )] in Fig. 7(c). In their works, over-damped ’s were chosen to fulfill the wideband input impedance matching condition while the even gain responses were achieved by post-input-stage peaking techniques to compensate the gain degradation at high frequencies. However, an over-damped inevitably leads to an uneven NF response because NF is dominated by the NF of the first stage and the uneven NF response

Fig. 8. (a) Schematic and (b) chip micrograph of the proposed wideband LNA.

cannot be remedied by peaking techniques. At , the input network of the LNA was close to the maximally flat was 0.888, giving a slightly response; the corresponding under-damped, but nearly maximally flat NF response. The advantage of our proposed circuit in Fig. 4(c) is that unlike conventional resistive shunt–shunt feedback amplifiers [12], [13], both the gain and NF response can simultaneously be tuned to approximate the maximally flat conditions. III. PROPOSED WIDEBAND LNA Fig. 8(a) shows the proposed wideband LNA, which uses the resistive shunt–shunt feedback topology described in Section II. and were added for dc decoupling. Since the Capacitors needed is small, it is implemented by the equivalent capacitor bottom metal (CBM) to substrate capacitance of . In and were set to be infinite for the previous discussion, ease of analysis. However, in a real design, only finite capaciand are allowed because the chip area is limited. tances At low frequencies, the input impedance will be very high, such that the input reflection coefficient lies outside the matching boundary ( -circle of 0.316). Consequently, in the case without

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and in Fig. 9(a), small input-matching bandwidth of only 11.8 GHz (0.3–12.1 GHz) was achieved because of the and at low frequencies and at high freeffects of quencies. As discussed in Section II-A, a series inductor and can be added to the input to compena parallel capacitor . That is, the third-order ladder-type sate for the effect of low-pass-filter-based input network can reduce the imaginary , delaying the intersection of with the boundary part of circle and increasing the input-matching bandwidth. Specifiof 0.32 nH and cally, to reduce NF at high frequencies, an of 163 fF that corresponds to a nearly maximally flat a (0.719) was chosen for the input matching network [see Fig. 7(c)]. This result unequivocally proves the effectiveness of and a parbroadband matching when a series input inductor is used in conjunction with an allel input capacitor resistive shunt–shunt feedback technique. was In Fig. 8(a), a small (50 pH) degeneration inductor added to achieve good linearity and stability [19], [20]. was implemented by a microstrip line (MSL) with a width of 5.77 m and a length of 105 m. A comparison with the smallinductance spiral inductor indicates that the small-inductance MSL inductor was less susceptible to process variation. This source degeneration effect also degrades the power gain of the LNA, and thus should be considered in (12) for real designs. is considered, impedance If the effect of should be replaced by . This, in (11a), and (12a) should be modified as in turn, results in follows:

(16)

Since the modified in (16) is slightly smaller than the in (11c), in terms of real design, a slightly larger should be . selected to consider the effect of Fig. 9(b) shows the calculated and simulated power gain of of the LNA with and without a series-peaking inductor 0.336 nH. As can be seen, the power gain of the LNA without began to roll off at 3 GHz due to the capacitive effect of large-sized M1 and M2. To counterbalance this gain degradawas added to flatten the gain response. Clearly, the tion, series inductive- peaking technique is useful for extending the bandwidth of the resistive shunt–shunt feedback LNA.

IV. EXPERIMENTAL RESULTS AND DISCUSSION The wideband LNA was implemented using 1P9M 90-nm CMOS technology provided by United Microelectronics Corporation (UMC), Hsin-chu, Taiwan. The main features of the back-end processes are as follows. Nine metal layers were to from bottom to top. The thickness of , named , and was 0.81, 0.5, and 0.22 m, respectively, and that of – was 0.25 m each. The oxide thickness and , 0.4 m between and was 0.34 m between (and and ), 0.32 m between other adjacent metal and the silicon substrate. layers, and 0.41 m between

Fig. 9. Calculated and simulated: (a) S and (b) S versus frequency of the wideband LNA. Calculated, simulated, and measured: (c) S and (d) S versus frequency of the wideband LNA.

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The interconnection lines as well as the inductors were realto minimize the resistive loss. Usually, the ized by layer ) should be larger than required midband gain ( 10 dB. In addition, a larger ( ) is preferred for lower resistive noise. Therefore, in our LNA dB ( ) is initially chosen to compendesign, sate gain degradation and to suppress resistive noise in real mS and implementation, leading to for . After several simulation iterations, mS and were determined. The other component parameters adopted were listed as folnH, nH, nH, lows: pF ( fF), pF, and k . Both transistors M1 and M2 had the same gate length of 90 nm and gatewidth per finger/finger number m . RF choke can be implemented by integrated of passive devices (IPDs) as has been previously shown by us [21]. However, in this work, a bias-T was used for the ease of measurement. Fig. 8(b) displays the die photograph of the wideband LNA. The die had a small chip area of only 0.34 0.41 mm including the testing pads, which was highly area efficient and consequently cost effective. On-wafer -parameter measurements were made using an Agilent HP8510C vector network V analyzer. The dc biases of the gate terminals were V. The wideband LNA drained 18-mA current at and ) of 1.2 V: It consumed 21.6-mW power. a supply voltage ( Fig. 9(c) shows the calculated, simulated, and measured versus frequency of the wideband LNA. Simulated results mean that the complete NMOS model given in Appendix B is used, while calculated results means the analytic formula are used. was 9.6 1.1 dB over the 1.6–28-GHz band, The measured and the corresponding 3-dB bandwidth was 29.3 GHz. The calculated result is consistent with the simulated and measured rewas below 10 dB from sults. Fig. 9(d) shows the measured 1.6 to 30.4 GHz, consistent with the calculated and simulated results. Besides, measured results show the wideband LNA was greater unconditionally stable (i.e., stability factors and than one [22]) over the frequency range of 0.5–60 GHz (not shown here). Fig. 10(a) shows the measured NF versus frequency characteristics of the LNA for frequencies from 3 to 26.5 GHz. The measured minimum and maximum NF for the LNA were 2.92 dB (at 9.5 GHz) and 4.4 dB (at 25.5 GHz), respectively. Compared with other results in the literature, this wideband CMOS LNA exhibits one of the flattest NFs (3.075 0.155 dB) across the 3.1–10.6-GHz UWB, to be explained as follows. According to (14a), the frequency responses (under, over, and . In this work, critical damped) of the NF are dominated by (0.719) was chosen for the input a nearly critical-damped matching network. According to the analysis in Section II, an intrinsic advantage of the proposed wideband LNA is that exceeds when constant the corresponding and and a under or nearly critical damped are is 0.888 being adopted. For example, the corresponding for . Since the frequency response of the NF was under damped, the LNA herein exhibited a wideband NF response. What is also shown from Fig. 10(a) is the calculated and

Fig. 10. (a) Calculated, simulated, and measured NF versus frequency of the wideband LNA. (b) Measured IIP3 of the wideband LNA for two-tone inputs of 12 and 12.1 GHz.

simulated NF versus frequency characteristics of the wideband LNA. As can be seen, the measured results are consistent with the simulated and calculated results due to the careful layout of the proposed LNA and the fact that it has fewer passive components in comparison with previous designs. It is interesting is included in the NF calculation or to note that whether not, a highly well-matched result between measurement and calculation is achieved. This is attributed to the fact that the . Moreover, based on the NF of the LNA is dominated by NMOS model provided by the foundry, the extracted (gate-to-drain capacitance of transistor M1) is 61.7 fF. should be corrected as if the Miller effect of is taken into account. If is considered, calculation shows should be modified from 0.719 to 0.68 [see (16)], and should be modified from 0.888 to 0.774 [see (14c)]. This means it is reasonable to neglect in the calculation of gain and and NF responses since the corresponding changes of are small.

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TABLE I SUMMARY OF THE IMPLEMENTED CMOS WIDEBAND LNA, AND RECENTLY REPORTED STATE-OF-THE-ART CMOS WIDEBAND LNAs

To characterize the nonlinear behavior, two-tone signals with equal power levels at 12 and 12.1 GHz were applied to the proposed wideband LNA. The measurements in Fig. 10(b) indicate that the LNA had an input third-order inter-modulation point (IIP3) of 4 dBm. This result demonstrates that the proposed wideband LNA also achieved good linearity. Table I summarizes the implemented CMOS wideband LNA and the state-of-the-art CMOS wideband LNAs, reported upon recently. Clearly, the performance of the proposed LNA is one of the best CMOS-based wideband LNAs. In addition, the chip size of our LNA including testing pads was only 0.139 mm , smaller than those in most published works. Note that since a larger mS, corresponding to a larger transistor size, is used in this work, the consumed power is also larger than those in [1] and [2]. Nevertheless, this work achieves much better NF performance than [1] and [2], and hence, overall speaking, the proposed topology and approach are very promising.

V. CONCLUSION A compact wideband LNA for wideband applications is demonstrated both theoretically and experimentally. The unique feature of the proposed circuit topology can achieve simultaneous wideband input impedance matching and low ) of the and flat NF by controlling the quality factor ( input-stage of the wideband LNA. A high and flat gain response can be obtained by a post-cascode gain peaking inductor. The measurements are highly consistent with the analytical and simulated results. Furthermore, comparisons with the CMOS-based wideband LNAs for 3–10-GHz applications in the literature show that the wideband LNA presented here exhibits one of the best NF flatness and the smallest chip size. The results indicate that the proposed LNA topology is very suitable for low-cost and high-performance wideband LNAs.

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APPENDIX A DERIVATION OF CASCODE AMPLIFIER WITH RESISTIVE SHUNT–SHUNT FEEDBACK AND POST-CASCODE INDUCTOR In this appendix, the NF response of the circuit shown in Fig. 4(c) is analyzed. Fig. 6(a) shows the complete equivalent circuit of the LNA in Fig. 4(c) for use in its NF calculation, where represents the thermal noise voltage of the source re, represents the thermal noise voltage of the sesistance of the gate inductor ; represents the ries resistance , thermal noise voltage of the transistor gate resistance represents the thermal noise voltage of the transistor source re, represents the thermal noise current of the sistance feedback resistance , represents the gate-induced noise current, and represents the channel resistance thermal noise current. The mean squared values of these noise sources are ,

as follows: ,

( ) represents the current-gain cutoff in which frequency of transistor M2, which is about 3.6 times the maximum frequency of interest ( 28 GHz) for the 90-nm CMOS technology used herein. Therefore, can be reasonably assumed over the frequency band of interest. For , holds. The noise factor of the LNA a small ) can be expressed as follows [23]: (

, ,

,

, and , to zero-bias drain conductance ) is where (ratio of about 0.85 in deep-submicron MOSFETs [23]; is the Boltzis the absolute temperature, and is fremann constant, quency. and are the coefficients of gate noise and channel noise. Based on the measurements in another work [7], a of 4.1 and a of 2.21 are adopted for the following NF calculation. Fig. 6(b) shows the simplified equivalent circuit of Fig. 6(a), is modified as to take into account the efin which fects of the cascode transistor ( ) and the peaking . Clearly, can be represented as follows: inductor

(17)

(18) where stands for the total noise current in the short-circuited output path originating from the noise current com, , , , , , and , which are ponents , , , produced by the noise generators from , , , and , respectively. ( ), ( ) and ( ) represent the corresponding noise factor contributions of , and to the LNA, respectively. A careful analysis yields (19a)–(19d), shown at the bottom of this page. Substituting (19a)–(19d) into for simplicity, (18), and assuming , which is usually the case for frequencies around 0 and it then yields (20), shown at the top of the following page. is a constant if the effect of Note that the first term of is disregarded, while it becomes a monotonously decreasing (equal to function for frequencies lower than 22 GHz in this work) if the effect of is taken into account. in the LNA can suppress the gate In a word, the addition of

(19a)

(19b)

(19c) (19d)

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forward gain ( ) of transistor M1 by using the BSIM4V4.3.0 model (provided by the foundry) and the simplified model [in Fig. 11(a)] for frequencies from 0.1 to 30 GHz. As can be seen, the simulated result of the simplified model conforms well to the BSIM4V4.3.0 model. Besides, in the estimation of the performances of the LNA in this work, it was found that only and (of the 11 parameters) need to be considered. This explains why the calculated results of the LNA in this and are taken into account for transistor work (only M1) are consistent with the simulated results (BSIM4V4.3.0 model is adopted for transistor M1). Detailed descriptions of the parameters and equations of the BSIM4V4.3.0 model can be found online.1 ACKNOWLEDGMENT The authors are very grateful for the supports from the United Microelectronics Corporation, Hsin-chu, Taiwan, and National Chip Implementation Center (CIC), Hsin-Chu, Taiwan, for chip fabrication and measurement. REFERENCES

Fig. 11. (a) Simplified small-signal equivalent-circuit model of transistor M1. (b) Simulated S , S , and S of transistor M1 both by the BSIM4V4.3.0 model (provided by the foundry) and the simplified model in Fig. 11(a).

noise over the frequencies of interest, as shown in the inset of Fig. 7(a). APPENDIX B NMOS MODEL In this appendix, we briefly introduce the NMOS model (provided by the foundry) used in the simulation. The modeling of the NMOS devices was based on the compact model of BSIM4V4.3.0. This large-signal model included 366 parameters, in which 108 parameters were dc (or low-frequency) related, and the others were high-frequency related. Fig. 11(a) shows the simplified small-signal equivalent-circuit model of the input transistor M1 in this work, in which only 11 important parameters (of the complete 366 parameters) were included. and ) and Fig. 11(b) shows the simulated return losses (

[1] M. Okushima, J. Borremans, D. Linten, and G. Groeseneken, “A DC-to-22 GHz 8.4 mW compact dual-feedback wideband LNA in 90 nm digital CMOS,” in Proc. IEEE RFIC Symp., Jun. 2009, pp. 295–298. [2] M. Chen and J. Lin, “A 0.1–20 GHz low-power self-biased resistive-feedback LNA in 90 nm digital CMOS,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 5, pp. 323–325, May 2009. [3] A. Bevilacqua and A. M. Niknejad, “An ultrawideband CMOS lownoise amplifier for 3.1–10.6-GHz wireless receivers,” IEEE J. SolidState Circuits, vol. 39, no. 12, pp. 2259–2268, Dec. 2004. [4] Y. Lu, K. S. Yeo, A. Cabuk, and J. Ma, “A novel CMOS low-noise amplifier design for 3.1- to 10.6-GHz ultra-wide-band wireless receivers,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 8, pp. 1683–1692, Aug. 2006. [5] C. F. Liao and S. I. Liu, “A broadband noise-canceling CMOS LNA for 3.1–10.6-GHz UWB receiver,” IEEE J. Solid-State Circuits, vol. 42, no. 2, pp. 329–339, Feb. 2007. [6] Y. Shim, C. W. Kim, J. Lee, and S. G. Lee, “Design of full band UWB common-gate LNA,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 10, pp. 721–723, Oct. 2007. [7] P. Heydari, “Design and analysis of performance-optimized CMOS UWB distributed LNA,” IEEE J. Solid-State Circuits, vol. 42, no. 9, pp. 1892–1905, Sep. 2007. [8] F. Zhang and P. R. Kinget, “Low-power programmable gain CMOS distributed LNA,” IEEE J. Solid-State Circuits, vol. 41, no. 6, pp. 1333–1343, Jun. 2006. [9] R. Liu, C. Lin, K. Deng, and H. Wang, “A 0.5–14-GHz 10.6-dB CMOS cascode distributed amplifier,” in VLSI Circuits Symp. Dig., Jun. 2003, vol. 17, pp. 139–140. 1[Online]. Available: http://www.simucad.com/products/analog/spicemodels/models/bsim4/bsim4.html

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[10] C. W. Kim, M. S. Kang, P. T. Anh, H. T. Kim, and S. G. Lee, “An ultra-wideband CMOS low noise amplifier for 3–5-GHz UWB system,” IEEE J. Solid-State Circuits, vol. 40, no. 2, pp. 544–547, Feb. 2005. [11] K. H. Chen, J. H. Lu, B. J. Chen, and S. I. Liu, “An ultra-wide-band 0.4–10-GHz LNA in 0.18-m CMOS,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 3, pp. 217–221, Mar. 2007. [12] C. W. Kim, M. S. Jung, and S. G. Lee, “Ultra-wideband CMOS low noise amplifier,” Electron. Lett., vol. 41, no. 7, pp. 384–385, Mar. 2005. [13] Q. Li and Y. P. Zhang, “A 1.5 V 2–9.6 GHz inductorless low-noise amplifier in 0.13 m CMOS,” IEEE Trans. Mircrow. Theory Tech., vol. 55, no. 10, pp. 2015–2023, Oct. 2007. [14] H. K. Chen, D. C. Chang, Y. Z. Juang, and S. S. Lu, “A compact wideband CMOS low-noise amplifier using shunt resistive-feedback and series inductive-peaking techniques,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 8, pp. 616–618, Aug. 2007. [15] Y. S. Lin, C. Z. Chen, H. Y. Yang, C. C. Chen, J. H. Lee, G. W. Huang, and S. S. Lu, “Analysis and design of a CMOS UWB LNA with dual-RLC -branch wideband input matching network,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 2, pp. 287–296, Feb. 2010. [16] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005, pp. 422–496. [17] S. Smith, Microelectronic Circuits, 5th ed. Oxford, U.K.: Oxford Univ. Press, 2004, pp. 818–830. [18] M. C. Chiang, S. S. Lu, C. C. Meng, S. A. Yu, S. C. Yang, and Y. J. Chan, “Analysis, design, and optimization of InGaP-GaAs HBT matched-impedance wide-band amplifiers with multiple feedback loops,” IEEE J. Solid-State Circuits, vol. 37, no. 6, pp. 694–701, Jun. 2002. [19] B. K. Ko and K. Lee, “A comparative study on the various monolithic low noise amplifier circuit topologies for RF and microwave applications,” IEEE J. Solid-State Circuits, vol. 31, no. 8, pp. 1220–1225, Aug. 1996. [20] B. Afshar and A. M. Niknejad, “X=Ku band CMOS LNA design techniques,” in IEEE Custom Integr. Circuits Conf., 2006, pp. 389–392. [21] H. K. Chen, Y. C. Hsu, T. Y. Lin, D. C. Chang, Y. Z. Juang, and S. S. Lu, “CMOS wideband LNA design using integrated passive device,” in IEEE MTT-S Int. Microw. Symp. Dig., 2009, pp. 673–676. [22] M. L. Edwards and J. H. Sinsky, “A new criterion for linear 2-port stability using a single geometrically derived parameter,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 12, pp. 2303–2311, Dec. 1992. [23] H. W. Chiu, S. S. Lu, and Y. S. Lin, “A 2.17 dB NF, 5 GHz band monolithic CMOS LNA with 10 mW DC power consumption,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 813–824, Mar. 2005. [24] M. T. Reiha and J. R. Long, “A 1.2 V reactive-feedback 3.1–10.6 GHz low-noise amplifier in 0.13 m CMOS,” IEEE J. Solid-State Circuits, vol. 42, no. 5, pp. 1023–1033, May 2007. [25] G. D. Nguyen, K. Cimino, and M. Feng, “A RF CMOS amplifier with optimized gain, noise, linearity and return loss for UWB applications,” in IEEE Radio Freq. Integr. Circuits Symp., 2008, pp. 505–508. [26] J. H. Lee, C. C. Chen, H. Y. Yang, and Y. S. Lin, “A 2.5-dB NF 3.1–10.6-GHz CMOS UWB LNA with small group-delay-variation,” in IEEE Radio Freq. Integr. Circuits Symp., 2008, pp. 501–504. [27] C. C. Chen, J. H. Lee, Y. S. Lin, C. Z. Chen, G. W. Huang, and S. S. Lu, “Low noise-figure P AA-mesh inductors for CMOS UWB RFIC applications,” IEEE Trans. Electron Devices, vol. 55, no. 12, pp. 3542–3548, Dec. 2008.

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Hsien-Ku Chen received the B.S. and M.S. degrees from Feng-Chia University, Taichung, Taiwan, in 2001 and 2003, respectively, and is currently working toward the Ph.D. degree in electronics engineering at National Taiwan University, Taipei, Taiwan. From 2003 to 2008, he was with the National Chip Implementation Center, Hsin-Chu, Taiwan, where he was involved with SiGe RF integrated circuit (RFIC) design. From 2008 to 2009, he was with Alcatel-Lucent, Bell Laboratories, Murray Hill, NJ, where he was involved in high-speed integrated circuits (ICs) for millimeter-wave applications. Since 2009, he has been the Realtek Semiconductor Corporation, Hsinchu, Taiwan, where he is currently engaged in analog RF circuits for wireless applications. His research interests are high-speed mixed-signal circuits, wireless transceivers for wireless sensor networks, and device modeling in nanometer CMOS technology for millimeter-wave application. Mr. Chen is a member of Phi Tau Phi.

Yo-Sheng Lin (M’02–SM’06) was born in Puli, Taiwan, on October 10, 1969. He received the Ph.D. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1997. His doctoral thesis concerned the fabrication and study of GaInP–InGaAs–GaAs doped- channel field-effect transistors and their applications to monolithic microwave integrated circuits (MMICs). In 1997, he joined the Taiwan Semiconductor Manufacturing Company (TSMC), as a Principle Engineer for 0.35/0.32-m DRAM and 0.25-m embedded DRAM technology development with the Integration Department, Fab-IV. Since 2000, he has been responsible for 0.18/0.15/0.13-m CMOS low-power device technology development with the Department of Device Technology and Modeling, Research and Development, and in 2001 became a Technical Manager. In August 2001, he joined the Department of Electrical Engineering, National Chi Nan University (NCNU), Puli, Taiwan, where he is currently a Professor. From June to September 2004, he was a Visiting Researcher with the High-Speed Electronics Research Department, Bell Laboratories, Lucent Technologies, Murray Hill, NJ. From February 2007 to January 2008, he was a Visiting Professor with the Department of Electrical Engineering, Stanford University, Stanford, CA. His current research interests are in the areas of characterization and modeling of RF active and passive devices (especially 30–100-GHz interconnections, inductors and transformers for millimeter-wave (Bi)CMOS integrated circuits), and RFICs/MMICs. Dr. Lin was a recipient of the 2006 Excellent Research Award presented by NCNU. He was also a recipient of the 2007 Outstanding Young Electrical Engineering Engineer Award presented by the Chinese Institute of Electrical Engineering.

Shey-Shi Lu (S’89–M’91–SM’99) was born in Taipei, Taiwan, on October 12, 1962. He received the B.S. degree from National Taiwan University, Taipei, Taiwan, in 1985, the M.S. degree from Cornell University, Ithaca, NY, in 1988, and the Ph.D. degree from the University of Minnesota at Minneapolis–St. Paul, in 1991, all in electrical engineering. His master’s thesis concerned the planar doped barrier hot electron transistor. His doctoral dissertation was focused on the uniaxial stress effect on AlGaAs/GaAs quantum well/barrier structures. In August 1991, he joined the Department of Electrical Engineering, National Taiwan University, where he is currently a Professor. Since August 2007, he has also been the Director of the Graduate Institute of Electronics Engineering, National Taiwan University. His current research interests are in the areas of RFIC/MMICs and micromachined RF components.

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Phase Noise of Distributed Oscillators Xiaofeng Li, O. Ozgur Yildirim, Wenjiang Zhu, and Donhee Ham

Abstract—In distributed oscillators, a large or infinite number of voltage and current variables that represent an oscillating electromagnetic wave are perturbed by distributed noise sources to result in phase noise. Here we offer an explicit, physically intuitive analysis of the seemingly complex phase-noise process in distributed oscillators. This study, confirmed by experiments, shows how the phase noise varies with the shape and physical nature of the oscillating electromagnetic wave, providing design insights and physical understanding. Index Terms—Distributed oscillators, oscillators, phase noise, pulse oscillators, soliton oscillators, standing-wave oscillators.

I. INTRODUCTION

P

HASE NOISE is among the most essential and interesting aspects of oscillator’s dynamics and quality [1]–[12]. The oscillator [see Fig. 1(a)], where noise perphase noise of an tank, is well turbs the voltage across, and the current in, the understood, owed to works developed until 1960s. For example, Lax’s 1967 work [1] provided a tremendous fundamental underoscillators. standing of phase noise in In contrast, phase-noise processes are harder to grasp in distributed oscillators, or wave-based oscillators, where energy storage components and/or gain elements are distributed along waveguides or transmission lines to propagate electromagnetic waves. The difficulty arises as a large or infinite number of voltage and current variables representing a wave are continually perturbed by noise sources distributed along waveguides or transmission lines. How can we visualize the complex perturbation dynamics and calculate phase noise of distributed oscillators? How does phase noise depend on waveforms, and how can we reconcile it with thermodynamic concepts? An explicit analysis of phase noise in distributed oscillators, which can offer physical understanding, remains to be carried out. In this paper, we conduct an explicit physically intuitive analysis of phase noise in distributed oscillators. The starting point is our realization that the comprehensive phase-noise framework established in 1989 by Kaertner [2], where he extended Lax’s phase-noise study to deal with any general number of Manuscript received October 20, 2009; revised March 07, 2010; accepted April 30, 2010. Date of publication July 19, 2010; date of current version August 13, 2010. This work was supported by the Army Research Office under Grant W911NF-06-1-0290, by the Air Force Office of Scientific Research under Grant FA9550-09-1-0369, and by the World Class University (WCU) Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology (R31-2008-000-10100-0). X. Li, O. O. Yildirim, and D. Ham are with the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138 USA (e-mail: [email protected]; [email protected]). W. Zhu was with the Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 USA. He is now with JP Morgan, New York, NY 10017 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.2010.2053062

voltage and current variables in oscillators, is applicable to the distributed oscillators. Our explicit application of Kaertner’s framework to phase-noise analysis in distributed oscillators provides physical understanding and design insight. The essence of our analysis is experimentally verified. Our analysis can be applied to distributed oscillators with arbitrary waveforms along waveguides or transmission lines. As demonstrational vehicles, we use three distributed oscillators, shown in Fig. 1(b)–(d). Each oscillator consists of a transmission line, an active circuit at one end of the line, and an open at the other line end. Both the open end and active circuit reflect an oncoming wave so that the wave can travel back and forth on the transmission line. The reflection by the active circuit comes with an overall gain, which compensates the line loss. Depending on the specific gain characteristic, different oscillation waveforms result. If the active circuit amplifies small voltages and attenuates large voltages as in standard oscillators, sinusoidal standing waves are formed on the transmission line [13], [14] [see Fig. 1(b)]. If the active circuit attenuates small voltages and amplifies large voltages,1 a bell-shaped pulse is formed and travels back and forth on the line [15] [see Fig. 1(c)]. If the normal, linear transmission line in Fig. 1(c) is replaced with a nonlinear transmission line, a line periodically loaded with varactors, as in Fig. 1(d), again a pulse is formed [16], [17], but the line nonlinearity sharpens the pulse into what is known as a soliton pulse, which is much sharper than the linear pulse [16]–[20]. Using these oscillators, we show how to calculate phase noise in distributed oscillators, a central outcome of this work. Another main outcome is that the calculation reveals (and experiments confirm) how phase noise of distributed oscillators depends on their waveforms: specifically, it is shown that the linear pulse oscillator [see Fig. 1(c)] has lower phase noise than the sinusoidal standing-wave oscillator [see Fig. 1(b)]. Not only is this result useful from the design point of view, but it offers fundamental physical understanding if reconciled with thermodynamic concepts. In the sinusoidal standing-wave oscillator, one osresonating mode is dominantly excited. Just like in the cillator, the single resonating mode possesses two degrees of freedom, namely, the electric and magnetic fields (voltage and current standing waves), each storing a mean thermal energy ( : Boltzmann’s constant; : temperature) according of to the equipartition theorem. Overall, a total thermal energy of perturbs the single resonating mode. This thermodynamic notion is in congruence with our analysis; thus, the sinusoidal oscillator standing-wave oscillator can be treated like the without having to resort to our analysis developed for general distributed oscillators. By contrast, in the pulse oscillator, the oscillating pulse contains multiple harmonic modes, thus, the equipartition theorem predicts that the pulse oscillator would 1If small voltages are attenuated, oscillation startup cannot occur. A special startup circuit is to be arranged in this case [15]–[17].

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Fig. 2. Oscillator’s limit cycle in 2N -D state space. For illustrative purposes, we only show two axes.

Fig. 1. Oscillator examples. (a) LC oscillator. (b) =2 sinusoidal standingwave oscillator. (c) Linear pulse oscillator. (d) Soliton pulse oscillator.

deal with noise sources, as in [3], which is another work by Kaertner, extending his original work [2] to include the effect of noise. Section II reviews the fundamental theories of phase noise by Lax [1] and Kaertner [2]. Section III analyzes direct phase perturbations and their effect on phase noise. Section IV examines indirect phase perturbations caused by amplitude-to-phase error conversion, and their effect on phase noise. Sections V and VI present measurements and their analysis. II. PHASE-NOISE FUNDAMENTALS: REVIEW OF LAX’S AND KAERTNER’S WORKS A. Lax’s Fundamental Theory of Phase Noise

have higher phase noise than the sinusoidal standing-wave oscillator, which is wrong and opposite the result of our analysis/experiment. The inapplicability of the equipartition theorem in the Fourier domain to the pulse oscillator arises as the multiple harmonic modes are inter-coupled or mode-locked together. The legitimate phase-noise calculation for the pulse oscillator thus calls for a general analysis like ours, which captures the correct physics: noise generated at a given point of a wave propagation medium affects a wave only when it passes through the point, thus, the noise’s chance to enter the phase-noise process is smaller for a spatially localized pulse than for a sinusoidal wave spread over the medium. Consequently, the linear pulse oscillator has lower phase noise. Yet another main outcome of this study involves the soliton pulse oscillator [see Fig. 1(d)]. As the soliton pulse oscillator has a smaller pulsewidth than the linear pulse oscillator [see Fig. 1(c)], according to the foregoing reasoning, the former would have lower phase noise than the latter. However, our analysis shows that this is not necessarily the case: due to the amplitude-dependent propagation speed of solitons, which is a hallmark nonlinear property of solitons, amplitude-to-phase-noise conversion can significantly contribute to phase noise of the soliton oscillator. Not only the waveform, but also the wave’s nonlinear nature, plays a role in the phase-noise process in the soliton oscillator. Our analysis can be readily applied to oscillators with distributed gains [21]. To show the essence simply, however, we do not include them in this paper. The essence is more easily tested with the three example circuits above. In addition, for mathematical simplicity, this paper focuses on phase noise incurred by white noise only, although our analysis can be extended to

The essence of Lax’s work [1] may be understood as follows. oscillator [see Fig. 1(a)]. The voltage across Consider an tank represent the oscillator’s state. and the current in the The steady-state oscillation follows a closed-loop trajectory, or ). limit cycle, in the 2-D – state space (Fig. 2 with Noise perturbs the oscillation, causing amplitude and phase errors. The amplitude error that puts oscillation off the limit cycle is constantly corrected by the oscillator’s tendency to return to its limit cycle. In contrast, the phase error on the limit cycle along its tangential direction accumulates without bound for no mechanism to reset the phase exists. In other words, the phase undergoes a diffusion along the limit cycle. Due to this phase diffusion, the oscillator’s output spectrum is broadened around the oscillation frequency, causing phase noise. Mathematically, in the presence of only white noise, the phase diffusion so occurs that the variance of the phase of the oscilgrows linearly with time lation (1) is the phase diffusion rate. This leads to the wellwhere known Lorentzian phase noise (2) and the

behavior for (3)

which is familiar from Leeson’s paper [22].

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As seen, once the phase diffusion rate is determined, phase noise is known. can be determined by evaluating the phase error along the tangential direction of the limit cycle for a given noise perturbation. The phase error depends not only on the noise level, but also on the oscillator state, or the limit cycle position where the perturbation occurs, due to the tangential projection [1]. This state-dependency is illustrated in Fig. 2: the same noise perturbation causes different phase errors at two different limit cycle positions and . Therefore, this state-dependent or time-varying phase error is averaged along the limit cycle, and thus will be a function of the oscillator’s waveform (limit cycle shape), as well as the noise level [1]. This state-dependent or time-variant property has been greatly exploited by the circuit community for low-noise oscillator design [4]. B. Kaertner’s Generalization Kaertner expanded Lax’s analysis to a general case where an oscillator state is described with variables in a -dimensional ( -D) state space (Fig. 2) [2]. Noise perturbations are decomposed into a component along the tangential direc-D state space, and tion of the oscillator’s limit cycle in the components orthogonal to the tangential direction. The former component, as in Lax’s work, corresponds to phase perturbation that directly drives the phase diffusion process. Besides, Kaertner considered how the latter components, corresponding to amplitude perturbations, indirectly contribute to phase diffusion through amplitude-to-phase-noise conversion determined by the oscillator’s dynamics. This calculation, like Lax’s theory, would follow the limit cycle and average the state-dependent phase errors in determining . Therefore, it holds true even in the general case treated by Kaertner that depends not only on the noise level, but also on the waveform or the shape of the limit cycle: this waveform dependency is a general hallmark property of oscillator’s phase diffusion. The general framework by Kaertner, including the orthogonal projection, is the basis of this work. We translate Kaertner’s mathematical language to what can be directly applied to calculating the phase noise of distributed oscillators. We decompose noise perturbations into phase and amplitude perturbations, and study their contributions to phase noise separately in Sections III and IV.

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Fig. 3. Artificial transmission line consisting of N LC sections. n = 1; 2;. . . ; N . Loss components R and G are included with associate noise.

in the -D state space.2 In the steady state, in the -D the state point evolves along a limit cycle, voltage and current variables of are space (Fig. 2). The perturbed by noise sources located along the transmission line, perturbations can be collectively represented by and these -D state space (Fig. 2). a single perturbation vector in the Overall, we have a single noise perturbation vector that collectively disturbs a single oscillator state point , the collection of voltage and current variables, which evolve all together along -D state space. Therefore, all voltage the limit cycle in the and current variables share exactly the same phase diffusion rate, and thus, the same phase noise. Our task is to calculate this common phase diffusion rate. When the transmission line is a continuous medium (e.g., coplanar stripline), the oscillator possesses infinitely many state variables. This can be dealt with as an extreme case of the arti. ficial line with Noise can be distributed along the line (e.g., thermal noise associated with the distributed loss in the line) or lumped (e.g., noise from the lumped active circuit). We will treat these two noise sources separately in Sections III-A and B. A. Direct Phase Perturbation Due to Distributed Noise We first consider the distributed noise from an artificial transstate variables (Fig. 3). The calculation mission line with of the direct phase perturbation by the distributed noise runs in two steps. First, by analyzing the oscillator dynamics in the presence of the noise, we identify the vector representing the -D state space. Second, we project noise perturbation in the this noise perturbation vector onto the tangential direction of the limit cycle to calculate the phase error. -D state To identify the vector of noise perturbation in the space, we apply Kirchhoff’s law in Fig. 33

III. PHASE NOISE DUE TO DIRECT PHASE PERTURBATION This section calculates the phase diffusion driven directly by the tangential projections of noise perturbations. We use the three transmission-line oscillators in Fig. 1(b)–(d) as demonstrational vehicles. Before going into detail, let us first visualize the phase-noise process in the transmission-line oscillator’s state space. The transmission line can be an artificial medium consisting of sections (Fig. 3). The corresponding oscillator has state variables, each corresponding to a voltage or current variables are variable of a capacitor or inductor. These collectively represented by a single point

(4a) (4b) and the dot above a variable represents where a time derivative. Resistance and conductance represent loss in the line. Their associated Nyquist voltage and current thermal noise and satisfy and . Other distributed noise sources, such as 2It is assumed that the oscillator’s active circuit is memoryless compared to the transmission line, not generating extra state variables. 3In case of the nonlinear transmission line, where is voltage dependent, the nominal -value is used in the following calculations for approximation.

C

C

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those from distributed gain elements [21], if any, can be similarly modeled. , , and Change of variables, with an impedance-unit parameter renders and dimensionless, and with the dimension of voltage. With the new variables, we rewrite (4) into

varying diffusion rate is usually time averaged over one oscillation period to yield a constant diffusion rate, which matters over a long run

(5a)

represents the time average. This concludes our calwhere culation of the direct phase perturbation and the resulting phase diffusion rate, arising from the distributed noise in the artificial transmission line. In the case of a continuous transmission line where with infinitesimally small sections, the phase diffusion rate can be directly obtained from (9) by replacing the summations and replacing , , with integrals and with their values per unit length, , , , and ( , , , , )

(5b) and are rescaled where noise terms that are independent Gaussian random variables with zero mean and variance [23]. Equation (5) identifies a random vector as the noise perturbation vector during . Also note that (5) treats voltage and the time interval current variables symmetrically, or, in the equal unit, thanks to . the change of variable, Now resorting to Kaertner’s scheme [2], we project onto the tangential direction of the limit cycle, or the direction of motion in the state space, which is defined by the state-space (Fig. 2). The inner product between and the unit velocity vector is the perturbation along the direction of motion. Division of this tangential perturbation by the magnitude of the velocity yields the timing error , which is converted to , where is the oscillation the phase error using frequency.4 In sum,

(9)

(10)

As stated earlier, Kaertner derived a more general expression for the phase error using the tangential projection [2], and (6) is an explicit reduction of Kaertner’s expression, suitable for and are independent the distributed oscillators. Since Gaussian random variables with zero mean, is also Gaussian with zero mean, and its variance is given by

where is the spatial coordinate along the line, is the partial is the wave velocity on the line time derivative, and (not to be confused with the state-space velocity ). Note that the time-averaged terms in (9) and (10) depend on the shape of the limit cycle or the oscillating waveform, which originates from the state dependency of the phase error [1], [2]. Therefore, distributed oscillators with differing oscillating waveforms will exhibit different phase noise. To see this concretely and to interpret the waveform dependency in thermodynamic terms, we now apply (9) or (10) to the three transmission-line oscillators [see Fig. 1(b)–(d)], but we start with a oscillator. lumped oscillator, Fig. 1(a): The voltage and current 1) Lumped in the tank of an almost sinusoidal oscillator are given and . Plugging by , while only taking into account these into (9) with with to model a parallel loss in the tank, we obtain

(7)

(11)

(6)

Since the phase error must accumulate, growing its variance linearly with time, (7) can be directly compared with the phase diffusion model (1) to find the phase diffusion rate (8)

Since and are periodic functions of time, above is an instantaneous rate at a given time, and it varies periodically with time. This is because the size of the phase error even for a fixed noise vector depends on the state of oscillation (where the state lies on the limit cycle), as indicated by (6), as illustrated in Fig. 2, and as mentioned as a hallmark property of the oscillator phase diffusion [1], [2] in Section II. The periodically 4The detailed projection procedure up to this part of the present paragraph is generally captured in (16) and (23b) in Kaertner’s paper [2].

where is the total oscillation energy stored in is the tank’s quality factor. This the tank and agrees with Lax’s work [1], and can be directly converted to the well-known Leeson’s formula [22]. Note that is the total thermal energy in the tank (corresponding to noise) because and each stores a mean thermal energy of according to the equipartition theorem. The ratio is the of the thermal energy to the oscillation energy noise-to-signal ratio. 2) Sinusoidal standing-wave oscillator, Fig. 1(b): Voltage and current standing waves in a continuous transand mission line are where is the wavenumber. Plugging these into (10) and performing the integrals over the line length , we find (12)

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where is the total energy stored in the transis the quality factor mission line, and of the line. Note that the phase diffusion rate of the sinusoidal standing-wave oscillator above conforms to that oscillator in (11). This is expected and of the lumped can be understood in the Fourier domain: although the distributed voltage and current variables in the transmission line correspond to a large (or infinite) number of degrees of freedom in general, in the sinusoidal standing-wave oscillator, only one resonance mode of the transmission line tank, this resonance is effectively excited. Like an mode possesses only two degrees of freedom, namely, the electric and magnetic fields (voltage and current standing . It waves), each storing a mean thermal energy of is then possible to treat sinusoidal standing-wave oscillaoscillators. In fact, using the tors effectively as lumped Fourier amplitudes as state variables, the dynamical equations of standing-wave oscillators can be reduced to those oscillators. Such reduction was already carof lumped ried out in the early stage of the phase-noise study for masers and lasers, whose microwave/optical cavities are indeed distributed waveguides that support microwave/optical standing waves [24]. The foregoing discussion may lead the reader to doubting the usefulness of our analysis. However, as we will show right below that this thermodynamic argument in the Fourier domain fails in the case of pulse oscillators, and our analysis becomes necessary. 3) Linear and soliton pulse oscillators, Fig. 1(c) and (d): The bell-shaped pulse in both linear and soliton pulse oscillators may be described by the same functional form with a full spatial width at half maximum , although the soliton pulse tends to be sharper (larger ). Therefore, their phase diffusion rates can be calculated together using (10). Since the pulse typically spans a length much shorter than the total length of the line, we can replace the integrals over the line length in (10) with integrals over the entire -axis,5 i.e., (13) to obtain (14) is the total energy carried by the where pulse. Equation (14) shows that phase noise of the pulse oscillator improves as the pulse gets sharper, a clear manifestation of the waveform dependency of captured by the shape factor (time-averaged term) in (10). This can be physically understood in time domain: noise perturbation generated at position affects the oscillator’s phase or the timing of the pulse only when the pulse passes through the point. Its chance to enter the phase-noise process thus becomes smaller as the pulsewidth decreases, yielding lower 5When the pulse reflects at the line ends, the instantaneous phase diffusion rate ( ) can be slightly modified due to superposition of the incident and reflected pulses. The result after time averaging, however, remains unaltered.

Dt

and better phase noise. Since is approximately the number of excited harmonic modes that constitute the pulse, (14) may be expressed as . Had we treated the problem in the Fourier domain assuming that the modes were independent of one another, each mode would store a thermal enaccording to the equipartition theorem, and ergy of would result. This, however, is an incorrect result, larger than the true value above by . The Fourier domain analysis fails because a factor of the harmonic modes are not independent, as their relative phases are coupled (or mode-locked) together. This consideration reveals the usefulness and necessity of our analysis in dealing with pulse oscillators. Now let us compare the phase diffusion rate of the sinusoidal standing-wave oscillator, (12), and that of the pulse oscillator, (14). Their evident difference is again the indication of the waveform dependency of . If the standing-wave oscillator and pulse oscillator have the same amplitude for the same noise level, we have (15) If the two oscillators have the same power dissipation for the same noise level, we have (16) . Therefore, for a very sharp pulse, This physically makes sense as explained above, i.e., noise at any given point on the transmission line has less chance to participate in the phase-noise process for a spatially localized pulse than for a standing wave spread over the transmission line, thus, the pulse oscillator is to have lower phase noise than the sinusoidal standing-wave oscillator. Also we note once again that the Fourier domain argument using the equipartition theorem would predict an opposite and wrong result, i.e., higher phase noise for the pulse oscillator that has a larger number of harmonic modes than the sinusoidal standing-wave oscillator. Since the harmonic modes in the pulse oscillator are inter-coupled, the equipartition theorem cannot be applied to the pulse oscillator, as discussed above. B. Direct Phase Perturbation Due to Lumped Noise The direct phase perturbation by the lumped noise from the active circuit in Fig. 1(b)–(d) can be calculated in a similar fashion. At the output of the active circuit (Fig. 4), Kirchhoff’s law gives (17) where describes the characteristics of the active circuit. Its associated noise source is lumped into and assumed to be white Gaussian with autocorrelation . The noise level, rep, is resented by the equivalent output noise conductance generally a function of time, which varies periodically with the oscillation [4].

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Comparing (21) with (22), we find (23) if the standing-wave and pulse oscillators have the same amplitude for the same lumped noise level, and

Fig. 4. Circuit model for the active circuit with lumped noise.

As in Section III-A, using change of variables , we rewrite (14) into

(24)

and

(18) is a Gaussian random variable with zero where . This equation identimean and variance fies as a noise perturbation vector in -D state space. By projecting along the tangential dithe rection of the limit cycle and following the same procedure in Section III-A, we obtain the following phase diffusion rate due to the direct phase perturbation by the lumped noise for the artificial transmission line case: (19)

In the case of continuous transmission line, it becomes (20) The time-averaged terms in (19) and (20) once again exhibit the waveform dependency, as a consequence of the state dependency of the phase error. We again apply these results to the distributed oscillators in Fig. 1(b)–(d). Evaluation of (19) and (20) requires knowledge of the specific form of the . In each of our experimental time-varying noise level circuits corresponding to Fig. 1(b)–(d), is almost a is used in the constant (Section V). Therefore, a constant following calculations. 1) Sinusoidal standing-wave oscillator, Fig. 1(b): and in (20) yield (21) 2) Linear and soliton pulse oscillators, Fig. 1(c) and (d): At the active circuit end of the line, an incident pulse is superposed with a reflected pulse. The joint voltage amplitude is about twice the incident pulse amplitude, or , assuming linear superposition.6 Using this and (13) in (20), we obtain (22) 6In the soliton oscillator case, V (0; t) may slightly differ from the linear superposition due to the nonlinear interaction between the oncoming and reflected pulses, but we do not expect this to significantly change the result.

if they have the same power dissipation for the same lumped noise level. These results are exactly the same as in the case of distributed noise perturbation: (15) and (16). The results again indicate that a sharper pulse experiences the lumped noise at for a shorter period of time, thus yielding a slower phase diffusion. We point out that even in the foregoing case of lumped noise, the analysis has still dealt with the distributed nature: the large set of voltage and current variables representing the oscillating electromagnetic wave are distributed along the transmission line, even if the noise source is lumped. The analysis has shown how a perturbation even at one fixed point affects the collective oscillation of the entire distributed system. Thus far we have considered only the direct phase perturbations along the tangential directions of the limit cycle, while not addressing the effect of the amplitude perturbation. As shown by Kaertner, the amplitude perturbation can contribute substantially to phase diffusion through amplitude-to-phase-noise conversion in certain oscillators [2]. This effect is not of great importance in the standing-wave and linear pulse oscillators for the oscillation frequency of the linear transmission line with reasonably high7 is essentially independent of oscillation amplitude. Therefore, the analysis above is sufficient for the standingwave and linear pulse oscillators, and it remains valid that the latter has less phase noise than the former. In contrast, in the soliton pulse oscillator, the amplitude perturbation will translate to timing perturbations through soliton’s amplitude-dependent propagation speed in the nonlinear transmission line, which can contribute significantly to phase noise. We now turn to this issue for the soliton oscillator.

IV. AMPLITUDE PERTURBATION AND ITS PHASE-NOISE EFFECT IN SOLITON OSCILLATORS This section examines how amplitude perturbation contributes to phase noise in the soliton oscillator through amplitude-to-phase error conversion. We again use Kaertner’s orthogonal projection scheme, but we do not carry it out in full to determine the exact amplitude-to-phase error conversion, which requires diagonalizing large matrices corresponding to oscillator’s dynamical equations linearized in proximity of its limit cycle. Instead, we evaluate the effect by devising an intuitive phenomenological approach that captures the essence of amplitude-to-phase error conversion in the soliton oscillator. A soliton pulse propagating in a nonlinear transmission line , can be described as 7As

seen in Section VI, measured Q is about 100.

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is a function of , this decaying amplitude perturbation will according to incur a velocity variation of

which will translate to a total position shift of

N

Fig. 5. Amplitude perturbation in the 2 -D state space. Amplitude error is translated to timing error due to the amplitude-dependent speed.

1t

1A

where both the spatial pulsewidth and wave velocity depend on the amplitude . In general, a taller soliton has a narrower width and propagates faster. The exact dependency is determined by the specific form of the line’s nonlinearity or the capacitance–voltage relation of the varactors. One particularly well-known example is provided in Appendix [25]. is now paWith the pulse waveform above, the limit cycle rameterized not only in terms of time , but also in terms of amgiving the direction of phase/timing plitude .8 Just like perturbation, defines the direction of amplitude perturbation in the state space. Following a procedure similar to that in Section III, we now calculate the amplitude error by projecting a noise perturbation vector along the direction defined by (Fig. 5).

This is equivalent to a timing error of

Therefore, the total contribution to the timing uncertainty by generated during is (27) Plugging (26) into (27) and comparing the result with (1), we find the phase diffusion rate due to the amplitude perturbation to be

(28) In continuous coordinate, this becomes

A. Indirect Phase Perturbation Due to Distributed Noise For distributed noise, in Section III-A acts again as the noise perturbation vector. It yields the following amplitude error:

(29) The phase diffusion rate depends on: 1) the sensitivity of amplitude to noise perturbations, captured by and ; 2) the sensitivity of velocity to amplitude ; and 3) lifetime , i.e., how fast the amplitude error is corrected by the oscillator.9

(25) B. Indirect Phase Perturbation Due to Lumped Noise Similar results can be obtained for the lumped noise source described in Section III-B

is Gaussian with zero mean and variance

(26) (30) Unlike phase perturbation, the amplitude perturbation puts the oscillation off the limit cycle and is corrected in finite time by the oscillator’s dynamics. Thus, the amplitude error cannot accumulate indefinitely, but decays exponentially as (Fig. 5) where is the initial amplitude error and is the life time of the decay, which is determined by the oscillator’s loss and gain characteristics. Since the soliton propagation velocity 8A similar parametrization of the limit cycle was used by Haus [26] in his study of the noise processes in soliton lasers.

for an artificial nonlinear transmission line, and

(31) for a nonlinear line approaching the continuous limit. 9In [2], the decay behavior of amplitude error and its translation to phase error are generally captured in (20) and the first term of (26).

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C. Indirect Versus Direct Phase Perturbations Let us first note that the total phase diffusion rate in a soliton caused by direct phase perturbations oscillator is a sum of induced by amplitude perturbations. This simple addiand tion is possible because the amplitude perturbation [e.g., (25)] and the phase perturbation [e.g., (6)] are orthogonal or uncorrelated to each other. This orthogonality can be checked by and in (25) are even functions of , noticing that and in (6) are odd. whereas Given this simple addition, we may directly compare and to see which process contributes more to the total phase diffusion rate. For a concrete comparison, we use the well-known soliton model that takes a waveform of with specific expressions for and found in the Appendix. With this soliton, (29) yields

(32) for distributed noise sources. Comparing this with of (14) that is caused by the same distributed noise sources, we have

(33) Exactly the same relationship holds in the case of lumped noise perturbations, according to (20) and (31) if is a constant, which is the case in our experiment. As seen, the ratio depends largely on the velocity-amplitude relation and the life time of can be far larger than , which is the amplitude errors. case in our experiment (see Section V). V. EXPERIMENT A. Oscillator Prototypes For experimental proof of concept, we designed the sinusoidal standing-wave, linear pulse, and soliton pulse oscillators of Fig. 1(b)–(d) in the lower microwave region. All three oscillators are represented by one schematic in Fig. 6(a). The operation of the active circuit, whose topology is shared by all oscillators, is found in [17]. By adjusting component parameters, the active circuit can amplify small voltages and attenuate large voltages, or vice versa, to produce sinusoidal standing-wave or pulse oscillations. The transmission line in Fig. 6(a) can be linear or nonlinear: the former is for the standing-wave and linear pulse oscillators; the latter is for the soliton pulse oscillator. The design was implemented on printed circuit boards using discrete components. The standing-wave and linear pulse oscillators were constructed on the same board as one physical structure [see Fig. 6(b)], as they share the same linear line (24 sections) and the same active circuit topology. The component parameters in the active circuit are adjusted to switch between standing-wave and linear pulse oscillation. The soliton pulse oscillator on a separate board [see Fig. 6(c)] using a nonlinear line consisting of 24 inductor–varactor sections is an adoption from [17], where we reported the operation of the soliton oscillator.

Fig. 6. (a) Schematic of a circuit that can work as a standing-wave, linear pulse, or soliton pulse oscillator, depending on active circuit component parameters and transmission line choice. (b) Implemented circuit that can work as a standing-wave or linear pulse oscillator, depending on active circuit component parameters. (c) Implemented soliton oscillator, adopted from [17].

The oscillators produced desired oscillation waveforms. A 6-V battery was used for a stable power supply. The waveforms were monitored using an Agilent Infiniium 54855A 6-GHz oscilloscope with Agilent 1156A 100-k active probes. Fig. 7 shows steady-state oscillation waveforms measured at different line positions in each of the three oscillators. Waveform parameters are summarized in Table I. The oscillation frequencies are around 100 MHz for all oscillators. The harmonic contents of the pulse oscillators extend well into the microwave regime. B. Phase-Noise Measurement Fig. 8 shows the phase-noise measurement setup. The transmission line’s open end is connected to an Agilent E5052B phase-noise analyzer via an -section impedance matching network and a Mini-Circuits ZFL-11AD amplifier. The matching network is to ensure that the line sees a high impedance at what is designed to be an open end so that almost total reflection can occur as intended. The amplifier enhances the weak signal coupled out from the oscillator. Fig. 9 shows the measured phase noise of the three oscillators. For offset frequencies higher than 10 kHz, the phasetrend. At lower offset frenoise data follow closely to an quencies, the slope is slightly greater than 20 dB/dec, but still closer to 20 dB/dec than 30 dB/dec. This indicates that the noise sources in all oscillators are dominantly white. As predicted, the linear pulse oscillator has a better phase noise than the standingwave oscillator. The soliton oscillator, despite the soliton pulse’s

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Fig. 8. Phase-noise measurement setup.

Fig. 9. Measured phase noise for the three oscillators.

Fig. 7. Measured oscillations. (a) Near-sinusoidal standing-wave oscillator with amplitude A V. (b) Linear pulse oscillator with pulse amplitude A : V and width W spanning nine and one-half LC sections. At the open end joint amplitude is A due to linear superposition of incident and : V and W spanning reflected pulses. (c) Soliton pulse oscillator with A three LC sections. At the open end joint amplitude is : A due to nonlinear superposition [17].

= 22

=2

2

18 24

VI. MEASUREMENT-THEORY COMPARISON We here compare the measured phase noise to phase noise calculated using our theory. For the calculation, we first determine the intensity of the noise sources in the oscillators. A. Intensity of Noise Sources

TABLE I WAVEFORM PARAMETERS

sharp width, has the worst phase noise due to its amplitude-tophase-noise conversion, as predicted. Thus, the measurements agree with the core of our theory, which we will further examine quantitatively in Section VI.

We start with the lumped noise from the active circuit. We observed in all three oscillators that the active circuit remains inactive, except at the rising edge of the voltage signal at the left end of the line [see Fig. 6(a)] (in a continuous coordinate, corresponds to ). The rising edge of triggers transistors , , and in Fig. 6(a) so that each pulls reaches maximum. a constant current of about 10 mA until During this process, energy is injected to provide gain, and also, appreciable noise is injected, as illustrated in Fig. 10. The intensity of the active noise injected during the rising edge could be measured at dc by biasing the circuit with the same currents (10 mA), as experienced at the rising edge. Such measurement, however, was stymied in our experiment, for 10-mA dc current exceeds the dc breakdown current of . Even if transistors that could survive the dc transistor current are chosen, the transistors’ temperature with the large continuing dc current would be higher than its actual value during the oscillator operation, thus, not necessarily providing a faithful replica of the actual active noise level. We thus chose to carry out the measurement scheme in SPICE simulation. The in Fig. 4 intensity of the active circuit noise (intensity of multiplied by ) during the rising edge of was estimated to V Hz for all three oscillators. be

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TABLE II CALCULATED PHASE DIFFUSION RATES AND PHASE NOISE

Fig. 10. Time-domain illustration of the lumped noise from the active circuits.

The active noise was negligibly smaller during the rest of dynamics (Fig. 10). For the phase-noise calculation, we can effectively reduce this time-varying active noise to a constant noise. Imagine we use V Hz throughout the entire dynamics in calculating the phase diffusion rate [(19)/(20)], as if it persisted all the time. During the “flat” part of the waveform, i.e., during the absence of pulse events or transition edges when there is no actual appreciable noise, the fictitious noise , thus, its use for does not contribute to (19)/(20), as the flat duration is justified. During the falling edges, the fictitious noise spuriously contributes the same amount to (19)/(20) as the real noise during the rising edges does, for the oscillation waveforms are almost symmetric about their maxima. Hence, if we use half of the actual noise intensity injected only during V Hz, as a the rising edge, i.e., constant noise intensity the entire time, it will yield the correct phase diffusion rate. We will use this constant noise in the following analysis for all three oscillators. The distributed noise in the transmission line is far smaller in our experimental circuits. For both linear and nonlinear lines, the measured is about 100 at the oscillation frequency and measured is 50 , thus, the total noise power of the lossy V Hz. This is three lines is orders of magnitude smaller than the lumped active noise. In the following calculation of phase noise, we only use the dominant lumped active noise. The oscillation is still distributed while the noise is injected at a fixed point. B. Calculated Phase Noise By plugging the relevant waveform parameters of Table I and the effective constant active noise level into (21) and (22), , due to direct tangential we find the phase diffusion rate, phase perturbation, for all three oscillators (see Table II). The standing-wave, linear pulse, and soliton pulse oscillators line up in the decreasing order of , owed to the decreasing waveform width in that order, as explained earlier. In the soliton oscillator, is substantially small, we must also consider the although due to the indirect phase perturbation phase diffusion rate

Fig. 11. Measured decay behavior of amplitude perturbation in the soliton oscillator. (a) Measurement setup. (b) Decay behavior after an amplitude perturbation A. (c)  extraction. (d) @T=@A extraction.

1

caused by amplitude perturbation and its conversion to phase error. It is shown in Table II as well. in the soliton oscillator is done as folThe calculation of lows. Equation (33) may be rewritten into (34) in the nonlinear by using soliton’s round-trip time and , we perturb the soliton’s ampliline. To estimate tude with a short pulse produced by an Agilent 81150A function generator [see Fig. 11(a)]. From the decaying behavior of and of the dethe amplitude perturbation, we measure caying pulse, as illustrated in Fig. 11(b). From the slopes of versus and versus [see Fig. 11(c) and (d)], we then obtain and ns, which are consistent with the standard soliton model (see the Appendix). Plugging these measured values in (34) Hz, yields which enter Table II. In the soliton oscillator, the indirect phase perturbation contributes about ten times more to the total phase noise than the direct phase perturbation. The corresponding calculated phase-noise values for all oscillators at 100-kHz offset are also shown in Table II. The measurement errors in and

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Fig. 12. Standard nonlinear transmission line model.

give a 1.2-dB variation in the phase-noise calculation for soliton pulse oscillator. C. Measurement-Theory Comparison Although the calculated and measured phase noise fall within proximity to each other (Fig. 9; Table II), their absolute comparison for each oscillator is not quite meaningful, as the active noise level was only estimated in simulation, because it could not be measured due to the transistor breakdown (see Section VI-A). Since the actual active noise level, albeit not exactly known, is expectedly the same for all three oscillators (see Section VI-A), it would be meaningful to compare the measured and calculated relative phase-noise differences amongst the oscillators. This is because (21) and (22) contain the same active noise term. The relative examination will also facilitate showing the essence of this study, i.e., how oscillation waveforms and their physical nature influence the phase noise. This relative examination we do in the following. First, when compared to the standing-wave oscillator, the soliton oscillator has a 2.8-dB worse phase noise in calculation (Table II) and a 3.9-dB worse phase noise in measurement (Fig. 9). They are close, consistently explaining that in the soliton oscillator, despite the soliton’s sharpness that would yield superb phase noise if only the direct phase perturbation existed, the indirect phase perturbation through amplitude-to-phase error conversion significantly contributes to phase noise, offsetting the benefit of soliton’s sharpness. The slight difference between 2.8–3.9 dB may be explained by measurement errors and that our analysis used constant varactor capacitance values for the nonlinear line in the soliton oscillator, while it varies by four times as the varactor voltage changes with the oscillation. Second, in comparison to the standing-wave oscillator, the linear pulse oscillator has a 3.8-dB better phase noise in calculation (Table II) and an 8.0-dB better phase noise in measurements (Fig. 9). They consistently show the essence, i.e., the phase-noise superiority of the linear pulse oscillator due to its short pulsewidth. The numerical difference may be attributed to that in the actual linear pulse oscillator, the injected noise could have been smaller due to the detailed difference in its operation. Nonetheless, the expected reduction of phase noise for a pulsed waveform is consistently confirmed. VII. CONCLUSION We studied in the time domain how noise perturbs an oscillating electromagnetic wave in distributed oscillators to determine phase noise. In addition to offering an explicit physically intuitive time-domain method to analyze phase noise in

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distributed oscillators, this study provides the following few findings. 1) While the thermodynamic argument in the Fourier domain fails, our time-domain analysis is suitable in analyzing phase noise in pulse oscillators, where a number of excited modes constituting a pulse are interdependent. 2) Phase noise depends on the shape of the electromagnetic wave: the linear pulse oscillator has less phase noise than the sinusoidal standing-wave oscillator, as a sharp electromagnetic pulse has a reduced time period to interact with noise at any given position. 3) The soliton oscillator, however, can have a larger phase noise than the standing-wave oscillator despite the soliton’s sharpness due to amplitude-to-phase error conversion. This study highlights a couple of useful design strategies. First, the linear pulse oscillator can achieve superb phase noise if its active circuit, solely responsible for pulse shaping, is designed to produce a very narrow pulse. Second, if the amplitude-to-phase-noise conversion in the soliton oscillator can be mitigated by a proper active circuit design to reduce amplitude error’s lifetime, due to the soliton’s sharpness, the soliton oscillator can yield superb phase noise.

APPENDIX SOLITON MODEL IN NONLINEAR TRANSMISSION LINE The nonlinear transmission line is constructed as an inductor–varactor ladder network (Fig. 12). A standard model for the voltage dependency of the varactor capacitance is [25]. The corresponding soliton with propagating on the line is

which indicates that a taller soliton pulse has a narrower width and travels faster. The sensitivity of the oscillation period to amplitude in the soliton oscillator is given by

Since

V from the SPICE model and V, sections, , and (taking sections and ), which are in good agreement with our measurements. ACKNOWLEDGMENT

The authors thank N. Sun, Harvard University, Cambridge, MA, for valuable discussions. REFERENCES [1] M. Lax, “Classical noise. V. Noise in self-sustained oscillators,” Phys. Rev., vol. 160, pp. 290–307, Aug. 1967.

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[2] F. X. Kaertner, “Determination of the correlation spectrum of oscillators with low noise,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 1, pp. 90–101, Jan. 1989. noise in oscillators,” Int. [3] F. X. Kaertner, “Analysis of white and f J. Circuit Theory Appl., vol. 18, pp. 485–519, 1990. [4] 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. [5] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: A unifying theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 5, pp. 655–674, May 2000. [6] A. Demir, “Phase noise and timing jitter in oscillators with colored noise sources,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 12, pp. 1782–1791, Dec. 2002. [7] A. Suarez, S. Sancho, S. Ver Hoeye, and J. Portilla, “Analytical comparison between time- and frequency-domain techniques for phasenoise analysis,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2353–2361, Oct. 2002. [8] T. Djurhuus, V. Krozer, J. Vidkjaer, and T. K. Johansen, “Oscillator phase noise: A geometrical approach,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 7, pp. 1373–1382, July 2009. [9] A. Blaquiere, Nonlinear System Analysis. New York: Academic, 1966. [10] K. Kurokawa, An Introduction to the Theory of Microwave Circuits. New York: Academic, 1969. [11] E. E. Hegazi, J. Rael, and A. Abidi, The Designer’s Guide to HighPurity Oscillators. Berlin, Germany: Springer, 2004. [12] A. Suarez, Analysis and Design of Autonomous Microwave Circuits. New York: Wiley, 2009. [13] F. O’Mahony, C. P. Yue, M. A. Horowitz, and S. S. Wong, “A 10-GHz global clock distribution using coupled standing-wave oscillators,” IEEE J. Solid-State Circuits, vol. 38, no. 11, pp. 1813–1820, Nov. 2003. [14] W. F. Andress and D. Ham, “Standing wave oscillators utilizing waveadaptive tapered transmission lines,” IEEE J. Solid-State Circuits, vol. 40, no. 3, pp. 638–651, Mar. 2005. [15] L. A. Glasser and H. A. Haus, “Microwave mode locking at X -band using solid-state devices,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 2, pp. 62–69, Feb. 1978. [16] D. Ricketts, X. Li, and D. Ham, “Electrical soliton oscillator,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 373–382, Jan. 2006. [17] O. O. Yildirim, D. Ricketts, and D. Ham, “Reflection soliton oscillator,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 10, pp. 2344–2353, Oct. 2009. [18] C. J. Madden, R. A. Marsland, M. J. W. Rodwell, D. M. Bloom, and Y. C. Pao, “Hyperabrupt-doped GaAs nonlinear transmission line for picosecond shock-wave generation,” Appl. Phys. Lett., vol. 54, no. 11, pp. 1019–1021, Mar. 1989. [19] M. Case, M. Kamegawa, R. Y. Yu, M. J. W. Rodwell, and J. Franklin, “Impulse compression using soliton effects in a monolithic GaAs circuit,” Appl. Phys. Lett., vol. 58, no. 2, pp. 173–175, Jan. 1991. [20] K. S. Giboney, M. J. W. Rodwell, and J. E. Bowers, “Traveling-wave photodetector theory,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 8, pp. 1310–1319, Aug. 1997. [21] B. Kleveland, C. H. Diaz, D. Vook, 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. [22] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE, vol. 54, no. 2, pp. 329–330, Feb. 1966. [23] J. A. Gubner, Probability and Random Processes for Electrical and Computer Engineers. Cambridge, MA: Cambridge Univ. Press, 2006, pp. 453–459. [24] M. Lax and W. H. Louisell, “Quantum noise IX: Quantum Fokker–Planck solution for laser noise,” IEEE J. Quantum Electron., vol. QE-3, no. 2, pp. 47–58, Feb. 1967. [25] M. Toda, Nonlinear Waves and Solitons. Norwell, MA: Kluwer, 1989. [26] H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: A linearized approach,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 7, no. 3, pp. 386–392, Mar. 1990.

Xiaofeng Li was born in Luoyang, China. He received the B.S. degree in electrical engineering from the California Institute of Technology, Pasadena, in 2004, and is currently working toward the Ph.D. degree in electrical engineering from the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA. His research interests lie in the transformative area of emerging circuits and devices in conjunction with statistical physics, condensed matter physics, and material sciences. His research includes nonlinear electrical soliton oscillators, phase noise of self-sustained electrical oscillators and biological oscillators, and microfluidic stretchable electronics. He is currently conducting an experiment to study gigahertz–terahertz plasmonic waves using 1-D nanoscale devices, and to demonstrate their engineering utility in circuit applications. Mr. Li was a Gold Medalist at the 29th International Physics Olympiad, Iceland, 1998. He ranked first in the Boston Area Undergraduate Physics Competition in both 2001 and 2002. He was also the recipient of the 2002 California Institute of Technology Henry Ford II Scholar Award, the 2004 Harvard University Peirce Fellowship, and the 2005 Analog Devices Outstanding Student Designer Award.

O. Ozgur Yildirim received the B.S. and M.S. degrees in electrical and electronics engineering from Middle East Technical University, Ankara, Turkey, in 2004 and 2006, respectively, and is currently working toward the Ph.D. degree in electrical engineering and applied physics at Harvard University, Cambridge, MA. In the summers of 2002 and 2003, he was with the Scientific and Research Council of Turkey (TUBITAK), Ankara, Turkey, where he was involved with the development of a dynamic power compensation card and a field-programmable gate array (FPGA) for synchronous dynamic RAM interface. For his Master’s research, he was involved with the development of on-chip readout electronics for uncooled microbolometer detector arrays for infrared camera applications. His research interests include ultrafast and RF integrated circuits and devices, soliton and nonlinear waves and their utilization in electronic circuits, and nanoscale and terahertz electronics. Mr. Yildirim was the recipient of the 2009 Analog Devices Outstanding Student Designer Award.

Wenjiang Zhu was born in Lanzhou, China. He received the B.S. degree in chemistry from Peking University, Beijing, China, in 2002, the M.A. degree in chemistry from Princeton University, Princeton, NJ, in 2004, and the Ph.D. degree in chemistry from Harvard University, Cambridge, MA, in 2008. He also received the A.M. degree in statistics from Harvard University, Cambridge, MA, in 2008. He is currently with JP Morgan, New York, NY. His scientific expertise lies in applying statistical and stochastic methods to a range of dynamical problems in physics, biology, and finance.

Donhee Ham received the B.S. degree (summa cum laude) in physics from Seoul National University, Seoul, Korea, in 1996, and the Ph.D. degree in electrical engineering from the California Institute of Technology, Pasadena, in 2002. His doctoral research examined the statistical physics of electrical circuits. He is currently the Gordon McKay Professor of Applied Physics and Electrical Engineering at Harvard University, Cambridge, MA, where he is with the School of Engineering and Applied Sciences. The intellectual focus of his research laboratory at Harvard University is electronic, electrochemical, and optical biomolecular analysis using silicon integrated circuits and bottom-up nanoscale systems for biotechnology and

LI et al.: PHASE NOISE OF DISTRIBUTED OSCILLATORS

medicine; quantum plasmonic circuits using 1-D nanoscale devices; spin-based quantum computing on silicon chips; and RF, analog, and mixed-signal integrated circuits. His research experiences include affiliations with the California Institute of Technology–Massachusetts Institute of Technology (MIT) Laser Interferometer Gravitational Wave Observatory (LIGO), the IBM T. J. Watson Research, and a visiting professorship with POSTECH. He was a coeditor of CMOS Biotechnology (2007). Dr. Ham is a member of the IEEE conference Technical Program Committees including the IEEE International Solid-State Circuits Conference and the IEEE Asian Solid-State Circuits Conference, Advisory Board for the IEEE International Symposium on Circuits and Systems, International Advisory Board for the Institute for Nanodevice and Biosystems, and various U.S., Korea, and Japan industry, government, and academic technical advisory positions on subjects including ultrafast electronics, science and technology at the nanoscale, and convergence of information technology and biotechnology. He was a guest editor

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for the IEEE JOURNAL OF SOLID-STATE CIRCUITS. He received the Valedictorian Prize, the Presidential Prize, ranked top first across the Natural Science College of Seoul National University, and also the Physics Gold Medal from Seoul National University. He was the recipient of the Charles Wilts Prize given for the best doctoral dissertation in electrical engineering at the California Institute of Technology. He was the recipient of the IBM Doctoral Fellowship, Caltech Li Ming Scholarship, IBM Faculty Partnership Award, IBM Research Design Challenge Award, Silver Medal in the National Mathematics Olympiad, and Korea Foundation of Advanced Studies Fellowship. He was recognized by MIT Technology Review as among the world’s top 35 young innovators (TR35) in 2008 for his group’s work on CMOS RF nuclear magnetic resonance biomolecular sensor to pursue disease screening and medical diagnostics in a low-cost hand-held platform.

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Analytical Modeling of Microwave Parametric Upconverters Blake Gray, Student Member, IEEE, Bob Melville, Member, IEEE, and J. Stevenson Kenney, Fellow, IEEE

Abstract—Parametric amplification is a well-studied phenomenon by which a nonlinear reactance mixes an RF large-signal (pump) with an IF small-signal (source) to generate mixing products with gain. In this paper, two analytical models are derived and validated that predict limitations in the gain and efficiency of a parametric upconverter associated with varactor tuning range and quality factor. The analytical models are validated by circuit simulations and by two breadboard upconverters. Index Terms—Amplifier, modeling, parametric amplifiers (paramps), parametric circuits.

I. INTRODUCTION

M

ICROWAVE parametric amplifiers (paramps) were a topic of active research up through the 1960s [1], [2]. These circuits typically employed varactor diodes excited by a pump signal source, at frequency , to either generate a negative resistance or act as a mixing element. With parametric upconverters, a low-frequency source signal, at frequency , applied to the varactor will mix with the pump signal generating mixing products with gain. Output filtering extracts the desired mixing product and suppresses the pump and unwanted terms [3], [4]. The pump and source signal frequencies are incommensurate and the pump upper sideband mixing product is usually selected as the output. Historically, the research in parametric amplifiers was focused on low-noise applications, and power amplification was largely ignored. However, the same reactive mechanisms which allow low noise amplification in paramps, are also attractive for power amplification. Moreover, developments in new power technologies such as gallium nitride (GaN) may allow for significant increases in attainable power output from paramps. For this reason, the authors believe that an investigation into efficient power amplification using parametric circuits is warranted. Analytical modeling of parametric power amplifiers was performed in previous work by the authors, and resulted in a closed-form mathematical expression for the linear gain of

microwave paramps [5]. The gain analytical model was expressed as a sole function of the frequency of excitation signals and the varactor’s maximum-to-minimum capacitance ratio. In this paper, we expand on the gain model by including the effects of the losses inherent to the varactor in terms of its series quality factor. In addition, an analytical model for efficiency is presented, which relates the conversion of pump power to the output signal power as a function of the frequency of the input signal, the varactor’s series quality factor, capacitance ratio, bias voltage, and peak source voltage. Derivation of both models results in real and reactive expressions, where the reactive equations present a description of reflective mismatch between the varactor and load. Both models provide an additional systems analysis tool to the designer, which show value in recognizing operational constraints and performance limitations of parametric upconverters. We validate these models using a commercially available harmonic balance simulator and with comparisons to measured results from two breadboard paramp power upconverters. II. BACKGROUND Manley and Rowe developed a closed-form mathematical description of the power in the mixing products of nonlinear reactances as a function of the frequency of excitation signals [6]. Their derivation, as applied to parametric upconverters, provided a figure of merit by establishing an upper bound for the achievable real power gain. Their analysis was independent of the circuitry surrounding the nonlinear reactance, as long as the excitation currents at the desired frequency or frequencies were incommensurate. Parametric amplifiers employing rationally related frequencies must be analyzed in a conceptually different manner [1]. A typical application of parametric amplification is the upconverter. The circuit is tuned such that currents at , , and only are allowed to flow in the varthe mixing product actor. Application of the Manley–Rowe relations to parametric upconverters yields (1)

Manuscript received January 25, 2010; revised May 20, 2010; accepted May 20, 2010. Date of publication July 15, 2010; date of current version August 13, 2010. This work was supported in part by the Defense Advanced Research Projects Agency (DARPA)/Defense Sciences Office (DSO) under Contract AFRL FA8650-09-1-7970. B. Gray and J. S. Kenney are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA (e-mail: [email protected]; [email protected]). B. Melville is with the Office of Polar Programs, National Science Foundation (NSF), McMurdo Station, Antarctica (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.2010.2053064

where and are the power in the upper sideband and source signals, respectively. Equation (1) provides the maximum achievable gain under ideal operating conditions. There are multiple ways to define system efficiency of parametric upconverters; however, one should avoid solving the Manley–Rowe relations for simply the ratio of the power in the upconverted output to that of the pump, as it results in an equation that predicts an efficiency greater than 100%. must be the sum of the The power in the upper sideband

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source and pump power and , respectively, as required by conservation of power for a lossless reactance

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several orders of magnitude greater than the source, and domican be expressed as a nates the terminal voltage such that time-varying sinusoidal function with frequency

(2) For this reason, an accurate way to represent the system efficiency is (3) which guarantees the maximum obtainable efficiency under ideal conditions is 100%; this is effectively equivalent to power-added efficiency for transconductance amplifiers. The Manley–Rowe equations lack a closed-form solution for , as defined by (3), which we will subsequently derive. It should be noted that system efficiency, as defined by (3), describes RF-RF parametric power conversion. Practical implementations of high-efficiency parametric upconverters mandates highly efficient generation of the pump signal. Broadband highly efficient transconductance power amplifiers have been demonstrated in [7], which shows promise as being implementable as the pump PA in parametric amplifiers. While not addressed in this paper, the system efficiency defined by dc–RF power conversion can only approach 100% when both the pump and parametric amplification mechanisms are ideal.

(4) where is proportional to the pump voltage and gives the coupling between the voltages at the two angular frequencies and , and is the average capacitance such that [10]. be a single-valued function Let the charge stored on . The of the terminal voltage charge can be expressed as a Taylor series in to obtain (5) . Since all powers where all derivatives are evaluated at of exist in (5), the frequencies of the charge coefficients will . Thus, the frequencies of current coefficients also span span , and the voltage developed across contains information on all possible mixing products. Consequently, the charge can be represented as a 2-D Fourier series (6) where the charge series coefficients

are expressed as

III. DERIVATION OF THE ANALYTICAL MODELS In the derivation performed by Manley and Rowe, they demonstrated the ideal mathematical relationship that exists between the mixing products of a nonlinear reactive element under excitation. Their relationships came as a result of the assumptions of incommensurable and periodic excitation signals. Under these two assumptions, it was never necessary for Manley and Rowe to solve the 2-D Fourier integrals for the mixing term coefficients. In doing so, and with a proper circuit model for the nonlinear reactance, analytical models can be developed describing the nonideal achievable gain and efficiency of a parametric power upconverter.

A. Maximum Gain and Gain-Degradation Varactor diodes, when operating well below their self-resonance frequency, can be modeled as a series variable resistance and nonlinear reactance , both a function of the varactor terminal voltage [8]. In the following derivation, it will be with terminal voltage is minassumed that the change in imal (assuming reverse bias operation); hence, it can be treated as a constant to a first-order approximation. It was demonstrated in [9] that the capacitive change in the varactor can be approximated as a linear function of the terminal voltage , where is dependent on the bias voltage, and is some constant of units Farads per volt. In the particular instance of high-gain parametric amplifiers, the pump voltage is

(7) and . with through The total current of the charge series coefficients

is the total time derivative

(8) However, the initial assumption of the incommensurability of and gives rise to . As a result, (8) reduces to the partial time derivative of the charge series terms (9) (10) The frequencies of the impedance coefficients , as with , in the Fourier dothe current coefficients, will span in (4), while a good mathemain. The representation of matical model for the time-dependent change in capacitance, is a result of large-signal excitation. Thus, (4) must be linearized , and therefore, can be accurately apabout the bias voltage to obtain the varactor proximated by the average capacitance impedance series terms (11)

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Fig. 1. Piecewise-linear approximation of the square-law region of the varactor junction capacitance, as normalized to its maximum value at 0-V bias, versus ideal characteristics. This figure demonstrates the difference between the varactors maximum available change in capacitance, as opposed to that observed under RF drive.

Fig. 2. Contour plot of the change in the gain-degradation factor versus var: ; : ; and . actor capacitance ratio with

=01 05

2-D Fourier synthesis can now be used to express the complex power of the varactor

1

Substitution of (19) into (15) and (16) provides a complete description of the achievable real and reactive gain of the parametric upconverter

(12) (13) Evaluation of (13) for the ratio of

to

(20)

results in (21) (14)

We are interested in both the real and reactive power being provided by the varactor such that (14) can be written as . Separating the real and reactive terms in (14) yields

(15) (16) where

Manley and Rowe predict the maximum achievable real . power gain of any parametric upconverter to be Therefore, term 2 in (20) can be considered a gain-degradation factor, and (22) as predicted by Manley and Rowe under ideal conditions. Conversely, (23)

For any appreciable value of and , the last term in (20) is approximately equal to 1. Gain degradation is, therefore, dominated by the change in capacitance in the varactor. Let (17) (24) (18)

In (4), it was assumed that the varactor’s capacitance was as sinusoidal. piecewise linear, as in Fig. 1, to describe Making the same assumption, the coupling factor in term 2 in (14) can be treated as a constant and written as a direct function of the change in capacitance experienced under RF excitation at a specified amplitude

. Fig. 2 shows a family of Then isolines of the gain-degradation term of (20) with and . The reactive power gain gives a measure of the mismatch between the varactor and load. An ideal varactor will deliver all available real power to the load, and all reactive power should should be be reflected back to the varactor; the reactive gain zero. Let

(19)

(25)

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Then it can be seen that

. Under ideal conditions,

(26)

Equations (20) and (21) do not include higher order terms that would account for the strong nonlinear effects of gain compression. Therefore, predicting gain degradation is limited to the linear region of the AM–AM distortion curve, and will begin to deviate from measured results as the output power begins to saturate.

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There is only one solution to the quadratic in (31): . With reference to Fig. 1, selecting a bias point in the middle of the linear approximation to the square-law region, equal to twice the bias and allowing a symmetric swing in voltage, ensures that the varactor will experience its available to ratio. In addition, the terminal voltage of the varactor will exceed its built-in potential, forcing saturated operating conditions. This confirms the derivational necessity of the varactor experiencing its maximum change in terminal capacitance, and suggests that driving the parametric upconverter into saturation may maximize the efficiency. Conversely, (32)

B. System Efficiency and Sensitivity The derivation of the achievable efficiency of a parametric upconverter proceeds similar to gain degradation and, therefore, will not be presented with as much detail. For system efficiency, the power series coefficients given in (13) are still valid. In the definition of efficiency presented in (3), it can be shown that

The reactive component of the efficiency provides a measure of the reflective mismatch between the varactor and load. In the ideal case, the parasitic self-resistance of the varactor goes to zero, and the quality factor at all frequencies of operation is infinite (33)

(27) As with gain, both the real and imaginary components of (27) are of interest. Equation (28), shown at the bottom of this page, shows the real component of the efficiency, and (29), shown at the bottom of this page, shows the imaginary component such . that Equations (28) and (29) cannot be simplified in such a way to contain an efficiency-degradation term as was done with the maximum gain. However, functional analysis of (28) and (29) can be used to confirm the correctness of the derivation. Under and . As a result, ideal conditions, (30) . Consequently, let , In (4), it was assumed that and equate (30) to 1, being the maximum obtainable efficiency. Then (31)

It is difficult to determine the dominant terms in (28) with uncertainty in its independent variables using a graphical means as performed with the gain analytical model. Instead, first-order variable sensitivity analysis can be employed to examine the uncertainty of the efficiency analytical model. The first-order sensitivity of a dependent function with respect to independent variable is defined as [11] (34) at a typical operating point demonstrated the Evaluation of general sensitivity of the efficiency analytical model, showing , , , the per unit change in with a per unit change in and was insignificant if all are sufficiently large. The sensi, and , tivity of is dominated by the uncertainty in , as expected with the choice of initial assumptions. A practical efficiency analytical model is thus described in (35) as follows: (35)

(28)

(29)

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Fig. 3. Circuit topology of the parametric upconverter.

Similar to the gain analytical model, the efficiency analytical model is limited to predicting system efficiency in backed-off operating conditions only. Equation (35) does not include higher order terms, and as a result cannot compensate for saturated operating conditions.

IV. CIRCUIT SIMULATIONS AND HARDWARE VALIDATION The authors designed a push–pull paramp upconverter circuit using a harmonic-balance simulator to demonstrate the achievable gain and efficiency, as seen in Fig. 3, using standard SPICE models for all lumped elements. The paramp generates currents in antiphase through two varactor diodes, creating a virtual ground between them. The pump and source signals can be introduced into the paramp at the virtual ground without shorting the upper sideband current [12]. The upconverter translates a 140 MHz IF to 1.3 GHz through parametric mixing with a 1.16-GHz pump source. The varactor used was the 1S2208, which has a junction capacitance range of approximately 3 pF at maximum reverse bias to 25 pF at zero bias. Simulations predict a power gain of 4 dB and a maximum dBm when practical efficiency of 30% in backoff at losses are included in the transmission lines. The breadboard parametric upconverter demonstrated a maximum gain and efapproached satficiency of 4.5 dB and 37%, respectively, as urated operating levels.

V. VALIDATION OF THE ANALYTICAL MODELS Both (20) and (35) can be validated using measurements made on the parametric breadboard. To compute the predicted responses from the gain and efficiency analytical models, the varactor terminal voltage swing must be known at both the pump and source frequencies. Simulated measurements were used to determine both the capacitance ratio of the varactor under excitation and the peak source voltage at the varactor terminals. Due to the accuracy of the simulated results, as compared to the measured results from the breadboard parametric upconverter, the simulated capacitance ratio and peak source voltage at the varactor terminals are considered to be indicative of that which could be physically measured on the breadboard parametric upconverter.

Fig. 4. Surface plot comparison of the measured gain of the breadboard parametric upconverter to that predicted by the analytical model of (20).

A. Gain Fig. 4 compares the surface plot of the measured gain of the breadboard parametric upconverter to that predicted by the analytical model of (20). The small error between the analytical prediction and measured response can be graphically explained. Consider a small perturbation in the terminal voltage of the varactor about an operating point near the breakpoint of the piecewise-linear C–V curve, as seen in Fig. 5(a). The resulting change in capacitance is greater for the piecewise-linear curve than that of the actual, as the slope of the actual curve is less than that of the approximation near the operating point. As a result, the gain analytical model overestimates the change in capacitance and returns a value greater than is measured. Conversely, in Fig. 5(b), the perturbation is applied about a bias point near zero volts, where the slope of the actual C–V curve is greater than the slope of the piecewise linear. The gain analytical model now underestimates the change in capacitance, returning a value that is less than measured. This error could be corrected by implementing a model for the C–V curve of the varactor similar to that used in [13], but the resulting equation is much more complex and is of little practical use for the purposes of circuit design. B. Efficiency Fig. 6 compares the surface plots of the measured efficiency of the breadboard parametric upconverter to that predicted by the analytical model of (35) under varying source and pump powers. As can be seen, (35) is a good predictor of paramp efficiency up to mild saturation. Fig. 6 additionally indicates that the peak efficiency may occur in saturation; however, the use of the linear time-varying capacitance model of (4) in the derivation of the efficiency analytical model of (35) neglects high-order terms, prohibiting accurate prediction of saturated effects. C. Validation of the Analytical Models Using a VHF Parametric Upconverter The work performed by the authors in [5] included the design and fabrication of a VHF breadboard parametric upconverter. This amplifier upconverted a 30-MHz signal to 300 MHz

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Fig. 7. Comparison of surface plots of the measured gain of the VHF breadboard parametric upconverter developed in [5] against that as predicted by the analytical model of (20).

Fig. 5. Mathematical reasoning behind the error between the prediction made by the gain analytical model of (20) and the measured results from the breadboard parametric upconverter.

Fig. 8. Comparison of surface plots of the measured system efficiency of the VHF breadboard parametric upconverter developed in [5] against that as predicted by the analytical model of (35).

VI. CONCLUSION

Fig. 6. Comparison of surface plots of the measured system efficiency of the breadboard parametric upconverter against the predicted efficiency of the analytical model of (35).

through parametric mixing with a 270-MHz pump. Measurements demonstrated the maximum gain to be 8.5 dB with an . To further demonefficiency of 51% when approaching strate the validity and accuracy of the analytical models, Figs. 7 and 8 compare (20) and (35) to that of the measured data for the VHF breadboard parametric upconverter. As can be seen, there is strong agreement between the measured gain and system efficiency and the analytical models. The small mismatch in the slopes of the gain planes in Fig. 7 can be explained using the same argument presented above.

In this paper, we developed analytical models that accurately predict the gain and efficiency of a parametric upconverter from the change in nonlinear reactance, quality factor, and terminal voltage of a microwave varactor. The strength of the gain analytical model lies in its strong dependence on the maximum-to-minimum capacitance ratio, which is a function of the applied pump voltage. The efficiency analytical model demonstrated a strong dependence on the terminal voltage of the varactor at the source frequency. Excellent agreement was obtained between the analytical models and simulated and hardware measurements. ACKNOWLEDGMENT The authors wish to acknowledge Dr. A. Suárez and Dr. F. Ramírez, both with the Universidad de Cantabria, Santander, Spain for useful suggestions regarding circuit simulations. REFERENCES [1] L. Blackwell and K. Kotzebue, Semiconductor Diode Parametric Amplifier. New York: Prentice-Hall, 1961. [2] D. P. Howson and R. B. Smith, Parametric Amplifiers. New York: McGraw-Hill, 1970.

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[3] H. Heffner and G. Wade, “Gain, band width, and noise characteristics of the variable-parameter amplifier,” J. Appl. Phys., vol. 29, no. 9, pp. 1321–1331, 1958. [4] M. Hines, “The virtues of nonlinearity-detection, frequency conversion, parametric amplification and harmonic generation,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 9, pp. 1097–1104, Sep. 1984. [5] B. Gray, J. Kenney, and R. Melville, “Behavioral modeling and simulation of a parametric power amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., 2009, pp. 1373–1376. [6] J. Manley and H. Rowe, “Some general properties of nonlinear elements—Part I: General energy relations,” Proc. IRE, vol. 44, no. 7, pp. 904–913, Jul. 1956. [7] P. Wright, J. Lees, J. Benedikt, P. Tasker, and S. Cripps, “A methodology for realizing high efficiency class-J in a linear and broadband PA,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 12, pp. 3196–3204, Dec. 2009. [8] R. Engelbrecht, “Parametric energy conversion by nonlinear admittances,” Proc. IRE, vol. 50, no. 3, pp. 312–321, Mar. 1962. [9] A. Suárez, Analysis and Design of Autonomous Microwave Circuits. New York: Wiley, 2009. [10] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: Wiley, 2000. [11] A. Saltelli, S. Tarantola, F. Campolongo, and M. Ratto, Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models, 1st ed. New York: Wiley, 2004. [12] B. Perlman, “Current-pumped abrupt-junction varactor power-frequency converters,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 2, pp. 150–161, Feb. 1965. [13] D. Xu and G. Branner, “An efficient technique for varactor diode characterization,” in Proc. 40th Midwest Circuits Syst. Symp., 1997, vol. 1, pp. 591–594. Blake Gray (S’08) received the B.S. degree in both electrical and computer engineering (with honors), and M.S. degree in electrical engineering from the University of Missouri—Rolla (UMR), Rolla, in 2004 and 2005 respectively, and is currently working toward the Ph.D. degree in electrical engineering at the Georgia Institute of Technology, Atlanta. While with UMR, he authored or coauthored multiple peer-reviewed publications on mine wall convergence monitoring systems using digital image processing of laser light scattering properties. His current research interests at the Georgia Institute of Technology are focused on microwave energy, particularly the use of advanced multiferroic materials and GaN varactor technology in parametric microwave power amplifier architectures. Mr. Gray is a student member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S).

Bob Melville (M’02) received the B.A. degree in computer science from the University of Delaware, Newark, in 1976, and the Ph.D. degree in computer science from Cornell University, Ithaca, NY, in 1981. He was with Bell Laboratories for 15 years, during which time he was involved in the areas of computer-aided design, numerical simulation of electronic circuits, and design and fabrication of RF integrated circuits. Most recently, he has taught electrical engineering at Columbia University and served for a year with the U.S. Antarctic Program at the Amundsen–Scott Base, South Pole, Antarctica. He is currently with the Office of Polar Programs, National Science Foundation (NSF), McMurdo Station, Antarctica.

J. Stevenson Kenney (S’84–M’85–SM’01–F’08) possesses over 14 years of industrial experience in wireless communications. He has held engineering and management positions with Electromagnetic Sciences, Scientific Atlanta, Pacific Monolithics, and Spectrian. He has authored or coauthored over 100 technical papers. Dr. Kenney has been an active member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He was an officer of the IEEE MTT-S Santa Clara Valley Chapter (1996–2000). He served three terms on the IEEE MTT-S Administrative Committee (AdCom). From 2001 to 2003, he was treasurer of the IEEE MTT-S, and was the IEEE MTT-S president in 2007. He was the recipient of the 2005 IEEE MTT-S Application Award.

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Poly-Harmonic Modeling and Predistortion Linearization for Software-Defined Radio Upconverters Xi Yang, Dominique Chaillot, Patrick Roblin, Member, IEEE, Wan-Rone Liou, Member, IEEE, Jongsoo Lee, Hyo-Dal Park, Member, IEEE, Jeff Strahler, Member, IEEE, and Mohammed Ismail, Fellow, IEEE

Abstract—This paper presents a new predistortion linearization scheme for single-sideband mixers to be used for removing unwanted harmonics and intermodulation products of the digital IF in an heterodyne transmitter. The proposed algorithm, called poly-harmonic predistortion linearization, relies on an orthogonal expansion in the frequency domain of the nonlinearities for the mixer modeling. It takes into account memory effects that are piece-wise quasi-memoryless and enables the independent cancellation of unwanted spurious sidebands of the digital IF harmonics. The poly-harmonic predistortion linearization scheme for the weak-nonlinear regime was implemented in a field-programmable gate array and experimentally investigated for the linearization of a four-path polyphase single-sideband upconverter. The ability of the poly-harmonic predistortion algorithm to linearize the four-path polyphase mixer for input signals with high envelope fluctuation is demonstrated. 70-dBc 62-dBc 60-dBc spurious rejection and 18-dB/10-dB/8-dB linearization improvement of the third-order distortions are achieved for a two-tone RF signal, a 64-tone 10-MHz bandwidth multisine signal and orthogonal frequency-division multiplexing signal, respectively. The combination of the polyphase multipath technique and the poly-harmonic predistortion linearization technique offers an attractive filterless approach for the development of multimode broadband software-defined radio. Index Terms—Broadband transmitter, linearization, memory effect, modeling, poly-harmonic predistortion, polyphase multipath mixer, software-defined radio. Manuscript received March 15, 2010; revised May 17, 2010; accepted May 17, 2010. Date of publication July 15, 2010; date of current version August 13, 2010. This work was support in part by the National Science Foundation (NSF) under GOALI Grant ECS-0622003 and IUCRC Grant IIP-0631286 and by the Samsung Corporation under a grant. X. Yang, P. Roblin, and M. Ismail are with the Department of Electrical and Computer Engineering, The Ohio State University (OSU), Columbus, OH 43210 USA (e-mail: [email protected]; [email protected]; [email protected]). D. Chaillot is with the Department of Electrical and Computer Engineering, The Ohio State University (OSU), Columbus, OH 43210 USA, on leave from the Commissariat à l’Énergie atomique (CEA)/ Centre d’études scientifiques et techniques d’Aquitaine (CESTA), Le Barp 33170, France (e-mail: [email protected]). W.-R. Liou is with the Graduate Institute of Electrical Engineering, National Taipei University, Taipei 10617, Taiwan (e-mail: [email protected]). J. Lee is with the Samsung Electronics Company Ltd., Suwon-si, Gyeonggi-do 443-742, Korea (e-mail: [email protected]). H.-D. Park is with the Department of Electronic Engineering, Inha University, Nam-gu, Incheon 402-751, Korea (e-mail: [email protected]). J. Strahler is with the Andrew Corporation, Columbus, OH 43235-4721 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2053068

I. INTRODUCTION INEARITY HAS become one of the most challenging issues in the design of broadband multicarrier RF transmitters. Indeed broadband multicarrier modulation schemes exhibit high envelope fluctuation, which induces a strong nonlinear response in RF mixers and RF amplifiers leading to spectral regrowth, inband distortion, which can never be filtered out, and adjacent channel interferences. Various linearization strategies are then employed to reduce the nonlinearity or suppress the distortion products generated while not degrading the power efficiency of the transmitter. Filtering is one of the most commonly used methods to remove the unwanted outband distortion products. The design of filters providing the desired selectivity and insertion loss at RF frequencies is not without challenge. Furthermore, different filters are required according to the different standards, and this becomes the bottleneck for developing wireless transceiver terminals with multiple applications and multiple standards. Another method for removing the distortion products is linearization by feedback [1]. However, this is not a preferred method for RF transmitters due to the instability risks and bandwidth limitation. The feed-forward technique [2] has demonstrated broadband linearization capability for multicarrier amplifiers and is also conceptually applicable to mixers. However feed-forward linearization increases the circuit complexity and cost and digital predistortion methods are being actively pursued as an alternative [3], [4] . The polyphase multipath technique has been recently proposed [5], [6] as an effective architecture for removing harmonics and intermodulation products. A well-known example is a balanced circuit, a two-path polyphase circuit, which is able of canceling the even order harmonics. Reference [5] demonstrates that by increasing the number of paths in the circuit more distortion products can be canceled. An -path system will suppress spurious bands out of a group of bands. The method could facilitate the development of filter-less software radios: one wideband integrated upconverter with no dedicated external filters [6]. However, this technique has its own limitation. As indicated, the number of distortion products canceled is related to the number of paths used. However, increasing the number of paths will also increase the circuit complexity. Furthermore, mismatches and memory effect between different paths will degrade the linearization performance and must be dealt with separately. Finally, the fundamental inband intermodulation prod-

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ucts of the digital IF, which are often the key issue in RF transmitters, cannot be removed by using a polyphase multipath architecture without canceling the desired signal [6]. In this paper, we shall introduce a new technique, the poly-harmonic predistortion linearization, to help reduce the remaining uncanceled digital IF baseband harmonics and intermodulation products in four (or a multiple of four)-path polyphase mixers or in single-sideband mixers in general. Unlike a simple cancellation scheme where the suppression of distortion products only holds for a single input signal power level, this new proposed predistortion method aims at addressing multicarrier modulation schemes having high envelope fluctuations, within a certain power range. Furthermore, the poly-harmonic predistortion technique will also be designed to account for both AM/AM and AM/PM nonlinearities[7], [8]. This paper is organized as follows. First we will present in Section II the general modeling of nonlinear single-sideband mixers for both AM/AM and AM/PM nonlinearities. Based on the modeling equations, the poly-harmonic predistortion algorithm is presented in Section III. In Section IV, we will present the implementation of this algorithm in a field-programmable gate array (FPGA) test-bed for the experimental investigation. Using this digital baseband test-bed, we will then experimentally demonstrate in Section V the RF performance of a fourpath polyphase mixer linearized with the proposed poly-harmonic predistortion linearization scheme. Finally, in Section VI we will summarize the results obtained.

(plus for the upper sideband and minus for the lower sideband) would be the tunable digital IF used before upconverand sion to RF. Next let us consider an ML nonlinear modulator. As the RF harmonics can be readily eliminated by filtering, our focus is placed on the nonlinear spurious terms (IF harmonics) generated . In this paper, the around the fundamental RF frequency ML nonlinear single-sideband upconverter can be represented and with an ideal modulator upconverting the distorted ML input signals as follows: (1) Hilbert

(2) (3)

Only the odd-order nonlinearities up to the th order are considered initially, assuming perfectly balanced mixers. and Before proceeding with the evaluation of , let us define the functions and with to be used for the orthogonal expansion in and the frequency domain of (4) The orthogonal components calculated in terms of and

and can then be directly using

II. NONLINEAR SINGLE-SIDEBAND MIXER MODELING In order to develop the linearization algorithm for the singlesideband mixer, we need first to develop a nonlinear model to represent it. We will first start with a nonlinear memoryless (ML) modulator model to account for ML AM/AM nonlinearities and then upgrade the model to a piece-wise ML model to account for separate nonlinear AM/PM effects for each IF harmonics.

and

A. ML Nonlinear Modulator Modeling The baseband input signal and to be upconverted by an ideal single-sideband RF mixer can be expressed as

Hilbert

with

where , and are the time-varying envelope, timevarying phase, and center frequency, respectively, of the input to be upconverted. In the single-sideband upconsignal verter scheme of a filterless broadband software radio transis plus or minus the Hilbert transform of mitter,

Note that in the limit , reduce to the Chebyshev polynomials of the first kind, and the reduce to of the Chebyshev polynomials of the first kind, for odd. These functions will now be used for simplifying the analysis of the nonlinear response of the ML modulator. To analyze up to the th odd order, the ML weakly nonlinear system described by (1), we need to evaluate the odd. To infer the form of the modeling equaorder powers of tions needed for such a system, it is beneficial to sort the nonlinear terms generated in terms of frequencies rather than the order of the nonlinearity generating them. Thus, the output of an ML weakly nonlinear system represented by an th odd-order

YANG et al.: POLY-HARMONIC MODELING AND PREDISTORTION LINEARIZATION FOR SOFTWARE-DEFINED RADIO UPCONVERTERS

power series, can be modeled with the help of the band harmonics previously defined as follows:

and

base-

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which allow us to express (8) and (9) in the following compact equations: (13)

(5) (14) (6)

The QML nonlinear single-sideband upconverter is then presented as follows:

(7) is the highest order of the odd nonlinear terms, , and is an integer. gives the contribution th-order nonlinearities to the th harmonic weight of the band . where

B. Quasi-Memoryless (QML) Nonlinear System Modeling The modulator model expressed by (5) and (6) is only appliare only depencable to ML systems as the functions dent on the instantaneous value of the envelope square of the baseband input pair and no phase shift is used to account for the modulator group delay. Models who account for a frequency-independent phase shift are referred to as quasi-ML [9]. A piece-wise ML model can be used as an approximation for a system with memory by dividing the frequency spectrum in different frequency bands. To account for memory effects, which are piece-wise quasi-ML in the modulator, it is therefore necessary to introduce an independent phase shift for each IF harmonic band. In the general case, these phase shifts are also a function of the instantaneous value of the envelope (AM/PM effect). By adding this phase contribution to square (5) and (6) of the ML nonlinear modulator considered thus far, we obtain the following modeling equations for a QML nonlinear modulator:

(15) This model aims at accounting for the unwanted IF harmonic , but it does not account for memory effects sidebands within each of the IF-harmonic sidebands themselves. This is partly justified because for the modulator used in Section V, the IF harmonic sidebands that are distributed over a wide-bandwidth (100 MHz, in this work) exhibit dominant memory effects compared to the baseband signal with its narrower bandwidth (10 MHz, in this work). However, if needed, the inband memory effects could be corrected by using memory polynomials [10], [11]. From (10) and (11), it is seen that the nonlinearities of the system originate from two nonlinear sources, which are: 1) the amplitude distortion nonlinearity represented by the functions and 2) the phase distortion nonlinearity represented by the functions . For low enough input signal power where the modulators remain in the weak-nonlinear regime, the funcand can be expanded in a Taylor series like tions in (7) (16)

(17) account for the phase shift of each sideband of where the digital IF harmonic .

(8) III. POLY-HARMONIC PREDISTORTION LINEARIZATION

(9) Now by using the functions

and (10) (11)

the following nonlinearly scaled and phase shifted modulation terms can be defined:

and

(12)

We shall now present a poly-harmonic predistortion linearization technique to remove the spurious frequency components generated by a general single-sideband mixer in the weak-nonlinear regime. Note that, unlike for a simple tone cancellation scheme at a given power level, the linearization pursued should ideally work for arbitrary input-power levels in the weak-nonlinear regime, and thus track well the variation of the unwanted harmonics and intermodulation products of the digital IF as the signal envelope varies in time within a specific bandwidth and prescribed power range. Specifically in this work, the linearization should work well with broadband multicarrier modulation signals with large envelope fluctuations such as in multisine or orthogonal frequency-division multiplexing (OFDM) signals. The proposed predistortion equations for the QML weaknonlinear system inherit the forms of the modulator modeling equations, as shown in (13) and (14), while the predistorted complex coefficients should be approximately the negative (see

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Fig. 1. Poly-harmonic predistortion linearization algorithm. The system level implementation is shown in Fig. 2.

equations below) of the modeling coefficients. Note that the linearization signals that are added to the desired inband data are jointly upconverted by the nonlinear modulator. This inband channel can itself be linearized in the QML case using predistortion as follows:

According to (16) and (17), for a seventh odd-order QML system in the weak-nonlinear regime, we have

and and with

Note that the subscript has been now added to differentiate the modulator coefficients from the predistortion coeffiis the targeted reduced linear modulator cients and that and upconversion gain. The inband envelopes are also those measured at the input of the predistortion stage and the IQ modulator, respectively, and should be a monotonous function of . Since the predisused to linearize the modulator are tortion functions and in the QML case, the still a function of the inband envelope predistortion signal is therefore of the form

(18) Note that since an IQ modulator is used for the single-sideband upconversion, then the input is just minus the Hilbert of for the lower sideband (L) with the index running from 1 to , and plus the Hilbert of for the dc term and the upper sideband (U) with running from 0 to . The poly-harmonic predistortion equation for is then

(19)

Fig. 1 shows the proposed poly-harmonic predistortion linearization algorithm. With a proper selection of the predistortion coefficients and , the odd-order harmonics and intermodulation products resulting from the nonlinear circuit are expected to be canceled for the desired input power range. Since different phase coefficients are used to deal with different harmonic bands centered on , the proposed poly-harmonic predistortion technique should be applicable to nonlinear systems with memory, which is piece-wise QML in the frequency domain. Note that the later assumption neglects the frequency dependence of the nonlinearity within each harmonic band , but accounts for the proper phase shifts between these different frequency bands. The method for determining the value of and is as follows: we first increase the amplitude of the fundamental input signal to drive the circuit into the weak-nonlinear regime: that is when the third-order nonlinearity is first detected. By changing the coefficients , the third-order intermodulation centered at can

YANG et al.: POLY-HARMONIC MODELING AND PREDISTORTION LINEARIZATION FOR SOFTWARE-DEFINED RADIO UPCONVERTERS

be canceled. By changing , the third-order harmonics and intermodulation products centered at can also be canceled. We keep the value of and while increasing the amplitude of and , until the fifth-order intermodulation products and , harmonics arise. We determine the coefficients , and such that the distortions centered at , , generating from the and disappear. Tuning will not affect the already canceled third-order nonlinearities owning to the orthogonal expansion and independent cancellation of the nonlinear terms in the weak nonlinear regime. By increasing the input signal amplitude, one can tune the seventh-order nonlinearity using the same method. Note that the predistortion are used to deal with the inband coefficients distortion. Within this extraction power range, which covers the weak-nonlinear regime, the nonlinearities should remain canceled using the same predistortion coefficients. Up to now we have derived the predistortion (18) and (19) for linearizing a QML system with odd-order nonlinearities in the weak-nonlinear regime. However, in the physical implementation of the nonlinear polyphase mixer, mismatches in the -path topology introduce even-order harmonics and an image band. What is more, the local oscillator (LO) leakage is modulated and may require proper compensation. Thus we enrich the poly-harmonic predistortion equations by adding the even terms to perform these corrections. The poly-harmonic predistortion equaand is still the same as in (18) and (19), extions for cept that the summation over the harmonics now includes both even and odd harmonics. Equations (16) and (17) will also remain the same, except that is the highest even order nonlinfor the even terms. The signals earity and and are still obtained from and using (12). can be generated by the following recurrence Note that equation: for for (20) where kind.

are generalized Chebyshev functions of the first are calculated using the recurrence equation for for for odd for even

(21)

To our knowledge, this is the first time these generalized Chebyshev functions are introduced. In the subsequent experimental results presented, the even harmonics are effectively reduced below the noise floor by the differential topology of the mixers, except for the second harmonic , the modulated LO leakage , and the image band. The second harmonic was effectively canceled by using a second-order expansion and

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Fig. 2. FPGA implementation.

Fig. 3. Digital test-bed.

Similarly for the modulated compensation, the following second-order approximation was used:

Note that we use instead of for and , as this yields a better linearization performance. IV. FPGA IMPLEMENTATION AND TEST-BED Figs. 2 and 3 show the FPGA implementation of the seventh-order poly-harmonic predistortion algorithm and the digital test-bed we used for the mixer linearization. and could be any In general, the input signals bandwidth-limited signals. In the FPGA test-bed developed, and signal generators rely on multiple cosine and the sine look-up tables (LUTs) to synthesize a multisine signal with arbitrary phase and constant amplitude, as presented in (22) and (23) as follows: (22)

(23) In this case, the input consists of signal upconverted at digital IF frequency

-tone multisine with amplitude

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Section III. At the end, another limiter is used to protect the system from possible high peak impulses. The differential IQ modulator (TRF3703) integrated in the TI DAC5682Z evaluation module (EVM) is used as a four-path single-sideband mixer in our experiments for predistortion linearization. The and output third-order intercept point (OIP3) of the TRF3703 are 9 and 23 dBm, respectively [14]. V. EXPERIMENTAL RESULTS

Fig. 4. CCDF plot of the 64-tone multisine signal.

parameter and phase distribution function . Multisine signals can exhibit high envelope fluctuations with high peak to average power ratio (PAPR), owing to the possibility of the tones superposing in phase. Thus, properly selecting the phase to have a realistic PAPR for and distribution function is of critical importance. In this study, we used a random of a 10-MHz 64-tone phase generator to vary the phases multisine signal and optimize its PAPR and complementary cumulative distribution function (CCDF). Fig. 4 shows (plain line, red in online version) the final CCDF selected, which features about 6.1-dB PAPR achieved with a probability of 0.01%. This corresponds to a 2.1-dB reduction in PAPR compared to a multisine with equally distributed phase (dashed line, blue in online version). Alternatively this multicarrier signal can be modulated using binary phase-shift keying (BPSK) and quadrature phase-shift keying (QPSK) modulation with adjustable symbol duration to generate an OFDM signal. In order to approximate a typical OFDM communication with proper PAPR, we use a 17-bit linear feedback shift-register [12] to generate the random phase modulation command. A hard clip limiter is also used after the signal generator, to avoid the high peak amplitude, which can result from the 64 tones OFDM signal. The amplitude and phase of the multisine tones are controlled by a MATLAB GUI. The signals are then sent to the poly-harmonic predistortion block. The baseband (IF) signal generator is implemented using a LUT to provide sine and cosine signals for the frequencies , , , , and . According to the algorithm expressed by (18) and (19), the predistortion block is then implemented to generate the predisand . The predistortion coeffitorted signal cients and are sent from the MATLAB GUI to the FPGA board via an RS232 connection. A frequency-selective IQ path imbalance correction [13] is implemented with four IQ balancing blocks for the fundamental, third-, fifth-, and seventh-order harmonics. The modulated LO leakage is addressed by using the LO leakage cancellation block, as indicated in

Fig. 5(a) and (b) shows the nonlinearized and poly-harmonic predistortion linearization of the four-path mixer output for a two-tone excitation. The baseband signal is upconverted to 3.5 GHz. The total span is 100 MHz. First, we use the dc-offset compensation provided by the TI DAC 5682Z evaluation board to reduce the dc LO leakage. The digital dc values are externally added to the in-phase (I) and quadrature (Q) data path in the evaluation board providing a system level offset adjustment capability independent of the input data. This external user-tunable dc-offset feature of the DAC 5682Z is used to completely cancel the dc LO leakage with no observed reduction in the mixer dynamic range. We then eliminate the modulated LO leakage by using the envelope compensation technique introduced in Section III. Note that the modulated LO leakage is relatively small (40 dB less than the dc LO leakage, as shown in the experimental results) such that this compensation does not significantly reduce the dynamic range of the DAC used to drive the modulator. Using the tuning technique introduced in Section III, we then linearize the output spectrum to independently suppress in the weak nonlinear regime the sidebands, intermodulation products, and image of the digital-IF harmonics by using the poly-harmonic predistortion and balancing blocks implemented in the FPGA test-bed. When the input signal power level remains within the limit of the weak-nonlinear regime, the distortions remain canceled without any further tuning of the predistortion coefficients. This is due to the fact that the linearization scheme tracks well the variation of the nonlinearities with the instantaneous power level via the model functional dependence on the envelope square. Within the weak-nonlinear regime, only third-order nonlinearities were observed for the modulator considered. An experimental test was also carried out for a 10-MHz bandwidth 64-tone multisine input signal, as shown in Fig. 5(c) and (d), and with QPSK modulated (OFDM signal), as shown on Fig. 5(e) and (f). In this linearization process, the same predisand are used through out the tortion coefficients experiments (a)–(f), which means that the poly-harmonic predistortion linearization works for multicarrier input signals with high envelope fluctuations, as long as the input signal power level remains inside the weak-nonlinear regime of the mixer. From our experimental investigation, the 5682Z EVM could handle input signal power range up to 3 dBm for two-tone and 6.2 dBm for 64-tone multisine excitation, respectively, while remaining in the weak-nonlinear regime. The phase-shift comsideband rejection, balpensations used for the , inband intermodulaancing, balancing at the fundamental sideband rejection are summarized tion products, and the in Table I.

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Fig. 5. Poly-harmonic predistortion linearization results of a four-path polyphase mixer (TRF3703) for: (a) and (b) a two-tone excitation, (c) and (d) a 64-tone multisine signal, and (e) and (f) QPSK modulated OFDM signal (0.198-ms refresh rate) measured with 30-kHz resolution bandwidth for (a)–(d) and 3 kHz for (e) and (f), respectively.

The linearization results are summarized in Table II. It is shown that the fewer the input tones, the higher the absolute value nonlinear rejection the system could achieve. This is as expected since the more input tones, the easier we can reach the limit of the weak-nonlinear regime of the circuit, due to the higher PAPR. This was the motivation for reducing the PAPR by a proper selection of the phase values for the input signal and . The PAPR reduction is usually performed with

the use of a soft limiter. Note that for the OFDM signal, the spectral leakage around the 10-MHz signal band could be reduced by using raised-cosine signaling in the OFDM signal generator. In summary, the poly-harmonic predistortion linearization technique optimizes the output spectrum of the polyphase multipath circuit. The advantage of this technique lies in the fact that the linearization is effective over a large bandwidth (100 MHz, in this work) for broadband multicarrier input

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TABLE I PHASE-SHIFT COMPENSATIONS FOR BALANCING AND SIDEBAND REJECTION OF THE DIGITAL IF

TABLE II SUMMARY OF SIDEBAND, INTERMODULATION PRODUCTS, IMAGE REJECTION OF THE DIGITAL IF HARMONICS AND LO REJECTION USING POLY-HARMONIC PREDISTORTION LINEARIZATION TECHNIQUE

signals (10 MHz with 6–8 dB PAPR, in this work) with only moderate degradation in spurious rejection. Further as was observed, once the predistortion linearization is set up for the circuit, no more adjustment in the predistortion coefficients needs to be carried out when the input signal changes. VI. CONCLUSION In this paper, we have developed a QML nonlinear model for single-sideband mixers to predict the harmonics and intermodulation products of the digital IF it generates when used in a heterodyne transmitter. Based on the modeling work for the weak-nonlinear regime, a novel predistortion scheme, the poly-harmonic predistortion linearization, was presented for the linearization of a single-sideband mixer. This linearization scheme finds application in the cancellation of the residual distortion products not addressed in polyphase mixers. It was demonstrated experimentally that the linearization performance was maintained for various input signals while using the same predistortion coefficients. A spurious rejection of 70 dBc with 18-dB linearization improvement for the third-order distortion products was achieved for a two-tone RF signal and a spurious rejection of 62 dBc 60 dBc with 10-dB/8-dB linearization improvement for a 10-MHz bandwidth 64-tone multisine/OFDM signal using the proposed linearization scheme. The polyphase multipath technique combined together with the poly-harmonic predistortion linearization should help satisfy the linearity requirement of broadband RF transmitters. They also offer an attractive approach for the development of filterless software-defined radios by providing a flexible broadband upconverter with acceptable distortion rejection for multistandard communications. ACKNOWLEDGMENT The authors would like to thank the Altera Corporation, San Jose, CA, for the donation of the DE3 FPGA test-bed, which was used in this study. The authors also thank the anonymous reviewers for their useful comments. REFERENCES [1] Y. Kim, Y. Yang, S. Kang, and B. Kim, “Linearization of 1.85 GHz amplifier using feedback predistortion loop,” in IEEE MTT-S Int. Microw. Symp. Dig., Baltimore, MD, Jun. 1998, pp. 1675–1678. [2] R. Meyer, R. Eschenback, J. Walter, and M. Edgerley, “A wide-band feedforward amplifier,” IEEE J. Solid-State Circuits, vol. 9, no. 12, pp. 422–428, Dec. 1974.

[3] P. Roblin, S. K. Myoung, D. Chaillot, Y. G. Kim, A. Fathimulla, J. Strahler, and S. Bibyk, “Frequency selective predistortion linearization of rf power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 1, pp. 65–76, Jan. 2008. [4] X. Yang, P. Roblin, D. Chaillot, S. Mutha, J. Strahler, J. Kim, M. Ismail, J. Wood, and J. Volakis, “Fully orthogonal multi-carrier predistortion linearization for rf power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig.., Boston, MA, Jun. 2009, pp. 1077–1080. [5] E. Mensink, E. A. Klumperink, and B. Nauta, “Distortion cancellation by polyphase multipath circuits,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 9, pp. 1785–1794, Sep. 2005. [6] R. Shrestha, E. A. Klumperink, E. Mensink, G. J. Wienk, and B. Nauta, “A polyphase multipath technique for software-defined radio transmitters,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2681–2692, Dec. 2006. [7] T. Vuong and A. F. Guibord, “Modeling of nonlinear elements exhibiting frequency-dependent AM/AM and AM/PM transfer characteristics,” Can. Elect. Eng. J., vol. 9, no. 3, pp. 112–116, 1984. [8] W. Bosch and G. Gatti, “Measurement and simulation of memory effects in predistortion linearizers,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1885–1890, Dec. 1989. [9] J. S. Kenney, W. Woo, L. Ding, R. Raich, H. Ku, and G. T. Zhou, “The impact of memory effects on predistortion linearization of RF power amplifiers,” in Proc. 8th Int. Microw. Opt. Technol. Symp., Montreal, QC, Canada, Jun. 2001, pp. 189–193. [10] J. Kim and K. Konstantinou, “Digital predistortion of wideband signals based on power amplifier model with memory,” Electron. Lett., vol. 37, no. 23, pp. 1417–1418, Dec. 2001. [11] L. Ding, G. T. Xhou, D. R. Morgan, Z. Ma, J. S. Kenny, J. Kim, and C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun., vol. 52, no. 1, pp. 159–165, Jan. 2004. [12] P. Alfke, “Efficient shift registers, LFSR counters, and long pseudo random sequency generators, version 1.1,” Xilinx, San Jose, CA, Appl. Note XAPP 052, Jul. 1996. [13] S. Myoung, X. Cui, P. Roblin, D. Chaillot, F. Verbeyst, M. V. Bossche, S. Doo, and W. Dai, “Large signal network analyzer with trigger for baseband and rf system characterization with application to k-modeling and output baseband modulation linearization,” in 64th ARFTG Conf. Dig., Orlando, FL, Dec. 2004, pp. 189–195. [14] “0.4-GHz to 4-GHz Quadrature Modulators, Product Manual SLWS184G,” Texas Instruments Incorporated, Dallas, TX, Dec. 2009.

Xi Yang was born in Xi’an, China, in February 1983. She received the B.S. degree in electrical engineering from Beihang University, Beijing, China, in 2005, the Ingenieur and M.S. degree in electrical engineering from Ecole Centrale de Lyon, Ecully, France, in 2007, and is currently working toward the Ph.D. degree in electrical and computer engineering at The Ohio State University, Columbus. While with the INL-ECL, her master’s work was focused on ultra-low-power double-gate MOSFET circuit design. Her research experience and current research include filterless software-defined radio transmitter architecture, digital predistortion linearization for RF power amplifiers and mixers, and RF integrated circuit (RFIC) design. Dominique Chaillot was born in Brive, France, in October 1963. He received the Maitrise de Physique degree from the Université des sciences, Laboratoire IRCOM, Limoges, France, in 1985, the Ph.D. degree in electrical engineering from IRCOM (now XLIM) Limoges, France, in 1989, and the Master of Business Administration degree from IAE Sorbonne Paris, Paris, France, in 1992. In 1990, he joined the Commissariat à l’énergie atomique (French Atomic Agency), Bruyère le Chatel, France, as a Research Engineer. In 1997, he joined the CESTA, Le Barp, France. He is currently on leave with the Department of Electrical and Computer Engineering, The Ohio State University (OSU), Columbus, where he is an Invited Scholar. His expertise is real-time signal processing in RF systems. His current research interests include the measurement design and linearization of nonlinear RF devices and power amplifiers.

YANG et al.: POLY-HARMONIC MODELING AND PREDISTORTION LINEARIZATION FOR SOFTWARE-DEFINED RADIO UPCONVERTERS

Patrick Roblin (M’85) was born in Paris, France, in September 1958. He received the Maitrise de Physics degree from the Louis Pasteur University, Strasbourg, France, in 1980, and the M.S. and D.Sc. degrees in electrical engineering from Washington University, St. Louis, MO, in 1982 and 1984, respectively. In 1984, he joined the Department of Electrical and Computer Engineering, The Ohio State University (OSU), Columbus, OH, where he is currently a Professor. His current research interests include the measurement, modeling, design, and linearization of nonlinear RF devices and circuits such as oscillators, mixers, and power amplifiers. He is the founder of the Non-Linear RF Research Laboratory, OSU. While with OSU, he developed two educational RF/microwave laboratories and associated curriculum for training senior undergraduate and graduate students. He coauthored the textbook High-Speed Heterostructure Devices (Cambridge Univ. Press, 2002).

Wan-Rone Liou (M’04) received the B.S. and M.S. degrees in electrical engineering from National Cheng-Kung University, Tainan City, Taiwan, in 1984 and 1986, respectively, and the Ph.D. degree in electrical engineering from The Ohio State University (OSU), Columbus, in 1993. From 1994 to 2006, he was the National Taiwan Ocean University . He is currently a Professor with the Graduate Institute of Electrical Engineering, National Taipei University, Taipei, Taiwan. He is also a consultant for several integrated circuit (IC) design houses in Taiwan. He has authored or coauthored over 80 papers in international journals and conference proceedings. He holds over ten patents in the U.S. and Taiwan. His current research interests include high-efficiency power management IC design, low electromagnetic interference (EMI) pulsewidth modulation IC design, RF voltage-controlled oscillators (VCOs) and mixers design, and A/D converters design. Dr. Liou was the three-time recipient of the Research Award presented by the National Science Council, Taiwan.

Jongsoo Lee was born in Dangjin, Korea, in 1974. He received the B.S. degree in physics from Chung-Ang University, Seoul, Korea, in 1999, and the M.S. and Ph.D. degrees in electrical engineering from The Ohio State University (OSU), Columbus, in 2003 and 2008, respectively. From 2000 to 2006, he worked under a Texas Instruments Incorporated Fellowship. As part of the requirement of his M.S. thesis, he developed inductor structures for a high- factor and designed a CMOS mixer. In 2008, upon completion of his doctoral degree, he joined the Samsung Electronics Company Ltd., Suwon-si, Gyeonggi-do, Korea, as a Senior Engineer. His recent research was focused on the design of a WCDMA transmitter and development of a linearization scheme for a WCDMA transmitter. He was also involved with 900-MHz RFID transceiver development and ubiquitous 8-GHz wireless transceiver development. His research interest is RF and analog integrated circuit design.

Q

Hyo-Dal Park (M’92) was born in EuiSung, Korea, in June 1952. He received the B.S. degree in electronic engineering from Inha University, Incheon, Korea, in 1978, the M.S. degree in electronic engineering from Yonsei University, Seoul, Korea, in 1981, and the M.S. and Ph.D. degrees in electronic engineering from the Ecole Nationale Supérieure de l’Electronique et de ses Applications (ENSEA), Cergy-Pontoise , France, in 1984 and 1987, respectively. From 1983 to 1984, he was with the Centre National de Recherche Scientifique. From 1984 to 1987, he was with the Centre

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National d’Etude Spatial. From 1987 to 1992, he was the Chief Researcher with the Korea Institute of Aerospace Technology. In 1992, he joined the Department of Electronic Engineering, Inha University, Incheon, Korea, as an Assistant Professor. He is currently a Professor with Inha University. He is the founder of the Microwave and Avionics Laboratory, Inha University. In 2007, he has joined the Non-Linear RF/Microwave Laboratory, Department of Electrical and Computer Engineering, The Ohio State University (OSU), Columbus, as a Visiting Scholar. He authored the textbook Microwave Technology (HongReung, 2004). His current research interests include the linearization of RF power amplifiers, RF front-end circuit design, avionic devices, electronics for transportation systems, and air traffic control and management system design.

Jeff Strahler (S’88–M’97) received the B.S.E.E. degree from the University of Cincinnati, Cincinnati, OH, in 1989, and the M.S.E.E. degree in electrical engineering from The Ohio State University (OSU), Columbus, in 1991. He was with the ElectroScience Laboratory, Electrical Engineering Department, OSU, where his master research concerned computational electromagnetics. From 1986 to 1990, he worked in various capacities with Comsat Laboratories, during which time he designed microwave circuits and antennas for communication satellite and earth station systems. In 1991, he joined AT&T Bell Laboratories (now Alcatel-Lucent) Columbus, OH, as a Member of the Technical Staff (MTS) and then as a Distinguished Member of the Technical Staff (DMTS). As part of his duties, he has been a team leader for the design and development of wireless base-station amplifiers for AMPS, TDMA, GSM, and CDMA systems. In June 2001, he joined the Celiant Corporation (acquired by the Andrew Corporation in June 2002), Columbus, OH, where he has recently become an Andrew Fellow. His research continues to focus on research and development activities for base-station power-amplifier products.

Mohammed Ismail (S’80–M’82–SM’84–F’97) received the B.S. and M.S. degrees in electronics and communications from Cairo University, Cairo, Egypt, in 1978 and 1979, respectively, and the Ph.D. degree in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 1983. He has held several positions in both industry and academia and has served as a corporate consultant to nearly 30 companies in the U.S., Europe, and the Far East. He is a Professor of electrical and computer engineering and the Founding Director of the Analog Very Large Scale Integration (VLSI) Laboratory, The Ohio State University (OSU), Columbus, and of the Radio and Mixed Signal Integrated Systems (RaMSiS) Group, Royal Institute of Technology (KTH), Stockholm, Sweden. He cofounded ANACAD–Egypt (now part of Mentor Graphics Inc.) and Firstpass Technologies Inc., a developer of CMOS radio and mixed-signal IPs for handheld wireless applications. He has authored or coauthored numerous publications. He holds 11 patents. He has advised the research work of 51 Ph.D. and over 90 M.S. students. He has coedited and coauthored several books, including Analog VLSI Signal and Information Processing (McGraw-Hill, 1994). His last book is Radio Design in Nanometer Technologies (Springer, 2007). He is the founder and the Editor-in-Chief of the International Journal of Analog Integrated Circuits and Signal Processing. He is on the International Advisory Board of several journals. He possesses over 25 years research and development experience in the fields of analog, RF, and mixed-signal integrated circuits. His current research interests include fully integrated CMOS radios, BIST and digital self-calibration of RFICs, yield enhancement design, and power management solutions. Prof. Ismail was an associate editor for several IEEE TRANSACTIONS. He was on the Board of Governors of the IEEE Circuits and Systems Society. He is the founder of the International Conference on Electronics, Circuits, and Systems (ICECS), the Circuit and Systems (CAS) flagship conference for IEEE Region 8. He was the recipient of several awards including the U.S. National Science Foundation (NSF) Presidential Young Investigator Award, the U.S. Semiconductor Research Corporation Inventor Recognition Awards (1992 and 1993), the College of Engineering Lumley Research Award (1992, 1999, 2002, and 2007), and a Fulbright/Nokia Fellowship Award (1995).

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Design of Doherty Power Amplifiers for Handset Applications Daehyun Kang, Jinsung Choi, Dongsu Kim, and Bumman Kim, Fellow, IEEE

Abstract—In this paper, we analyze the power drive of a Doherty power amplifier (PA), and introduce a technique for proper input dividing without a coupler. For the proper Doherty operation, we place a phase compensation circuit at the input of the carrier amplifier. We also propose an output matching of Doherty PA to reduce the number of matching components, and to match the output impedances to enhance efficiency and linearity in consideration of the uneven input drive. The PA circuit is fabricated using a 2- m InGaP/GaAs heterojunction bipolar transistor process and combined for Doherty operation using merged lumped components. For the IEEE 802.16e m-WiMAX signal, which has a 9.54-dB crest factor and 8.75-MHz bandwidth, the PA has an error vector magnitude of 3% and a power-added efficiency of 40.2% at an output power of 26 dBm. Index Terms—Doherty, efficient, handset, heterojunction bipolar transistors (HBTs), linear, monolithic microwave integrated circuit (MMIC), power amplifier (PA).

I. INTRODUCTION

A

S THE communication systems evolve, the information content increases, which leads to modulation systems that have wider bandwidths and higher crest factors. Despite these increasing demands, power amplifiers (PAs) are still required to be linear and efficient. The conventional approach of power back-off becomes an unacceptable solution as efficiency further degrades [1]. Doherty and envelope elimination and restoration (EER) techniques are the most popular schemes for high-efficiency operation at a back-off power region. The EER modulates a collector or drain bias voltage according to the envelope of a signal, and the PA operates in the saturation mode for a constant envelope signal, which delivers high efficiency [2]–[7]. However, the PA takes phase information of the signal as an input whose bandwidth is about ten times wider, and is required to be

Manuscript received April 13, 2009; revised May 24, 2010; accepted May 31, 2010. Date of publication July 15, 2010; date of current version August 13, 2010. This work was supported by the World Class University (WCU) Program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (Project R31-2008-000-10100-0), and by The Ministry of Knowledge Economy (MKE), Korea, under the Information Technology Research Center (ITRC) Support Program supervised by the National IT Industry Promotion Agency (NIPA) [NIPA-2010-(C1090-10110011)]. D. Kang, D. Kim, and B. Kim are with the Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Gyeongbuk 790-784, Korea (e-mail: [email protected]; [email protected]). J. Choi is with the Samsung Advanced Institute of Technology (SAIT), Yongin-si, Gyeonggi 446-712, 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.2010.2053074

operated across the wide bandwidth [2]. The envelope tracking (ET) technique allows the modulated signal to be the input of the PA, which relieves the bandwidth burden. The serious AM–PM problem at a low drain/collector bias can also be solved with the ET technique [4]. However, a few microHenry inductors for the dc–dc converter of the bias modulator is still an obstacle for integration. The Doherty technique is a load modulation technique developed in the 1930s as a means for achieving high efficiency from tube amplifiers [8]. Since the 1980s, however, the Doherty technique has been applied to microwave transistors [9]–[24]. The Doherty PAs efficiently amplify modulated signals with high crest factors since only a half of the power cell is operated in a low power region. Previous research attempts to further increase the efficiency in the back-off regions were primarily focused on -way [11]–[13] and -stage [14], [15] Doherty techniques. The PAs for handset applications frequently operate near a 0-dBm power level, which is over 20-dB back-off from the peak power. Thus, the Doherty PA incorporating the ET technique is applicable for highly efficient handset PAs at the low power region [16]. Since the peaking amplifier is operated at a low current due to the lower bias, an uneven drive [17], [18] and a bias adaption technique [19], [20] were introduced for better power-handling capability. Integrating the Doherty structure into a single chip is a difficult task as the structure normally requires an input coupler and a quarter-wavelength transformer, which are bulky. A number of studies have overcome the size constraint for handset applications while improving linearity [18] and enhancing efficiency [21]. We previously proposed the direct-input power-dividing technique of Doherty PAs in [22]. In this paper, we analyze the Doherty operations under differing input drives, and discuss solutions for enhancing the efficiency and linearity. We further analyze the nonlinear input impedance, and describe the input dividing with and without a coupler in greater depth. In addition, we propose output matching for uneven power driving, using a reduced number of components for handset applications. We analyze the circuit using an HBT, but the same concept can be applied to the MOSFET since behaviors of the nonlinear input impedances are identical. II. DOHERTY OPERATION CONSIDERING INPUT DRIVE The Doherty technique is based on load modulation at the output [8]. The output load is modulated by the ratio of currents between the carrier and peaking amplifiers. In this section, we will propose the uneven drive, which properly modulates the output load, and has the advantage of gain, efficiency, and linearity. We have published the uneven drive amplifier to

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TABLE I INPUT POWER DRIVING CONDITIONS

Fig. 1. Collector currents of the amplifiers, extracted from the HBT model.

Fig. 2. Uneven drive conditions according to input power level.

improve the linearity and peak power level in [17]. In this case, the input power to the carrier amplifier is reduced while that to the peaking amplifier is enhanced in the overall power level. This uneven drive reduces gain and efficiency of the Doherty PA at the back-off power region. Fig. 1 shows the collector currents of the carrier and peaking amplifiers, extracted from the HBT model without a Doherty connection. The quiescent bias currents are 50 mA for the carrier amplifier (class AB) and 0.5 mA for the peaking amplifier of the peaking amplifier does (class C). The collector current not reach to that of the carrier amplifier at the high input due to the lower bias level. Fig. 2 shows the input driving conditions for Doherty operation, Uneven-I denotes the uneven drive shown in [17]. Uneven-II denotes the uneven drive that more input power is delivered to the carrier amplifier in low power regions and to the peaking amplifier in high power regions. The input power driving conditions are categorized in Table I. The fundamental components of the current are calculated according to the driven input voltage in Fig. 2, and depicted in Fig. 3(a), where the currents of the even drive are the same as in Fig. 1. For the uneven drives, the carrier curthe currents rents are decreased and the peaking currents are increased, reducing the gaps compared to the even drive. Fig. 3(b) shows the

Fig. 3. (a) Calculated collector currents for each case of input dividing. (b) Calculated output load impedance modulation.

calculated load impedance modulations. The optimum load impedances for the carrier and peaking amplifiers are chosen to deliver the maximum output power

(1) where is the collector dc-bias voltage, is the minimum collector operating voltage similar to knee voltage of the fieldeffect transistor (FET), and is the maximum collector current. The load impedances are modulated as functions of the collector currents of the carrier and peaking amplifiers. In the case of Uneven-II, is chosen to be 3.8 , and the load is modulated from 7.6 to 3.8 . Since the carrier current of Uneven-II in Fig. 3(a) is lower than the others at the point, of Uneven-II (3.8 ) is higher than that of Even (3.4 ) and that of Uneven-I (3.5 ). Fig. 3(b) clearly shows that Uneven-II

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Fig. 5. HBT nonlinear equivalent-circuit model.

Fig. 4. (a) Calculated collector voltage. (b) Comparison of calculated efficiency and gain for each case of input dividing.

provides the proper load modulation. Fig. 4(a) shows the fundamental component of the collector voltage given by multiplying and the fundamental load impedance. at for and are assumed to be 0.4 and 0.32 V, respectively. The voltage swing of Uneven-II at the is larger due to the higher , lower , and late turn-on of the peaking amplifier. Fig. 4(b) shows the calculated efficiencies and gains. Uneven-II gives the highest efficiency at the back-off power region, and good linearity performance since the high drive to the peaking amplifier leads to the proper third-order intermodulation (IM3) cancellation. The gain curve also indicates the linearity of the Doherty PAs. Uneven-I delivers the lowest efficiency in the back-off power region because the peaking amplifier turns on earlier than the others, but the linearity is the best among the driving cases.

III. INPUT OF DOHERTY PAS The proposed uneven drive is the solution for enhancing efficiency and linearity. However, even though the input power divider provides the desired power to the carrier and peaking amplifiers, the nonlinearity of the amplifier’s input capacitances disrupts the power dividing, thereby causing imperfect load modulation. Thus, the nonlinear behavior of the input impedances of the Doherty PAs is explored.

Fig. 6. (a) Extracted g C .

and g (R kR

). (b) Extracted input capacitance

A. Nonlinear Input Impedances The parameters of the HBT nonlinear equivalent circuit shown in Fig. 5 are extracted from the HBT model described in is given by [23]. The extracted (2) where is the output matching impedance. For both the carrier and peaking amplifiers, and increase as the input power increases. However, because of the class-C bias, and of the peaking amplifier increase the large-signal more rapidly than those of the carrier amplifier. Thus, as shown of the carrier amplifier inin Fig. 6, the input capacitance creases only 10%, but that of the peaking amplifier increases dramatically (over 200%) as the input increases. The input impedances of the HBT are depicted in Fig. 7. The power-level-dependent input impedances are matched to 50 at a high power

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Fig. 9. (a) Coupled power contours according to the trace of Z and Z with a coupler. P and P are independent of each other. (b) Power dividing ratio contours (P : P ) according to the trace of Z with direct is matched to Z . Impedances are normalized by Z . input dividing. Z Fig. 7. Simulated input impedances of HBT model and the impedances after matching to 50 by LPF and HPF type circuits. Input impedances are varied with the increased input.

Fig. 8. (a) Wilkinson power divider for conventional input of the Doherty PA. Two output ports are isolated. (b) Direct input dividing.

using low-pass filter (LPF) and high-pass filter (HPF) type circuits. The matched input impedance variations are simulated and are the impedances and also shown in Fig. 7. of the carrier and peaking amplifiers, respectively. As expected, the carrier amplifier is matched to 50 , but the peaking amplifier shows significant mismatches. B. Input Dividing With Coupler Wilkinson power dividers or 90 hybrid couplers are normally used for the input divider of Doherty PAs. The two output ports are connected to the carrier and peaking amplifiers, respectively. With the power divider or coupler in Fig. 8(a), the ports of the carrier and peaking amplifiers are isolated, and the input drive power is determined by the coupler. The power to the PA is of the carrier and coupled by the reflection coefficients peaking amplifiers. The coupled power to the carrier amplifier is given by

(3) The coupled power to the peaking amplifier is the same as (3), but is determined by . The coupled power contours according to and are depicted in Fig. 9(a). If the input impedances of the carrier and peaking amplifiers are matched to at the maximum output power, the input power is driven as varies, as depicted in Fig. 7. shown in Fig. 10(a) because

Fig. 10. (a) Input power driving with a coupler, when both input impedances of carrier and peaking amplifiers are matched to Z at the maximum output power. (b) Uneven power driving, which can be achieved by the coupler [Z is matched to the 3.5-dB circle shown in Fig. 9(a)].

0

Similarly to the even input drive, the input drive of Fig. 10(a) cannot be optimum for the Doherty operation because the current of the peaking amplifier is lower due to the lower bias. For the proper input drive, the input power driving should be an uneven drive shown in Fig. 10(b). When the input impedance of and that of carrier amthe peaking amplifier is matched to plifier to a 3.5-dB circle, as shown in Fig. 9(a), the carrier input power is reduced by 1 dB, and 1 dB more power is delivered to the peaking amplifier at the maximum output power, realizing the uneven drive for optimum operation of the Doherty PA. C. Direct Input Dividing Without Coupler The Doherty PAs for handset applications should be compact, and the Wilkinson power divider, normally adopted for

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Fig. 11. Input power dividing determined by the input impedances. Impedances are normalized by Z . For 1:1, 1:2, and uneven power dividing. Fig. 12. Doherty structure. (a) Conventional. (b) Proposed.

base-station PAs, cannot be applied. We propose a direct input dividing circuit considering the impedance variations of the carrier and peaking amplifiers. Unlike the Wilkinson power dividers, the direct input power divider does not isolate the carrier and peaking amplifiers since the carrier and peaking amplifiers share the same voltage node at the input, as shown in Fig. 8(b). The power contours determined by input impedances are depicted in Fig. 9(b). When the input impedances of the amplifiers and , respectively, the power at the are assumed to be junction is given by

(4) The input currents are divided by the inverse ratio of the impedances. The divided currents and the voltage at the junction determine the input powers of the carrier and peaking amplifiers, respectively. The divided powers are given by (5) (6)

Fig. 13. Load impedance for each amplifier. (a) At a low output region power before the peaking amplifier turns on. (b) At a high output power region after the peaking amplifier turns on.

power dividing, with the matched carrier amplifier, the transformed input impedance of the peaking amplifier is

The matched impedances of the amplifiers are given by

(9) (7)

The divided power is equal to the ratio of input admittances of the amplifiers

(8) The input impedance of the carrier amplifier remains almost constant for the input power variation, while that of the peaking amplifier changes significantly because of the class-C bias. For

where

and

are normalized by

. Equation (9) becomes (10)

which is an equation of the circle centered at and with radii , and is represented by a conductance , and circle on the Smith chart. In the case of are on the conductance circle on the Smith chart. Fig. 9(b) shows power dividing ratio contours according to the trace of , where is matched to . Fig. 11 shows the input impedances of the peaking amplifier for 1:1 dividing for even and 1:2 dividing for uneven in [17]. The uneven drive,

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Fig. 14. Schematic of the proposed Doherty PA.

located inside the 1:1 circle, delivers more power to the peaking matched by the HPF circuit, the trace amplifier. Following drive is depicted in Fig. 11. The trace indicates of the uneven that the carrier amplifier receives more power at a low input power region; the peaking amplifier has more power at a high input power region, which is a desired input drive.

and , respectively. The load impedance of the carrier amto . Thus, plifier varies from (11) The impedance mined by

IV. PROPOSED DESIGN METHOD OF DOHERTY PAS The input impedance of the peaking amplifier varies significantly according to the input power level, and can be utilized for the uneven input dividing for the Doherty PAs [17], [18], [21]. As shown in Fig. 7, the HPF circuit can divide the input power properly for the uneven drive because the input circuit creates the proper impedance traces for the uneven drive. However, the phase compensation network of the quarter-wavelength line at the input of the peaking amplifier can generate an improper input dividing. The input impedances with the network are also shown by the line (— —) in Fig. 11. In this case, large power is delivered to the carrier amplifier at the low power region, and the gain at the low power region becomes much higher than that at the high power region, deteriorating the gain flatness and linearity of the Doherty PA. The output circuit of a conventional Doherty structure shown in Fig. 12(a) has an offset line and a quarter-wavelength transformer at the carrier path. The transformer modulates the load impedance, and the offset line compensates the imperfect load modulation caused by parasitics of the transistors. In our proposed Doherty structure, the output matching circuit takes the role of a quarter-wavelength transformer including parasitics, and the offset line is not employed for the carrier amplifier. As a result, the phase compensation network is employed at the carrier amplifier path, as shown in Fig. 12(b), and the proper input drive is possible. The detailed topology of the output circuit is depicted in Fig. 13. The output matching circuit at the peaking path converts to , and the offset line is employed for high impedance at the cutoff. As depicted in Fig. 3(a), in a real design where the currents of the carrier and peaking amplifiers differ, the optimum load impedances at the maximum output power for the carrier and peaking amplifiers should be determined differently as

at the junction in the peaking path is deter-

(12) where and are currents from the carrier and peaking amplifiers to the junction, respectively. The characteristic impedance of a quarter-wavelength transformer for the peaking amplifier is calculated by following: (13) where is fully turned on.

when the carrier and peaking amplifiers are is further calculated to be

(14) and are currents when the carrier and where peaking amplifiers are fully turned on. works as a matching circuit, and provides dual functions of a matching and a load modulation circuit. The offset line at the peaking amplifier assists the impedance to be near infinity before the peaking amplifier turns on. The offset line has the characteristic impedance of when the peaking amplifier is turned on. The capacitances of the offset line can be merged into and so only one inductor is additionally needed. Fig. 14 shows the full circuit schematic of the proposed Doherty PA. Two-stage carrier and peaking amplifiers are employed for the PA. The power stage of the carrier amplifier is biased at deep class AB, and that of the peaking amplifier is biased at class C. To preserve the uneven drive characteristic using the HPF circuit at the peaking input, the phase compensation network is inserted at the input of the carrier amplifier. The second and third harmonics are controlled to enhance the

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Fig. 15. (a) Measured and simulated input impedances of the carrier and peaking amplifiers as the input power increases. (b) Measured and simulated CW tests for the PA.

efficiency of both the carrier and peaking amplifiers [24]. Moreover, the second and third harmonics control circuits are also utilized for the quarter-wavelength transformer by connecting the capacitor , forming a -network. The junction parasitic capacitance of the transistor, second harmonic control circuit, and bonding inductance at the bias line are considered as one capacitor for the quarter-wavelength transformer. The third harmonic control circuit, the parallel and , is inductive at the fundamental frequency, and can be an inductor for the quarter-wavelength transformer. The capacitors of the offset line are merged to ’s to reduce the number of components. Since the input power is driven unevenly to the carrier and peaking amplifiers by the HPF type input matching, the larger portion is delivered to the carrier amplifier at a low power region, achieving higher gain and power-added efficiency (PAE). In a high power region, a larger input power is delivered to the peaking amplifier, compensating for the smaller currents due to the low bias, and the linearity can be improved [17]. The load impedances for the carrier and peaking amplifiers, respectively, are matched to the optimum impedances determined by the uneven current ratio of the amplifiers. V. EXPERIMENTAL RESULTS The Doherty PA shown in Fig. 14 is implemented using two class-AB/F PA chips fabricated using an InGaP/GaAs 2- m

Fig. 16. (a) Measured performance of the Doherty PA for 9.54-dB crest factor 8.75-MHz bandwidth m-WiMAX signal. (b) Measured output spectrum.

HBT process [24]. The off-chip lumped components are used for the input and output matching, as depicted in Fig. 14. However, all the lumped components can be integrated on a chip, using metal–insulator–metal (MIM) capacitors and bond wires as inductors. The offset line is placed at the output of the peaking amplifier. The phase compensation network is placed at the input of the carrier amplifier. The input is divided unevenly by the proposed concept without using a coupler. Fig. 15(a) shows the measured and simulated input impedances with a continuous wave (CW). At a low and high input power level, more power is delivered to the carrier and peaking amplifiers, respectively, indicating the proper uneven drive. Therefore, the PA provides high gain and PAE at the low power level, and linear characteristic at the high power level by the proper load modulation. The fabricated Doherty PA delivers an output power of 33 dBm with a PAE of 51% at the P1 dB output power, as shown in Fig. 15(b). The PAE at the 6-dB back-off power is 45%. Above the 27-dBm output power region where the peaking amplifier starts to turn on, the power generated from the peaking amplifier compensates the saturated power from the carrier amplifier, and the gain remains flat due to the appropriate input dividing and load modulation. Fig. 15 also shows that the simulation and measurement are in good agreement. For a 9.54-dB crest factor 8.75-MHz IEEE 802.16e m-WiMAX signal, the Doherty PA presents PAE of 40.2% at the output power of 26 dBm with the error vector magnitude

KANG et al.: DESIGN OF DOHERTY PAs FOR HANDSET APPLICATIONS

Fig. 17. Measured EVM performance at 26-dBm average output power.

(EVM) of 3%, as shown in Fig. 16(a). The measured spectrum at the output power of 26 dBm satisfies the linearity specification of commercial handset PAs for the WiMAX application, as shown in Fig. 16(b). Fig. 17 shows the measured EVM performance at 26-dBm average output power. VI. CONCLUSIONS We have investigated the Doherty operation considering the input drive. We have also analyzed the input power drive of the Doherty PA, and proposed the proper input dividing technique without a coupler. A phase compensation network is placed at the input of the carrier amplifier for proper uneven Doherty operation. We have also proposed the output matching of the Doherty PA to reduce the number of matching components, as well as to optimally match the output impedances considering the uneven input drive. The output matching with harmonics control circuits are applied to both the carrier and peaking amplifier to enhance the efficiency. The components of a quarter-wavelength network and an offset line are merged. The proposed uneven drive achieves a high efficiency at a back-off power region, and good linearity by driving a larger power to the peaking from the 6-dB back-off power level. For a one-tone signal, the PA delivers the maximum output power of 33 dBm with a PAE of 51%, and has 45% PAE at the 6-dB back-off power. For the m-WiMAX application, having a 9.54-dB crest factor and 8.75-MHz bandwidth, the PA has EVM of 3% and the PAE of 40.2% at the output power of 26 dBm. The PA delivers excellent efficiency and linearity performance for handset applications without employing a complicated circuit structure or an additional linearization. ACKNOWLEDGMENT The authors would like to thank Wireless Power Amplifier Module (WiPAM) Inc., Seongnam, Gyeonggi, Korea, for the advice and the chip fabrication. REFERENCES [1] S. C. Cripps, Advanced Techniques in RF Power Amplifier Design. Norwood, MA: Artech House, 2002.

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[2] F. Wang, D. F. Kimball, J. D. Popp, A. H. Yang, D. Y. C. Lie, P. M. Asbeck, and L. E. Larson, “An improved power-added efficiency 19-dBm hybrid envelope elimination and restoration power amplifier for 802.11 g WLAN applications,” IEEE Trans. Microw. Theory Tech, vol. 54, no. 12, pp. 4086–4099, Dec. 2006. [3] F. Wang, D. F. Kimball, D. Y. Lie, P. M. Asbeck, and L. E. Larson, “A monolithic high-efficiency 2.4-GHz 20-dBm SiGe BiCMOS envelopetracking OFDM power amplifier,” IEEE J. Solid-State Circuits, vol. 42, no. 6, pp. 1271–1281, Jun. 2007. [4] F. Wang, A. H. Yang, D. F. Kimball, L. E. Larson, and P. M. Asbeck, “Design of wide-bandwidth envelope-tracking power amplifiers for OFDM applications,” IEEE Trans. Microw. Theory Tech, vol. 53, no. 4, pp. 1244–1255, Apr. 2005. [5] J. Choi, D. Kang, D. Kim, J. Park, B. Jin, and B. Kim, “Power amplifiers and transmitters for next generation mobile handset,” J. Semiconduct. Technol. Sci., vol. 9, no. 4, pp. 249–256, Dec. 2009. [6] I. Kim, Y. Y. Woo, J. Kim, J. Moon, J. Kim, and B. Kim, “High-efficiency hybrid EER transmitter using optimized power amplifier,” IEEE Trans. Microw. Theory Tech, vol. 56, no. 11, pp. 2582–2593, Nov. 2008. [7] J. Choi, D. Kim, D. Kang, and B. Kim, “A polar transmitter with CMOS programmable hysteretic-controlled hybrid switching supply modulator for multistandard applications,” IEEE Trans. Microw. Theory Tech, vol. 57, no. 7, pp. 1675–1686, Jul. 2009. [8] W. H. Doherty, “A new high efficiency power amplifier for modulated waves,” Proc. IRE, vol. 24, no. 9, pp. 1163–1182, Sep. 1936. [9] F. H. Raab, “Efficiency of Doherty RF power amplifier systems,” IEEE Trans. Broadcast., vol. BC-33, no. 3, pp. 77–83, Sep. 1987. [10] Y. Yang, J. Yi, Y. Woo, and B. Kim, “Optimum design for linearity and efficiency of microwave Doherty amplifier using a new load matching technique,” Microw. J., vol. 44, no. 12, pp. 20–36, Dec. 2001. [11] Y. Yang, J. Cha, B. Shin, and B. Kim, “A fully matched N-way Doherty amplfier with optimized linearity,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 986–993, Mar. 2003. [12] J. Moon, J. Kim, I. Kim, J. Kim, and B. Kim, “Highly efficient 3-way saturated Doherty amplifier with digital feedback predistortion,” IEEE Microw. Compon. Lett., vol. 18, no. 8, pp. 539–541, Aug. 2008. [13] M. Iwamoto, A. Williams, P. F. Chen, A. G. Metzger, L. E. Larson, and P. M. Asbeck, “An extended Doherty amplifier with high efficiency over a wide power range,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2472–2479, Dec. 2001. [14] W. C. E. Neo, J. Qureshi, M. J. Pelk, J. R. Gajadharsing, and L. C. N. de Vreede, “A mixed-signal approach towards linear and efficient -way Doherty amplifier,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 5, pp. 866–879, May 2008. [15] M. J. Pelk, W. C. E. Neo, J. R. Gajadharsing, R. S. Pengelly, and L. C. N. de Vreede, “A high-efficiency 100-W GAN three-way Doherty amplifier for base-station applications,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 7, pp. 1582–1591, Jul. 2008. [16] J. Choi, D. Kang, D. Kim, and B. Kim, “Optimized envelope tracking operation of Doherty power amplifier for high efficiency over an extended dynamic range,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 6, pp. 1508–1515, Jun. 2009. [17] J. Kim, J. Cha, I. Kim, and B. Kim, “Optimum operation of asymmetrical-cells-based linear Doherty power amplifiers—Uneven power drive and power matching,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1802–1809, May 2005. [18] M. Nick and A. Mortazawi, “A Doherty power amplifier with extended resonance power divider for linearity improvement,” in IEEE MTT-S Int. Microw. Symp. Dig., 2008, pp. 423–426. [19] J. Moon, J. Kim, I. Kim, J. Kim, and B. Kim, “A wideband envelope tracking Doherty amplifier for WiMAX systems,” IEEE Microw. Compon. Lett., vol. 18, no. 1, pp. 49–51, Jan. 2008. [20] Y. Yang, J. Cha, B. Shin, and B. Kim, “A microwave Doherty amplifier employing envelope tracking technique for high efficiency and linearity,” IEEE Microw. Compon. Lett., vol. 13, no. 9, pp. 370–372, Sep. 2003. [21] D. Yu, Y. Kim, K. Han, J. Shin, and B. Kim, “Fully integrated Doherty power amplifiers for 5 GHz wireless-LANs,” in IEEE Radio Frequency Integr. Circuits Symp. Dig., 2006, pp. 177–180. [22] D. Kang, J. Choi, D. Yu, K. Min, M. Jun, D. Kim, J. Park, B. Jin, and B. Kim, “Input power dividing of Doherty power amplifiers for handset applications,” in IEEE MTT-S Int. Microw. Symp. Dig., 2009, pp. 601–604. [23] Y. Zhao, A. Metzger, P. Zampardi, M. Iwamoto, and P. Asbeck, “Linearity improvement of HBT-based Doherty power amplifiers based on a simple analytical model,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4479–4488, Dec. 2006.

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[24] D. Kang, D. Yu, K. Min, K. Han, J. Choi, B. Jin, D. Kim, M. Jeon, and B. Kim, “A highly efficient and linear class-AB/F power amplifier for multi-mode operation,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 1, pp. 77–87, Jan. 2008.

Daehyun Kang received the B.S. degree in electronic and electrical engineering from the Kyungpook National University, Daegu, Korea, in 2006, and is currently working toward the Ph.D. degree in electrical engineering at the Pohang University of Science and Technology (POSTECH), Pohang, Gyungbuk, Korea. His main interests are RF circuits for wireless communications, especially highly efficient and linear RF transmitters and RF PA design.

Jinsung Choi received the B.S. and Ph.D. degrees in electrical engineering from the Pohang University of Science and Technology (POSTECH), Pohang, Gyungbuk, Korea, in 2004 and 2010, respectively. He is currently with the Samsung Advanced Institute of Technology (SAIT), Yongin-si, Korea. His main research interests are analog/RF circuit design in ultra-deep-submicrometer CMOS technology, mixed-mode signal-processing integrated-circuit design, and digitally assisted RF transceiver architectures.

Dongsu Kim received the B.S. degree in electrical engineering from the Pohang University of Science and Technology (POSTECH), Pohang, Gyungbuk, Korea, in 2007 and is currently working toward the Ph.D. degree at POSTECH. His research interests are CMOS RF circuits for wireless communications, with a special focus on highly efficient and linear RF transmitter design.

Bumman Kim (M’78–SM’97–F’07) received the Ph.D. degree in electrical engineering from Carnegie–Mellon University, Pittsburgh, PA, in 1979. From 1978 to 1981, he was engaged in fiber-optic network component research with GTE Laboratories Inc. In 1981, he joined the Central Research Laboratories, Texas Instruments Incorporated, where he was involved in the development of GaAs power FETs and monolithic microwave integrated circuits (MMICs). He has developed a large-signal model of a power FET, dual-gate FETs for gain control, high-power distributed amplifiers, and various millimeter-wave MMICs. In 1989, he joined the Pohang University of Science and Technology (POSTECH), Pohang, Gyungbuk, Korea, where he is a Namko Professor with the Department of Electrical Engineering and Director of the Microwave Application Research Center involved in device and circuit technology for RF integrated circuits (RFICs). In 2001, he was a Visiting Professor of electrical engineering with the California Institute of Technology, Pasadena. He has authored over 200 technical papers. Dr. Kim is a member of the Korean Academy of Science and Technology and the Academy of Engineering of Korea. He was an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and a Distinguished Lecturer of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S).

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A CMOS Wideband RF Front-End With Mismatch Calibrated Harmonic Rejection Mixer for Terrestrial Digital TV Tuner Applications Hyouk-Kyu Cha, Member, IEEE, Kuduck Kwon, Student Member, IEEE, Jaeyoung Choi, Student Member, IEEE, Hong-Teuk Kim, and Kwyro Lee, Senior Member, IEEE

Abstract—A wideband direct-conversion RF front-end for Advanced Television Systems Committee terrestrial digital TV (DTV) tuner applications is realized in a 0.18- m CMOS technology. In order to effectively solve the critical local oscillator (LO) harmonic mixing problem in an ultra-wideband frequency environment of 48–860 MHz, the combination of a mismatch calibrated harmonic rejection mixer (HRM) and a simple preceding integrated thirdorder passive RF tracking filter with an external inductor is utilized to obtain 60–80 dB of harmonic rejection for all odd-order harmonic mixing within the DTV spectrum. In addition, an efficient novel calibration algorithm for the HRM is proposed in order to simplify the compensation process. The RF front-end also includes a broadband noise-canceling low-noise amplifier, an attenuator to cover the wide dynamic range, an LO multiphase generator, and peripheral circuits such as an I2 C serial interface for digital control. The implemented CMOS RF front-end achieves a total gain of 40 dB, output third-order intercept point of 30 dBm, and noise figure of 5.5 dB while consuming a low power of 140 mW from a 1.8-V supply voltage. Index Terms—Advanced Television Systems Committee (ATSC), CMOS RF front-end, digital TV (DTV) tuner, direct conversion receiver, harmonic rejection mixer (HRM), mismatch calibration, tracking filter, wideband receiver.

I. INTRODUCTION

A

WIDEBAND receiver for digital TV (DTV) tuner applications has gained much attention in recent years due to the demand for integrated silicon solutions to support various DTV standards all over the globe [1]. In comparison to narrowband wireless systems, the unique ultra-wideband environment of 48–860 MHz in DTV makes it a major challenge in

Manuscript received July 15, 2009; revised March 16, 2010; accepted June 03, 2010. Date of publication July 19, 2010; date of current version August 13, 2010. This work was supported in part by the LG Electronics Institute of Technology. H.-K. Cha was with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea. He is now with the Institute of Microelectronics, A*STAR, Singapore 117685 (e-mail: [email protected]). K. Kwon was with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305701, Korea. He is now with Samsung Electronics, Gyeonggi-do 446-711, Korea. J. Choi and K. Lee are with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea. H.-T. Kim is with System IC, LG Electronics Institute of Technology, Seoul 137-724, 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.2010.2053072

designing a high-performance highly integrated receiver. One of the critical issues that impede the ease development of the tuner receiver is the local oscillator (LO) harmonic mixing in the VHF band of 48–300 MHz. Due to its wideband frequency spectrum, the unwanted in-band RF signals located at odd-order harmonic frequencies of the square-wave LO signal in the RF path are also down-converted to baseband along with the desired channel, degrading the signal-to-noise ratio (SNR) of the receiver. Many previous papers targeted for terrestrial/cable DTV standards have addressed this issue and architectural approaches such as single-conversion architecture with external tracking filters [2] and up-down dual-conversion architecture with an external surface acoustic wave (SAW) filter [3], [4] were used to avoid this problem. However, these architectures have some disadvantages in terms of poor integration, large bill-of-material, and high power consumption. On the other hand, circuit level approaches employing a harmonic rejection mixer (HRM) [5] for down-conversion have been introduced [6]–[9]. However, the degradation in harmonic rejection performance due to mismatch results in 30–40 dB, which may not be sufficient to meet the system requirement, and higher order harmonic rejection performances have not been addressed in these previous publications. In this paper, a high-performance HRM with mismatch calibration [10] and a simple integrated preceding passive RF tracking filter are utilized in the wideband direct-conversion RF front-end targeted for Advanced Television Systems Committee (ATSC) terrestrial standard to completely solve the harmonic mixing problem within 48–860 MHz with over 60 dB of harmonic rejection for all odd-order harmonics. Section II discusses the method for harmonic rejection, while Section III briefly addresses the implemented RF front-end architecture. Section IV describes the circuit design in detail with an emphasis on the HRM with mismatch calibration circuitry. The novel calibration algorithm for the HRM is discussed in Section V. Section VI presents the experimental results followed by conclusions in Section VII. II. HARMONIC REJECTION APPROACH In order to meet the SNR requirement at the receiver output for DTV, a harmonic rejection ratio (HRR) of over 60 dB must be guaranteed within the band of 48–860 MHz. To solve the harmonic mixing problem in direct-conversion or low-IF receivers without external high- passive RF tracking filters, an HRM is required. Depending on the number of sub-mixer paths and

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Fig. 1. Block diagram of RF front-end for terrestrial DTV tuner.

phase-shifted LO signals, harmonic mixing can ideally be prevented by only using the HRM, but because the hardware complexity becomes too complicated as the number of sub-mixer paths increase and since the effect of gain and phase error due to process, voltage, and temperature (PVT) variations greatly degrades its rejection performance to 30–40 dB [5], [10], it is more or less practical to use three sub-mixers with 45 phase-shifted LO signals to reject third-, fifth-, 11th-, and 13th-order harmonics and pass the burden to a preceding high-order tracking RF filter in the receiver path for additional 30-dB rejection to meet the HRR specification [9], [11], [12]. However, satisfying the tracking filter’s requirement of large attenuation, wide frequency tuning range, consistent frequency response at different tuned frequencies, high linearity, and good noise performance is not trivial. On the other hand, if the mismatch of the HRM can be compensated and its rejection performance improved, then the stringent attenuation and tuning range requirement of the preceding tracking filter can be alleviated, resulting in a possibility to employ a simpler filter and ultimately improve the total system performance. The harmonic rejection scenario for the implemented RF front-end is as follows: the HRM maximizes the third-order HRR, which is the largest, closest, and therefore, the most difficult harmonic mixing to prevent, without the assistance of the tracking filter thanks to the proposed mismatch calibration method. For fifth-order and higher order harmonics, both the tracking filter and HRM are used to meet the HRR specification of over 60 dB. Since the tracking filter only has to attenuate fifth and higher harmonics within 48–860 MHz, its center frequency can be tunable from 48 to just 180 MHz. For 180 MHz and higher frequencies, the filter is bypassed and only the mismatch calibrated HRM is utilized to obtain over 60 dB of HRR. This is possible since only the third-order harmonic is in-band, while fifth-order and higher harmonics fall out-of-band beyond 860 MHz. III. RF FRONT-END ARCHITECTURE Fig. 1 shows the block diagram of a designed wideband RF front-end that employs the direct-conversion architecture for its simple architecture and high integration capability. Since the channel characteristics are not as tightly packed as the cable standard, single-ended paths are employed for the low-noise amplifier (LNA), attenuator, and the tracking filter in order to

Fig. 2. Simplified schematic of LNA.

eliminate the external wideband input balun and reduce redundant power consumption and die area. After the antenna, either the LNA or the attenuator is activated depending on the signal power to amplify or attenuate the received signal, respectively. A source–follower buffer is inserted between the LNA/attenuator and the tracking filter for isolation. For signal frequencies between 48–180 MHz, the tracking filter is enabled to assist the HRM while it is bypassed for the 180–860-MHz range. A single-to-differential amplifier (SDA) follows the tracking filter to change the single-ended signal to differential and to provide high impedance to both the filter output and the LNA/attenuator output (in bypass mode) to prevent gain loss due to loading. A differential source–follower buffer is placed after the SDA to drive the HRM and provide fine discrete-step gain control. The HRM with mismatch calibration is used to down-convert the RF signal to baseband while preventing LO harmonic mixing. For multiphase LO generation from an external LO signal, a divide-by-4 circuit along with LO buffers are integrated to drive the mixing stage of the HRM. Lastly, peripheral blocks such as current reference and I2C serial interface are included. IV. CIRCUIT IMPLEMENTATION A. Wideband Noise-Canceling LNA and Attenuator For successful reception of terrestrial DTV signals, some amount of gain control is required in the RF front-end to meet the large dynamic-range specifications. In this work, a discrete-control variable-gain LNA, an attenuator, and a differential buffer are employed, as shown in Fig. 1. The circuit topology of the LNA is shown in Fig. 2. It uses a common-gate (CG) input stage to obtain broadband input impedance matching while a common-source stage is added in parallel to facilitate noise cancellation [13] of the CG transistor and . The simulated LNA noise at the drain nodes of figure within 48–860 MHz is 2.5–3.5 dB where noise cancellation provides 3–4 dB of improvement. A large external inductor is used to provide the bias current path for . In order to

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Fig. 3. Schematic of attenuator.

Fig. 4. Simplified schematic of the passive tracking filter.

enhance the linearity of the LNA, the linearity of must be improved since the signal is large at the drain of . A source is initially placed in the source of degeneration resistor to improve the linearity. In addition, the subsidiary transistor biased in the subthreshold region is added to minimize the third-order intermodulation distortion [14] and further enhance the linearity of the LNA. In order to obtain a higher bandwidth and wide flat gain frequency response without employing on-chip spiral inductors for high integration, a cascode tranis placed at the output node of the LNA. A discrete sistor and . For fine gain step control is possible by controlling transistor is either turned on or off a coarse gain step, the by controlling its gate bias voltage. A gain range from 2 to 20 dB is obtained in 3-dB discrete control steps. The resistive-divider type attenuator is connected in parallel to the LNA and its circuit diagram is shown in Fig. 3. In order to obtain a dB-linear attenuation characteristic and good input matching over the operating frequency, careful simulations are carried out to find appropriate resistor values. Due to the chance of signal coupling through large parasitic capacitances of the MOS switch transistor when large high-frequency signals enter the attenuator input, an additional stack is added for cascode switch topology to sufficiently prevent the signal coupling. An attenuation from 1 to 22 dB is obtained in 8-bit 3-dB discrete control steps. B. RF Passive Tracking Filter For the integrated tracking filter, both the active and passive type implementations are possible. Considering its inherent high linearity and simple design, the passive type tracking filter is chosen. The RF tracking filter, shown in Fig. 4, is a third-order filter tunable from 48 to 180 MHz to attenuate passive undesired signals at fifth and higher harmonics frequencies in the input spectrum. A source–follower buffer is placed in front of the tracking filter for isolation from the preceding and are implemented LNA. The tunable shunt capacitors with on-chip metal–insulator–metal (MIM) capacitors, while . a high- external component is used for the inductor For frequency tuning from 48 to 180 MHz, 6-bit control MOS

Fig. 5. Schematic of HRM with mismatch calibration circuitry.

switches are used to program the capacitor array. Assuming that the following HRM provides 40 dB of fifth-order HRR and 18 dB of seventh-order HRR, the tracking filter is designed to attenuate the signals at the fifth- and seventh-order harmonic frequencies in the RF path by over 20 and 42 dB, respectively, so that over 60 dB of total HRR is obtained. In order to tune the filter to the desired frequency with sufficient accuracy and acceptable delay with PVT variations, an effective calibration is required. This work does not include the tracking filter calibration. However, for this purpose, the calibration method in [15] can be adopted at startup and during channel selection to tune the tracking filter to the wanted channel frequency. C. HRM Fig. 5 shows the schematic of the HRM with mismatch calibration circuitry. The designed HRM is divided into two stages where the first stage consists of four sub-mixers driven by eight differential phase-shifted LO signals to generate in-phase (I) and quadrature (Q) baseband signals. In this stage, mixing between the RF input and the 45 phase-shifted LO signal takes place in each of the sub-mixer paths. Each mixing stage of the sub-mixer path consists of a source-degenerated passive mixer terminated by transimpedance amplifiers [16]. The passive mixer topology is chosen for high linearity and low flicker is inserted in series noise [17]. The degeneration resistor with the switches to further improve the linearity and lessen the , which is the effect of spread of the nonlinear resistance of turn-on resistance of mixer switches, to obtain a high level of device matching among the mixer switches, and thus reduce the probability of gain mismatch among the sub-mixer paths. The value of the degeneration resistance is a tradeoff between noise

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performance, linearity, and matching. The conversion gain of this mixing stage can be derived as

(1) where the term is related to the fundamental term of the is the feedback resistor in the operational square-wave LO, is the source-degeneration resistor amplifier (opamp), and and are added for in the passive mixer. Capacitors low-pass filtering of distant interferers to mitigate the linearity burden of the following baseband analog blocks. The second stage consists of gain coefficient scaling and current summing stage where the weighted gain ratio of and the summation of the three 45 phase-shifted paths take place at nodes and shown in Fig. 5 in order to prevent third- and fifth-order harmonic mixing. In comparison to previous HRMs [5], [6], [9], the gain coefficient scaling stage is placed in the baseband instead of in the RF domain in the form of resistor ratios in order to minimize the probability of gain mismatch due to parasitic capacitances, and thus improve the initial harmonic rejection performance. The weighted gain ratio is accurately achieved among the down-converted baseband signals and through each resistor ratio between the series resistance . The values of and the opamp feedback resistance should be sufficiently high to lessen the probability of gain mismatch. However, a tradeoff exists between matching and noise . The gain of the gain performance with the varying value of coefficient stage can be derived as (2)

Fig. 6. Vector phase sum diagram: (a) without mismatch and (b) with gain and phase mismatch.

fifth-order and higher order harmonic frequencies in the input spectrum to obtain over 60 dB of total HRR. The two orthogonal calibration vectors to compensate both gain and phase mismatch can be realized in the analog domain and be directly applied to the HRM. In order to create 0 and 90 calibration vectors (and its differential counterparts of 180 and 270 for opposite directional vectors), cross-coupled digitally programmable resistors are inserted in the summing stage of the HRM, as shown in Fig. 5, and are utilized to compensate both gain and phase mismatch errors to maximize the third-order HRR. The relationship between the calibration vectors and calibration resistors can be derived as follows: (3) (4)

D. Mismatch Calibration of HRM In an ideal situation, the fundamental signal at the output of the HRM is constructively summed while a perfect cancellation of the third- and fifth-order harmonics will result, as is illustrated using a vector phase diagram in Fig. 6(a). However, the harmonic rejection performance of the HRM is degraded by gain and phase mismatch. Gain errors affecting the resulting HRR will depend upon the mismatch in the 1:1:1 resistor ratio , , and in the mixing of of the switches), as well as any destage (also affected by ratio of , , viation in the and in the summing stage. On the other hand, phase mismatch in the sub-mixer switches, as well as in the 45 phase-shifted LO signal generated by the multiphase generator, additionally degrade the harmonic rejection performance. In order to calibrate both gain and phase errors by completely canceling the remains of the uncancelled third-harmonic term, two orthogonal calibration vectors are required, as shown in Fig. 6(b). The compensation of the third-order harmonic term can slightly degrade the fifth-order harmonic term since the same set of calibration vectors will direct the resulting vector in a different direction for the fifth-order harmonic term. However, this is not a problem since much attenuation is easily possible using the preceding tracking filter for the signals located at

and their differential counterparts for opposite directional vectors (5) (6) , , , and , respectively. The two orthogonal calibration vectors are independent to each other, and they are controlled separately. Depending on the sign (positive or negative gain and phase) and the extent of mismatch and errors, all four vectors can be controlled by controlling , and the noncancelled terms due to gain and phase mismatch are added or subtracted, reducing the harmonic mixed term. The relationship between the amount of mismatch, calibration vectors, and the calibration resistors can be shown as where

Mismatch

or

(7)

in which an increase in the amount of mismatch will require a decrease in the value of the calibration resistors. The programmable discrete resistances are realized with resistors and

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Fig. 7. Block diagram of the LO generator.

MOS switches. The size and the range of the resistors are decided by the degree of mismatch the calibration system is targeting to cover. Higher resolution of the unit calibration resistor enables more accurate mismatch cancellation, and thus results in a higher calibrated HRR. In this design, up to 2 of phase mismatch and 1% gain mismatch can be compensated. Converting this amount of mismatch to HRR is around 30 dB. The size of the MOS switches should be large enough so that the on-resistance has negligible effect on the unit calibration resistor. E. Multiphase LO Generation In order to obtain a high initial HRR before third-order HRR calibration, both gain and phase error must be minimized. Gain mismatch is dominant in the HRM, while phase mismatch depends primarily on the phase performance of the multiphase LO generator. In addition, a symmetric LO waveform with accurate phases is critical in ensuring the balanced switching operation of the mixer. Fig. 7 shows the complete block diagram of the multiphase LO generator that includes a divide-by-4 circuit and LO buffers. In this design, a divide-by-four circuit using cascaded MOS current mode logic master–slave D flip-flops is used to produce accurate 50% duty differential multiphase LO signals to minimize LO asymmetries [18]. By cascading two divide-by-2 stages connected in a master–slave configuration, the eight differential phases of 0 , 45 , 90 , 135 , 180 , 225 , 270 , and 315 for differential I/Q paths can be generated. The internal LO frequency ranges from 48 to 860 MHz, while the divider operates four times the internal frequency. The first LO buffer in Fig. 7 acts as the input buffer to reshape the high-frequency waveform that is distorted by the packaged pin, bond wire, and pad parasitic [19]. Since a single LO generator has to cover a wide frequency range, the LO buffer paths following the divide-by-4 are divided into two different paths depending on the divided frequency in order to maintain equal rise and fall times between the wide frequency range LO signal. These buffers are utilized to provide rail-to-rail LO swings to drive the switching quad of the mixer. Larger LO swing expedites the switching transition and helps improve the mixer noise figure and linearity performance [20]. Fig. 8 illustrates the schematic of the complete divide-by-4 along with its timing diagram where a 45 phase shift is present between each successive differential D flip-flop output. V. CALIBRATION ALGORITHM In this section, a simple calibration algorithm is introduced for the HRR mismatch calibration. The calibration procedure

Fig. 8. Schematic and timing diagram of divide-by-4 circuit.

is carried out by bypassing the tracking filter and externally applying a single-tone signal that is three times the desired channel frequency to the RF front-end input. The signal gets down-converted by the third-order LO harmonic and the output of the HRM is observed using a spectrum analyzer, while the calibration code is shifted successively until the maximum HRR is achieved. The best way to obtain the highest calibration performance in any working condition is to carry out this calibration process for every input frequency at startup and store the calibration code for each channel frequency in a nonvolatile memory [21]. This, however, is an inefficient and very time-consuming process. In order to reduce the calibration time and the number of times the calibration procedure must be carried out, an efficient calibration algorithm is essential. If it is possible to estimate the relationship between input frequency of 48–300 MHz and the required calibration code at that certain frequency in a simple first-order, second-order, or perhaps a higher order function/equation, then by simply updating the calibration code in reference to the incoming channel frequency using this relationship, the calibration procedure will be less complicated. For example, let us assume the relationship between input frequency and calibration code can be estimated by shown in Fig. 9(a). a first-order function Code If the frequency is 50 MHz, then the required calibration code . If the frequency is changed will be Code to 300 MHz, then the code can simply be updated by using the . Therefore, if above function Code the relationship between input frequency and calibration code can be found, the calibration procedure at startup will only have to be repeated by the number of required unknown variables to form that relationship function. If the relationship is a first-order and , exist. Thus, function, two unknown variables, i.e., the calibration procedure must be repeated twice. Afterwards, no more calibration steps are required and the calibration code can be updated using the estimated calibration code function. In order to find out the ultimate relationship between the frequency and calibration code, the relationship between the frequency and calibrated HRR is first found from the previous measurement in [10]. Assuming that gain mismatch is independent versus freof frequency, the relationship of phase mismatch quency must be known and this is obtained from the Cadence SpectreRF simulation. From this preliminary information, the can be found. Finally, with relationship of frequency versus

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Fig. 10. Chip microphotograph of implemented RF front-end.

Fig. 9. (a) Frequency versus required calibration code or resistor. (b) Measured relationship of frequency versus required calibration resistor.

this information and the relationship of versus the calibration code (resistance) obtained from the simulations, the desired relationship between the frequency and calibration code can be obtained. This is shown in Fig. 9(b) with the dotted lines that the relationship between the frequency and calibration code can roughly be estimated as a first-order linear function. The frequency and calibration resistors show a proportional relationship. The reason for this relationship is because phase mismatch in the LO generator and the mixer switches increase with the increase in frequency (i.e., due to the increase in effect of the parasitic). Since the amount of mismatch increases, the values of the calibration resistors need to be proportionally changed in order to provide larger compensation current to cancel out the remaining uncancelled harmonic mixed term. Since it is a first-order function, two unknown variables exist. To find out the relationship between frequency and calibration code for the whole frequency band of interest, the calibration procedure need to be carried out only twice to obtain the values for the unknown variables, which greatly simplifies the calibration process. VI. MEASUREMENT RESULTS Fig. 10 shows the chip microphotograph of the implemented DCR RF front-end using a 1P 6M 0.18- m CMOS process. The chip size is 3.1 mm 2.9 mm including electrostatic discharge (ESD) protection circuits and pads. Fig. 11 shows the measured for both attenuator and LNA modes. dB for 48–860 MHz with different attenuation/gain configurations are obtained. The programmable RF front-end gain measurement for different input frequencies in attenuation mode is plotted in Fig. 12 in which a consistent decibel-linear response can be seen. In Fig. 13, the measured frequency response of the

Fig. 11. Measured S 11 in the: (a) attenuator mode and (b) LNA mode.

tracking filter is presented with different cutoff frequency settings at 50, 100, 150, and 200 MHz. The filter attenuates the signals in the frequencies that are five and seven times higher than the cutoff frequency by over 37 and 42 dB, respectively. To test the performance of the HRM with and without calibration, the tracking filter is bypassed and the two-tone signals at 201 and 601.1 MHz are applied to the LNA input, and its down-converted signals are shown in Fig. 14. The fundamental and third-order harmonic mixed signals are down-converted to 1 and 1.1 MHz, respectively. Through calibration, the HRR is improved from 45 to 73 dB in this measurement example. Fig. 15 shows the measured third- and fifth-order HRR with and without third-order mismatch calibration from over 20 die samples at various signal frequencies. An improvement of 20–30-dB rejection is achieved, resulting in an average of 75 dB

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Fig. 12. Measured programmable RF front-end gain at different input frequencies in attenuation mode (LNA disabled and tracking filter bypassed).

Fig. 15. Measured HRR at 1-MHz IF for 23 die samples with and without third-order calibration. (a) Third-order HRR. (b) Fifth-order HRR (tracking filter bypassed). Fig. 13. Measured frequency response of passive tracking filter.

Fig. 14. Measured third-order HRR at HRM output with and without mismatch calibration (tracking filter bypassed).

of third-order HRR by using the proposed mismatch calibration method for normal operating conditions. With the third-order HRR calibration, an average degradation of 5 dB is seen for the fifth-order HRR. However, since the preceding tracking filter attenuates the signal located at the fifth-harmonic frequency in the RF path, over 70 dB of combined HRR is still obtained. The supply voltage and temperature variation measurements for the third-order HRR with and without calibration are shown in Fig. 16. For supply voltages of 1.7 and 1.9 V, over 70 dB of calibrated HRR is maintained without changing the calibration code used for normal operating conditions. For temperatures at 0 C and 60 C, over 60 dB of calibrated HRR is maintained using the same calibration codes for normal temperature of 27 C. The reason that the calibrated HRR is over 60 dB even for varying supply voltage and temperature conditions is because the HRM is robust to PVT variations and that the calibra-

Fig. 16. Measured average third-order HRR for 23 die samples with and without calibration at different: (a) supply voltages and (b) temperatures (tracking filter bypassed).

tion codes do not have to be updated for varying environmental conditions. Table I summarizes the measured performances of the implemented direct-conversion RF front-end. It includes the

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TABLE I MEASURED PERFORMANCES OF RF FRONT-END

a higher burden is placed on the preceding tunable filter due to the insufficient performance of the HRM. In addition, this paper also does not address the higher order harmonic rejection issue. Thanks to the proposed HRM with mismatch calibration and a simple preceding tracking filter, it shows that only our work achieves over 60 dB of HRR for all odd-order harmonic mixing. VII. CONCLUSION

TABLE II PERFORMANCE COMPARISON OF RF FRONT-END WITH OTHER REPORTED WORK

In this paper, a wideband highly integrated direct-conversion RF front-end for ATSC terrestrial DTV is implemented using 0.18- m CMOS technology. To effectively solve the critical harmonic mixing problem, the mismatch calibrated HRM with high third-order HRR performance and a simple preceding passive tracking filter are utilized to obtain over 60 dB of HRR for all odd-harmonic mixing within 48–860 MHz. The fabricated wideband RF front-end achieves a total gain of 40 dB, output third-order intercept point (OIP3) of 30 dBm, and a noise figure of 5.5 dB while consuming 140 mW from a 1.8-V supply voltage. ACKNOWLEDGMENT The authors thank Assistant Prof. I. Nam, Pusan National University, Nam-gu, Busan, Korea, and the members of the RF Group, System IC, LG Electronics Institute of Technology, Seoul, Korea, for technical discussions. The authors also thank the associate editor and reviewers for valuable comments and suggestions. REFERENCES

total measured HRR for third-, fifth-, seventh-, and ninth-order harmonic mixing. For 48–180 MHz, both the tracking filter and HRM are enabled, whereas only the HRM is activated for over 180 MHz. At least over 60 dB of HRR is achieved for all oddharmonic mixing within 48–860 MHz. Table II compares this work with previous receivers with harmonic rejection functions. In [6], a conventional HRM is used and only 38–40 dB of thirdand fifth-order HRR are reported, while over 60 dB of thirdand fifth-order HRR are reported using a two-stage HRM in [7]. However, higher order HRR is not addressed in this paper. In [8], over 80 dB of third- and fifth-order HRR (not simultaneously) are achieved utilizing adaptive interference cancellation (AIC) in the digital domain. This work is similar with our work with regard to the fact that compensation is used to enhance the HRR performance of the HRM. Another similarity is that either the third or fifth HRR can be improved at one time and not simultaneously since the compensation conditions are different. However, two additional ADCs and DSP are required to employ the AIC technique and can only calibrate one harmonic signal at one time, which means that other order harmonic responses will be poor. On the other hand, in [9], over 70 dB of third-order HRR is presented utilizing the combination of a fourth-order tunable filter and a conventional HRM. However,

[1] M. Dawkins, A. Burdett, and N. Cowley, “A single-chip tuner for DVB-T,” IEEE J. Solid-State Circuits, vol. 38, no. 8, pp. 1307–1317, Aug. 2003. [2] J. Tourret, S. Amiot, M. Bernard, M. Bouhamame, C. Caron, O. Crand, G. Denise, V. Fillâtre, T. Kervaon, M. Kristen, L. Lo Coco, F. Mercier, J. M. Paris, F. Pichon, S. Prouet, V. Rambeau, S. Robert, and J. van Sinderen, “SiP tuner with integrated LC tracking filter for both cable and terrestrial TV reception,” IEEE J. Solid-State Circuits, vol. 42, no. 12, pp. 2809–2821, Dec. 2007. [3] I. Mehr, S. Rose, S. Nesterenko, D. Paterson, R. Schreier, H. L’Bahy, S. Kidambi, M. Elliott, and S. Puckett, “A dual-conversion tuner for multi-standard terrestrial and cable reception,” in IEEE VLSI Cicuits Symp. Tech. Dig., Jun. 2005, pp. 340–343. [4] J. M. Stevenson, P. Hisayasu, A. Deiss, B. Abesingha, K. Beumer, and J. Esquivel, “A multi-standard analog and digital TV tuner for cable and terrestrial applications,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2007, pp. 210–211. [5] J. A. Weldon, R. S. Narayanaswami, J. C. Rudell, L. Li, M. Otsuka, S. Dedieu, T. Luns, K. C. Tsai, C. W. Lee, and P. R. Gray, “A 1.75-GHz highly integrated narrow-band CMOS transmitter with harmonic-rejection mixers,” IEEE J. Solid-State Circuits, vol. 36, no. 12, pp. 2003–2015, Dec. 2001. [6] R. Bagheri, A. Mirzaei, S. Chehrazi, M. E. Heidari, M. Lee, M. Mikhemar, W. Tang, and A. A. Abidi, “An 800-MHz–6-GHz software-defined wireless receiver in 90-nm CMOS,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2860–2876, Dec. 2006. [7] Z. Ru, E. A. M. Klumperink, G. J. M. Wienk, and B. Nauta, “A software-defined radio receiver architecture robust to out-of-band interference,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2009, pp. 232–234. [8] N. Moseley, Z. Ru, E. A. M. Klumperink, and B. Nauta, “A 400-to–900 MHz receiver with dual-domain harmonic rejection exploiting adaptive interference cancellation,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2009, pp. 232–234. [9] S. Lerstaveesin, M. Gupta, D. Kang, and B.-S. Song, “A 48–860 MHz CMOS low-IF direct-conversion DTV tuner,” IEEE J. Solid-State Circuits, vol. 43, no. 9, pp. 2013–2024, Sep. 2008.

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[10] H.-K. Cha, S.-S. Song, H.-T. Kim, and K. Lee, “A CMOS harmonic rejection mixer with mismatch calibration circuitry for digital TV tuner applications,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 9, pp. 617–619, Sep. 2008. [11] K. Kwon, H.-T. Kim, and K. Lee, “A 50–300-MHz highly linear and low-noise CMOS Gm-C filter adopting multiple gated transistors for digital TV tuner ICs,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 2, pp. 306–313, Feb. 2009. [12] P. Delizia, M. De Matteis, S. D’Amico, A. Baschirotto, C. Azeredo-Leme, and R. Reis, “Design procedure for DVB-T receivers large tuning range LP filter,” in IEEE Int. Circuits Syst. Symp., May 2008, pp. 2913–2916. [13] C.-F. Liao and S.-I. Liu, “A broadband noise-canceling CMOS LNA for 3.1–10.6-GHz UWB receivers,” IEEE J. Solid-State Circuits, vol. 42, no. 2, pp. 329–339, Feb. 2007. [14] T. W. Kim and B. Kim, “A 13-dB IIP3 improved low-power CMOS RF programmable gain amplifier using differential circuit transconductance linearization for various terrestrial mobile D-TV applications,” IEEE J. Solid-State Circuits, vol. 41, no. 4, pp. 945–953, Apr. 2006. [15] O. Jamin, V. Rambeau, F. Mercier, and I. Meliane, “On-chip auto-calibrated RF tracking filter for cable silicon tuner,” in IEEE Eur. SolidState Circuits Conf., 2008, pp. 158–161. [16] M. Notten, J. van Sinderen, F. Seneschal, and F. Mounaim, “A low-IF CMOS double quadrature mixer exhibiting 58 dB of image rejection for silicon TV tuners,” in IEEE RF Integr. Circuits Symp., Jun. 2005, pp. 171–174. [17] M. Valla, G. Montagna, R. Castello, R. Tonietto, and I. Bietti, “A 72-mW CMOS 802.11a direct conversion front-end with 3.5-dB NF and 200-kHz 1=f noise corner,” IEEE J. Solid-State Circuits, vol. 40, no. 4, pp. 970–977, Apr. 2005. [18] K. Kivekas, A. Parssinen, J. Ryynanen, and J. Jussila, “Calibration techniques of active BiCMOS mixers,” IEEE J. Solid-State Circuits, vol. 37, no. 6, pp. 766–769, Jun. 2002. [19] N. Poobuapheun, W.-H. Chen, Z. Boos, and A. M. Niknejad, “A 1.5-V 0.7–2.5-GHz CMOS quadrature demodulator for multiband direct-conversion receivers,” IEEE J. Solid-State Circuits, vol. 42, no. 8, pp. 1669–1677, Aug. 2007. [20] D. Manstretta, M. Brandolini, and F. Svelto, “Second-order inter-modulation mechanisms in CMOS downconverters,” IEEE J. Solid-State Circuits, vol. 38, no. 3, pp. 394–406, Mar. 2003. [21] H.-K. Cha, I. Yun, J. Kim, B.-C. So, K. Chun, I. Nam, and K. Lee, “A 32-KB standard CMOS antifuse one-time programmable ROM embedded in a 16-bit microcontroller,” IEEE J. Solid-State Circuits, vol. 41, no. 9, pp. 2115–2124, Sep. 2006. Hyouk-Kyu Cha (S’05–M’09) received the B.S. and Ph.D. degrees in electrical engineering and computer science from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2003 and 2009, respectively. His doctoral dissertation was focused on CMOS RF integrated circuit design for DTV tuner applications. In 2009, he joined the Institute of Microelectronics, Agency for Science, Technology, and Research (A*STAR), Singapore, as a Senior Research Engineer, where he is currently engaged in the development of ultra-low-power and low-voltage CMOS RF/analog integrated circuits (ICs) for implantable/wearable biomedical applications. His research interests include CMOS RF/analog IC/system design for wireless communications.

Kuduck Kwon (S’07) received the B.S. and Ph.D. degrees in electrical engineering and computer science from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2004 and 2009, respectively. His doctoral research concerned DTV tuners and dedicated short-range communications (DSRC) systems. In 2009, he was a Post-Doctoral Researcher with KAIST. In 2010, he joined Samsung Electronics, Gyeonggi-do, Korea, as a Senior Engineer. His research interests include CMOS RF/analog integrated circuits and RF system design for wireless communications.

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Jaeyoung Choi (S’07) received the B.S. and M.S. degrees in electrical engineering and computer science (EECS) from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2007 and 2009, respectively, and is currently working toward the Ph.D. degree in EECS from KAIST. His research interests include CMOS RF/analog integrated circuits and RF system design for wireless communications.

Hong-Teuk Kim received the M.S. degree in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1993, and the Ph.D. degree from Seoul National University, Seoul, Korea, in 2004. In 2002, he joined the LG Electronics Institute of Technology, Seoul, Korea, where, until 2004, he designed high-efficiency and linear power amplifier MMICs for CDMA mobile phones, RF sensor systems for volatile organic compound (VOC) detection, and compact wideband mobile internal antennas. He is currently a Team Leader for DTV CMOS RF integrated circuit (RFIC) development and a Technical Consultant for plasma display panel (PDP) electromagnetic interference (EMI). He is also a member of the LG Technical Expert Council in the RF area. His research has concerned monolithic microwave integrated circuit (MMIC) designs for 6–18-GHz power amplifier, Ka=Q-band phase shifters, 60-GHz active antenna transmitters, and many pioneering designs for millimeter-wave low-loss microelectromechanical systems (MEMS) circuits such as new transmission lines, tunable filters, and phase shifters for 60-GHz beam-forming systems.

Kwyro Lee (M’80–SM’90) received the B.S. degree in electronics engineering from Seoul National University, Seoul, Korea, in 1976, and the M.S. and Ph.D. degrees from the University of Minnesota at Minneapolis–St. Paul, in 1981 and 1983, respectively. Upon graduation, from 1983 to 1986, he was an Engineering General Manager with GoldStar Semiconductor Inc. Korea, where he was responsible for the development of the first polysilicon CMOS products in Korea. In 1987, he joined the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, as an Assistant Professor with the Department of Electrical Engineering. He is currently a Professor with KAIST. Since 1997, he has been the Director of the Micro Information and Communication Remote-object Oriented Systems (MICROS) Research Center, an Engineering Center of Excellence supported by the Korea Science and Engineering Foundation. From 2005 to 2007, he was an Executive Vice President with LG Electronics Inc., where his responsibility was to direct the LG Electronics Institute of Technology as the Executive Vice President. LG Elite is the LG Electronics cooperate-wide central basic research center, which consists of the Device and Material, Information Technology, and Communication Technology Research Laboratories. He has authored or coauthored over 200 publications in major international journals and conferences on semiconductor devices and wireless circuits. He is the principal author of Semiconductor Device Modeling for VLSI (Prentice–Hall, 1993). He is a codeveloper of AIM-SPICE, the world’s first SPICE run under windows. His research has concerned numerous pioneering studies for characterization and modeling of AlGaAs/GaAs heterojunction field-effect transistors. Dr. Lee is a Life Member of the Institute of Electronics Engineers of Korea (IEEK). From 1998 to 2000 and from 2004 to 2005, he was the KAIST Dean of Research Affairs. From 1998 to 2000, he was also chairman of the IEEE Korea Electron Device Chapter, as well as an elected member of the IEEE Electron Device Society (EDS) Administrative Committee (AdCom).

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Waves in a Semiconductor Periodic Layered Resonator Aleksey A. Bulgakov, Associate Member, IEEE, and Oksana V. Shramkova, Senior Member, IEEE

Abstract—A study is made of the TM- and TE-wave propagation in a periodic layered resonator composed of alternating layers of two different semiconductors. The dispersion dependence for a finite medium and field distributions are obtained. The effect of the dissipation processes on the wave dispersion is considered. The influence of dielectric permittivities of uniform half-spaces on the dispersion dependence of the waves is demonstrated. Index Terms—Periodic structures, semiconductor waveguides, surface waves.

I. INTRODUCTION O DATE, great advances have been made in the technology of fabrication of periodic multilayered structures. That is why such heterogeneous dielectric, semiconductor, metallic, oxide, and superconducting structures have been thoroughly studied. The propagation of electromagnetic waves in periodic layered media has been intensively examined in the literature. The properties of frequency bandgaps and transmission band-edge resonances are presented in [1]–[4]. It was shown that increasing the number of the layers in each unit cell of the -layer superlattice formed out of a periodic repetition of different slabs gave rise to a number of the minigaps and surface modes [5]. The problem of electromagnetic wave propagation in a finite layered medium is not only of interest from a theoretical point of view, but also offers many possible applications. It is well known that multilayer waveguides are widely used in the fabrication of different microwave and optical devices, such as modulators, switches, directional couplers, Bragg deflectors, spectrum analyzers, and semiconductor lasers. The results from investigating the finite layered structures should be very useful in the study of wave propagation in solids. The propagation characteristics of the modal spectrum of the planar layered waveguides were examined in [1], [2], [6]–[10]. In the papers by Li and Lit [11]–[13], the concepts of many characteristics of a three-layer waveguide, such as half-phase shifts, effective thickness, group index, and general dispersion properties, were extended to a general multilayer waveguide. The contribution of a

T

Manuscript received January 15, 2009; revised May 18, 2010; accepted May 25, 2010. Date of publication July 15, 2010; date of current version August 13, 2010. This work was supported in part by the National Academy of Sciences (NAS) of Ukraine under Grant N6/07-H. The authors are with the Department of Solid-State Radiophysics, Institute of Radiophysics and Electronics, National Academy of Sciences (NAS) of Ukraine, 61085 Kharkiv, Ukraine (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2052865

periodic stratified structure to the guiding properties of a multilayer planar waveguide has been analyzed in [14]. The numerical method to determine the bound modes of multilayer planar dielectric waveguides was described in [15]. The characteristics of the slab waveguide structure that use a 1-D photonic crystal as a horizontal guiding region is studied in [16]. The coupled-plasmon-resonant waveguide in a 1-D all-evanescent multilayer metal–dielectric nanofilm structure was studied in detail in [17]. In [18], the transmission properties of the periodic dielectric waveguide formed by aligning a sequence of dielectric cylinders in air are investigated theoretically. The 3-D finite-difference time-domain (FDTD) analysis of the transmission and the waveguiding properties of different types of waveguide in photonic crystals with a finite height are presented in [19]. The present-day technology is focused on the formation of objects with dimensions of several micrometers or less. Impressive advances in the technology have been attained in the area of producing the nanostructures with sizes of thousandths and hundredths of a micrometer. It is known that the analog between the wave processes in the periodic structures and properties of wave function of electron moving in a periodic potential of quantum superlattice takes place. The energy spectrum of waves in the infinite periodic media has a zonal structure. In the case of periodic structures with a finite number of periods, the wave vector component perpendicular to the boundaries of layers takes a discrete series of values. In our earlier published works, we investigated the eigenwaves of an infinite layered semiconductor and dielectric media [20]–[23]. In this paper, we look into a finite periodic layered structure with layers exhibiting frequency dispersion. We calculate the dispersion characteristics of eigenwaves and the field distributions. The effect of attenuation on the dispersion properties of electromagnetic waves is considered. II. STATEMENT OF THE PROBLEM AND BASIC EQUATIONS We analyze a finite periodic waveguide consisting of alternating layers of two semiconductors having different dielectric and . We place our ficonstants and different thicknesses nite periodic structure between the uniform media with dielectric permittivities and . We assume that the -axis is parallel to the boundaries of the layers and the –axis is perpendicular to the layers. Suppose that the structure is bounded by the perfect conductive half-planes that are perpendicular to the -axis. Now combine the origin of coordinates with the surface of the periodic structure. The multilayer resonator structure is shown in Fig. 1. The electromagnetic processes in this structure are described by Maxwell equations and by the continuity equations. We seek

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are the lattice-related components of the dielectric constants and , are the Langmuir frequencies, and are the frequencies of collisions between the charge carriers. To take the periodicity of the structure into account, we apply the Floquet theorem. As a result, we get the following dispersion relation: (3)

Fig. 1. Geometry of the problem.

the variables in these equations in the form of , where and are the longitudinal and transversal components of the wave vector with respect to the periodic structure. Since the structure is uniform in the -direction, we can ; in this case, Maxwell equations are split into two set sets of equations for two types of waves with different polarizations. Consider the polarization with the nonzero components , , and (TM polarization). Now we will examine an infinite-in-the -direction periodic semiconductor structure enclosed between two perfect conductive half-planes perpendicular to the layers. Assume that is a distance between these planes. Using the variable separation method and boundary conand , we obtain a set ditions for metal surfaces at of eigenvalues for the longitudinal wavenumber

Here, and is the Bloch wavenumber. Now consider the finite periodic waveguide structure. The uniform half-space with dielectric permittivity is assumed to . The thickness of the periodic medium correspond to ( is the number of waveguide periods). will be corresponds to The half-space with dielectric permittivity . If we consider the periodic waveguide, the transversal wavenumbers for media and must be imaginary values (4) In this case, the fields of propagating waves fall off from the boundaries of the waveguide. To describe the finite periodic structure, we use the Abeles theory [24] and raise the transfer matrix to the th power, where is the number of periods of the waveguide under study. Using the boundary conditions for and tangential components of the electromagnetic field at , we arrive at the dispersion relation

(1) To obtain the dispersion relation, we make use of the method of the transmission matrix (which relates the fields at the beginning of a wave period and at its end) and the boundary conditions at the boundaries between the layers

where the components of the transmission matrix form [20]

take the

(5) Here, are the components of the transmission matrix (2). The Bloch wavenumber is determined from the dispersion relation (3) for the infinite periodic structure. Relation (5) is the dispersion equation for eigenwaves of the semiconductor periodic waveguide that is infinite in the -direction. In this case, we have a continuous set of wavenumbers . . Consider the TE polarization with components As in the case of the TM polarization, eigenvalues (1) correspond to the eigenfunctions of the problem, but in contrast to . the TM polarization, the index can be equal to zero The components of the transmission matrix can be written as

(2) Here bers of layers;

are the transversal wavenum,

(6)

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Fig. 2. Dispersion dependence for the infinite periodical structure (TM polarization, the transmission bands are indicated by hatching).

The dispersion relation for TE modes of the investigated periodic resonator takes the form

(7) It can be shown that dispersion relations (5) and (7) have roots for each value of longitudinal wavenumber (1) in each allowed band of the spectrum. The propagation of surface electromagnetic waves in the finite periodic semiconductor structure occurs, i.e., the wavenumber , for these waves is complex and their dispersion curves lie in the forbidden bands. The energy of such oscillations is localized near the edges of the periodic medium. III. NUMERICAL RESULTS Fig. 2 shows the band structure of the TM-eigenwave spectrum for the infinite (in the - and -direction) periodical structure in the region around the plasma frequencies. The calculation was carried out for the lattice consisting of an InSb semicm, conductor as the first layer inside the unit cell ( , s , ) and a GaAs semicm, , conductor as the second slab ( s , ). One can see three hatched bands where eigenwaves can propagate [20]. The two lower bands, the so-called “acoustic” and “optical” zones for plasma polaritons, asymptotically tend to the line , are the frequencies where of surface plasmons propagating at the interface between the semiconductor layers. The upper band is the one for “dielectric” modes. This zone asymptotically tends to the light line for the . The group velocity of the “optical” dielectric layer modes is negative.

+

Fig. 3. Dispersion dependence for the periodic layered waveguide (“ ”) and resonator (“”), TM polarization.

In Fig. 3, the dispersion curves for the periodic layered waveguide that is finite in the -direction and infinite in the -direction are denoted by the symbols “ .” The symbols “ ” indicates the dispersion dependence for the periodic layered resonator shown in Fig. 1. The bold lines are the boundaries of the allowed zones. The calculations were performed for five periods , (i.e., the uniform media are vacuum), , cm. The oblique solid line is the light line for vacuum . According to (1), the eigenwave spectrum of the periodical semiconductor resonator represents the points for each value of longitudinal set of points. We get wavenumber (1) in each allowed band of the spectrum. Note that the total number of dispersion relation’s roots depends on the distance between two perfect conductive planes. Some of these points correspond to the dispersion dependence for the surface waves. The generation of the surface waves and occurs close to the boundaries of “acoustic” allowed band, therefore the penetration depth of these waves is high. The penetration depth of the surface wave is lower. The phase velocity of this wave tends to zero with an increase of . It is interesting to note that, in the case of surface waves and , the waves in the layers are ; for the surface wave , the wave of surface type . This wave may be in the layers is of the bulk type of interest in terms of interaction with active waves (e.g., with drift waves) because the phase velocity of the wave may be far less than the velocity of light. Now consider the effect of the collision frequencies on the wave dispersion properties and solve dispersion relation (5) (we consider the case of a periodic layered waveguide that is finite in the -direction and infinite in the -direction) numerically for , assuming that real frequencies lying close to the frequency . In Fig. 4, the dependence of on the real part of the wavenumber is denoted by symbols “ ” and the dependence is shown by symbols “ ”. The calculations were of on s , . The spatial damping carried out for of waves with the negative group velocity is seen to be negative:

BULGAKOV AND SHRAMKOVA: WAVES IN SEMICONDUCTOR PERIODIC LAYERED RESONATOR

Fig. 4. Dispersion dependence with allowance for dissipation: ! (k ) (“”) and !(k ) (“+”).

Fig. 5. Dispersion dependence with allowance for dissipation:  and  = 5 1 10 s ,  = 0 (“+”).

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Fig. 6. Dispersion dependencies for curve c in Fig. 3 (1: 2: " = 1, " = 12; 3: " = " = 12; 4: " = " = 25).

"

=

"

= 1;

= 0 (“”)

the corresponding eigenwave is damped in the energy propagation direction. These plots differ from those for a nondissipative takes on a medium in that they have a turning point where maximum value; in this case, the phase velocity has a minimum . Interest[22], [23]. The turning point corresponds to ingly, the dispersion curves for the polaritons in the “acoustic” branch goes over to the curves for the waves in the “optical” branch. ( is related to dissipation). Assume that Fig. 5 shows the dispersion curves with allowance for collision frequency for the case of the periodic layered waveguide. The curves marked by symbols “ ” indicate the frequency versus . The curves marked by symbols “ ” the wavenumber at correspond to the dependence at s ,

Fig. 7. Distribution of resonator.

H

component of the field for the five-period

. For these curves, the imaginary part of the frequency is s . The dispersion dependence for the resonator with dissipative layers is the set of points of the dispersion curves for the periodic waveguide for the specified values of longitudinal wavevector (1). It is evident that the dispersion of waves is dependent upon the dielectric permittivities of the uniform media. It was shown that only curve (Fig. 3) becomes deformed with a variation

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odical structure. The field distribution at point F of the spectrum in Fig. 8 is given in Fig. 9. This point satisfies the condition of the dimensional resonance, under which the layer thickness is equal to an integer number of half-waves. The wave propagates and are in the semiconductor layers as in a waveguide ( real). IV. CONCLUSION

Fig. 8. Dispersion dependence for TE waves in the finite and infinite (inset) periodic structures.

The dispersion of waves of two polarizations in the semiconductor periodic layered resonator has been studied theoretically. We have found that for a stratified waveguide, we obtain a set of dispersion curves whose number is defined by that of structure periods. For a periodic layered resonator bounded by two perfect conductive planes, we get a set of points on the dispersion curves for the waveguide corresponding to the eigenvalues for the longitudinal wavenumber. Dispersion characteristics are calculated with regard to collisions in the semiconductor layers and changes in dielectric permittivities of the uniform half-spaces. Dissipation is shown to limit the phase velocity by a certain minimum value, which is collision frequency dependent. The field distributions for the stratified resonator are examined. From a practical standpoint, the investigations reported herein are essential in terms of using stratified semiconductor structures for development of different nanoelectronic devices. These structures can be used to develop multilayer devices for data transfer in computers. REFERENCES

Fig. 9. Distribution of real part of E

component.

in and . The dispersion dependencies of the eigenwaves of the periodical semiconductor waveguide for different dielectric permittivities of uniform half-spaces are shown in Fig. 6. The steepness of the dispersion curve increases with the dielectric permittivities and . This feature is attributed to the change in the penetration depth of this surface wave. , where is the Let us call the modes of resonator . Fig. 7 shows the field number of modes -field comdistribution of a real and an imaginary part of the wave (parameters ponent in the layered resonator for the of the wave correspond to the point F in Fig. 3). Note that the and are imagiwave in the layers is of the surface type ( nary). The propagating wave field decreases exponentially with a distance from the boundaries of layers. Consider the TE polarization. Fig. 8 shows the dispersion dependence for TE waves in the finite semiconductor periodic structure (parameters of the structure are the same as in Fig. 2). The inset in Fig. 8 points to the band structure of the TE-eigenwaves spectrum for the infinite (in the - and -direction) peri-

[1] A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation. New York: Wiley, 1984. [2] P. Yeh, Optical Waves in Layered Media. New York: Wiley, 1988. [3] A. Figotin and I. Vitebskiy, “Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 72, no. 3, 2005, Art. ID 036619. [4] A. A. Chabanov, “Strongly resonant transmission of electromagnetic radiation in periodic anisotropic layered media,” Phys. Rev. A, Gen. Phys., vol. 77, no. 3, 2008, Art. ID 033811. [5] E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, and L. Dobrzynski, “Theory of surface and interface transverse elastic waves in N -layer superlattices,” Phys. Rev. B, Condens. Matter, vol. 54, no. 20, pp. 14 728–14 741, 1996. [6] R. Wu, T. Zhao, P. Chen, J. Xu, and X. Ji, “Periodic layered waveguide with negative index of refraction,” Appl. Phys. Lett., vol. 90, no. 8, pp. 082 506–082 509, 2007. [7] F. Mesa and M. Horno, “Computation of proper and improper modes in multilayered bianisotropic waveguides,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 1, pp. 233–235, Jan. 1995. [8] R. Rodriguez-Berral, F. Mesa, and F. Medina, “Appropriate formulation of the characteristic equation for open nonreciprocal layered waveguides with different upper and lower half-spaces,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1613–1623, May 2005. [9] C. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Exp., vol. 7, no. 8, pp. 260–272, Oct. 2000. [10] K. Halterman, J. M. Elson, and P. L. Overfelt, “Characteristics of bound modes in coupled dielectric waveguides containing negative index media,” Opt. Exp., vol. 11, no. 6, pp. 521–529, Mar. 2003. [11] Y.-F. Li and J. W. Y. Lit, “Contribution of low-index layers to mode number in multilayer slab waveguides,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 4, no. 12, pp. 2233–2239, 1987. [12] Y.-F. Li and J. W. Y. Lit, “Effective thickness, group velocity, power flow, and stored energy in a multilayer dielectric planar waveguide,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 7, no. 4, pp. 617–635, 1990. [13] Y.-F. Li and J. W. Y. Lit, “Generalized dispersion properties of a multilayer dielectric planar waveguide,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 9, no. 1, pp. 121–131, 1992.

BULGAKOV AND SHRAMKOVA: WAVES IN SEMICONDUCTOR PERIODIC LAYERED RESONATOR

[14] Y.-F. Li, K. Iizuka, and J. W. Y. Lit, “Periodic stratified structure in a multilayer planar optical waveguide,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 9, no. 4, pp. 559–568, 1992. [15] L. Li, “Determination of bound modes of multilayer planar waveguides by integration of an initial-value problem,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 11, no. 3, pp. 984–991, 1994. [16] H. Taniyama, M. Notomi, and Y. Yoshikuni, “Propagation characteristics of one-dimensional photonic crystal slab waveguides and radiation loss,” Phys. Rev. B, Gen. Phys., vol. 71, no. 15, 2005, Art. ID 153103. [17] S. Feng, J. M. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Exp., vol. 13, no. 11, pp. 4113–4124, May 2005. [18] P.-G. Luan and K.-D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Exp., vol. 14, no. 8, pp. 3263–3272, Apr. 2006. [19] M. Kafesaki, M. Agio, and C. M. Soukoulis, “Waveguides in finiteheight two-dimensional photonic crystals,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 19, no. 9, pp. 2232–2240, 2002. [20] F. G. Bass and A. A. Bulgakov, Kinetic and Electrodynamic Phenomena in Classical and Quantum Semiconductor Superlattices. New York: Nova Sci., 1997. [21] A. A. Bulgakov and O. V. Shramkova, “Dispersion and instability of electromagnetic waves in layered periodic semiconductor structure,” Tech. Phys., vol. 48, no. 3, pp. 361–367, 2003. [22] A. A. Bulgakov and O. V. Shramkova, “Slow plasma waves in a finitestratified periodic semiconductor structure,” Plasma Phys. Rep., vol. 31, no. 10, pp. 818–823, 2005. [23] O. V. Shramkova, “Attenuation of electromagnetic waves in a semiconductor superlattice in a magnetic field,” Tech. Phys., vol. 49, no. 2, pp. 232–237, 2004. [24] M. Born and E. Wolf, Principles of Optics. Oxford, U.K.: Pergamon, 1965.

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Aleksey A. Bulgakov (A’09) was born in Kharkov, Ukraine. He received the M.S. degree in radiophysics and Ph.D. (with a speciality in radiophysics), and D.Sc. (with a speciality in physics of semiconductors and dielectrics) degrees from Kharkov State University, Kharkov, Ukraine, in 1960, 1971, and 1988, respectively. Since 1965, he has been a Member of the Research Staff with the Institute of Radiophysics and Electronics, National Academy of Sciences (IRE NASU), Kharkov, Ukraine. From 1989 to 2003, he was a Leading Researcher with the IRE NASU. Since 2003, he has been a Senior Scientist. He has authored or coauthored three books and over 200 journal and conference papers. His research interests include nonlinear phenomena in solid-state structures, electromagnetic and acoustic waves propagation in solid-state, instabilities in semiconductors, physical properties of superlattices, and mathematical simulation.

Oksana V. Shramkova (M’07–SM’08) was born in Kharkov, Ukraine. She received the M.S. degree in solid-state physics from the National Technical University “Kharkov Polytechnic Institute” (NTU-KhPI), Kharkov, Ukraine, in 1998, and the Ph.D. degree in radiophysics from the Institute of Radiophysics and Electronics, National Academy of Sciences (IRE NASU), Kharkov, Ukraine, in 2001. Since 2001, she has been a Member of the Research Staff with the IRE NASU. She is currently a Senior Scientist with the Department of Solid-State Radiophysics, IRE NASU. She has authored or coauthored over 70 journal and conference papers. Her research interests include electromagnetic waves in layered and periodic media, waveguides, instabilities in semiconductors, and nonlinear phenomena. Dr. Shramkova was the recipient of the 2008 Prize of the President of Ukraine (top prize for young researchers).

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Fields at a Finite Conducting Wedge and Applications in Interconnect Modeling Thomas Demeester and Daniël De Zutter, Fellow, IEEE

Abstract—The fields at a finite conducting 2-D wedge are studied by means of the surface admittance operator, and compared to the case of a perfect conductor. This technique, applied to a number of numerical examples, allows a thorough investigation of the singular behavior of the fields near the edge, including nonsingular fields such as the longitudinal current distribution. Special attention is devoted to the validity of the quasi-TM approximations, when edge singularities are taken into account. The studied field properties lead to the formulation of an approximative local surface impedance for conductors, and are finally used to show how some differences in the resistive and inductive behavior of conductors with a different geometry are due to edge effects. Index Terms—Conductor, edge effect, finite conductivity, inductance, resistance, skin effect, surface impedance, transmission line parameters, wedge.

I. INTRODUCTION OR MANY years, researchers have been looking for accurate descriptions of the loss mechanisms in interconnect structures. As modern technological applications in very large scale integration (VLSI) circuits push the limits of speed and miniaturization, conductor losses more than ever remain an important issue, by far more relevant than radiation or dielectric losses. Not only heat generation needs to be kept under control, the losses also have an important impact on the signal integrity due to attenuation and dispersion. In a period of almost a hundred years, many authors have paid attention to the topic discussed in this paper, and in general, to the skin effect and losses in rectangular conductors [1]–[19]. We thank the reviewer for bringing some of these papers to our attention. The earliest research on interconnect losses is well summarized in [11], mentioning, for instance, Wheeler’s incremental inductance rule, where the magnetic field generated by the axial current flow is used to calculate the losses, assuming an equal real and imaginary part of the high-frequency internal impedance per unit length (p.u.l.). The resistive properties of coupled lines with finite conductivity were more rigorously studied in [16] and [17], using the method of moments (MoM) with a boundary discretization, respectively, a volume discretization of the field quantities. Many other numerical

F

Manuscript received July 09, 2009; revised March 02, 2010; accepted March 02, 2010. Date of publication July 19, 2010; date of current version August 13, 2010. The authors are with the Department of Information Technology, Ghent University, B-9000 Gent, Belgium (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2053061

approaches were used for analogous purposes, e.g., finite-elements methods [12], hybrid methods based on the “filament technique” at low frequencies, and a surface integral equation at high frequencies [19], or a combination of the MoM and the method of lines [20], just to mention a few. Generally, the boundary integral-equation techniques appear to be more suited in terms of both computation time efficiency and accuracy than methods based on a volume discretization of the currents, especially at the highest frequencies. The importance of the edge effect in the current profile of polygonal (mostly rectangular) conductors became clear with the (sometimes mutually inconsistent) results that were found from internal inductance calculations [21]–[24], and which clearly showed an important deviation from Wheeler’s rule. In parallel with the research on the effect of the finite conductivity on the circuit level properties (resistance, inductance) of the lines, another topic of investigation was the singular field behavior at edges. In [25], and further in [26], the cases of perfectly electric conducting (PEC) wedges and wedges with a dielectric contrast were treated. A more detailed analysis and further references can be found in [27]. The theory of the singularity exponent, as formulated in [26], was extended to finite conducting wedges in [28]. Although the specific field behavior at conductors’ edges (both in the PEC and in the finite conducting case) and the current profile (relevant to the resistive and inductive properties of the lines) are intrinsically linked, both aspects were not examined simultaneously thus far. On the one hand, “circuit oriented” papers such as, for example, [15], [18], [21], concentrate on the interconnect behavior, with no specific attention devoted to edge effects and their influence on the circuit parameters. On the other hand, [26] and [28] focus on the edge singularities only, not paying particular attention to the properties of the longitudinal field components, such as the current density, as these do not exhibit a singular behavior at the edges. This paper describes the behavior of a finite conducting wedge, as a function of its opening angle , in combination with the longitudinal current profile. As opposed to the singularity exponent technique of [26] and [28], the applied method enables the description of the total edge field quantities, not restricted to the strongest singularity only. Although the technique is a numerical approximation obtained by the MoM, it is well suited for an accurate description of the fields near an edge from low to very high frequencies, as it makes use of a boundary integral-equation formulation, in combination with a field expansion that exactly describes the current crowding phenomenon inside the conductor. Essential in this technique is the surface admittance matrix, which relates the electric field to the equivalent surface current

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DEMEESTER AND DE ZUTTER: FIELDS AT FINITE CONDUCTING WEDGE AND APPLICATIONS IN INTERCONNECT MODELING

densities that replace the conductors. The surface admittance matrix is calculated by means of a discretization of the Dirichlet to Neumann (DtN) operator of the considered conductor’s cross section. The method was first introduced for conductors in [29] and extended to dielectrics and semiconductors in [30]. In [31] and [32], the original method for rectangular conductors was extended to general polygonal shapes. In Section II, a single wedge is considered. First, the relationship between Meixner’s [26] work on field singularities and the quasi-TM approximations underlying the coupled transmission line model presented in [30] is elucidated. Next, the equivalent , as introduced in [29], and the cursurface current density rent profile inside the wedge are studied. Subsequently, an approximative local surface impedance model for conductors is proposed, intended to demonstrate the principle physical properties of . In Section II-E, the properties of the wedge current profile are used to reveal the influence of varying conductor angles on the p.u.l. resistance and inductance for conductors with a high, but finite, conductivity. Finally, Section III summarizes the results.

II. INVESTIGATION OF THE EDGE EFFECT This section is intended to provide the reader with some insight into the field distribution and essential phenomena that occur near edges. Focus is on the physics of the current and field distribution near a single edge, and the validity of the quasi-TM approximations (which is confirmed by numerical results). The considered configuration consists of one triangular metallic nonmagnetic conductor with conductivity , placed in free space. It is assumed that no other materials (dielectric or semiconducting substrates) are around, in order to study the edge effect in its most basic configuration, although the results are valid for more complicated structures as well. In order to avoid the proximity effect in the current distribution, the reference conductor is considered infinitely far away. All simulation results shown here are obtained by using the numerical method described in [31]. For the discretization of the boundary quantities, piecewise linear basis functions are used over a nonuniform grid. In this way, a very fine grid can be used near the corner tips. When focusing on the edge effect only, the simulation frequency will be chosen sufficiently high, such that the influence of the side, opposite to the corner of interest, is negligible. In practice, this means the skin depth has to be much smaller than the distance between that corner and its opposite side. A. Equivalent Surface Current Density Consider the triangle shown in Fig. 1 with an opening angle at corner , with the permeability of free space, and a “high” ). As explained above, we are conductivity (such that only interested in the edge effect in the neighborhood of . As mentioned in Section I, the volume current flowing through is replaced by an equivalent surface current density source in free space on the boundary of . This equivalent source is

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Fig. 1. Basic configuration of a conducting wedge (as part of a triangle) placed in free space.

found from the requirement that outside , and it generates the original fields [29]

(1) being the outward pointing normal unit vector on . The with superscripts and are used for the outer limit, respectively, the inner limit of the fields at , and the subscript “0” indicates field quantities inside in the equivalent configuration in which the material properties of are replaced by free space. To obtain the last part of (1), the continuity of the tangential magnetic field in terms of boundary was invoked, which allows to write quantities inside in the original, respectively, the equivalent configuration. The quasi-TM approximations [30] dictate that for the deter(to determine the resistive and inductive propmination of of the total equivalent erties), the longitudinal component current density suffices (as briefly motivated in the Appendix). dependence of the fields, the transverse magFor an is found from Faraday’s law netic field (2) in which written as

. Hence,

, concisely

, is given by (3)

with and being the outward normal derivative. In [26], Meixner presents an expansion of the fields near the edge in order to investigate the field singularities. He shows that and do not display a singularity the longitudinal fields at the edge, and if there is no magnetic contrast (as is the case field, for here), the total magnetic field remains finite. The example, can be expanded in polar coordinates and as (4) omitting higher order terms in , and with (5)

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with being a complex integration constant. The transverse tangential and normal electric field components exhibit a singular . In [28], it is shown that, in the case term that behaves as of a good conductor, can be well approximated by

(6) which exactly dictates the singular behavior of in the case of a PEC conductor. If the field expansions from [26] are inserted in (3), the sinand cancel each other out, as is gular terms in and . Furthermore, these exalso the case for and pansions allow to compare the singular terms in . Inside the wedge, satisfies (7) with tuted in (7) and yields

. The expansion (4) of

is substiFig. 2. Surface current density jj j for a 50 copper wedge (see inset). Solid = 57:2 MS/m) with indication of the lines: jj j for finite conductivity ( j for PEC wedge. distance  from the corner top. Dashed line: jj

(8) to zero, confirms the behavior Setting the coefficient of in (5), which is independent of the material parameters. of in . In the The same remark holds for the term in , quasi-TM limit, the right-hand side of (7) becomes and as seen from (8), this diffusion term is not relevant very close to the corner tip. A completely similar reasoning can be , which satisfies put forward for (9) . By construction, has the same boundary with by in (7) has no value on as , and because replacing influence on the singular behavior, the two highest order terms are identical. As a consequence, of the expansions of and is cancelled out, as well as the the singularity in . first higher order term The above reasoning shows that all four field components in the right-hand side of (3) contain the same singular term. Thus, . Let leaving these singular terms out has no influence on us indicate the fields in (3) without their singular term with the caret symbol “ .” Within the quasi-TM limit, the terms and are both negligible with respect to (condue to the cursidering the fact that rent crowding effect). The reason for this is twofold. On the one hand, the longitudinal wavelength is much larger than a typical cross-sectional distance over which the fields extend, and there” for a certain field fore we can, in general, say that “ . Taking these two arquantity. On the other hand, guments into account leads to (10) (11)

This proves that in the quasi-TM approximation expression (11), already put forward in [29] for the -independent TM case, still remains valid when the singular field behavior at an edge is accounted for, as not only the singular terms in in (11) cancel out, but also the nonsingular terms in and a will become zero at the tip , as will possible constant term be confirmed by the numerical examples. for a finite conductor will be compared to In the sequel, on a PEC wedge. This is motithe surface current density vated by the following observation. As the fields inside the PEC conductor are zero, the inside can be substituted by free space, provided proper surface charges and surface currents are placed on the boundary. If these sources are equal to the original surface charge and current on the PEC conductor, the fields in both configurations are the same and the boundary conditions are is, hence, the “equivalent current source” for the PEC met. problem and is, in this paper, compared to the finite conducting can be obtained by solving a static potential problem case. with the longitudinal magnetic vector potential on . This result is obtained from the general relationship (12) , , and on the combined with boundary of the considered perfect conductor in free space. near a In Fig. 2, the equivalent surface current density copper wedge of 50 is compared with the PEC case for various frequencies. In the simulation, the wedge was the top corner of an equilateral triangle (with both legs 60- m long, such that the edge effects of the different corners do not interfere at the shown frequencies), with the electric boundary potential put to V. To get an idea of the frequency relative to the dimensions, the point where the distance to the corner tip equals one skin is indicated as well. Notice that, as explained earlier, depth

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behavior for a 90 angle. For each value of , the proportionality becomes one in constant is chosen such that . the limit for B. Electric Boundary Potential This paragraph is intended to demonstrate the validity of the assumption that, in the quasi-TM limit, the electric potential on the boundary of a good conductor remains constant over its edges. With Meixner, the potential can be written as (13) The expansions in [26] only describe the singular behavior of the fields (or their normal derivatives), and here, (13) is completed with a constant term , as motivated in [33]. The coefficient is found as (14)

Fig. 3. Normalized surface current densities for different angles  and at and (b) j =Cr with C chosen such that 10 GHz. (a) jj j=j . j =Cr becomes one at r

=0

vanishes at the tip, and therefore, does not have as its because is always singular limiting case for (for an wedge angle smaller than 180 ). It is clear though that and more than a few skin depths away from the edge, become almost identical. In Fig. 3, a similar wedge is treated as in Fig. 2, but at a fixed frequency (10 GHz) and for a varying top angle . In order with respect to for difto investigate the behavior of ferent values of , Fig. 3(a) shows the normalized current dennear the corner tip. As the solution of the diffusity instead of both sion equation only depends on the product factors separately, the abscis is normalized by the skin depth . The deviation of the curves in copper in Fig. 3(a) from unity shows the influence of the finite conductivity. The equivalent current density at the sharpest angles than at the wider angles. This can be deviates more from explained by the diffusion at skin-effect frequencies. Near the edge of a narrow wedge, the adjacent sides are more tightly coupled for a wider wedge, and the current crowding effect starts appearing further away from the tip as compared to the wide wedge case. Therefore, the edge effect is more important for narrower wedges. In [31], this phenomenon appears to be the reason for the slower convergence of the iterative combined waveguide modes (ICWM) procedure for sharper angles, where the coupling between the sides of the conductor is gradually taken into account. , Fig. 3(b) As a verification of the singular behavior of for the same wedge, normalized by a factor shows with given by (6). For , the singularity exponent lays between 0 and 1/2, with, for example, an

and the term has a singular normal derivative at . From (5) and (12), we see that neglecting this term with respect to the total potential corresponds to neglecting the term in . This means that an excitation with a constant does not give rise to a singularity in boundary value . This approximation is acceptable within the quasi-TM limit, as is briefly discussed in the Appendix. We will demonstrate with a numerical example how accurate this approximation really is by comparing the approximative with the term constant voltage excitation on the boundary of a wedge (see Fig. 1). Near the dominates edge, where the singular term proportional to (see Appendix), is, with the results the surface charge from [26], given by (15) such that (16) Note that is a very good approximation of the actual surface charge even though calculated with the approximative excitation , as it is very similar to the PEC case where (see itself is proportional to , and the Appendix). Note that tends to zero as . hence, that For the copper 50 wedge from Fig. 2 and at the same freis compared with the voltage , quencies, the term the constant excitation voltage used in the quasi-TM simulations. Fig. 4 shows the results, and it appears that even at the remains many orders of maghighest shown frequencies, m nitude smaller than , for the shown region where the approximation (16) can be assumed to be valid. C. Electric Field Distribution Inside the Wedge The longitudinal electric field on a conductor’s boundary field is can be found with the MoM [30], and the inside readily determined as well by means of an expansion in terms of parallel-plate waveguide modes, as described in [31]. In Fig. 5, the normalized longitudinal electric field distribution

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1

Fig. 4. Term  in the boundary potential expansion near the edge, in absolute value and normalized by  for the 50 wedge shown in Fig. 2.

Fig. 6. (a) Normalized electric field je =j  j on the boundary of a copper wedge  : MS/m for different angles  and at 10 GHz. (b) Phase of V). the corresponding electric field e (for 

(

Fig. 5. Normalized electric field : MS/m with an angle 

57 2

)

j

e =j 

= 50

j inside a copper wedge shown for r 2 ; m .

[0 15 ]

= 57 2

)

=1

( =

is shown inside the 50 copper wedge of Fig. 2 and for a radial m at 10 MHz, 100 MHz, 1 GHz, and 10 GHz. length of For these frequencies, the skin depth is given, respectively, by 21, 6.6, 2.1, and 0.66 m. In order to visualize the influence of the wedge angle on the current distribution, Fig. 6(a) again shows the normalized elec, and Fig. 6(b) shows the phase of on . tric field The same geometries are used as for the simulations shown in toward the edge is much more proFig. 3. The increase of nounced for the sharpest angles, and almost nonexisting for the for the sharp corners obtuse angle of 135 . The phase displays a large deviation of the plane-wave limit of 45 over a distance of many skin depths away from the tip. As a verification of the numerical results, the boundary value of on a rectangular conductor is compared with results found in [17]. The simulated configuration consists of a golden microstrip line above a ground plane. Fig. 7 shows the geometry (see inset), as well as the results calculated by means of the

Fig. 7. Normalized longitudinal electric field je =j j on the boundary of the m, t m, h m, and microstrip, shown in the inset with w MS/m. The simulations were performed for different frequencies  : GHz. with f

= 41 = 4 367

= 10

=2

=2

MoM in combination with the DtN operator in solid lines at different frequencies. At the frequency , the reference data from [17] are indicated as well. Note that in [17], a golden ground plane was used, whereas here just a PEC ground is considered. Yet the results seem to match quite accurately. In [17], the fields were considered to be independent, and excited by means of , which corresponds with our term the external field in (12). In Fig. 7, the normalized field is given. At , such that the current disthe lowest frequencies, . At skin-effect tribution is almost uniform and

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frequencies, decreases because the difference between and goes to zero for , as the fields approach the . PEC field distribution with D. Local Surface Impedance Approximation and near a wedge, we have a sufficient After studying understanding on the wedge effects as to propose the following on the boundary of an arbitrary 2-D approximation for conductor in free space (hence, not restricted to the wedge only): (17) The approximate surface current density contains the actual boundary value of , multiplied with the normal derivative of a function , which satisfies (18) itself), but is 1 on . The idea beinside the conductor (as hind this approximation is the separation of the two major phenomena that play a role. On the one hand, we have the value of on , increasing toward the corners as governed by the (outside) magnetic field, and on the other hand, the current crowding phenomenon inside the conductor, which would behave like if the boundary field were a constant. At the higher frequencies and far enough away from the corners, the local plane-wave behavior confirms that the approximation becomes is approximately conaccurate. At the lower frequencies, stant over the cross section, and the approximation holds as well. With (17), we have, therefore, constructed a local surface as follows: impedance (19)

Fig. 8. Real and imaginary part of j and j at: (top) 1 MHz, (middle) 100 MHz, and (bottom) 10 GHz for a rectangular copper conductor in MS/m, width m). m, height free space (

= 58

= 200

= 30

angular reference conductor with the same cross-sectional area despite their longer circumference. In a last numerical example, the field distribution for these situations is shown in direct relationship with the corresponding circuit properties. The starting point is the following telegrapher equation for a single line: (21)

with the correct low- and high-frequency limits. The approximation is acceptable because the total current is, for any frequency, found as (20) This can be proven by invoking Green’s theorem in combination with (17) and (18) and the same diffusion equation for . As a numerical verification, consider a rectangular copper MS/m with dimensions 200 m 30 m conductor in free space. At 1 MHz, 100 MHz, and 10 GHz, the real and and are shown in Fig. 8. For imaginary part of both these frequencies, the skin depth in copper is, respectively, 66, and 6.6, and 0.66 m. At the low and high frequencies, are very close to one another as expected, but also at the intermediate frequencies the behavior is quite similar, confirming the physical ideas behind the equivalent surface current density. E. Influence of the Edge Effect on the p.u.l. Resistance and Inductance In [31], it was found that the high-frequency resistance for trapezoidal or triangular conductors is higher than for a rect-

on the conductor’s cross section, or its normal Studying , not derivative at the boundary, only yields information on directly on the resistance and inductance of the line. Therefore, we will transform (21) into (22) with the excitation voltage chosen real and positive, but such A (or, alternatively, ). The integrand that of the tangential magnetic equals the complex conjugate could field in the quasi-TM limit. The contribution of in the integrand (as it does not be included as well, to obtain contribute to the integration), but this would make the graphical results less transparent. The first considered configuration consists of a rectangular m and height m golden conductor with width placed above a PEC ground plane with a separation of m. This is the configuration used in Fig. 7 and operated at 10 GHz. Secondly, a symmetric trapezoidal golden conductor is considered, also placed above a PEC ground plane, with a m and a top width m, and bottom width hence, the same area as its rectangular counterpart.

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Fig. 9. Resistance and inductance p.u.l. for a golden conductor (solid lines: m above a rectangular; dashed lines: trapezoidal) placed a distance h m, t m, B m, PEC ground plane g . The dimensions are w m. and b

= 10

=8

=2

=2

= 12

previous investigations at wedges. For the bottom sides, there is a larger deviation, primarily due to the proximity effect as well. As the bottom side of the trapezoidal conductor is larger than the corresponding side for the rectangle, and given the fact that the total current flowing through both conductors remains fixed at 1 A and is primarily located in the lower part of the conductor, due to the proximity effect, the average bottom field away from the corners is smaller in the trapezoidal case, as compared to the rectangular one. This translates into value. The authors have verified that the current a lower density distribution shows a similar behavior. However, near the sharp 45 angles, the phase shift in the current density [see also Fig. 6(b)], and therefore, also in the tangential magnetic is no field, has the effect that the resistive term longer smaller than at the 90 angles of the rectangle, but even slightly higher, resulting in an overall higher p.u.l. resistance at skin-effect frequencies. III. CONCLUSION The quasi-TM techniques for modeling the resistive and inductive properties of conductors remain valid near conductor edges. The specific field behavior of a finite conducting wedge was investigated and illustrated with a number of numerical examples to clarify the underlying physical mechanisms, which led to the formulation of an approximative local surface impedance description of conductors. Finally, the studied phenomena were used to explain some differences in the -behavior of microstrips with different geometries. APPENDIX This appendix deals with some issues related to the quasi-TM approximations. In the frequency range where they are valid, . The reason is that with (1) (23)

~

Fig. 10. Comparison of j V h on the boundary of the rectangular versus the trapezoidal conductor (each above a PEC ground plane) used in Fig. 9.

Fig. 9 shows the resistance and the inductance of both configurations. The inductance of the trapezoidal conductor is considerably lower than that of the rectangle, and the resistance displays the opposite behavior. and are According to (22), the circuit parameters and obtained by integration of the quantities shown in, respectively, Fig. 10(a) and (b). The between the bottom and top side main difference in is governed by the proximity effect. There is a very close resemblance on the top side of both conductors with a considerably lower tangential magnetic field than on the bottom side. It means that the 135 angles of the trapezoidal conductor have very little influence on the field distribution, confirming

(with the subscript denoting the transverse part of the total tangential field component), as has a static contribution, is only of second order in . However, to keep whereas cannot be considered the quasi-TM equations consistent, zero, as it is relevant for the capacitive behavior. Indeed, the , associated with the total equivalent current surface charge density, is found as (24) The surface charge, associated with the normal electric field, is singular at the edges, as is , while rein the direction mains finite. Moreover, the variation of is small, compared to variations in the cross section. Therefore, is essential in (24). In practice, the influence the term in is well taken care of if a surface charge source is of directly imposed on the surface, together with the current source . This charge distribution can be determined by solving a quasi-static potential problem replacing the actual conductor with a PEC one, as motivated in [30]. This approach remains has valid at the edges as well because the singular term in

DEMEESTER AND DE ZUTTER: FIELDS AT FINITE CONDUCTING WEDGE AND APPLICATIONS IN INTERCONNECT MODELING

an behavior, and for a good conductor, approximately equals its value for a PEC conductor [28]. corresponds As shown earlier, the approximation to omitting the term in . That (finite) term keeps up for the “slow” longitudinal variation of the (singular) transverse current because the curl of the total electric field has to remain is associated with the transverse electric finite. The field field (more specifically, its transverse rotation), and similarly, with the part of without the term the much larger (by means of its normal derivative). Within the quasi-TM limit , (23), we can therefore say that near the edge and hence, the approximation is sufficiently accurate (confirmed also in Fig. 3). REFERENCES [1] A. Press, “Resistance and reactance of massed rectangular conductors,” Phys. Rev., vol. 8, no. 4, pp. 417–422, Oct. 1916. [2] H. Schwenkhagen, “Untersuchungen über Stromverdrängung in rechteckigen Leitern,” Arch. Elektrotech., vol. 17, no. 6, pp. 537–589, 1927. [3] D. Cockcroft, “Skin effect in rectangular conductors at high frequencies,” Proc. Roy. Soc., vol. 122, no. 790, pp. 533–542, Feb. 1929. [4] S. J. Haefner, “Alternating-current resistance of rectangular conductors,” Proc. IRE, vol. 25, no. 4, pp. 434–447, Apr. 1937. [5] H. G. Groß, “Die Berechnung der Stromverteilung in zylindrischen Leitern mit rechteckigem und elliptischem Querschnitt,” Arch. Elektrotech., vol. 34, no. 5, pp. 241–268, May 1940. [6] F. Lettowsky, “Eine Methode zur Berechnung des Hochfrequenzwiderstandes zylindrischer Leiter allgemeiner Querschnittsform,” Arch. Elektrotech., vol. 41, no. 1, pp. 64–72, Jan. 1953. [7] P. Silvester, “Modal network theory of skin effect in flat conductors,” Proc. IEEE, vol. 54, no. 9, pp. 1147–1151, Sep. 1966. [8] P. Silvester, “The accurate calculation of skin effect in conductors of complicated shape,” IEEE Trans. Power App. Syst., vol. PAS-87, no. 3, pp. 735–742, Mar. 1968. [9] L. Pouplier, “Berechnung des komplexen Wechselstromwiderstandes von zylindrischen Leitern mit rechteckigem Querschnitt,” ETZ-A, vol. 89, no. 22, pp. 611–617, 1968. [10] P. Hammond and J. Penman, “Calculation of eddy currents by dual energy methods,” Proc. Inst. Elect. Eng., vol. 125, pp. 701–708, 1978. [11] E. Denlinger, “Losses of microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-28, no. 6, pp. 513–522, Jun. 1980. [12] L. Olson, “Application of the finite element method to determine the electrical resistance, inductance, capacitance parameters for the circuit package environment,” IEEE Trans. Compon., Hybrids, Manuf. Technol., vol. CHMT-5, no. 4, pp. 486–492, Dec. 1982. [13] M. Krakowski and H. Morawska, “Skin effect and eddy currents in a thin tape,” Arch. Elektrotech., vol. 66, no. 2, pp. 95–98, Mar. 1983. [14] P. Waldow and I. Wolff, “The skin-effect at high frequencies,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 10, pp. 1076–1082, Oct. 1985. [15] A. Djordjevic, T. Sarkar, and S. Rao, “Analysis of finite conductivity cylindrical conductors excited by axially-independent tm electromagnetic field,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 10, pp. 960–966, Oct. 1985. [16] R.-B. Wu and J.-C. Yang, “Boundary integral equation formulation of skin effect problems in multiconductor transmission lines,” IEEE Trans. Magn., vol. 25, no. 4, pp. 3013–3015, Jul. 1989. [17] R. Faraji-Dana and Y. Chow, “The current distribution and ac resistance of a microstrip structure,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 9, pp. 1268–1277, Sep. 1990. [18] R. Faraji-Dana and Y. Chow, “Ac resistance of two coupled strip conductors,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 138, no. 1, pt. H, pp. 37–45, Feb. 1991. [19] M. Tsuk and J. Kong, “A hybrid method for the calculation of the resistance and inductance of transmission lines with arbitrary cross sections,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 8, pp. 1338–1347, Aug. 1991. [20] G. Plaza, R. Marques, and F. Medina, “Quasi-TM MoL/MoM approach for computing the transmission-line parameters of lossy lines,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 198–209, Jan. 2006.

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[21] G. Antonini, A. Orlandi, and C. Paul, “Internal impedance of conductors of rectangular cross section,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 7, pp. 979–985, Jul. 1999. [22] F. Medina and R. Marques, “Comments on ’internal impedance of conductors of rectangular cross section’,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 8, pp. 1511–1512, Aug. 2001. [23] G. Antonini, A. Orlandi, and C. R. Paul, “Authors’ reply,” , vol. 48, no. 8, pp. 1512–1513, Aug. 2001. [24] W. Heinrich, “Comments on “internal impedance of conductors of rectangular cross section”,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 580–581, Mar. 2001. [25] A. Maue, “Zur Formulierung eines allgemeinen Beugungs—Problems durch eine Integralgleichung,” Z. Phys. A, At Nuclei, vol. 126, no. 7–9, pp. 601–618, Jul. 1949. [26] J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag., vol. AP-20, no. 4, pp. 442–446, Jul. 1972. [27] J. Van Bladel, Singular Electromagnetic Fields and Sources. New York: Oxford Univ. Press, 1991. [28] J. Geisel, K.-H. Muth, and W. Heinrich, “The behavior of the electromagnetic field at edges of media with finite conductivity,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 1, pp. 158–161, Jan. 1992. [29] D. De Zutter and L. Knockaert, “Skin effect modeling based on a differential surface admittance operator,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 8, pp. 2526–2538, Aug. 2005. [30] T. Demeester and D. De Zutter, “Quasi-TM transmission line parameters of coupled lossy lines based on the Dirichlet to Neumann boundary operator,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 7, pp. 1649–1660, Jul. 2008. [31] T. Demeester and D. De Zutter, “Construction of the Dirichlet to Neumann boundary operator for triangles and applications in the analysis of polygonal conductors,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 1, pp. 116–127, Jan. 2010. [32] T. Demeester and D. De Zutter, “Modeling the broadband resistive and inductive behavior of polygonal conductors,” in Int. Electromagn. Adv. App. Conf., Turin, Italy, Sep. 2009, pp. 205–208. [33] A. Omar and K. Schunemann, “Application of the generalized spectral-domain technique to the analysis of rectangular waveguides with rectangular and circular metal inserts,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 6, pp. 944–952, Jun. 1991.

Thomas Demeester was born in Gent, Belgium, in 1982. He received the M.Sc. degree in electrical engineering from Ghent University, Gent, Belgium, in 2005, and is currently working toward the Ph.D. degree at Ghent University. He spent a one-year period with ETH Zürich, during which time he worked on his master thesis in the field of time-domain electromagnetics. He is currently a Research Fellow of the Fund for Scientific Research, Flanders, Belgium, with Ghent University. His research concerns electromagnetic field calculations in the presence of highly lossy media and the development of transmission line models for interconnects. Daniël De Zutter (M’92–SM’96–F’01) was born in 1953. He received the M.Sc. degree in electrical engineering and Ph.D. degree (equivalent to the French Aggrégation or the German Habilitation) from Ghent University, Gent, Belgium, in 1976 and 1981, respectively. From 1976 to 1984, he was a Research and Teaching Assistant with Ghent University. In 1984, he completed his doctoral thesis. From 1984 to 1996, he was with the National Fund for Scientific Research of Belgium. He is currently a Full Professor of electromagnetics. over the past four years, he has been Dean of the Faculty of Engineering, Ghent University. Most of his earlier scientific work dealt with the electrodynamics of moving media. He has authored or coauthored over 150 international journal papers. His research currently focuses on all aspects of circuit and electromagnetic modeling of high-speed and high-frequency interconnections, packaging, on-chip interconnects, and numerical solutions of Maxwell’s equations. Dr. De Zutter is an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.

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Robust Trust-Region Space-Mapping Algorithms for Microwave Design Optimization Slawomir Koziel, Senior Member, IEEE, John W. Bandler, Life Fellow, IEEE, and Qingsha S. Cheng, Senior Member, IEEE

Abstract—Convergence is a well-known issue for standard space-mapping optimization algorithms. It is heavily dependent on the choice of coarse model, as well as the space-mapping transformations employed in the optimization process. One possible convergence safeguard is the trust region approach where a surrogate model is optimized in a restricted neighborhood of the current iteration point. In this paper, we demonstrate that although formal conditions for applying trust regions are not strictly satisfied for space-mapping surrogate models, the approach improves the overall performance of the space-mapping optimization process. Further improvement can be realized when approximate fine model Jacobian information is exploited in the construction of the space-mapping surrogate. A comprehensive numerical comparison between standard and trust-region-enhanced space mapping is provided using several examples of microwave design problems. Index Terms—Computer-aided design (CAD), electromagnetic (EM) optimization, microwave design, space mapping, trust-region methods.

I. INTRODUCTION

S

PACE-MAPPING technology is exploited both in microwave engineering and other areas [1]–[16] to deal with computationally expensive objective functions. The main idea behind space-mapping optimization is to shift the optimization burden from an expensive “fine” (or high-fidelity) model to a cheap “coarse” (or low-fidelity) model by iteratively optimizing and updating a surrogate model, which is built using the coarse model and available fine model data. If the coarse model is a sufficiently accurate representation of the fine model, space-mapping optimization may yield a satisfactory solution after only a few fine model evaluations. This enjoys a

Manuscript received February 18, 2010; revised May 10, 2010; accepted May 21, 2010. Date of publication July 01, 2010; date of current version August 13, 2010. This work was supported in part by the Reykjavik University Development Fund under Grant T09009 and by the Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant RGPIN7239-06 and Grant STPGP 381153-09. S. Koziel is with the Engineering Optimization and Modeling Center, School of Science and Engineering, Reykjavik University, 101 Reykjavik, Iceland (e-mail: [email protected]). J. W. Bandler is with the Simulation Optimization Systems Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1, and also with Bandler Corporation, Dundas, ON, Canada L9H 5E7 (e-mail: [email protected]). Q. S. Cheng is with the Simulation Optimization Systems Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1 (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.2010.2052666

substantial advantage over other techniques in terms of computational cost, such as direct optimization of the fine model using gradient-based methods. The idea of surrogate-based optimization is also exploited by other researchers [17]–[20], although many of them construct a surrogate model by direct approximation of the fine model data with no underlying physically based coarse model. Probably the most serious issue for standard space mapping is convergence, depending on the similarity between the fine model and the space-mapping surrogate [3], [21]. Also, because the zero- and first-order consistency conditions [17] between the fine and surrogate models are not necessarily satisfied (i.e., the surrogate model may not match the fine model with respect to value and first-order derivative at any given iteration points), and subsequent iterations are accepted regardless of objective function improvement, there is no guarantee that the specification error will be reduced from iteration to iteration. In fact, after a few successful iterations, the space-mapping algorithm may produce a design that is worse than ones found so far [22]. Given an optimization problem, the convergence properties and overall performance of the space-mapping algorithm can be improved to some extent by a proper choice of the coarse model and the space-mapping surrogate [21], [23]. On the other hand, given the models and the mapping type, good algorithm performance (measured by convergence and final design quality) is not guaranteed and has to be verified experimentally by executing the optimization process. Trust-region methodology [24] can be used to amend the convergence properties of space-mapping algorithms [23], [25]. Formally, using trust-region methods with space-mapping algorithms is not well justified because first-order consistency [17] between the fine model and space-mapping surrogate does not usually hold, and, therefore, trust-region methods for space mapping become heuristic rather than rigorous. More specifically, first-order consistency guarantees fine model objective function improvement provided that the trust-region radius is small enough [26]. Using trust-region methods in the space-mapping algorithm is justified by the fact that a physics-based surrogate model reflects the general features of the fine model so that their local behavior is similar. Nevertheless, the objective function improvement is not guaranteed regardless of how small the trust region is. First-order consistency can be enforced in the space-mapping surrogate by explicit use of sensitivity information [3]; however, this increases the overall cost of the space-mapping optimization process. In this paper, we expand a systematic treatment of the trustregion-enhanced space-mapping algorithms originated in [22],

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KOZIEL et al.: ROBUST TRUST-REGION SPACE-MAPPING ALGORITHMS FOR MICROWAVE DESIGN OPTIMIZATION

where we gave a heuristic explanation of why the trust region actually works with space-mapping algorithms, as well as proposed a modification of the trust-region-enhanced space-mapping algorithm, which uses an approximate fine model Jacobian to enhance the surrogate model. Here, alternative ways of creating surrogate models exploiting the fine model Jacobian estimation are described. Also, additional in-depth analysis of the trust-region-based space-mapping algorithms is presented. In particular, an analytical argument is given to support the expected performance improvement for the algorithm using the approximated fine model Jacobian. A comprehensive numerical comparison between the standard and the three variants of the trust-region-enhanced spacemapping algorithm is provided based on several examples of microwave design optimization problems. It is demonstrated that the trust-region-enhancement indeed improves the robustness of the space-mapping optimization process with respect to all performance measures: the quality of the optimized design, the convergence properties, and the computational cost of the optimization process. II. SPACE MAPPING WITH TRUST-REGION CONVERGENCE SAFEGUARDS

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B. Robustness Issues regardAs the algorithm (2)–(4) accepts a new design less of the specification error improvement, convergence of the space-mapping algorithm is not guaranteed [1]. Moreover, and at is not ensured as a perfect match between (with respect to value and/or first-order derivatives), there is no guarantee for the space-mapping algorithm to locate the (local) fine model optimal solution [21]. Existing theoretical results for algorithm (2)–(4) or some of its sub-classes provide convergence conditions, which are, however, difficult to verify beforehand [3], [21]. Moreover, conditions for convergence are typically different from conditions for convergence to the fine model optimal solution (i.e., its firstorder stationary point) [21]. This is because fine model sensitivity information is not utilized by current space-mapping algorithms. Excellent results reported in the literature [1]–[13], obtained with space-mapping algorithms, are largely dependent on carefully chosen coarse models and properly selected space-mapping type. No safeguard is offered in the standard space mapping for a not-so-well-selected coarse model or/and a mapping type.

A. Standard Space-Mapping Algorithm

C. Trust-Region Enhanced Space-Mapping Algorithm

, denote the response vector Let of a fine model of the device of interest. Our goal is to solve

A trust-region approach [24] can be used to improve the convergence properties of the space-mapping algorithm. In particular, the surrogate optimization process (2) can be constrained to a neighborhood of , defined as , as follows

(1) where is a given objective function, e.g., minimax [1]. Direct optimization of the fine model is replaced by an itera, tive procedure generating a sequence of designs , and a family of surrogate models , , so that (2) Let , denote the response vectors of the coarse model that describes the same object as the fine model: less accurate, but much faster to evaluate. Surrogate in (2) are constructed as follows: models (3) is a generic space-mapping surrogate where model, which is composed with some suitable space-mapwith being the paping transformations, and rameter space of these transformations. A vector of space-map, is obtained using the parameter extraction ping parameters, procedure (4) An example of the generic surrogate model is an input space mapping of the form , where . A variety of other space-mapping surrogates can be found in [1]–[4], [27].

(5) where is a trust-region radius at iteration . The trust-region radius is reduced if the improvement of the fine model objective function is not sufficient, i.e., if is too small, or if , in which case the new design is rejected. Typically, the standard trust-region radius updating rules [24] are used. , the surrogate model satisfies the If, for all zeroth- and first-order consistency conditions of the form (6) (7) where denotes the Jacobian of the respective model, then, under mild assumptions concerning smoothness of the models, algorithm (5) is convergent to the local fine model optimum [26]. The fundamental reason is that (6) and (7) ensure that if the trust-region radius is sufficiently small. Unfortunately, (6) and (7) are not necessarily satisfied by the space-mapping surrogate model. In particular, the space-mapping optimization process may get stuck at some point as the does not bring any imreduction of the trust-region radius provement to the fine model objective function, which results in termination of the algorithm. In other words, the trust-region method applied to the space-mapping algorithm, as in (5), is a

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heuristic procedure that may improve robustness of the algorithm, but still does not ensure sufficient performance. On the other hand, an important prerequisite of space-mapis physics based so that the surrogate ping algorithms is that ; in particular, the local model reflects the general features of behavior of both models is similar. This, in combination with the multipoint parameter extraction (4), ensures that (6) and (7) may be satisfied approximately. Moreover, condition (6) can be easily enforced by means of the output space mapping [3] using the surrogate

formula for estimating the fine model Jacobian) [2], [29], i.e., we have

(13) where the Broyden-based fine model Jacobian estimation

(14)

(8) with

with

obtained by (4) and (9)

Numerical results presented in Section IV demonstrate that the trust-region-enhanced (output) space-mapping algorithm (8) and (9) indeed exhibits improvement over the standard algorithm. III. ROBUST TRUST-REGION SPACE-MAPPING ALGORITHMS Relations (6) and (7) can be enforced if the space-mapping surrogate explicitly uses fine model sensitivity information, e.g., as in the following model: (10) where ters

is any space-mapping surrogate with parameobtained with (4) and

,

, and zero matrix, where is

( is an the dimension of the model response). Unfortunately, model (10), (11), and (13) will not work well contains information about in practice. The reason is that the fine model Jacobian at only for directions from , i.e., the subspace of spanned by to . As (14) is a rank-one formula, we need at vectors least iterations of the space-mapping algorithm in order to get an estimate of the fine model Jacobian that would be valid for may actually mislead all directions. Before that, the term the algorithm because is a zero vector for all directions not belonging to so that the directional derivative of (10) for such directions may substantially differ from the actual derivative of the fine model.1 Moreover, even if is sufficiently large, so that spans the entire space, estimate (14) may not be sufficiently accurate as many of the points used in the . Broyden update may be located too far from

(11)

A. Surrogate Model With Restricted Broyden Update

(12)

The trust-region space-mapping algorithm with Broyden upterm (13) is restricted to the date can be improved if the subspace so that for all from , the orthogonal complement of , defined as for all . In this case, the derivative of at will coincide with the derivative of for all directions from , but it will use the Jacobian estimation of the fine model for all directions from . An appropriate formula is as follows:

Convergence of algorithm (5) with surrogate model (10)–(12) is guaranteed under standard assumptions concerning the smoothness of the fine and coarse models [26]. Note, however, that the computational cost of space-mapping optimization may substantially increase in this case because of the necessity of calculating the fine model Jacobian at all iterations. Jacobian estimation using finite differences exploits extra fine model evaluations per estimation, unless some other techniques are used, such as adjoint sensitivity [28], which, however, is not yet generally available in commercial electromagnetic (EM) simulators. In this section, we propose some modifications of the standard trust-region space-mapping algorithm that aim at ensuring robustness of the algorithm without compromising the computational cost of the algorithm, i.e., avoiding explicit calculation of the fine model Jacobian. All the proposed algorithms are based on modifications of the surrogate model (10)–(12) so that the exact fine model sensitivity data is replaced by a suitable approximation. Before we turn to specific models, let us discuss a naive approach, according to which the surrogate model Jacobian is calculated using finite differences, whereas the fine model Jacobian in (12) is replaced by a Broyden-based estimation (the iterative

(15) where

is given by (14), whereas

projection onto

is an orthogonal

given by (16)

with , , being the orthonormal basis of that can be obtained from , , using the Gram–Schmidt procedure. 1In fact, the directional derivative of (10) equals zero at x for all directions not belonging to X and it is close to zero in the neighborhood of x .

KOZIEL et al.: ROBUST TRUST-REGION SPACE-MAPPING ALGORITHMS FOR MICROWAVE DESIGN OPTIMIZATION

Equation (15) basically says that we trust the fine model Jacobian in directions where fine model data is available, however, the fine model Jacobian should match the surrogate Jacobian in directions of “no data.” It can be further improved by so that inusing only points that are sufficiently close to stead of considering , we can that satisfy consider the subspace spanned only by those of , where is a user-defined threshold; can be , or it can be related to the distance beeither fixed, i.e., , tween the latest iteration designs, i.e., and are user-defined positive numbers. where

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Fig. 1. Third-order Chebyshev bandpass filter: geometry [30].

B. Surrogate Model With Broyden-Based Jacobian Estimation of the Fine and Surrogate Model [22] term (13) to the subspace can also Restriction of the be obtained if the Jacobians of the fine model and the spaceare estimated jointly using the mapping surrogate Broyden update, i.e., with the following formula [22]: (17) where

(18) with ,

, and

.

In this case, we do not need to project onto as the condition for all from is automatically fulfilled. As before, it is recommended to consider local updates using , where is a userpoints satisfying defined threshold (either fixed or relative to ). It should be noted that (15)–(18) are two alternative methods in (10); however, it is difficult to of determining the term say beforehand which is better in terms of algorithm performance. The numerical experiments of Section IV indicate that both methods are comparable. C. Robustness of Trust-Region Space-Mapping Algorithms Note that in contrast to our standard trust-region space-mapping algorithm (5), the modifications proposed in this section ensure improvement of the fine model objective function for sufficiently small trust-region radius, which is a fundamental property behind the trust-region methodology. In particular, for the algorithms using the term defined in Sections III-A and III-B. and are we have two consecutive iteration points) provided that the trust-region is sufficiently small. A sketch of this property is proradius vided in the Appendix. IV. VERIFICATION EXAMPLES We provide a comprehensive numerical verification of the trust-region-enhanced space-mapping algorithms. In

Fig. 2. Third-order Chebyshev bandpass filter: coarse model (Agilent ADS).

particular, we compare the performance of the standard algorithm (2)–(4), the trust-respace-mapping algorithm (5) with gion-enhanced space-mapping the output space-mapping model (8), (9), and the trust-region-enhanced algorithms using models (10), (11), (15) and (10), (11), (17) . In all cases, the algorithm was terminated if one of the following conditions , , or was satisfied: , , only). and A. Third-Order Chebyshev Bandpass Filter [30] Consider the third-order Chebyshev bandpass filter [30] shown in Fig. 1. The design variables are mm. Other parameters are mm. The is simulated in Sonnet [31]. The coarse fine model model (Fig. 2) is implemented in Agilent Advanced Design System (ADS) [32]. The design specifications are dB for GHz GHz, and dB for GHz GHz and GHz GHz. The initial design is the coarse mm model optimal solution dB). (specification error Table I shows the optimization results. Two types of spacemapping surrogate models were tested: input space-mapping [2] and the combination of input model in and frequency space-mapping model is the coarse model evaluated at frequencies difwhich ferent from the original sweep according to the linear mapping [parameters and are obtained using the usual parameter-extraction process (4)] [3]. For this example, the standard space-mapping algorithm converges for both surrogate model types, however, especially for , the final design is not the input space-mapping model as good as for trust-region-enhanced algorithms, particularly the ones using the Jacobian estimation. Fig. 3 shows the initial fine

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TABLE I THIRD-ORDER CHEBYSHEV FILTER: OPTIMIZATION RESULTS [22]

Fig. 4. Fourth-order ring resonator bandpass filter: geometry [33].

Fig. 3. Third-order Chebyshev filter: Initial (dashed line) and optimized (solid line) jS j versus frequency; optimization using SM algorithm with the R (x + c) model [22].

TABLE II FOURTH-ORDER RING RESONATOR FILTER: OPTIMIZATION RESULTS

Fig. 5. Fourth-order ring resonator bandpass filter: coarse model (Agilent ADS).

model response and the optimized fine model response obtained algorithm with the model. using the B. Fourth-Order Ring Resonator Bandpass Filter [33] As the next example, consider the fourth-order ring resonator bandpass filter [33] shown in Fig. 4. The design parameters are mm. The fine model is simulated in FEKO [34]. The coarse model (Fig. 5) is implemented in Agilent ADS [32]. The design specifications are dB for GHz GHz, and dB for GHz GHz and GHz GHz. The initial design is the coarse model optimal solution mm dB). (specification error Table II shows the optimization results. Here, we consider the following space-mapping surrogate models: the input space[2] and the implicit spacemapping model in which is the coarse mapping model model with the substrate height and dielectric constants used as preassigned parameters [27] to improve the matching between the surrogate and fine model.

Fig. 6. Fourth-order ring resonator filter: initial (dashed line) and optimized algorithm (solid line) jS j versus frequency; optimization using SM with the R (x + c) model. (a) Full frequency range. (b) Magnification from 1.4 to 2.6 GHz and 022 to 0 dB.

Fig. 6 shows the initial fine model response and the optimized algorithm fine model response obtained using the model. Fig. 7 shows the convergence plot with the for the standard algorithm and the algorithm both . working with the implicit space-mapping surrogate For this example, the standard space-mapping algorithm does not converge for either surrogate model type. Also, the final design is worse than the best one found in the course of optimiza-

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Fig. 7. Fourth-order ring resonator filter: convergence plots for SM (o) and SM (3), both using surrogate model R (x), versus iteration index. Fig. 9. Wideband ring resonator bandpass filter: coarse model (Agilent ADS).

TABLE III WIDEBAND RING RESONATOR FILTER: OPTIMIZATION RESULTS

Fig. 8. Wideband ring resonator bandpass filter: geometry [35].

tion, which is because the standard algorithm does not ensure specification error improvement from iteration to iteration, as discussed in Section II. Trust-region-enhanced algorithms exhibit clear performance improvement. C. Wideband Ring Resonator Bandpass Filter [35] Our third example is the wideband ring resonator bandpass filter [35] shown in Fig. 8. The design parameters are mm. The fine model is simulated in FEKO [34]. The coarse model (Fig. 9) is implemented in AgdB ilent ADS [32]. The design specifications are GHz GHz, and dB for for GHz GHz and GHz GHz. The initial design is the coarse model optimal solution mm dB). (specification error For this example, the following space-mapping surrogate models were utilized: the combination of the input and frein which quency space-mapping model is the frequency-mapped coarse model [3], and the imin which plicit space-mapping model is the coarse model with the substrate height and dielectric constants used as preassigned parameters [27]. Table III shows the optimization results. The standard spacemapping algorithm does not perform well. For the input/frequency space-mapping surrogate, it even fails to find a solution satisfying the design specifications. The trust-region-enhanced algorithms perform well ensuring both algorithm convergence and a high-quality final design. The fine model responses: the algoinitial and optimal designs obtained using the rithm with the model are shown in Fig. 10. Conver-

Fig. 10. Wideband ring resonator filter: initial (dashed line) and optimized (solid line) jS j versus frequency; optimization using SM algorithm with the R (x) model. (a) Full frequency range. (b) Magnification from 2.5 to 6.0 GHz and 02 to 0 dB.

gence plots for and plicit space-mapping surrogate

working with the imare shown in Fig. 11.

D. Open-Loop Ring Resonator Bandpass Filter [36] Our final example is the open-loop ring resonator bandpass filter [36] (Fig. 12). The design parameters are mm. Other parameter values mm, mm. is simulated in FEKO are [34]. The coarse model (Fig. 13) is implemented in Agilent dB ADS [32]. The design specifications are for GHz GHz, and dB for

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Fig. 11. Wideband ring resonator filter: convergence plots for SM (o) and SM (3), both using surrogate model R (x), versus iteration index.

Fig. 13. Open-loop ring resonator bandpass filter: coarse model (Agilent ADS).

TABLE IV OPEN-LOOP RING RESONATOR FILTER: OPTIMIZATION RESULTS[22] Fig. 12. Open-loop ring resonator bandpass filter: geometry [36].

GHz GHz and GHz GHz. The initial design is the coarse model optimal solution mm (specification error 11.8 dB). Here, we use the following surrogate models: the frequency , and the implicit space-mapping model space-mapping model with the substrate height and dielectric constants used as preassigned parameters [22]. The optimization results are shown in Table IV. For this problem, the standard space-mapping algorithm does not perform well when the frequency space-mapping surrogate is used; however, it is as good as the trust-region-enhanced algorithms for the implicit space-mapping surrogate. This indicates that, in a way, the trust-region mechanism is being “turned on” only if necessary; otherwise, it does not interfere with the space-mapping optimization process. Fig. 14 shows the initial and final algorithm with the fine model responses for the model. Fig. 15 shows the evolution of the specification and algorithm using the error for model. This figure illustrates a quite common behavior of the standard space-mapping algorithm: after initial success, the algorithm may not be able to yield further improvement if the surrogate model type is not properly selected. E. Discussion The numerical results presented in Sections IV-A–IV-D indicate the advantages of the trust-region-enhanced space-mapping algorithms , and over the standard . The results can be summaspace-mapping algorithm rized as follows. , , and ensure (a) Algorithms convergence for all considered test problems, which is not the case for the standard space-mapping algorithm. (b) In many cases, the basic trust-region algorithm performs as well as the algorithms using the Jacobian and ), however, in estimation (

Fig. 14. Open-loop ring resonator filter: initial (dashed line) and optimized (solid line) jS j versus frequency; optimization using the SM algorithm with the R (x) model [22].

Fig. 15. Open-loop ring resonator filter: evolution of the specification error for SM (o) and SM (2), both using surrogate model R (x) versus iteration index [22].

some cases, the latter approach prevails: the algorithm may simply get stuck because the property may not hold even for very small trust-region radius values (cf. Section III-C). and can be considered (c) Algorithms as equally good; the small differences in the final design

KOZIEL et al.: ROBUST TRUST-REGION SPACE-MAPPING ALGORITHMS FOR MICROWAVE DESIGN OPTIMIZATION

quality and the computational cost of the optimization process come from different ways of implementing the -term in model (10) (cf. Section III). (d) In the cases when the does well, both and have little or no effect, which is exactly what we require. (e) Using trust-region-enhancement makes the space-mapping algorithm less sensitive to the selection of surrogate may perform well for certain surromodel type: gate model types and poorly for others, see Tables I–IV; performance differences are much less pronounced in trust-region enhanced algorithms. These conclusions are in agreement with theoretical predictions. In particular, the convergence of the trust-region-enhanced algorithms, as well as iteration-to-iteration improvement with respect to the specification error value (cf. Fig. 15) is ensured by the trust-region mechanism itself. On the other hand, extra information in the form of the fine/surrogate model Jacobian estimation allows further improvement of the final design quality because the fundamental property is ensured for sufficiently small trust-region radius. V. CONCLUSION A systematic treatment of trust-region-enhanced space-mapping algorithms is presented. A basic trust-region space-mapping algorithm and its extensions exploiting the fine model Jacobian estimation are considered. An extensive performance comparison with the standard algorithm indicates that the trust region approach discussed in this paper can be considered as an important step toward improving the robustness of the spacemapping optimization process. APPENDIX We provide an analytical argument showing that the modified trust-region-enhanced space-mapping algorithms of Sections III-A and III-B ensure improvement of the fine model , proobjective function, i.e., [cf. (5)] is sufficiently vided that the trust-region radius small. We assume, for simplicity, that the merit function is smooth. Suppose that the optimization of the surrogate produces a new iteration point such that model . Design must be then will be updated [e.g., using (10), rejected and model (11), (15) or (10), (11), (17)] so that the following relation holds for a sufficiently small . In particular, we have that , where denotes the . Now, again directional derivative along being sufficiently small, the for the trust-region radius of the updated surrogate will be located optimal solution almost on the line defined as , . with More specifically, . We have the following relation denotes the gradient of at ): (

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, which gives , and finally, , as we as.

sumed

ACKNOWLEDGMENT The authors thank Sonnet Software Inc., Syracuse, NY, for available, and Agilent Technologies, Santa Rosa, making CA, for making ADS available. REFERENCES [1] S. Koziel, Q. S. Cheng, and J. W. Bandler, “Space mapping,” IEEE Microw. Mag., vol. 9, no. 6, pp. 105–122, Dec. 2008. [2] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Sondergaard, “Space mapping: The state of the art,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 337–361, Jan. 2004. [3] S. Koziel, J. W. Bandler, and K. Madsen, “A space mapping framework for engineering optimization: Theory and implementation,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3721–3730, Oct. 2006. [4] D. Echeverria and P. W. Hemker, “Space mapping and defect correction,” Int. Math. J. Comput. Methods Appl. Math., vol. 5, no. 2, pp. 107–136, 2005. [5] M. A. Ismail, D. Smith, A. Panariello, Y. Wang, and M. Yu, “EMbased design of large-scale dielectric-resonator filters and multiplexers by space mapping,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 386–392, Jan. 2004. [6] K.-L. Wu, Y.-J. Zhao, J. Wang, and M. K. K. Cheng, “An effective dynamic coarse model for optimization design of LTCC RF circuits with aggressive space mapping,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 393–402, Jan. 2004. [7] J. E. Rayas-Sánchez, F. Lara-Rojo, and E. Martínez-Guerrero, “A linear inverse space mapping (LISM) algorithm to design linear and nonlinear RF and microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 960–968, Mar. 2005. [8] M. Dorica and D. D. Giannacopoulos, “Response surface space mapping for electromagnetic optimization,” IEEE Trans. Magn., vol. 42, no. 4, pp. 1123–1126, Apr. 2006. [9] S. Amari, C. LeDrew, and W. Menzel, “Space-mapping optimization of planar coupled-resonator microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2153–2159, May 2006. [10] D. Echeverria, D. Lahaye, L. Encica, E. A. Lomonova, P. W. Hemker, and A. J. A. Vandenput, “Manifold-mapping optimization applied to linear actuator design,” IEEE Trans. Magn., vol. 42, no. 4, pp. 1183–1186, Apr. 2006. [11] G. Crevecoeur, P. Sergeant, L. Dupre, and R. Van de Walle, “Two-level response and parameter mapping optimization for magnetic shielding,” IEEE Trans. Magn., vol. 44, no. 2, pp. 301–308, Feb. 2008. [12] M. F. Pantoja, P. Meincke, and A. R. Bretones, “A hybrid geneticalgorithm space-mapping tool for the optimization of antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 777–781, Mar. 2007. [13] P. Sergeant, R. V. Sabariego, G. Crevecoeur, L. Dupre, and C. Geuzaine, “Analysis of perforated magnetic shields for electric power applications,” IET Elect. Power Appl., vol. 3, no. 2, pp. 123–132, Mar. 2009. [14] V. K. Devabhaktuni, B. Chattaraj, M. C. E. Yagoub, and Q.-J. Zhang, “Advanced microwave modeling framework exploiting automatic model generation, knowledge neural networks, and space mapping,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp. 1822–1833, Jul. 2003. [15] J. C. Rautio, “A space mapped model of thick, tightly coupled conductors for planar electromagnetic analysis,” IEEE Microw. Mag., vol. 5, no. 3, pp. 62–72, Sep. 2004. [16] L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2752–2767, Sep. 2005. [17] N. M. Alexandrov and R. M. Lewis, “An overview of first-order model management for engineering optimization,” Optim. Eng., vol. 2, no. 4, pp. 413–430, Dec. 2001.

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[18] A. J. Booker, J. E. Dennis, Jr., P. D. Frank, D. B. Serafini, V. Torczon, and M. W. Trosset, “A rigorous framework for optimization of expensive functions by surrogates,” Struct. Optim., vol. 17, no. 1, pp. 1–13, Feb. 1999. [19] J. E. Dennis and V. Torczon, “Managing approximation models in optimization,” in Multidisciplinary Design Optimization, N. M. Alexrov and M. Y. Hussaini, Eds. Philadelphia, PA: SIAM, 1997, pp. 330–374. [20] N. V. Queipo, R. T. Haftka, W. Shyy, T. Goel, R. Vaidynathan, and P. K. Tucker, “Surrogate-based analysis and optimization,” Progr. Aerosp. Sci., vol. 41, no. 1, pp. 1–28, Jan. 2005. [21] S. Koziel, J. W. Bandler, and K. Madsen, “Quality assessment of coarse models and surrogates for space mapping optimization,” Optim. Eng., vol. 9, no. 4, pp. 375–391, 2008. [22] S. Koziel, J. W. Bandler, and Q. S. Cheng, “Trust-region-based convergence safeguards for space mapping design optimization of microwave circuits,” in IEEE MTT-S Int. Microw. Symp. Dig, Boston, MA, 2009, pp. 1261–1264. [23] S. Koziel and J. W. Bandler, “Space-mapping optimization with adaptive surrogate model,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 3, pp. 541–547, Mar. 2007. [24] A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust Region Methods, ser. MPS-SIAM Optimization. Philadelphia, PA: , 2000. [25] M. H. Bakr, J. W. Bandler, R. M. Biernacki, S. H. Chen, and K. Madsen, “A trust region aggressive space mapping algorithm for EM optimization,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2412–2425, Dec. 1998. [26] N. M. Alexandrov, J. E. Dennis, R. M. Lewis, and V. Torczon, “A trust region framework for managing use of approximation models in optimization,” Struct. Multidisciplinary Optim., vol. 15, no. 1, pp. 16–23, 1998. [27] J. W. Bandler, Q. S. Cheng, N. K. Nikolova, and M. A. Ismail, “Implicit space mapping optimization exploiting preassigned parameters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 378–385, Jan. 2004. [28] N. K. Nikolova, Y. Li, Y. Li, and M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1598–1610, Apr. 2006. [29] C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput., vol. 19, pp. 577–593, 1965. [30] J. T. Kuo, S. P. Chen, and M. Jiang, “Parallel-coupled microstrip filters with over-coupled end stages for suppression of spurious responses,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 10, pp. 440–442, Oct. 2003. [31] em. ver. 11.54, Sonnet Softw. Inc., North Syracuse, NY, 2008. [32] Agilent ADS. ver. 2008, Agilent Technol., Santa Rosa, CA, 2008. [33] M. H. M. Salleh, G. Prigent, O. Pigaglio, and R. Crampagne, “Quarter-wavelength side-coupled ring resonator for bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 1, pp. 156–162, Jan. 2008. [34] “FEKO® User’s Manual,” EM Software & Syst. S.A. (Pty) Ltd., Stellenbosch, South Africa, 2008. [35] S. Sun and L. Zhu, “Wideband microstrip ring resonator bandpass filters under multiple resonances,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 10, pp. 2176–2182, Oct. 2007.

[36] C. Y. Chen and C. Y. Hsu, “A simple and effective method for microstrip dual-band filters design,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 5, pp. 246–248, May 2006. Slawomir Koziel (M’03–SM’07) received the M.Sc. and Ph.D. degrees in electronic engineering from Gdansk University of Technology, Gdansk, Poland, in 1995 and 2000, respectively, and the M.Sc. degrees in theoretical physics and in mathematics and Ph.D. degree in mathematics from the University of Gdansk, Gdansk, Poland, in 2000, 2002, and 2003, respectively. He is currently an Associate Professor with the School of Science and Engineering, Reykjavik University, Reykjavik, Iceland. His research interests include computer-aided design (CAD) and modeling of microwave circuits, surrogate-based optimization, space mapping, circuit theory, analog signal processing, evolutionary computation, and numerical analysis.

John W. Bandler (S’66–M’66–SM’74–F’78– LF’06) studied at Imperial College, London, U.K. He received the B.Sc. (Eng.), Ph.D., and D.Sc.(Eng.) degrees from the University of London, London, U.K., in 1963, 1967, and 1976, respectively. In 1969, he joined McMaster University, Hamilton, ON, Canada, where he is currently a Professor Emeritus. He was President of Optimization Systems Associates Inc., which he founded in 1983, until November 20, 1997 (the date of acquisition by the Hewlett-Packard Company). He is President of Bandler Corporation, Dundas, ON, Canada, which he founded in 1997. Dr. Bandler is a Fellow of several societies including the Royal Society of Canada. He was the recipient of the 2004 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Microwave Application Award.

Qingsha S. Cheng (S’00–M’05–SM’09) was born in Chongqing, China. He received the B.Eng. and M.Eng. degrees from Chongqing University, Chongqing, China, in 1995 and 1998, respectively, and the Ph.D. degree from McMaster University, Hamilton, ON, Canada, in 2004. In 1998, he joined the Department of Computer Science and Technology, Peking University, Beijing, China. In 1999, he joined the Department of Electrical and Computer Engineering, McMaster University, where he is currently a Research Associate with the Department of Electrical and Computer Engineering and a Lecturer with the Faculty of Engineering. His research interests are surrogate modeling, computer-aided design (CAD), modeling of microwave circuits, software design technology, and methodologies for microwave CAD.

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Large Overlapping Subdomain Method of Moments for the Analysis of Frequency Selective Surfaces Arya Fallahi, Alireza Yahaghi, Habibollah Abiri, Member, IEEE, Mahmoud Shahabadi, and Christian Hafner

Abstract—A new set of basis functions is presented for the analysis of frequency selective surfaces (FSSs) by the method of moments (MoM). Each of the separate patches in the unit cell of an FSS is covered with some large overlapping sub-patches. Each sub-patch has a shape that is simple enough for obtaining an appropriate set of basis functions analytically from an eigenvalue problem. This technique is called the large overlapping subdomain MoM. It is shown that the proposed method has a better convergence than the standard MoM with rooftop basis functions in terms of both required basis functions and Fourier modes. Furthermore, this approach offers a much easier way to model patches with curved boundaries. The extraction of the proposed basis functions involves mostly analytical procedures. Therefore, this kind of moment method is computationally more efficient than versions with entire domain basis functions, in which the basis is obtained from the boundary integral resonant mode expansion technique. Nonetheless, this is accompanied by a small decay in the convergence rate, i.e., the large overlapping subdomain is clearly placed between the standard subdomain and entire domain versions of the MoM. It is shown that the developed method is advantageous for usual FSSs and unit cell configurations with several patches. Index Terms—Frequency selective surfaces (FSSs), large overlapping subdomain basis functions (LOSBFs), method of moments (MoM), periodic structures, spectral-domain analysis.

I. INTRODUCTION REQUENCY selective surfaces (FSSs) as a subclass of periodic structures are mainly planar arrays of metallic patches printed on dielectric substrates. Their unprecedented capability to show frequency selective properties has made them promising for many applications. Hence, they have been the subject of much research for many years [1], [2]. They found a variety of applications such as reflector antennas [3], [4] antenna radomes [5], [6], and polarizers [7]. In recent years, along with the development of metamaterials to fabricate structures

F

Manuscript received April 09, 2009; revised May 10, 2010; accepted May 10, 2010. Date of publication July 23, 2010; date of current version August 13, 2010. This work was supported by ETH Zürich. A. Fallahi and C. Hafner are with the Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zürich, Zürich CH-8092, Switzerland (e-mail: [email protected]). A. Yahaghi is with the Department of Electrical Engineering, Shiraz University, Shiraz, Iran, and also with the Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zürich, Zürich CH-8092, Switzerland (e-mail: [email protected]). H. Abiri is with the Department of Electrical Engineering, Shiraz University, Shiraz, Iran. M. Shahabadi is with the Faculty of Engineering, School of Electrical and Computer Engineering, University of Tehran, Tehran 14395-15, Iran. Digital Object Identifier 10.1109/TMTT.2010.2053790

with properties not found in nature, some new applications like planar absorbers [8], [9] and artificial magnetic conductors [10] were introduced. To design an FSS for specific applications, various schemes were proposed [2], and some studies were done on their optimization [10], [11]. For this purpose, one requires a method that should be accurate, efficient, and robust at the same time. Methods like the periodic method of moments (MoM) [1], finite difference time domain [12], [13], finite element method [14], transmission line matrix [15] were applied to analyze FSS. It is widely accepted that the methods based on the MoM are the most suitable algorithms for the analysis of layered printed structures [16]. A periodic MoM is indeed the modified version of the MoM devised for periodic structures. The properties of the basis functions, which are used to expand the excited current on the patches in a unit cell, strongly influence the efficiency of the method. Moreover, in the formulation of the periodic MoM, Fourier transforms of the basis functions are involved. Therefore, the efficiency of the method should be investigated based on two main aspects, namely, convergence in terms of the number of required basis functions and convergence in terms of the required Fourier modes. The search for appropriate basis functions in solving electromagnetic problems using the moments method is not limited to FSS simulation. There has been wide research on establishing proper basis functions for other cases. This led to the development of the characteristic basis functions (CBFs) [17], the adaptive basis functions (ABFs) [18], and the extension of the use of the ABFs [19] for arrays. The above basis functions are mainly proposed for the analysis of large and finite problems such as antenna arrays. However, in the FSS problem, a single patch is treated with periodic boundary conditions. Therefore, the problem is inherently different and consequently the above basis functions are not applicable to the FSS simulation. “Classical” basis functions can be categorized into two main classes: sub-domain and entire domain basis functions. Each of these classes has its own advantages and drawbacks. Sub-domain basis functions are easily implementable but they are local functions and one needs to consider a large set of these functions to model the whole induced current. Additionally, their slowly decaying spectra lead to slow convergence of the method [20], which is critical in the case of having periodic substrates [21]. For modeling complicated patches with or without curved boundaries, one usually needs to use very fine meshes, which, in turn, deteriorates the efficiency of the method. Due to these problems, entire domain basis functions are the superior choice. In the early years of FSS studies, entire domain functions were introduced merely for some

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limited and relatively simple patch shapes. In [22] and [23], a general procedure based on the boundary integral-resonant mode expansion (BI-RME) technique was introduced to produce this set of functions for arbitrary patch shapes. However, implementing the BI-RME to extract the entire domain basis functions—especially when there are several separate patches in the unit cell—causes additional computational costs. In the MoM/BI-RME, every separate patch in the unit cell is thought of as a cross section of a metallic waveguide [22], [23], which obviously leads to an eigenvalue problem to be solved. The transverse magnetic fields of the waveguide modes have the same boundary conditions as the excited currents on the patches, and therefore they create a set of proper entire domain basis functions. In this paper, the idea of waveguide modes is essentially followed for obtaining basis functions, but patches of complicated shape are subdivided into subpatches with overlapping cross sections. This method is called the large overlapping subdomain MoM (LOS-MoM). In practice, a small number of well-known waveguides (rectangular, circular, coaxial, wedge, and sectorial) suffices to obtain the desired basis. The spectra of these functions usually decay faster than standard sub-domain functions such as rooftop functions. Furthermore, there is no need for solving waveguide problems with complicated cross sections. However, to obtain an acceptable result, one usually needs to use more basis functions than in the entire domain MoM. This slightly weakens the efficiency of the method, but is not the dominant effect. Consequently, a procedure is developed, which essentially combines the advantages of the two groups of MoM basis functions. This paper presents the whole numerical procedure of the LOS-MoM. To verify the proposed method and investigate its efficiency, some sample FSSs are analyzed and their results are compared with the MoM/rooftop and MoM/BI-RME. A comparison of the convergence of the different methods regarding both the number of basis functions and Floquet modes is presented. It is shown that the introduced approach outperforms the other ones for modeling FSSs. Nonetheless, it is not claimed that the LOS-MoM is the optimal choice and future study on the subject may yield more suitable basis functions. A crucial problem in the analysis of FSSs is the singularity of the induced current at sharp edges of the printed patch. In this study, some special basis functions are introduced to overcome this problem and their effects are explored as well.

Fig. 1. Typical geometry of an FSS with periodic substrate.

is fulfilled by considering the multiconductor transmission line model for each layer. In the next step, the resulting equation is solved by the periodic MoM. To this end, the large overlapping subdomain basis functions (LOSBFs) should be first extracted. In the following, a very brief summary of periodic MoM formulation and a detailed explanation of extracting the mentioned basis functions is presented. The detailed description of the periodic MoM can easily be found in the literature. However, a short summary can help to better understand the numerical procedure. A. Periodic MoM The MoM is accepted to be an efficient technique to solve the integral-equation formulation of the electric field in planar metallic structures. The reason lies in the analytical formulation of the Green’s function for the mentioned patterns. Due to the periodicity of FSS structures, Floquet’s theorem leads to a discrete spectrum for every quantity in the spectral domain. Therefore, the integral equation reduces to a series equation consisting of Fourier components of currents [1]. This method is usually referred to as the periodic MoM. The boundary conditions that must hold on the patches are

(1) where and ( ) are tangential components of the scattered and incident electric field on the patches, respectively, is the surface impedance of the metallic patches. and The scattered electric field is obtained from induced electric currents using the dyadic Green’s function concept [21]

II. FORMULATION OF THE PROBLEM Fig. 1 illustrates a typical geometry of an FSS. The structure consists of arrays of arbitrarily shaped metallic patches arranged in a 2-D lattice and printed on a substrate, which may be periodic. The periodic substrate shown in Fig. 1 is obtained by drilling holes in a homogenous substrate. The goal of the analysis is the determination of the reflected and transmitted electromagnetic fields when the structure is illuminated by a plane wave with a given angular frequency , and direction ( , ) (Fig. 1). The LOS-MoM procedure consists of the following steps. In the first step, the incident field is related to the induced currents on the patch out of which the scattered field is evaluated. This

(2) where

and

are given by

(3) (4) is the wave vector component of the incident deterplane wave in the transverese plane and the pair

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Fig. 2. Typical geometry of an FSS is divided to some sub-patches. The waveguide modes of the sub-patches are used as the basis functions.

mines the lattice constants of the periodic structure in both the - and -directions, respectively. is a row is a zero mamatrix containing the exponential terms and trix with the same size as . In this paper, the transpose sign is a superscript , which is used to distinand are column guish between row and column vectors. vectors obtained from the Fourier coefficients and in the basis. The matrix is a diagonal matrix with diagonal elements equal to . Finally, is the Green function matrix. In case of homogeneous substrates, it consists of four diagonal matrices, including the values of dyadic Green’s funcin the spectral domain. tion components In case of periodic inhomogeneous substrates, can be found from the multiconductor transmission line model for the substrate [21]. For solving (2) using the concept of the MoM, electric currents excited on the patches should be expanded in terms of some basis functions

(5) and are row vectors containing the basis functions where used for expanding and , respectively. The unknown coefficients of these functions are arranged in the column vector . Using Galerkin’s method and after some algebraic operations the following system of equations is obtained:

Fig. 3. Cross section of five basic waveguides. The modes of these waveguides generate our so-called LOSBFs. (a) Rectangular waveguide. (b) Circular waveguide. (c) Wedge waveguide. (d) Coaxial waveguide. (e) Sectoral waveguide.

(6)

, and is the incident elecwhere and are matrices whose th columns tric field vector. are Fourier coefficients of th corresponding basis functions. The signs and stand for the complex and Hermitian conjuand in (2) are related to and gate, respectively. through the following equation:

Using the obtained coefficients , all the desired quantities such as reflection and transmission coefficients can easily be calcuis considered as a phase factor lated [1]. The term in all basis functions. Therefore, the Fourier coefficients are calbasis. culated in the Using the presented formulation, an arbitrary FSS can be simulated. As previously mentioned, the efficiency of the method is strongly affected by the properties of applied basis functions. In Section II-B, the LOSBFs as a proper set of basis Functions are introduced. B. LOSBFs

(7)

The basis functions in (5) should be chosen properly to be able to expand the excited current correctly. In other words,

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TABLE I FOURIER COEFFICIENTS OF THE GUIDED MODES IN A RECTANGULAR WAVEGUIDE

TABLE II FOURIER COEFFICIENTS OF THE GUIDED MODES IN A CIRCULAR WAVEGUIDE

they should be a complete set of functions with the same boundary conditions as the electric current on the patch. The boundary condition is basically zero normal component of the current at the boundaries. It is known from the waveguide theory that transverse magnetic fields of different guided modes in a metallic waveguide with the same cross section as the patch shape satisfy these requirements. In the MoM/BI-RME approach, this assumption is used to construct a basis for the whole patch for expanding the induced current. This leads to a low cost for the computation of the basis only for very simple patches. As seen in [23], this calculation may cause more costs than the FSS simulation itself. The idea of an LOS-MoM is to cover each separate patch in the unit cell with some large overlapping sub-patches whose shapes are the cross sections of well-known waveguides such as rectangular and circular waveguides. Hence, the expensive computation of waveguide modes in the BI-RME may be avoided also for relatively complicated shapes. The transverse magnetic fields of the guiding modes in all the waveguides construct a set of functions that are able to expand the excited current on the patch and can be thought as LOSBFs. Each interior boundary of each sub-patch should be covered by at least one other sub-patch in order to avoid vanishing of the normal current component to the corresponding boundary. Fig. 2 shows how a patch can be covered with some simple sub-patches. In this example, the waveguide modes of two rectangular waveguides and four sectoral waveguides are used as the required basis functions.

Most of the practical FSSs can be constructed using the five fundamental sub-patches shown in Fig. 3. The waveguide modes with these cross sections can be evaluated analytically [24]. As seen from (6), the Fourier coefficients of the basis functions are involved in the formulation of the periodic MoM. The derivation of the Fourier coefficients for transverse magnetic fields of all the guided modes are lengthy. For the sake of brevity, only the final results are presented in Section II-C. As an example, the main steps for calculating the Fourier coefficients of the guided modes in a circular waveguide are explained in the Appendix. C. Fourier Coefficients of Basis Functions The Fourier coefficients of a two variable function such as are calculated from

(8)

and . The funcwhere tion is considered as the guiding mode of the involved waveguides. The Fourier coefficients of the guided modes in the considered waveguides are listed in Tables I–V. In all these tables, denotes the th Fourier coefficient of the compomode. nent of the magnetic field corresponding to the

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TABLE III FOURIER COEFFICIENTS OF THE GUIDED MODES IN A WEDGE WAVEGUIDE

Similar notation is used for other polarizations and other directions. The basis functions can be multiplied by a constant factor, which only affects the amplitude of each basis function. Moreover, in the tabulated equations, the center of the unit cell is assumed to lay on the origin of the coordinate system. In all the , , and are defined presented tables, the variables as the following:

(9) Table I refers to the modes of the rectangular waveguide shown in Fig. 3(a). In this table, the sinc function is defined as . The Fourier coefficients of the guided modes in a circular waveguide [see Fig. 3(b)] are tabulated in is the Bessel function of the first Table II. In this table, kind with the order ( for the TM case and for the TE case). and are the th zeroes of the corresponding Bessel function and its derivative, respectively. The terms even and odd refer to the even and odd azimuthal dependencies of the guided mode, namely, and , respectively. In Table III the same task is done for the wedge waveguide [see Fig. 3(c)]. The definitions of all the parameters are the same as the previous tables. The integrals for this waveguide should be computed numerically. In this paper, a five-point Gaussian quadrature method is utilized to calculate all the integrals. In Table IV, the considered coefficients regarding a coaxial waveguide [see Fig. 3(d)] is presented. In

and are the th-order Bessel functions this table, for of the first and second kinds, respectively ( for the TE case). and the TM case and are the th roots of the two following equations: (10) (11) Similarly, the corresponding coefficients for the sectoral waveguide [see Fig. 3(e)] are listed in Table V. and are obtained in the same way as the coaxial case. Using the presented tables, one can set up appropriate basis functions to analyze an FSS. Obviously, there are different ways to cover a given patch with the different sub-patches. The corresponding choice directly affects the efficiency of the method. An improper selection of sub-patches might even cause a higher computation cost compared with the previously developed procedures. As a rule of thumb, the largest possible sub-patches should be considered, which means that one pushes the LOS-MoM toward the entire domain MoM. III. NUMERICAL RESULTS To illustrate the advantages of the proposed method, four different examples are outlined. Three of these examples deal with frequently used patches: cross shaped, tripod, and double square loop patches, and the final one contains a patch with a rather complicated shape. The validity of the method is investigated by comparing the obtained results with the results of other methods or measured data. The convergence of the method in

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TABLE IV FOURIER COEFFICIENTS OF THE GUIDED MODES IN A COAXIAL WAVEGUIDE

TABLE V FOURIER COEFFICIENTS OF THE GUIDED MODES IN A SECTORAL WAVEGUIDE

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Fig. 5. Magnitude of the reflection coefficient of the cross-shaped FSS in terms of frequency. A plane wave is vertically incident on the FSS shown in Fig. 4. The problem is solved using three different kinds of basis functions: LOSBFs, entire domain basis functions from BI-RME, and rooftop basis functions.

Fig. 4. FSS with cross-shaped patch printed on a homogeneous substrate with " . (a) Cross-shaped patch, which is printed in each unit cell of the structure. (b) Side view of the FSS. (c) Geometry of the considered sub-patches.

=4

terms of number of basis functions, as well as number of Floquet modes, in comparison with other methods is also presented. A. Cross Shape As the first example, an FSS consisting of cross-shaped (Fig. 4) is conpatches, printed on a substrate with sidered. As it is shown in Fig. 4(c), two rectangles and four wedges are used to set up the LOSBFs for the whole patch. The magnitude of the reflection coefficient is calculated using three different kinds of basis functions: large overlapping subdomain, entire domain (using the BI-RME), and rooftop basis functions (Fig. 5). As one can see, a very good agreement is observed between the results. Note that throughout this paper, the very good agreement of the results of the different techniques may prevent the reader from distinguishing the corresponding curves. Both components of the induced current on the patch are singular at sharp corners. To model this using only the modes of two rectangular sub-patches, one needs to consider a large number of basis functions, which, in turn, causes the weak performance of the method. If some basis functions contain singularities at the sharp corners, a better convergence may be achieved. Here, the wedge-type basis functions are responsible for better modeling the strong variation of the excited current in the sharp corners. In Fig. 6, the effect of these basis functions is investigated. It is observed that the wedge-type basis functions have an strong effect on the results. On the other hand, because of the small dimensions of the wedge sub-patches, considering all the modes of these basis functions deteriorates the convergence in terms of Fourier modes. Fig. 6 shows that taking one

Fig. 6. Comparison of reflection coefficient versus frequency for a vertically incident plane wave on the FSS of Fig. 4. Three different versions of LOSBFs are considered and their results are compared. The dashed line and the line with hollow circular markers coincide. Note that the wedge-type basis functions have a strong effect on the result.

singular TE and one singular TM mode (solid line) into account is sufficient. The convergence of the developed method in terms of the number of basis functions in comparison with the MoM/BI-RME is shown in Fig. 7. To this end, the magnitude of the reflection coefficient at the first resonance frequency is drawn in terms of the considered basis functions. The reason to select resonance frequencies is that the worst convergence usually is observed at resonance points because of the high sensitivity of the results at these frequencies. According to this figure, good accuracy is achieved for the LOS-MoM with 44 basis functions and the MoM/BI-RME with 30 basis functions. These results are used to investigate the convergence of the ). To method in terms of the Fourier truncation order ( show this aspect, a comparison between the convergence of in this paper stands these methods is shown in Fig. 8.

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Fig. 7. Convergence of two different methods: LOS-MoM and MoM/BI-RME in terms of the number of basis functions.

Fig. 9. FSS consisting of curved boundary patches printed on a homogeneous . (a) Curved boundary patch printed in each unit cell of substrate with " the structure. (b) Side view of the FSS. (c) Configuration of the sub-patches.

=4

Fig. 8. Convergence of two different methods: LOS-MoM and MoM/BI-RME in terms of the number of considered Fourier modes.

for the relative error when Fourier orders are considered and is defined by the following: (12) is the magnitude of the reflection coefficient and is an upper limit for the truncation order. As seen from the figures, the convergence in terms of basis functions in the MoM/BI-RME is slightly better than LOS-MoM and the convergence in terms of Fourier truncation order is almost similar in the two methods. However, extracting the introduced basis functions is much easier than extracting the entire domain basis functions. In this study, all the codes for the different methods are written in MATLAB and run on an AMD Dual Core Processor @ 2.61 GHz. To draw the curve in Fig. 5, the overall CPU time for the MoM/BI-RME with as the truncation order 30 basis functions and is 78 s (63 s for the calculation of basis functions and 15 s for the computation of reflection coefficient in 300 frequency points). When the LOS-MoM with 44 basis functions is used, where

the computation time reduces to 21 s (20 s for the calculation of reflection coefficient at 300 frequency points and less than 1 s for the calculation of the basis functions). As deduced from the above results, the computation cost for the basis functions is drastically reduced, while the cost of the FSS analysis increases only slightly. B. Cross Shape With Curved Boundaries and Square Hole In this example, the FSS consists of a more complicated patch with curved boundaries printed on a low-loss RO3010 1.27-mm (50 mil) substrate (Fig. 9). For analyzing this structure using a MoM/rooftop, one needs to divide the patch surface into a large number of small rectangles. This considerably decreases the efficiency of the method, especially when the substrate contains periodic inhomogeneities. In [21], the MoM/BI-RME is successfully applied using the entire domain basis functions. For the LOS-MoM solution, four rectangular, four sectoral, and four wedge shaped sub-patches are used to extract the required basis functions [see Fig. 9(c)]. The transmitted power for a vertically incident plane wave is simulated using both the entire domain basis function and LOSBF. The simulated and also the measured data are shown in Fig. 10. As in the first example, wedge-type basis functions have a remarkable effect on the result. The convergence of the method compared with the MoM/BI-RME in terms of both the number of basis functions and number of Floquet modes is presented in Figs. 11 and 12, respectively. The computation time for the MoM/BI-RME

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Fig. 10. Magnitude of the transmission coefficient for a vertically incident plane wave on the FSS shown in Fig. 9 versus frequency using MoM/BI-RME and LOS-MoM. The simulation results are compared with measurement. Applying wedge-type basis functions has a strong effect on the result.

Fig. 13. FSS consisting of double square loop patches printed on a homoge. (a) Double square loop patch, which is printed in neous substrate with " each unit cell of the structure. (b) Side view of the FSS. (c) Configuration of the utilized sub-patches.

=4

for the LOS-MoM with 68 basis functions and the same truncation order in the Fourier series is 18 s (2 s for calculations concerning the basis functions and 16 s to compute the power transmission coefficient in 300 frequency points). Thus, one has the same effect as in the previous example. The introduced technique requires more basis functions, but its computation time is considerably shorter. Fig. 11. Convergence of two different methods: MoM/BI-RME in terms of the number of basis functions.

Fig. 12. Convergence of two different methods: MoM/BI-RME in terms of the number of Floquet modes.

LOS-MoM

and

C. Double Square Loop

LOS-MoM

and

with 48 basis functions and to obtain the curve in Fig. 10 is 68 s (57 s for the calculation of basis functions and 11 s for the FSS analysis in 300 frequency points). The time

The FSS in the third example is a 2-D lattice of double square loop patches printed on both sides of a homogeneous substrate (Fig. 13). As shown in Fig. 13(c), LOSBFs are obtained using eight rectangular and eight wedge-shaped sub-patches. The reflected power for a vertically incident plane wave is simulated using the MoM/rooftop, MoM/BI-RME, and LOS-MoM (Fig. 14). A good agreement between the results is observed. Solving the problem using the MoM/BI-RME takes nearly 11 s to calculate the basis functions related to each of the patches and 0.13 s to solve for the reflected field in each frequency point. This results in the total CPU time of 74 s to calculate the reflection coefficient in 400 frequency points and draw the curve in Fig. 14. The computation time for the LOS-MoM is 0.17 s for each frequency point and less than 1 s to perform the calculation of basis functions, which leads to 51 s for the whole simulation, i.e., the speed-up of the developed method is not as strong as in the previous examples. Note that because of the flexibility of the definition of large overlapping subdomains, it might be possible to find another discretization with better performance. The same simulation using roof-top basis functions takes 11 s for each

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Fig. 14. Frequency dependence of the reflection coefficient for a vertically incident plane wave on the FSS shown in Fig. 13 is obtained using MoM/rooftop, MoM/BI-RME, and LOS-MoM.

Fig. 16. Frequency dependence of the reflection coefficient for a vertically incident plane wave on the FSS shown in Fig. 16 obtained using MoM/BI-RME and LOS-MoM.

. Therefore, in previous equations, the following changes should be applied:

Fig. 15. FSS consisting of tripod patches printed on a periodic substrate with " and the thickness equals to d : mm. (a) Sample part of the FSS. (b) Unit cell of the structure. (c) Configuration of utilized large overlapping sub-patches.

=4

= 0 64

frequency point. Thus, the MoM/rooftop performs much worse than the LOS-MoM although rooftop discretization seems to be very appropriate for the double square loop structure. D. Tripod An FSS consisting of tripod-shaped patches printed on a periodic substrate is considered as the last example (Fig. 15). The periodic substrate is obtained by drilling holes in a homogenous substrate. As seen in the figure, the two primitive translation vectors of the lattice are and with mm and the skew angle

In this case, three rectangular and three wedge-shaped sub-patches are used for constructing the LOSBFs [see Fig. 15(d)]. The simulated results for reflected power using the MoM/BI-RME and LOS-MoM are shown in Fig. 16. Again, the agreement between the results is quite good. In Fig. 17, the reflected power versus angle of incidence is calculated for both the TE and TM cases at 78 GHz. The computation cost for the MoM/BI-RME with 51 basis functions is 41 s (27 s to obtain the basis functions and 14 s for the FSS analysis), and for the developed method with 78 basis function it is 27 s. Since the geometry is not well suited for the MoM/rooftop, this method was not applied for solving the tripod configuration. From the presented numerical results, one can set up the following conclusion about the numerical complexity of each approach. Using rooftop basis functions leads to a numerical technique with a cost of , where is the number of basis functions and is the number of required Fourier coefficients. The computation cost of the MoM/BI-RME is , where stands for the computation cost required for finding the waveguide modes, i.e., developing the basis functions. Note that and are much smaller for the MoM/BI-RME compared to the MoM/rooftop and that is the main advantage of using this entire domain basis function. The computation cost of the MoM with LOSBFs is similar to the MoM/BI-RME with a negligible , while and are kept in the same order.

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the MoM/BI-RME in terms of both the number of basis functions and also the number of Floquet modes is usually better than the introduced method. Thus, the LOS-MoM requires more basis functions for obtaining the same accuracy as the MoM/BIRME. Furthermore, the way that one chooses the sub-patches can highly affect the convergence of the method. With a reasonable discretization, it is usually possible to outperform the MoM/BI-RME considerably because the reduction of the computation time for the basis function is dominant. In all of the presented test cases, the LOS-MoM outperformed the MoM/BIRME. In this paper, LOSBFs are used only for modeling FSSs. However, they can easily be applied for other planar devices such as microstrip antenna and microstrip circuits. Moreover, in comparison with the MoM/BI-RME to optimize FSS structures, this method is more promising. In optimization, one needs to calculate the results for a wide variety of unit cell configurations, which may contain multiple patches. This makes the computation cost of each test case simulation very high since one needs to solve the waveguide problem for every patch when using the BI-RME, while the introduced method only requires the solution of a small set of simple waveguide problems. APPENDIX CALCULATION OF THE FOURIER COEFFICIENTS OF LOSBFS

Fig. 17. Magnitude of the reflection coefficient, obtained for different angles of incidence using MoM/BI-RME and LOS-MoM at 78 GHz. Different polarizations are assumed: (a) TE case and (b) TM case.

IV. CONCLUSION A new set of LOSBFs is proposed to be used in the MoM. These basis functions may be obtained from waveguide theory, i.e., solving an eigenvalue problem. The procedure is essentially the same as in the MoM/BI-RME approach, but solving the eigenvalue problem is usually much easier because the large overlapping subdomains may have a much simpler shape than the entire domain ones. In comparison with rooftop basis functions, with nearly the same implementation, the proposed basis functions are much more general and allow one to model almost all the practical structures in a very efficient way. Furthermore, the convergence of the LOS-MoM in terms of number of Floquet modes is much better than the MoM/rooftop. Finally, the total computation time for this method is always considerably shorter than for the MoM/rooftop. Compared to the entire domain basis functions, LOSBFs can be extracted much easier without any noticeable extra computation cost, which leads to considerable reduction of the computation time for the basis functions. However, the convergence of

The transverse magnetic field of different modes of some basic waveguides is used for obtaining the required basis functions in the LOS-MoM (and the MoM/BI-RME). For periodic problems, i.e., the periodic MoM, the corresponding Fourier coefficients are required. The procedure for calculating the Fourier coefficients of the basis functions is outlined here. The TM and TE modes of a uniform waveguide can be obtained from the solutions of the 2-D Helmholtz equation with different boundary conditions [25]

TM Modes in the cross section on the boundary

(A1) (A2)

in the cross section

(A3)

on the boundary

(A4)

TE Modes

For waveguides that contain TEM modes, the Laplace equation should be solved

in the cross section on the boundary

(A5) (A6)

The transverse magnetic fields of each mode is found from the following equations:

(TEM modes)

(A7)

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(TM modes)

(A8)

ACKNOWLEDGMENT

(TE modes)

(A9)

The contributions of authors A. Fallahi and A. Yahaghi to this work has been an equal collaboration.

These functions are indeed the required basis functions. The coupling integrals that are introduced in [22] should be used to calculate the Fourier coefficients of these functions

(A10)

(A11) (A12) (A13) represents the transverse where magnetic field vectors of different TM, TE, and TEM modes , , and are defined as of a metallic waveguide. before. As an example, for a circular waveguide in Fig. 3(b), one can write

(A14) (A15) where and . With the aid of the following integral identities and after some algebraic calculations, one obtains the Fourier coefficient of the circular basis functions as in Table II:

(A16) where

is defined previously.

REFERENCES [1] T. K. E. Wu, Frequency Selective Surface and Grid Array. New York: Wiley, 1995. [2] B. A. Munk, Frequency Selective Surfaces Theory and Design. New York: Wiley, 2000. [3] F. O’Nians and J. Matson, “Antenna feed system utilizing polarization independent frequency selective intermediate reflector,” U.S. Patent 3 231 892, Jan. 25, 1966. [4] J. Encinar, “Design of two-layer printed reflectarrays using patches of variable size,” IEEE Trans. Antennas Propag., vol. 49, no. 10, pp. 1403–1410, Oct. 2001. [5] B. A. Munk et al., “Transmission through a two-layer array of loaded slots,” IEEE Trans. Antennas Propag., vol. AP-22, no. 6, pp. 804–809, Nov. 1974. [6] “High bandpass structure for the selective transmission and reflection of high frequency radio signals,” U.S. Patent 5 103 241, Jul. 4, 1992. [7] R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys., vol. 7, no. 1, pp. 37–50, 1967. [8] N. Engheta, “Thin absorbing screens using metamaterial surfaces,” in Proc. IEEE AP-S Int. Symp., San Antonio, TX, 2002, pp. 392–395. [9] G. Kiani, K. Ford, K. Esselle, A. Weily, and C. Panagamuwa, “Oblique incidence performance of a novel frequency selective surface absorber,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2931–2934, Oct. 2007. [10] D. Kern, D. Werner, A. Monorchio, L. Lanuzza, and M. Wilhelm, “The design synthesis of multiband artificial magnetic conductors using high impedance frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 8–17, Jan. 2005. [11] A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1340–1349, May 2008. [12] P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for fss structures,” IEEE Trans. Antennas Propag., vol. 42, no. 9, pp. 1317–1324, Sep. 1994. [13] M. Karkkainen and P. Ikonen, “Finite-difference time-domain modeling of frequency selective surfaces using impedance sheet conditions,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2928–2937, Sep. 2005. [14] I. Bardi, R. Remski, D. Perry, and Z. Cendes, “Plane wave scattering from frequency-selective surfaces by the finite-element method,” IEEE Trans. Magn., vol. 38, no. 2, pp. 641–644, Mar. 2002. [15] M. Sobhy, M. El-Azeem, K. Royer, R. Langley, and E. Parker, “Simulation of frequency selective surfaces (FSS) using 3D-TLM,” in 3rd Int. Comput. Electromagn. Conf. , 1996, pp. 352–357, Conf. Pub. 420. [16] M. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3644–3653, Nov. 2005. [17] R. Maaskant, R. Mittra, and A. Tijhuis, “Fast analysis of large antenna arrays using the characteristic basis function method and the adaptive cross approximation algorithm,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3440–3451, Nov. 2008. [18] M. L. Waller and S. A. Rao, “Application of adaptive basis functions for a diagonal moment matrix solution of arbitrarily shaped three-dimensional conducting body problems,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1445–1452, Oct. 2002. [19] I. A. Eshrah and A. A. Kishk, “Analysis of linear arrays using the adaptive basis functions/diagonal moment matrix technique,” IEEE Trans. Antennas Propag., vol. 53, no. 3, pp. 1121–1125, Mar. 2005. [20] C. Chan and R. Mittra, “On the analysis of frequency-selective surfaces using subdomain basis functions,” IEEE Trans. Antennas Propag., vol. 38, no. 1, pp. 40–50, Jan. 1990. [21] A. Fallahi, A. Yahaghi, H. Abiri, M. Shahabadi, C. Hafner, and R. Vahldieck, “Analysis of frequency selective surfaces on periodic substrates using entire domain basis functions,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 876–886, Mar. 2010.

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[22] M. Bozzi, L. Perregrini, J. Weinzierl, and C. Winnewisser, “Efficient analysis of quasi-optical filters by a hybrid MoM/BI-RME method,” IEEE Trans. Antennas Propag., vol. 49, no. 7, pp. 1054–1064, Jul. 2001. [23] M. Bozzi and L. Perregrini, “Analysis of multilayered printed frequency selective surfaces by the MoM/BI-RME method,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2830–2836, Oct. 2003. [24] R. F. Harrigton, Time–Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [25] G. Conciauro, M. Guglielmi, and R. Sorrentino, Advanced Modal Analysis. New York: Wiley, 2000.

Arya Fallahi received the B.S. degree in electrical and electronics engineering from the Sharif University of Technology, Tehran, Iran, in 2004, the M.Sc. degree in fields and waves communication engineering from the University of Tehran, Tehran, Iran, in 2006, and is currently working toward the Ph.D. degree from the Swiss Federal Institute of Technology (ETH) Zürich, Switzerland. For his Master’s thesis, he has performed theoretical studies on analysis of photonic crystal structures using a transmission line formulation. From September 2004 to March 2005, he was with the Iran Telecommunication Research Center on characterization of optical fibers. From November 2005 to June 2006, he was involved in a microwave project with the University of Tehran, in which he designed microstrip antennas with SIW feeds. In November 2006, he joined the ETH Zürich, where he is currently with the Laboratory for Electromagnetic Field Theory and Microwave Electronics (IFH). His research interests are design and numerical simulation of metamaterials. His current research is about metamaterials in microwave frequencies with focus on radar absorbers, artificial magnetic conductors, and electromagnetic bandgap surfaces.

Alireza Yahaghi received the B.S. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 2002, the M.Sc. degree in fields and waves communication engineering from the Iran University of Science and Technology, Tehran, Iran, in 2004, and is currently working toward the Ph.D. degree from Shiraz University, Shiraz, Iran. Since June 2008, he has been with the Laboratory for Electromagnetic Fields and Microwave Electronics (IFH), ETH Zürich, Zürich, Switzerland, as an Academic Guest. His research interests are numerical methods in electromagnetics and optics. His current research is about applications of planar metamaterials in microwave frequencies.

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Habibollah Abiri (M’05) received the B.S. degree in electrical engineering from Shiraz University, Shiraz, Iran, in 1978, and the D.E.A. and Doctor d’ Ingenieur degrees from National Polytechnique Institute of Grenoble (INPG), Grenoble, France, in 1981, and 1984, respectively. In 1985, he was an Assistant Professor with the University of Savoie, Chambery, France. Since 1985, he has been with the Electrical Engineering Department, Shiraz University, where he is currently a Professor. In 1995, he was on sabbatical leave with the Electrical Engineering Department, Colorado State University, Fort Collins. His research interests include numerical methods in electromagnetic theory, microwave circuits, and integrated optics. Dr. Abiri is a member of the Iranian Association of Electrical and Electronics Engineers.

Mahmoud Shahabadi received the B.Sc. and M.Sc. degrees from the University of Tehran, Tehran, Iran, in 1988 and 1991, respectively, and the Ph.D. degree from the Technische Universitaet Hamburg–Harburg, Hamburg, Germany, in 1998, all in electrical engineering. Since 1998, he has been an Assistant Professor and then an Associate Professor with the School of Electrical and Computer Engineering, University of Tehran. From 2001 to 2004, he was with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada, as a Visiting Professor. He is also a co-founder and CTO of MASSolutions Inc., Waterloo, ON, Canada, a Waterloo-based company with a focus on advanced low-profile antenna array systems. His research interests and activities encompass various areas of microwave and millimeter-wave engineering, as well as photonics. Computational electromagnetics for microwave engineering and photonics are his special research interest. He currently conducts research and industrial projects in the field of antenna engineering, terahertz engineering, photonic crystals, plasmonics, left-handed materials, and holography. Dr. Shahabadi was the recipient of the 1998/1999 Prize of the German Metal and Electrical Industries, Nordmetall, for his contribution to the field of millimeter-wave holography and spatial power combining.

Christian Hafner was born in Zurich, Switzerland, in 1952. He received the Diploma and Ph.D. degree in electrical engineering and the Venia Legendi degree in analytical and numerical calculations of electromagnetic fields from ETH Zürich, Zürich, Switzerland, in 1975, 1980, and 1987, respectively. In 1999, he was given the title of Professor. He is currently the Head of the Computational Optics Group (COG), Laboratory for Electromagnetic Fields and Microwave Electronics (IFH), ETH Zürich. Since 1976, he has been a Scientific Assistant and Lecturer with ETH Zürich, where he has studied different topics of electricity and magnetism. Since 1980, he has been developing the MMP code for numerical computations of dynamic fields for a large range of applications [electrostatics, guided waves on different structures, scattering, electromagnetic compatibility (EMC)]. In addition, he is involved with philosophical and historical concepts of physics and engineering, evolutionary and genetic strategies for optimization, computer graphics and animation, and on “strange” theories (chaos theory, fractals in classical electromagnetics, cellular automata for electromagnetics, irregular grid worlds, etc.).

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Theory of Coupled Resonator Microwave Bandpass Filters of Arbitrary Bandwidth Smain Amari, Member, IEEE, Fabien Seyfert, and Maged Bekheit, Member, IEEE

Abstract—The paper presents a new theory of coupled resonator bandpass microwave filters of arbitrary bandwidth. Constraints on the scattering parameters of lossless, time-invariant, and reciprocal two-port networks, expressed as rational functions of the physical frequency, are presented. Filtering functions exhibiting arbitrarily positioned transmission zeros and locally symmetric or asymmetric responses with respect to the center of the passband, and satisfying these constraints are then introduced. A direct synthesis technique to extract the elements of the transversal equivalent circuit to yield a prescribed response is presented. Response-preserving similarity transformations are then applied to the circuit to force a desired topology. As opposed to the prevailing narrowband approximation, any topology other than the transversal will have off-diagonal elements that depend on frequency. Interesting features that are not found in the narrowband approximation are reported. Examples demonstrating the soundness of the theory are presented. Index Terms—Bandpass filters, broadband, narrowband, resonator filters, synthesis.

I. INTRODUCTION HE SYNTHESIS and design techniques of general coupled resonator filters are based on narrowband circuit models such as the coupling matrix. The narrowband approximation seems to have originated in the desire to design unsymmetrical bandpass filters through low-pass prototypes, as described by Baum [1]. Baum argued that, as long as only a narrowband description is desired, we can neglect the poles and zeros of the driving point or transfer impedance function that are far away from the passband, especially those around the negative band [1]. He also showed that the introduction of frequency-independent reactances combined with a linear frequency transformation, or equivalently a frequency shift, allows the synthesis of low-pass LCX networks with asymmetrical frequency responses [1]. Naturally, the frequency-independent elements are not physical; Baum refers to them as fictitious

T

Manuscript received May 28, 2009; revised May 28, 2009; accepted April 25, 2010. Date of publication July 12, 2010; date of current version August 13, 2010. This work was supported in part by the Natural Science and Engineering Research Council of Canada (NSERC). S. Amari is with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON, Canada K7K 7B4 (e-mail: [email protected]). F. Seyfert is with the Institut National de Recherche en Informatique et en Automatique (INRIA), Sophia-Antipolis 06902, France (e-mail: [email protected]). M. Bekheit is with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON, Canada K7K 5B3 (e-mail: mbekheit@ece. queensu.ca). Digital Object Identifier 10.1109/TMTT.2010.2052874

“mathematical” components [1]. It is, however, known that such components can be successfully approximated over a narrow band of frequencies in the microwave region. They have been used extensively in microwave filter theory. Another equivalent circuit that has been widely used in the design of narrowband microwave bandpass filters is the coupling matrix [2]. Within the narrowband approximation, the coupling coefficients between the resonators are assumed frequency independent and the resonant frequencies clustered around the center of the passband. Deviations of the resonant frequencies from the center of the passband are represented by constant frequency shifts in the diagonal elements of the coupling matrix [3]. Applications of the same model to design filters with asymmetrically placed transmission zeros were reported by many researchers. Unfortunately, the extension of the coupling matrix concept to moderate and wide bandwidths seems to have been hampered by the fact that actual coupling coefficients, instead of being constant, do in fact depend on the frequency. Propagation-based models have been used to design broadband filters by many researchers [4]–[8]. For commensurate TEM transmission lines, the frequency dependence is fully described by Richard’s transformation over a wide frequency band. This feature was central to the seminal work of Wenzel [6]–[8]. These techniques are limited to commensurate lines or structures where simple propagation-based models are available such as coupled uniform sections of waveguide or transmission lines. Another technique to design broadband filters is to cascade low- and high-pass filters [9]. No systematic way to calculate the ideal response of the low- and high-pass sections to achieve an optimal bandpass response is known. To overcome this problem, an active isolator (amplifier) was used in [10]. However, the complexity of designing a broadband filter within this approach is now shifted to that of designing a broadband isolator (amplifier), this is arguably an even more complex task. Another design was presented in [11]. It is based on a combination of sections of lines some of which are twice as long as the remaining ones. A bandstop section was used to suppress the spurious response of a broadband filter. Other techniques based on multimode resonators have also been reported [12]. In this paper, we propose a comprehensive examination of the issue of coupled resonator filters beyond the narrowband approximation. To this end, we start from equivalent circuit that is obtained directly from Maxwell’s equations. This paper then presents a discussion of the constraints that the scattering parameters of a lossless, reciprocal, causal, and time-invariant network must satisfy when these are rational functions of the physical frequency. Filtering functions, which yield scattering

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AMARI et al.: THEORY OF COUPLED RESONATOR MICROWAVE BANDPASS FILTERS

parameters that satisfy these constraints, are then presented. An exact synthesis technique to extract the elements of the transversal admittance or impedance matrix is then introduced. The transversal form is not convenient for an initial design although it is very useful during the optimization of the filter. A key question is how to reduce the transversal circuit to a sparse topology. Within the narrowband approximation, this is achieved by applying a succession of similarity transformations (rotation) to the normalized transversal coupling matrix [3]. The result is another coupling matrix whose coupling coefficients are frequency independent. However, it is known that the coupling coefficients between the resonators do depend on frequency. Interestingly, when a rotation is applied to the transversal form of the equivalent circuit used in this work, the off-diagonal elements are indeed dependent on frequency. II. EQUIVALENT CIRCUIT OF LOSSLESS RESONANT STRUCTURES The synthesis and design of coupled resonator filters requires the knowledge of the port parameters of a given resonating structure in terms of its resonant modes. In existing models, a given filtering structure is divided into smaller elements that act either as resonators or coupling elements. Although this approach is very effective in understanding how narrowband filters can be designed as a set of coupled resonators, it does not provide any explicit information on the functional form of the port parameters of the structure as functions of frequency. As such, it does not lend itself to extension to responses that are not narrowband. In order to handle bandpass filters of arbitrary bandwidths, we need the general form of the port parameters as a function of frequency. This can be achieved by viewing the entire filtering structure as a single unit with well-defined resonant modes. The frequency dependence of the entries of the generalized admittance matrix is derived by expanding the electromagnetic field inside the filtering structure in a series over its resonant modes. The details are not repeated here since exhaustive analyses can be found in [13]–[16]. It can be shown that [13]–[15]

(1)

Here, , , and with , are real constants related to the coupling integrals of the feeding waveguide mode to the th eigenresonance of the entire structure with the input and output coupling elements replaced by short circuits. The resonant frequency of the th such eigenresonance is . The explicit expressions of these coupling integrals can be found in [13]–[16]. An equivalent circuit based on (1) in terms of resonators and inverters was given in [17]. We further assume that for the filters under consideration, only resonances are responsible for the power transport in the passband and its vicinity. For simplicity, and since we are interested only in bandpass filters in this paper, we assume that the term with a pole at zero frequency in the admittance in (1) is not present or can be neglected. Its presence in

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N

Fig. 1. Equivalent transversal circuit of th-order bandpass filter used in this work. Only resonances are taken into account.

N

i

Fig. 2. Equivalent circuit of the branch through th resonator in Fig. 1. The . and capacitance is

C=1

L = 1=!

some structures, such as non-TEM or non-TEM-fed structures, is not discussed in this paper. However, the structure will ex, or one of hibit a first-order transmission zero at dc an odd order, unless the residue matrix of the pole at dc is of rank 1. This is equivalent to the existence of a resonance with a zero resonant frequency such as a parallel-plate mode. We, therefore, assume that the filter has at least one transmission zero at dc. Under these conditions, the equivalent circuit of a bandpass coupled-resonator filter is as given in Fig. 1; it is referred to as the transversal circuit. The equivalent circuit consists of parallel paths between that the source and load. The th path contains an inverter connects the input to a parallel LC resonator with resonant angular frequency . The same resonator is connected to the . The capacitance of output by another admittance inverter the LC circuit is set to unity for convenience; the corresponding value of the inductance is determined from the resonant frequency . The equivalent circuit of the th path is shown in Fig. 2. Note that a dual circuit that is based on the impedance matrix may be more convenient for certain structures such as planar filters [18]. The resulting equivalent circuit, which is the dual of the circuit in Fig. 1, is not discussed here since the governing equations of both circuits are identical. For a given closed or nonradiating structure, the order of the is determined by the number of resonances in the frefilter quency range of interest (passband) in the complete structure when the reference planes at the ports are placed right at the first discontinuities. These are the global resonances of the filter. resonances that follow from a These are also the relevant model-order reduction scheme, e.g., [19]. However, only those resonances that are coupled to the input and output should be included in determining . The assumption that higher order resonances are located far enough from the passband to be ignored is central to the success of the present theory. The location of the higher order resonances determines the maximum achievable bandwidth. Be-

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fore starting a design, the filter designer must make sure that the chosen technology allows the placement of the spurious resonances far enough from the frequency band of interest. For example, using uniform half-wavelength resonators in waveguide technology to design filters of arbitrary bandwidths is simply impossible. The theory presented in this paper does not offer a solution to such a problem. Instead, the waveguide resonators must be modified to push their spurious response away from the intended passband by using metallic ridges for example. The concept of stepped-impedance resonators can also be fruitful in this respect [20]. In certain constrained structures, such as commensurate TEM lines, the presence of the spurious bands can be included in the design process through Richard’s transformation [6]. The response of the structure in this case is a periodic function of the frequency. For other technologies, the nature of the spurious response is more complicated, thereby making the inclusion of the higher order resonances in the design extremely complex if not impossible. The theory presented in this paper does not provide ways to remove the spurious passbands. Instead, it assumes that the higher order resonances are already far enough from the passband and then provides the ideal circuit elements that the design must achieve. A. Analysis of Equivalent Circuit

, regroups the voltages . The source and load voltages are and , respectively. The excitation vector is denoted by since only the source node is assumed excited. In order to avoid writing out large matrices repeatedly, to an , we associate two arbitrary square matrix of size defined by matrices of size

.. .

.. .

..

.

.. .

.. .

(4)

and (5)

and in (3) are defined from The matrices and through (4). The response of these two-port networks will be taken as the 2 2 symmetric scattering matrix whose elements are

The parallel arrangements in Fig. 1 are fed by a unit current source of internal conductance . The voltages across the parallel LC resonator, with resonant frequency , are denoted by . These are related to the voltages at the source and the load and by nodes

(6) where the network matrix

is defined in (3).

III. POLYNOMIAL STRUCTURE OF REAL RATIONAL SCATTERING MATRICES (2) These equations can be rewritten in matrix form as follows:

(3) Here, is an square diagonal matrix whose elements are all zero, except the first and last one, which are . The matrix is equal to the identity matrix. The matrix is a real diagonal matrix such that , . The square symmetric and frequency independent real matrix , of size , contains the inverters in its first and last rows and is not columns, its remaining entries are all zero. Note that transversal coupling matrix identical to the since all of its diagonal elements are zero. It does not contain a source–load coupling element either. The vector , of length

The next objective is the frequency-domain synthesis of Chebyshev-like responses that are compatible with the circuit realizations of Section II. We first investigate the polynomial structure of such 2 2 scattering matrices. Let be a 2 2 rational scattering matrix. We assume the following. (i) derives from a reciprocal admittance matrix . is real, i.e., for is real. (ii) is strictly proper (i.e., its value at infinity is 0). (iii) is lossless. (iv) (v) has no pole at zero. This implies the exclusion of circuits with a direct dc path between the input the output as in some planar circuit [18]. Cases where a pole is at zero, with a residue matrix of rank 2, is present are relevant, but are not discussed here. From (ii)–(v), we get that the MacMillan degree of , and . Now (iii) hence, of , is necessarily an even number, say, indicates that the value of at infinity is the 2 2 identity matrix. Finally, (i) and (ii) yield immediately that is real and reciprocal. These conditions on combined with classical results on the structure of 2 2 unitary rational matrices yields the following general form of [21]: (7)

AMARI et al.: THEORY OF COUPLED RESONATOR MICROWAVE BANDPASS FILTERS

where the following hold. , , and are polynomials with real coefficients of the • . variable • and are monic (coefficient of is equal to unity) of . degree • is the unique Hurwitz polynomial solving the Feldkeller equation (8) •

is an odd polynomial of order strictly less than particular, has a zero at dc of odd multiplicity.

. In

IV. FILTERING FUNCTIONS We first need to determine the polynomials , , and such that the scattering matrix in (7) meets the given specifications of the bandpass filter. These are related to the filtering function by (9)

The construction of the filtering function to meet a given set of specifications builds on the classical approach, which is now summarized. A. Classical Low-Pass to Bandpass Transformation In the large body of literature on analog filters, the filtering function of a bandpass filter is most often taken as a transformation of a low-pass prototype such as a Chebyshev or pseudoelliptic filter. A common transformation, which has its roots in the theory of lumped LC filters, is the mapping (10) where is the center of the passband whose edges are at and . Other transformations have also been reported [22]. Within this approach, the filtering function is first determined in the normalized low-pass frequency variable . Once the filtering function is known, the polynomials and (and ) are determined from the poles-and-zeros technique. In the narrowband case and for even MacMilan degrees, the coefficients of these polynomials are not all real; even in the symmetric case, the ) has purely imaginary coefficients. polynomial (of An example of filtering functions that has found wide application in bandpass microwave filters is the generalized Chebyshev prototype [23]

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filter whose response is obtained from a low-pass symmetric Chebyshev filter of odd degree through this transformation is real, but necessarily has the same number of transmission zeros at dc and infinity. This is clearly shown from the fact that the satisfies what is called magnitude of its transfer function the geometrical-symmetry condition [25]. The corresponding scattering matrix satisfies the condition . Unfortunately, it is rarely the case that a microwave bandpass filter has this property. While this point is of little importance in narrowband microwave bandpass filters, it becomes more noticeable as the bandwidth increases. We, therefore, need a technique to compute filtering functions with one or an odd number of transmission zeros at dc and arbitrarily placed transmission zeros at finite frequencies. A new class of filtering functions, which meet this requirement, is introduced in this paper. B. New Filtering Functions for Broadband Filters In view of Section III, the filtering functions we are interested in must satisfy a number of requirements. (i) The polynomials and have real coefficients in . (ii) The degree of is even and is an odd polynomial. . (iii) In particular, has at least one zero at In order to derive filtering functions that meet these requirements and present a passband characteristic for frequencies in (and, of course, also in the reflected band ), the two following frequency transformations are introduced: (12a) and (12b) These two mappings satisfy (13) The function in (11) is modified to include a simple transmission zero at and generate two passbands and instead . The result is the following of the single normalized band filtering function:

(11) (14) where are the normalized locations of the transmission zeros in the complex plane, including infinity. Recursion formulas to calculate are known [24]. It is important to keep in mind that not all bandpass responses can be mapped onto a low-pass prototype through (10). In fact, the transformation in (10) produces only a limited class of bandpass responses from a low-pass prototype. Indeed, a bandpass

Here, the transmission zeros , are not in and and occur in complex pairs in the complex -plane. in (14) satisfies the following The filtering function properties. (i) It is single valued. . (ii) It is a rational function of , i.e.,

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(iii) The denominator of

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is (up to a normalization factor)

(15) (iv) The roots of are . When complex, the roots appear in complex conjugate pairs. is an even polynomial of degree . (v) are all in or . (vi) The roots of (vii) is equiripple (between the value 1 and 1 in and . To compute this filtering function, once the positions of the are known, we establish a recursion retransmission zeros lation for its numerator . To this end, we introduce the no. The following recursion relations hold: tation

V. SYNTHESIS OF BANDPASS FILTERS The basic idea of the synthesis is to write the admittance or parameters in two ways as in the narrowband case [28]. The first one is obtained directly from the equivalent circuit in Fig. 1, i.e., (19) The second one is obtained from the scattering parameters that meet the specifications as given in (7). The admittance paramecan be expressed in terms of the polynomials in (7) as ters

(20) The polynomial is determined from the Feldkeller (8). To deal with numerical problems that arise for high-order filters, the Feldkeller equation is solved in the transformed variable , as explained in the details in the first two examples. The entries of the matrix in (20) are now rewritten as partial fraction expansions, which, thanks to the lossless and reciprocal nature of , takes the following form: (21)

(16) With

and , follow The resonant frequencies and the constants by matching the residues and poles of the expansions in (19) and (21): (22)

(17) Numerical instabilities arise for very narrow bandwidths where the coefficients of can be very large. To deal with these numerical pathologies, we introduce new polynomials in the through . Recursion forvariable mulas for the new polynomials can be deduced from those of ’s and are given by (18) as follows:

(18) The recursion relations in (18) are very similar to those of the narrowband case given in [13]. Naturally, it is possible to determine the filtering function by numerically locating its poles and zeros directly in the physical frequency [26], [27]. However, we are not aware of closed-form expressions such as (14).

This completes the synthesis of the network. Note that the signs of and are not uniquely determined by (22). A negative sign through the th resonator can be attributed either to or without affecting the response of the circuit. In order to avoid any confusion, we always take the coupling coefficients to the as positive. In what follows, the more common nosource tation and instead of and will be used to denote the inverters or coupling coefficients from the source and load to the resonators of the transversal circuit. VI. TRANSFORMATION OF TRANSVERSAL CIRCUIT The equivalent circuit shown in Fig. 1 provides a representation of a bandpass coupled-resonator filter within its global eigenresonances [29]. This representation is valid for any coupled-resonator filter, and not necessarily of a narrowband response, as long as only global resonances are taken into account and the zero-frequency pole (e.g., parallel-plate mode in planar circuits) in (1) is not present. Although the transversal form is very useful as an optimization tool, a circuit with a sparse topology, such as folded or inline when applicable, is more intuitive. Sparse topologies result when localized resonances are used as basis. It is well known that localized resonances provide an accurate representation in the narrowband approximation, as well as a basis for an initial design. A very important question is whether the transversal circuit can be transformed into a sparse topology when the bandwidth

AMARI et al.: THEORY OF COUPLED RESONATOR MICROWAVE BANDPASS FILTERS

is not necessarily narrow. In order to answer this question, we consider two classes of transformations that do not change the response of the circuit.

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sponse of the circuit is preserved. The corresponding rotations are represented by orthogonal matrices of the form (25)

A. Scaling The scaling of the transversal circuit consists in multiplying the th row and the th column of the matrix in (3) by a real constant . Note that, in order to preserve the symmetry of , both the column and row are multiplied by the the matrix same constant. The scaling operation transforms the matrix according to [30], [31]

Here, is an orthogonal matrix of size and is a square real matrix of size . From the form of the matrix and the orthogonality of the matrix , it follows that is itself orthogonal, i.e., identity matrix. Under the orthogonal transformation in (25), the state variables in (3) are transformed as follows: (26)

(23)

From the form of the matrix , it can be easily seen that (27)

where

.. .

.. .. .

.

.. .

(24)

The matrices in (23) do not commute in general, it is important to keep the correct order in which the multiplication is performed. We also do not scale the source and load nodes beyond the impedance normalization that is already assumed, i.e., . Obviously, the scaling operation does not change the topology of the matrix . In other words, if an element of this matrix is zero before the scaling, it remains zero after the scaling operation. B. Rotations Rotations, or similarity transformations through orthogonal matrices, have been used extensively in the theory of coupled resonator filters within the narrowband approximation. A series of such rotations is used to force a desired topology starting from a canonical coupling matrix such as the transversal or the folded topology [3]. The resulting sparse topology, such as inline, is then implemented directly by arranging physical resonators in accordance with the topology. It was recently pointed out that an alternative interpretation of similarity transformations allows the representation of coupled-resonator filters by coupling matrices whose topology does not correspond to the physical arrangement [29]. In particular, the transversal coupling matrix results when the global eigenmodes are used as basis. Within the narrowband approximation, the transversal coupling matrix provides a universal representation of coupled resonator filters [29]. The class of similarity transformations that leave the response of the transversal circuit unchanged is the same as the one used in the narrowband approximation. As long as the matrix elements in (6) are not changed under the transformation, the re-

When (26) and (27) are used in (3), we get a new representation of the original circuit in which the original matrix is transformed according to the standard relation

(28) Here, we used the fact that, by construction of the matrix , the is not affected by the similarity transformation. matrix Although some structures may have other symmetries, these two operations, scaling and similarity transformations, are the only ones considered here. A crucial question is then the following: Do these two operations commute? In other words, if in (3) and apply first a rotation we start from a given matrix and then a scaling would the result be identical to first scaling the same matrix and then applying the rotation? Mathematically, this is equivalent to whether the following equation holds: (29) , the matrices It turns out that, except for trivial forms of and do not commute in general. As will be shown later, this simple observation has very important consequences: reversing the order in (29) results in frequency dependent coupling coefficients even in the narrowband approximation. C. Class of Transformations Let us assume that the matrix form

describing a filter is of the

(30) is the matrix defined in (3), and where symmetric real positive definite matrices, and

are is a real sym-

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metric matrix of size . The matrices and are related to and , respectively, through (4). We would like to determine the set of transformations that leave the response of the network unchanged by reducing it to a is real and symmetric, transversal circuit. Since the matrix the diagit can always be diagonalized. Let us denote by . We also denote onal matrix that contains the eigenvalues of by a square orthogonal matrix whose columns are the nor. We now apply the similarity transmalized eigenvectors of to the matrix in (30) to get formation represented by

while preserving the response. We conclude that the response is preserved under transformations of the form

(34) where and are two orthogonal real matrices (rotations) and is a scaling operation. The overall operation in (34) involves transformations, which are represented by matrices of the form (35)

(31)

Using this matrix, we can rewrite (34) as (36)

and . We now scale the matrices in (30) by . This is equivathe square roots of the eigenvalues of lent to multiplying (31) from the left and right by the matrix with

. Under this operation, the matrix is invariant and transformed into

becomes

. Therefore, (31) is

(32) in (32) is real and symmetric and can also The matrix whose be diagonalized. We introduce the diagonal matrix and whose diagonal elements are the eigenvalues of columns are the corresponding normalized eigenvectors. If the in (32) is multiplied by from the left and by matrix from the right, we get

It is straightforward to show that (35) implies (37) The result in (37) is very interesting and equally important. It is [32]. Since the singular value decomposition of the matrix has strictly nonzero diagonal elements, the diagonal matrix is not singular, i.e., invertible. We therefore reach the matrix the very important result that the response is invariant under the is any nonsingular, i.e., inverttransformation (36) where . It is also important to ible, real square matrix of order apnote the form of (36) and especially how the matrix pears in it. This matrix is not necessarily orthogonal; its transpose is not necessarily equal to its inverse. Under a similarity transformation, represented by a change of coordinate matrix , a matrix transforms according to . It reduces to (27) when is orthogonal. The transformation in (36) is not of this form because of the inclusion of scaling operations in the set of acceptable transformations. This has very important consequences in coupled-resonator filter theory, especially in taking into account the frequency dependence of the coupling coefficients. Pure rotations, or orthogonal transformations, and scaling operations are particular cases of (37). A scaling operand in (37) equal to the ation corresponds to having and either identity matrix. A rotation corresponds to having or equal to the identity matrix. Also, adjusting the signs of the inverters connected to a given resonator is represented by an orthogonal matrix, which differs from an identity matrix only by the signs of selected elements.

(33) VII. SYNTHESIS EXAMPLES It is obvious that (33) has the same form as the transversal cirin (3) is equal to , cuit in (3). The diagonal matrix in (30) is assumed positive defwhich is well defined since in (3) is equal to the matrix in (33). inite. The matrix We have, therefore, succeeded in reducing the system matrix in (30) to the general transversal form in (3) while preserving the response of the system. Inversely, the transversal form in (3) can be brought into the form in (30) by reversing the order of the transformations. For the general case, we need two rotations and one scaling operation to reduce (30) to (3), and vice-versa,

In the following, we present a few examples that highlight the effect of wider bandwidths and the transformation of the transversal circuit to sparse topologies with frequency-dependent coupling coefficients. A. Sixth-Order Chebyshev Filter The first example is a sixth-order Chebyshev filter with 20-dB return loss and a passband extending from 4 to 10 GHz (86% relative bandwidth). The filter has one transmission zero at dc and the remaining ones at infinity.

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of order , solving the equation directly in terms of would reinstead of quire finding the roots of a polynomial of degree when is used. More specifically, we need the roots of (41) For a filter of order 6, this equation has 12 roots. These are

and These roots are then related to their positions plane by inverting the transformation . in the From each root , we calculate the roots Fig. 3. Response of transversal circuit (54), inline circuit (55), and filtering function. The three are indistinguishable.

1) Filtering Function and Scattering Parameters: The filtering function is generated from the recurrence relations in (18). A plot of the scattering parameters is shown in Fig. 3. These parameters are obtained by retaining only the left-half is equal, within a norplane poles of (8). The numerator of malization constant, to the denominator of the filtering function given in (15). The normalization constant is determined as in the narrowband case, i.e., by forcing the return loss to be equal or to the specified value at the edge of the passband . The frequencies are measured in units of 1 GHz and the ’s are actual frequencies and not angular frequencies. With and , the generalized low-pass variable is

(42) If we keep only those roots with a negative real part (stable), we have the following 12 roots (these are the roots of the deand in (7)): , nominator of , , , , and . and in (7) can be formed from The denominator of these roots yielding

(43) (38) The parameters in the recursion relation are , , , and . From the : recursion relation, we get the following expressions of

Since we have a first-order transmission zero at , the numerator in (7) is proportional to . The final expression of is, therefore, (44) The constant can be determined from the condition that at the edge of the passband, say, , we have (45)

(39) The filtering function is constructed from the relation

With the present specifications, we get . of in (7) is found from the roots of The numerator the filtering function. These are all on the imaginary axis in the and negative fretwo passbands at positive frequencies quencies . We get

(40) To construct the scattering parameters, we need to solve the Feldkeller equation and keep only the roots that are in the lefthalf -plane. Here it becomes important to realize that we can locate these roots by solving this equation first in the generalized low-pass variable , and not in the variable . For a filter

(46) The scattering parameters obtained from these polynomials are shown in Fig. 3. All the specifications are met.

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2) Admittance Matrix: With the polynomials , , and known, we can determine the admittance matrix through (20). The common denominator of the admittance parameters is

the source and different resonators. With the convention that these coupling coefficients be taken as positive, we get , , , , , , , , , , , and . is Similarly, the partial fraction expansion of

(47) The numerator of

is (52) The partial fraction expansion of cuit is

as obtained from the cir-

(48) Finally, the numerator of

is (53) (49)

3) Transversal Circuit: To get the elements of the transversal circuit, we use the partial fraction expansion of the admittance matrix in the form (21) to get

(50) This expression must be identical to the one obtained form the circuit in (19)

(51) By direct comparison of these two expansions, we get the values of the resonant frequencies and coupling coefficients between

By direct comparison of these two expansions, and using the above, we get , , values of , , , and . of the With these values, we form the network matrix transversal circuit in (3), shown in (54) at the bottom of this page. The response of this network versus frequency is shown in Fig. 3. The equiripple character of the in-band response and the transmission zero at dc are evident. The response is not symmetric about its center because of the larger number of transmission zeros at infinity. 4) Inline Circuit: The transversal circuit can be transformed into an inline topology, i.e., direct-coupled resonators by applying a series of transformation, as given in (36). A very interesting result that emerges from this is the dependence of the coupling coefficients on frequency. For this particular case where only one pole appears at dc, the only exact inline topology involves off-diagonal elements (coupling coefficients), which are inversely proportional to frequency. More specifically, an inline network with a matrix of (55), shown at the bottom of

(54)

AMARI et al.: THEORY OF COUPLED RESONATOR MICROWAVE BANDPASS FILTERS

this page, form results. By comparing (55) and (54), we see in (34) cannot that the required transformation matrix change the linear diagonal term in . This implies that it must be orthogonal, exactly as in the case of narrowband filters [3]. The procedure in this case, which is well documented in [3], is not repeated here. The corresponding rotation matrix is given by (56), shown at the bottom of this page. By using this matrix in (36), we get the following parameters of the inline net, , , work: , , , , , , , and . Its response is also plotted in Fig. 3, but is indistinguishable from that of the transversal network in (54). The following few points are worth mentioning. (1) The resonant frequencies of the resonators in the inline configuration are , , , , , and . They are not all equal. This implies that this bandpass response cannot be obtained from a symmetric low-pass prototype through the classical mapping in (10). (2) The off-diagonal elements (coupling coefficients) are inversely proportional to frequency. Note, however, that the coupling to the input and output does not depend on frequency. This results from the choice of the reference planes at the input and output, as discussed by Kurokawa [15]. It can be shown that the case when these two coupling coefficients are proportional or inversely proportional to the frequency is equivalent to a pole at dc or infinity in the admittance matrix. This case is not treated here. (3) The signs of the off-diagonal elements can be changed without affecting the insertion and return loss of the filter.

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At this point, one might wonder whether it is possible to transform the transversal network into an inline one where the off-diagonal elements are linear functions of frequency. This is not possible for the present response because of the number of transmission zeros at dc that would result. Indeed, if an inline direct-coupled topology with such coupling coefficients were possible, a transmission zero would appear whenever a direct coupling coefficient vanishes. This would happen at dc for each of the five coupling coefficients in the inline configuration. However, our initial response has only one transmission zero at dc. The two responses are not identical. This statement has been verified on actual suspended stripline (SSL) filters where it was not possible to match the full-wave response to a purely inline network with coupling coefficients that are linear functions of frequency. However, if one is interested only in the response far away from dc, inline solutions with either capacitive or inductive or a mixture of capacitive and inductive coupling elements are possible, especially as the bandwidth decreases. This situation is well known in direct coupled cavity filters where both inductive and capacitive irises can be used. It is also well known that the out-of-band response of a filter with inductive irises is different from that of a filter with capacitive irises. Within the present theory, this is reflected in the number of transmission zeros at dc and infinity. 5) General Topology: Naturally, it is possible to transform the transversal circuit into a circuit with a full matrix, but with the same response. The off-diagonal elements will depend on the frequency as , where and are two constants. In some cases, it is possible to use the variation of the off-diagonal elements to implement transmission zeros in purely inline configurations. This happens when the direct coupling co. efficient vanishes at a specific frequency given by

(55)

(56)

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For the narrowband case, filters based on this observation have been reported [33], [34]. The direct coupling coefficient is then viewed as a mixed coupling with two parallel coupling paths. One path is magnetic and the other electric. A transmission zero results when the electric and magnetic coupling paths cancel each other. However, this approach frequently increases the insertion loss and reduces the power-handling capability of the filter. If the bandwidth of the filter is not too wide, the coupling coefficients can be approximated by a Taylor expansion around the center of the passband [35]. The slopes in those expansions can be used to control the symmetry of the response, e.g., [35]. Circuits that include capacitive, inductive or “nonsimple” coupling coefficients were given for simple topologies by Wenzel [27]. A general procedure to transform the transversal circuit to one with off-diagonal elements which depend on frequency in “nonsimple” patterns requires solving (36) for . This has been done successfully the nonsingular matrix and exhaustively for low-order examples. An extension of the technique presented in [36] to this cases is under investigation. In the case where the off-diagonal elements are all of one type, i.e., all depend on frequency linearly or are all inversely proportional to frequency, the transformation of the transversal circuit reduces to applying rotations as in the narrowband case. A complete solution for such cases is well known [36]. B. Fourth-Order Filter With Two Transmission Zeros The second example is a fourth-order filter with two transmission zeros at finite frequencies (in addition to the transmission zero at dc). The passband extends from 8 to 12 GHz (40% relative bandwidth) with an in-band return loss of 20 dB. The transmission zeros at located at 7 and 13 GHz. The ripple constant is . The elements of the transversal circuit of this fourth-order filter can be obtained as in the previous example. The obtained , , , values are , , , , , , , , and . With these values, we form the network matrix of the transversal circuit in (3), shown in (57) at the bottom of this page. The response of this network is shown in Fig. 4. It is indistinguishable from the scattering parameters that are obtained from the filtering function.

Fig. 4. Response of transversal circuit (57), inline circuit (58), and filtering function. The three are indistinguishable.

1) Circuit Transformation: The transversal circuit can be transformed into other topologies by applying a series of transformations, as in the previous case. In this specific case, it is well known that a quadruplet implements such a symmetric response in the limit of narrow bandwidths. It is, therefore, interesting to see whether the same topology (quadruplet) can implement a broadband response with two symmetrically placed transmission zeros, as in the present example. We consider the case where the off-diagonal elements are inversely proportional to frequency and seek a network matrix of the form of (58), shown at the bottom of the following page. In the narrowband approxivanishes for a symmetric response. This is mation, the term not the case as the bandwidth increases. For the given specifica, tions, the following values are obtained: , , and , , , , , and . The response of this network is also plotted in Fig. 4, but is indistinguishable from that of the transversal circuit. From these values, the following few points are worth mentioning. (1) The resonant frequencies of the resonators in this topology are , , , and . They are not all equal. (2) A coupling between resonators 2 and 4 is necessary, but is much weaker than the other ones. (3) The sign of the coupling between resonators 1 and 4 is negative (assuming that the other relevant ones are taken

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positive). Realization of such a filter requires a phase reversal or a coupling of a different nature. (4) If the relative bandwidth of the filter decreases, the extra goes to zero. In the same limit, coupling coefficient the resonant frequencies of the resonators all approach the center frequency of the passband. VIII. REALIZATION EXAMPLES

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Fig. 5. Circuit used to implement broadband sixth-order filter taken in [37]. : nH, L : nH, L : nH, L : nH, C (L : pF, C : pF, C : pF). : pF, C

= 1 28 = 1 419 = 0 976 = 1 30 = 0 522 = 0 687 = 0 902 0 936

=

We give two examples of broadband filters that can be accurately described by the theory in this paper. A. Sixth-Order Filter A sixth-order filter with three transmission zeros at dc was recently presented in [37]. The filter was realized in multilayer polymer technology. The bandpass covers the frequency range from 3.2 to 9.76 GHz and corresponds to the 3-dB unlicensed Federal Communications Commission (FCC) band. The in-band return loss is 18.5 dB. 1) Transversal Circuit: The network matrix of the transversal network is of the same form as the one in (54). Application of the synthesis technique introduced in this paper yielded the fol, , , lowing values: , , , , , , , , , , , , , , and . 2) Inline Circuit: The inline circuit used to realize this filter is shown in Fig. 5, which was taken from [37]. In order to bring this circuit into the general form that can be eventually represented transversal circuit, we introduce unit by an inverters at the input and the output. These inverters do not affect the insertion and return loss of the filter. It is straightforward to show that the loop currents satisfy a matrix equation of the form (54). It can be brought into the transversal form by applying two rotations and one scaling operation, as discussed previously. Inversely, one can start from the transversal circuit and apply the inverse of the two rotations and the inverse of the scaling operation to obtain the inline circuit. Another approach for the matrix of the present circuit is to first determine the filter from the transversal circuit and then extract the elements of the circuit in Fig. 5 analytically. A comparison of the response of the transversal circuit with the response of the circuit in [37] is shown in Fig. 6. The dashed lines show the response of the transversal circuit in this paper and the solid lines show the response of the circuit in [37]. It is evident that an excellent agreement is obtained between the

Fig. 6. Response of circuit in Fig. 5 with values given in the text (dashed lines) and the circuit given in Fig. 1 in [37] (solid lines).

two circuits. This confirms the validity of the model and the synthesis technique. This filter was realized and measured with good results in [37]. This example provides a good illustration of the effect of the transmission zeros at dc. It is important to know beforehand the total number of these transmission zeros since they affect not only the stopband, but also the details in the passband. For example, if we assume that only one pole is present at dc in a filter with the same passband and in-band return loss, we obtain the response shown as dashed lines in Fig. 7. The solid lines show the response of the inline filter designed in [37]. It is obvious that the agreement between the two responses is not as good as in Fig. 6. If we assume that we have five transmission zeros at dc, instead of three, we get the response shown as dashed lines in Fig. 8. The solid lines again show the response of the inline circuit in [37]. The differences between the two responses in Figs. 7 and 8 demonstrate that the transmission zeros at dc have a strong influence on the performance of broadband filters. Another consequence of these results is the impossibility of mapping the response of this filter from a classical Chebyshev filter

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Fig. 9. Layout (top view and cross section) of fifth-order SSL bandpass filter.

Fig. 7. Response of bandpass filter with only one attenuation pole at dc (dashed lines) and that of the inline circuit in Fig. 1 in [37] (solid lines).

Fig. 8. Response of bandpass filter with five attenuation pole at dc (dashed lines) and that of the inline circuit in Fig. 1 in [37] (solid lines).

by using the classical low-pass-to-bandpass mapping in (10). Such a mapping always results in the same number of transmission zeros at dc and infinity, in this case, 6. However, we have already established that this filter has exactly three transmission zeros at dc.

B. Fifth-Order SSL Bandpass Filter The second example is a fifth-order SSL bandpass filter. It exhibits one transmission zero at dc, and a passband from 5 to 9 GHz with an in-band return loss of 14.5 dB. The remaining transmission zeros are at infinity. This example is used to test the localization of the individual resonators in this inline configuration, as well as the veracity of the assumption that only one transmission zero at dc is possible in this structure. The layout of the filter is shown in Fig. 9 with a top view and the cross section. The dielectric constant of the substrate is and its thickness is mm. The width of the metallic enclosure is 5 mm and height of the air-filled regions above and below the substrate is 2 mm. Each resonator in Fig. 9 consists of two low-impedance sections that act as quasi-lumped capacitive parts, which are connected by a high-impedance section that acts as quasi-lumped inductance. Stronger coupling between two elements of the filters is implemented by placing one element on one side of the substrate and the other on the opposite side. 1) Transversal Circuit : The transversal network matrix is of the form of (59), shown at the bottom of this page. For the given specifications, we obtain the following values: , , , , , , , , , , , , , , and . 2) Inline Circuit: The transversal circuit can be transformed into an inline topology with a network matrix of the form of

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Fig. 10. Response of filter in Fig. 9. The dashed lines show the response of the matrices in (59) and (60) with one transmission zero at dc, and the solid lines are the response of the filter as obtained from Sonnet. The dotted–dashed lines show the prototype with three transmission zeros at dc.

(60), shown at the bottom of this page. For this filter, the pa, rameters of this inline circuit are , , , , and . Note that the resonant frequencies of the resonators in the inline configuration are not all equal although the differences are small compared to the bandwidth of the filter. The initial design of the filter starts by forcing the resonators to have the resonant frequencies given in (60). The dimensions of the coupling elements between them are then calculated by forcing their eigenresonances of the structure to be identical to those obtained from the equivalent circuit. In fact, this process yields to the same expressions given in [38]. The initial design is then optimized by exploiting the transversal circuit as an intermediary step. The dimensions of the optimized filter are mm, , mm, mm, mm, mm, mm, and mm. The negative values of , , and indicate that the two metal layers are overlapping. The response of the matrices in (59) and (60) is shown in Fig. 10 via the dashed lines. The full-wave simulation, as obtained from the commercial software package Sonnet, is shown

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Fig. 11. Response of filter in Fig. 9 (dashed lines) and a transversal circuit of the form in (59) with additional transmission zeros away from the passband.

via the solid lines. Relatively good agreement between the two, especially in the passband, is achieved. The two responses deviate in the stopbands because of the presence of higher order modes. The behavior of the response at very low frequencies is purposely shown in Fig. 10. The dotted–dashed lines show the response of a filter with three transmission zeros at dc. It is obvious that this filter exhibits a single transmission zero at dc. Consequently, this response cannot be obtained from a classical Chebyshev low-pass prototype through the classical low-pass-to-bandpass frequency transformation in (10). This also implies that an inline direct-coupled resonator arrangement where the coupling coefficients are linear functions of frequency cannot accurately represent this filter for it would generate more than one transmission zero at dc. It is, however, possible to match the response of this filter to a transversal circuit of the form in (59), as shown in Fig. 11. The transversal circuit achieves this by mimicking the effect of higher order modes through the addition of transmission zeros away from the passband [35]. The approximation of the filter response by the transversal circuit, which is equivalent to the approximation of the scattering parameters by rational functions of the physical frequency, is very useful in identifying and optimizing microwave filters [17], [29].

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TABLE I FRACTIONAL CHANGE IN INLINE PARAMETERS OF FIFTH-ORDER FILTER VERSUS PERTURBATION OF GEOMETRIC DIMENSIONS

The inline circuit assumes that the resonators are well defined, or localized, in space. To test the localization of the resonances in this filter, we perturb the controlling dimensions , , , , , and in Fig. 9 and determine the variations in the elements of the inline circuit in (60). Each dimension is separately perturbed by 20 m. The results are summarized in Table I. From these results, we see that the resonators are still relatively well localized for this filter since a geometric parameter affects mostly the resonators that are directly connected to it. For exaffects mainly the ample, changing the size of the first gap resonant frequency of the first resonator and the coupling . The localization of the resonators allows simple initial designs that are based on extension of the techniques used in narrowband filters such as the coupling bandwidth. Finally, one should keep in mind that the degree of localization depends on the compactness and the layout of the structure. IX. CONCLUSIONS A theory of coupled resonators bandpass filters that is valid for both narrowband and broadband was presented. A welldesigned bandpass filter with the first spurious response sufficiently removed in frequency from the primary passband is represented by a transversal circuit consisting of resonators that are connected to the input and the output by constant inverters. The transformation of the transversal circuit to sparse topologies such as inline or folded leads inherently to frequency-dependent coupling coefficients. The class of transformations that do not change the response of the circuit is shown to include orthogonal transformations (rotations) and scaling operation. The transformation of network matrices, such as the coupling matrix, involves matrix multiplication from the left by the transpose of a nonsingular square matrix and by the same matrix from the right. This is a significant difference from the traditional similarity transformations of linear systems where the inverse of the matrix multiplies the network matrix from the left. New filtering functions, which are valid for a wider class of bandpass functions with arbitrarily placed transmission zeros, were introduced. It was shown that the number of transmission zeros at dc can have a significant effect on the response of broadband microwave bandpass filters. Examples of filters of orders four, five, and six were presented. REFERENCES [1] F. Baum, “Design of unsymmetrical bandpass filters,” IRE Trans. Circuit Theory, vol. CT-4, pp. 33–40, Jun. 1957. [2] I. C. Hunter, Theory and Design of Microwave Filters. London, U.K.: IEE Press, 2000.

[3] 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. [4] R. Levy, “Theory of direct-coupled cavity filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-15, no. 6, pp. 340–348, Jun. 1967. [5] L. Young, “Direct-coupled cavity filters for wide and narrow bandwidths,” IEEE Trans. Microw. Theory Tech., vol. MTT-11, no. 6, pp. 162–178, Jun. 1963. [6] R. J. Wenzel, “Exact design of TEM microwave networks using quarter-wave lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-12, no. 1, pp. 94–111, Jan. 1964. [7] R. J. Wenzel, “Synthesis of combline and capacitively loaded interdigital bandpass filters of arbitrary bandwidth,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, no. 8, pp. 678–686, Aug. 1971. [8] M. C. Horton and R. J. Wenzel, “General theory and design of optimum quarter-wave TEM filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 5, pp. 316–327, May 1965. [9] W. Menzel, “Broad-band filter circuits using an extended substrate transmission line configuration,” in Eur. Microw. Conf. Dig., Oct. 1992, vol. 1, pp. 459–463. [10] R. Gomez-Garcia and J. I. Alonso, “Systematic method for the exact synthesis of ultra-wideband filtering responses using high-pass and low-pass sections,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3751–3764, Oct. 2006. [11] C. W. Tang and M. G. Chen, “A microstrip ultra-wideband bandpass filter with cascaded broadband bandpass and bandstop filters,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 11, pp. 2412–2418, Nov. 2007. [12] L. Zhu, S. Sun, and W. Menzel, “Ultra-wideband (UWB) bandpass filters using multiple-mode resonator,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 796–798, Nov. 2005. [13] R. Muller, “Theory of cavity resonators,” in Electromagnetic Waveguides and Cavities, G. Goubeau, Ed. New York: Pergamon, 1961, ch. 2. [14] K. Kurokawa, An Introduction to the Theory of Microwave Circuits. New York: Academic, 1969. [15] K. Kurokawa, “The expansion of electromagnetic fields in cavities,” IRE Trans. Microw. Theory Tech., vol. MTT-6, no. 4, pp. 178–187, Apr. 1958. [16] G. Conciauro, M. Guglielmi, and R. Sorrentino, Advanced Modal Analysis, CAD Techniques for Waveguide Components and Filters. New York: Wiley, 2000. [17] M. Bekheit, S. Amari, and W. Menzel, “Modeling and optimization of compact microwave bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 2, pp. 420–430, Feb. 2008. [18] T. Okoshi, Planar Circuits for Microwaves and Lightwaves. New York: Springer Verlag, 1985. [19] L. Kulas and M. Mrozowski, “Reduced-order models in FDTD,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 10, pp. 422–444, Oct. 2001. [20] M. Morelli, I. Hunter, R. Parry, and V. Postoyalko, “Stopband performance improvement of rectangular waveguide filters using steppedimpedance resonators,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 7, pp. 1657–1664, Jul. 2002. [21] V. Belevitch, “Topics in the design of insertion loss filters,” IRE Trans. Circuit Theory, vol. CT-2, pp. 337–346, Dec. 1955. [22] G. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures. Norwood, MA: Artech House, 1980. [23] R. J. Cameron, C. M. Kudsia, and R. Mansour, Microwave Filters for Communication Systems. New York: Wiley, 2007. [24] S. Amari, “Synthesis of coupled resonator filters using a gradient-based optimization technique,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1559–1564, Sep. 2000. [25] N. Balabanian, Network Synthesis. New York: Prentice-Hall, 1958, p. 413. [26] G. C. Temes and S. K. Mitra, Modern Filter Theory and Design. New York: Wiley, 1973. [27] R. J. Wenzel, “Exact design of wideband equal-ripple bandpass filters with non-adjacent resonator couplings,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1976, pp. 125–127. [28] A. Atia and A. E. Williams, “New types of bandpass filters for satellite transponders,” COMSAT Tech. Rev., vol. 1, pp. 21–43, Fall, 1971. [29] S. Amari and M. Bekheit, “Physical interpretation and implications of similarity transformations in coupled resonator filter design,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 7, pp. 1139–1153, Jul. 2007.

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[30] E. A. Guillemin, Synthesis of Passive Networks. New York: Wiley, 1957. [31] R. J. Wenzel, “Theoretical and practical applications of capacitance matrix transformations to TEM network design,” IEEE Trans. Microw. Theory Tech., vol. MTT-14, no. 12, pp. 635–647, Dec. 1966. [32] G. H. Golub and C. F. Van Load, Matrix Computations. London, U.K.: The Johns Hopkins Univ. Press, 1989. [33] K. Ma, J. G. Ma, K. S. Yeo, and M. A. Do, “A compact size coupling controllable filter with separate electric and magnetic coupling paths,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1113–1119, Mar. 2006. [34] W. Menzel and A. Belalem, “Quasi-lumped suspended stripline filters and multiplexers,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 10, pp. 3230–3237, Oct. 2005. [35] S. Amari, M. Bekheit, and F. Seyfert, “Notes on bandpass filters whose inter-resonator coupling coefficients are linear functions of frequency,” in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, Jun. 2008, vol. 5, pp. 1207–1210. [36] R. J. Cameron, J. C. Faugere, F. Rouillier, and F. Seyfert, “Exhaustive approach to the coupling matrix synthesis problem and application to the design of high degree asymmetric filters,” Int. J. RF Microw. Comput.-Aided Eng., vol. 17, pp. 4–12, 2007. [37] Z. C. Hao and J. S. Hong, “Ultra wideband bandpass filter using multilayer liquid-crystal-polymer technology,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 9, pp. 2095–2100, Sep. 2008. [38] J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. Smain Amari (M’98) received the DES degree in physics and electronics from Constantine University, Constantine, Algeria, in 1985, and the Masters degree in electrical engineering and Ph.D. degree in physics from Washington University, St. Louis, MO, in 1989 and 1994, respectively. From 1994 to 2000, he was with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada. From 1997 to 1999, he was a Visiting Scientist with the Swiss Federal Institute of Technology, Zurich, Switzerland, and a Visiting Professor in Summer 2001. In 2006, he was a Visiting Professor with the University of Ulm, Ulm, Germany. Since November 2000, he has been with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON, Canada, where he is currently a Professor. He is interested in numerical analysis, numerical techniques in electromagnetics, applied physics, applied mathematics, wireless and optical communications, computer-aided design (CAD) of microwave components, and the application of quantum field theory in quantum many-particle systems.

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Fabien Seyfert received the Engineering degree from the Ecole Superierure des Mines (Engineering School), St. Etienne, France, in 1993 and the Ph.D. degree in mathematics from the Ecole Superierure des Mines de Paris, Paris, France, in 1998. From 1998 to 2001, he was with Siemens, Munich, Germany, as a Researcher who specialized in discrete and continuous optimization methods. Since 2002, he has been a Researcher with INRIA, Nice, France (French agency for computer science and control). His research interests focus on the development of efficient mathematical procedures and associated software for signal processing including computer-aided techniques for the design and tuning of microwave devices.

Maged Bekheit (M’03) received the B.S.c degree from Ain Shams University, Cairo, Egypt, in 1999, and the M.S.c and Ph.D. degrees in electrical engineering from Queen’s University Kingston, ON, Canada in 2005 ad 2010, respectively. He has been an RF Planning and Optimization Engineer for a number of cellular companies. His research is focused on the design and optimization techniques of microwave components. Dr. Bekheit was the recipient of the Natural Sciences and Engineering Council (NSERC) Canada Graduate Scholarship (CGS) and an Ontario Graduate Scholarship (OGS).

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Dual-Band and Wide-Stopband Single-Band Balanced Bandpass Filters With High Selectivity and Common-Mode Suppression Jin Shi and Quan Xue, Senior Member, IEEE

Abstract—Novel dual-band and wide-stopband single-band balanced bandpass filters with high selectivity and common-mode suppression are presented in this paper. Stepped-impedance resonators (SIRs) are usually used for designing dual-band bandpass filters; however, they have a strong common-mode response when designing a balanced filter. To suppress the common-mode signal, a half-wavelength SIR loaded by a capacitor or a resistor has been introduced and theoretically analyzed. It is found that the capacitor can minimize the common-mode external quality factor, and the resistor can reduce the common-mode unloaded quality factor. With the use of this property, the common-mode response can be suppressed, whereas the differential-mode response is almost unaffected. This property can be easily verified by comparing the results of the balanced bandpass filters with and without loaded elements. To demonstrate the design idea, one balanced dual-band bandpass filter operating at 2.4 and 5 GHz and another balanced single-band bandpass filter with a wide stopband are designed. It was found that the common-mode suppression level of both filters can be greatly improved, and high selectivity is obtained by giving two differential-mode coupling paths. Index Terms—Balanced filter, capacitor or resistor loaded, common-mode suppression, dual band, half-wavelength stepped-impedance resonator (SIR), high selectivity, wide stopband.

I. INTRODUCTION ALANCED filters have become more and more important in the modern communication systems because of many advantages. They are able to produce the desired differential-mode frequency response and reduce the common-mode signal at the same time. As such, the signal-to-noise ratio in the receiver can be increased and the efficiency of the dipole antenna of the transmitter can then be improved. Much effort has been paid to improve the selective and common-mode suppression levels of the balanced bandpass filter. Although previous reported works [1], [2] on balanced filters suffer from

B

Manuscript received November 05, 2009; revised March 29, 2010; accepted April 22, 2010. Date of publication July 15, 2010; date of current version August 13, 2010. This work was supported by the Shenzhen Science and Technology Planning Project for the Establishment of the Key Laboratory in 2009 (Project CXB200903090021A) and by the Science and Technology Development Fund of Macao SAR under Grant 020/2009/A1. The authors are with the State Key Laboratories of Millimeter Waves, City University of Hong Kong, Kowloon, Hong Kong (e-mail: jinshi0601@hotmail. com; [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.2010.2052959

poor common-mode suppression, the balanced filter that was designed using the double-sided parallel-strip line [3] was able to overcome this shortcoming with the cost of having larger circuit size. In [4], designed using the coupled line resonators, the balanced single-band bandpass filter has high selectivity and good common-mode suppression across the passband. With the use of coupled stepped-impedance resonators (SIRs) [5] and multisection resonators [6], the stopband can be extended, which further improves the selectivity. In [7], a wideband balanced filter is realized using the microstrip branch-line. However, relatively little research has been done on the balanced dual-band bandpass filter. With the ever-increasing demands for dual-band and multiband wireless systems, the balanced dual-band bandpass filter has also become a key passive component because it can replace either two dual-band baluns combining with a single-ended dual-band bandpass filter or two single-band balanced bandpass filters in the balanced RF front-ends. There are two popular methods for designing the single-ended dual-band bandpass filter. One is to combine two resonators with the common input and output ports [8], [9]. Another is to utilize the dual-frequency feature of the SIR [10]–[13]. By controlling the impedance and length ratios of the SIRs, the desired operating frequencies of the two desired passbands can be easily obtained. In [5] and [6], the SIRs of different sizes were used to design wide-stopband balanced single-band bandpass filters. However, the common-mode passband cannot be removed while designing the balanced dual-band bandpass filter because the SIRs have to be identical. Even for the wide-stopband balanced single-band bandpass filter, the common-mode response of the half-wavelength SIR will affect the suppression level and bandwidth of the stopband. In [14], using the coupled SIRs, a balanced dual-band bandpass filter is realized; however, it shows poor insertion loss, differential-mode selectivity, and common-mode rejection ability, which are crucial in the balanced filter design. In this paper, dual-band and wide-stopband single-band balanced bandpass filter that are designed using the half-wavelength SIR, loaded by a capacitor or resistor, are introduced. It is found that the capacitor can minimize the common-mode external quality factor, and the resistor can reduce the common-mode unloaded quality factor. Benefitting from this feature, a balanced dual-band bandpass filter with good common-mode suppression is realized, and the stopband of the balanced single-band bandpass filter is further extended. Meanwhile, to improve the selectivity, the configuration with four proposed resonators including two coupling paths is used.

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B. Common-Mode Analysis When a Capacitor is Loaded As common-mode excitation is applied, the circuit can be split with open circuits at the middle to obtain the equivalent . When the load circuit in Fig. 1(c). The load admittance is is a capacitor, (4) where is the capacitance of the capacitor. The common-mode input admittance is given by

Fig. 1. (a) Structure of the proposed resonator, a half-wavelength SIR with a load placed at the center. (b) Differential mode half equivalent circuit. (c) Common-mode half equivalent circuit.

Two prototype balanced filters, a balanced dual-band bandpass filter operating at 2.4 and 5 GHz and a balanced single-band bandpass filter with wide stopband, are designed and fabricated. The experimental results agree well with the theoretical predictions and simulation results. II. PROPOSED RESONATOR The configuration of the proposed resonator is shown in . Fig. 1(a). It consists of a half-wavelength SIR and a load In our case, the load can be a capacitor or a resistor. The half-wavelength SIR is symmetric and has different characterand and electric lengths and . The istic impedances load is placed at the center of the SIR. Since this resonator is symmetrical in structure, it is convenient to analyze it using the differential- and common-mode equivalent half circuits. In the analysis, it is assumed that the transmission line is lossless. A. Differential-Mode Analysis As differential-mode excitation is applied to the two ends of the loaded SIR, there is a voltage null at the SIR center. Therefore, the load can be removed and the center point is a virtual short, leading to the equivalent circuit in Fig. 1(b), which is identical to a quarter-wavelength shorted SIR. The resulted resonance condition can be written as follows: [15] (1) When , the electric length of the high- or low-impedance line is given as [15] (2)

(5) The common-mode resonance condition can be described as follows:

(6) From (6), it is found that the common-mode resonance frequency changes with the capacitance. Therefore, in balanced filter design, the lowest common-mode external quality factor can always be found by tuning the capacitance to change the common-mode resonance frequency, while the differential-mode external quality factor will not change. This characteristic can be used to reduce the common-mode response to some extent in the balanced filter. When is reduced to 0 pF, (6) can be written as (7) which is identical to the common-mode resonance condition of the unloaded SIR. When , the common-mode resonance . When is infinite, (6) gives the condition is same resonance condition as the differential-mode one. Hence, the common-mode external quality factor is almost identical to that of the differential-mode one. Actually, the common-mode external quality factor is varied with the capacitance. Fig. 2(a) derives the common-mode external quality factor of a capacitor loaded SIR. There exists an optimum capacitance that can make the proposed resonator to have the maximum common-mode external quality factor. Therefore, the capacitor can be used to suppress the common-mode response in the balanced filter design. C. Common-Mode Analysis When a Resistor is Loaded When the load is a resistor, the common-mode input admittance can be written as

and the ratio of the first and second differential-mode frequency is given by [15] (8)

(3) where when , and It is obvious that when . Based on this feature, the differential-mode dualband response and the differential-mode single-band response with a wide stopband can be obtained by giving the appropriate . impedance ration

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Fig. 3. Configuration of the proposed balanced dual-band bandpass filter designed using the proposed resonator.

Fig. 2. (a) Common-mode external quality factor versus capacitance. (b) Common-mode unloaded quality factor versus resistance.

Common-mode resonance happens when the imaginary part is equal to zero, namely,

reduce the common-mode noise level and achieve the desired differential-mode bandpass response simultaneously. The filter in Fig. 3 is implemented using the microstrip structure fabricated on a Taconic RF-60A-0310 substrate (with a thickness of 0.82 mm and a dielectric constant of 6.03). In the filter design, the simulation is accomplished using the full-wave simulator IE3D and ADS, and the differential- and commonmode response can be obtained by using the mixed-mode -parameters [16], [17]. As for the measurement, the balanced filter, as a four-port device, is measured using the Agilent N5230A network analyzer, which is able to measure the two-port difand ferential- and common-mode -parameters, i.e., directly. A. Differential-Mode Behavior

(12) When is reduced to 0 , (12) gives the same resonance condition as the differential-mode one. When is infinite, (12) has the identical condition with that of an unloaded SIR. It can be seen from (12) that the common-mode resonance frequency will also change with the resistance. In addition to that, for the resistor-loaded SIR, the common-mode unloaded quality factor varies with the resistance. Fig. 2(b) gives the simulated common-mode unloaded quality factor of a resistor loaded SIR. There exists an optimum resistance that can make the proposed resonator have the minimum common-mode unloaded quality factor. Therefore, the resistor can be used to suppress the common-mode response in the balanced filter design. III. BEHAVIOR OF THE BALANCED DUAL-BAND FILTER To verify the analytical results of the proposed resonator, a prototype balanced dual-band bandpass filter is designed. The configuration of the filter is shown in Fig. 3. It is composed of four half-wavelength SIRs with their respective loads. With the use of the identical structure, the proposed filter has the same equivalent circuit for both the differential- and common-mode operations shown in Fig. 1(b) and (c). Thus, it is possible to

In differential-mode excitation, the equivalent circuit is irrelevant to the loaded elements, as can be seen in Fig. 1(b). The small blocks (with dimension of , ) on the inner two SIRs are used to tune the coupling coefficient, with almost no effect on the SIRs. The four SIRs can be regarded as identical for a differential-mode dual-band response. Assuming that the low- and high-impedance sections to have the same electric length, the frequency ratio of the two differential-mode passbands and the electric lengths is decided . In this work, the proposed balby the impedance ratio anced dual-band filter is intended to be used in the 2.45- and 5.55-GHz wireless local area network (WLAN) bands. For these is found to be 2.06 calculated using (3) and frequencies, at 2.45 GHz using (2). The widths of the highand low-impedance lines are set to 0.35 and 1.67 mm, respectively, which are equivalent to 84.9 and 41.2 of characteristic impedances. After obtaining the SIRs, the differential-mode external [18]. quality factor is determined by the tap position The feeding and coupling schemes are illustrated in Fig. 4. There are two coupling paths for the differential-mode passbands. One path is labeled according to the order, resonator 1-2-3-4, and another is directly from resonator 1 to resonator

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Fig. 4. Feeding and coupling schemes for the proposed balanced dual-band bandpass filter designed using the proposed resonator.

Fig. 6. Common-mode response of the proposed balanced dual-band bandpass filter. (a) Loads are capacitors. (b) Loads are resistors.

elements is shown in Fig. 5(b). There is a common-mode passband near 4.2 GHz with almost no suppression. To improve the common-mode suppression level and verify the theory in Section II while keeping the differential-mode response unchanged, the common-mode performance of the filter is studied by having the SIR with different loads. Fig. 5. Simulated results of the proposed balanced dual-band filter. (a) Differential-mode response (b) Common-mode response without loaded elements.

4. Therefore, four differential-mode transmission zeros can be obtained in addition to the two differential-mode passbands, which can greatly improve the selectivity of the filter. The two short lines ( , ) in Fig. 3 are used to provide the coupling between resonators 1 and 4. By changing the gaps ( , , , , and ), the coupling coefficient can be conveniently tuned to the desired value. After optimization, mm, the dimensions are determined as follows: mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, and mm. The simulated differential-mode response is given in Fig. 5(a). There are two differential-mode passbands at 2.45 and 5.55 GHz, respectively. And four transmission zeros are obtained besides the two passbands. The common-mode response for the configuration without the loaded lumped

B. Common-Mode Behavior With the Loaded Capacitors When the loads are all capacitors, the simulated commonmode response is analyzed in Fig. 6(a). The first example is to have the four capacitors identical. The function of the capacitors is to find a common-mode resonance frequency, where the filter has the lowest common-mode external quality factor. This can be achieved when pF. The capacitors have introduced an improvement of about 10 dB in the common-mode suppression. The second example is to have the outer two resonators with different capacitance with the inner two, which will weaken the coupling between the outer and inner resonators because of their different resonance frequencies. However, both examples make the frequency of the common-mode second harmonic move down, and then reduce the common-mode suppression level near the second differential-mode passband. C. Common-Mode Behavior With the Loaded Resistors Fig. 6(b) shows the common-mode response of the proposed balanced dual-band bandpass filter when the loads are resistors.

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Fig. 8. Configuration of the modified resonator.

Fig. 7. Common-mode response of the proposed balanced dual-band bandpass filter when the via is considered.

It can be seen from Fig. 6(b) that the filter have the highest and common-mode suppression when . This is because that the inner two resistors only affect the common-mode unloaded quality factor, while the outer two change both the common-mode unloaded quality factor of the resonator and the common-mode external quality factor of and are identical to the resistance the filter. Therefore, of the lowest common-mode unloaded quality factor in Fig. 2, and are not. In a word, the loaded resistors can while improve the common-mode suppression level of the proposed filter, and the common-mode signal is mainly absorbed by the resistors. The aforementioned analysis is performed by assuming an ideal grounding. However, in the conventional microstrip structure, the circuits are usually connected to the ground by the vias. Hence, the inductance of the vias can affect the common-mode response of balanced filter, which can be clearly visualized in Fig. 7. It was found that the common-mode suppression level of the filter was reduced about 7 dB comparing with that for ideal grounding. In this paper, all the capacitors and resistors are 0603, and the radius of the via-holes is 0.4 mm. D. Modification of the Resonator Configuration To cancel out the effect of the vias and to utilize both the advantages of the loaded capacitors and resistors in reducing the common-mode response, a modified resonator is proposed in this filter. Fig. 8 shows the modified resonator configuration, where a capacitor and a resistor are loaded in series at the center of the SIR. Fig. 9 shows the common-mode response of the balanced dual-band bandpass filter designed using the modified resonator. Compared with the filter with only resistors, the common-mode suppression level of the filter with the modified resonator was further improved about 10 dB when components , , pF, and pF are used. The value of from 3 and 6 GHz is small as the common-mode signal is absorbed by the resistors. In this filter, the common-mode signals are suppressed three ways, i.e., the reflection caused by the poor external quality factor, the low coupling coefficient owing to different resonance frequencies, and the absorption of the resistors.

Fig. 9. Common-mode response of the proposed balanced dual-band bandpass filter designed using the modified resonator.

IV. BEHAVIOR OF THE BALANCED SINGLE-BAND FILTER WITH WIDE STOPBAND A. Filter Structure The dual-band feature of the SIR can also be adopted to realize the balanced single-band bandpass filter with wide stopof the SIR should be band. To have a wide stopband, the low enough to push up the second differential-mode passband. Since this filter is single band, the inner- and outer-pair SIRs can have different sizes while having the same fundamental differential-mode frequency. As shown in Fig. 10, the outer two SIRs have a different size from the inner two. This brings about two advantages: one is that it is easy in the layout to realize two coupling paths of this filter, liking the coupling scheme in Fig. 4, another is that it can reduce the coupling at the common-mode resonance frequency and the differential-mode harmonic response frequency because these frequencies of the outer and inner SIRs are different. According to [19], the stopband bandwidth of the filter using SIRs is mainly determined by the inner two resonators. To push up to a higher frequency such as , the parameters of the inner and at two SIR should be set as . B. Differential-Mode Behavior By carefully arranging the tap position and gaps, the differential-mode passband can be realized. The dimensions are given as follows: mm, mm, mm, mm, mm mm, mm, mm, mm, mm, mm, mm, and mm. Fig. 11(a) shows the differential-mode response of the filter. The differential-mode passband

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Fig. 10. Configuration of the proposed balanced single-band bandpass filter with wide stopband.

is located at 893 MHz with 8.2% of 1-dB relative bandwidth. Two transmission zeros near to the differential-mode passband are obtained because the filter has two coupling paths. The stopband bandwidth with a 35-dB suppression level is up to 6.5 GHz, . which is about C. Common-Mode Behavior Fig. 11(b) shows the simulated common-mode response of the balanced single-band filter designed using the unloaded SIR. The fundamental common-mode frequency is about with an amplitude of 17.5 dB, which is lower than that of the balanced dual-band bandpass filter designed using the unloaded SIRs. This is because the common-mode coupling between the SIRs of the single-band filter with different SIRs is weaker than that of the dual-band one with identical SIRs. However, the common-mode suppression level of 17.5 dB can be further improved. To improve the common-mode suppression level, the capacitor-loaded SIRs are used. By adding in a loading capacitor, the external quality factor of filter can be lowered, and the coupling between the inner and outer resonators can be reduced further. Unlike the balanced dual-band bandpass filter, for the balanced single-band one with a wide stopband, the second commonmode harmonic need not be considered because it is not inside the differential-mode stopband frequency range. Therefore, the SIR loaded with only a capacitor without a resistor can also make the filter obtain a good common-mode suppression level. Fig. 11(c) shows the simulated common-mode response of the balanced single-band filter designed using the capacitor-loaded pF and pF. The SIRs when common-mode suppression level is higher than 35 dB from 0 to . 6.33 GHz, which is about Like the balanced dual-band bandpass filter, the resistor-loaded SIRs can also reduce the common-mode response in the balanced single-band bandpass filter by tuning the resistance to make the resistor-loaded SIRs achieve the minimum common-mode unloaded quality factor. However, this filter does not need to use the resonator in Fig. 8. This is because the

Fig. 11. Simulated results of the proposed balanced single-band bandpass filter. (a) Differential-mode response. (b) Common-mode response without loads. (c) Common-mode response with capacitor loaded. (d) Common-mode response with resistor loaded.

effect of the vias is negligible as the frequency shift caused by its inductance is very small comparing to the wide stopband.

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Fig. 12. Photograph of the fabricated balanced dual-band filter.

Another factor is that the different size between the inner and outer SIRs has already reduced the coupling between them. Due to the resistors, further reducing the coupling and external quality factor does not affect the filter performance further. Obviously, the SIR that is only loaded with a resistor can also make the filter achieve a good common-mode suppression level. Fig. 11(d) shows the simulated common-mode response of the filter designed using the resistor-loaded SIRs when the resistance is 33 . The common-mode suppression level is . higher than 35 dB from 0 to 6.22 GHz, which is about V. MEASUREMENT RESULTS To verify the theory and analysis, two prototypes are fabricated and measured. One is the balanced dual-band bandpass filter in Section III, and another is the balanced single-band bandpass filter with wide stopband in Section IV. For the measurement setup, the proposed filters with two pair of ports are connected to the four-port network analyzer with four identical cables. The differential- and common-mode response can then be directly seen on the analyzer.

Fig. 13. Measurement and simulation results of the proposed balanced dualband bandpass filter designed using the modified resonator. (a) Differentialmode response. (b) Common-mode response.

A. Balanced Dual-Band Bandpass Filter The design parameters of the fabricated balanced dual-band bandpass filter are the same as those in Section III. Fig. 12 shows a photograph of the fabricated balanced dual-band filter. The measured differential- and common-mode results of the proposed balanced dual-band filter are shown as in Fig. 13, when , , pF, . The lower passband is centered and at 2.46 GHz with 1-dB bandwidth of 400 MHz or 16.26%. The minimum insertion loss including subminiature A (SMA) connectors is measured to be 0.96 dB. The upper passband is located at 5.56 GHz. The 1-dB bandwidth is 370 MHz or 6.65%. The minimum insertion loss including SMA connectors is measured to be 1.9 dB. The insertion loss of a pair of SMA connecters is 0.2 and 0.5 dB at 2.45 and 5.55 GHz, respectively. Four transmission zeros are created at 1.76, 3.42, 4.9, and 6.36 GHz. The common-mode suppression level is higher than 31 dB from 1 to 7 GHz. The common-mode rejection rations insides the two differential-mode passbands are larger than 36.2 and 31.1 dB, respectively. The total size of the filter is about (18.53 mm 24.42 mm) at 2.45 GHz. The slight difference between the differential-mode measurement and simulation bandwidth can be attributed to the

Fig. 14. Photograph of the fabricated balanced single-band filter.

fabrication tolerance of the gap. The measurement frequency of common-mode return loss is lower than the simulation one, which is caused by the difference between the ideal and actual capacitors. B. Balanced Single-Band Filter With Wide Stopband To demonstrate the design idea, the resistor-loaded SIR is also used to build the balanced single-band filter. Fig. 14 shows a photograph of the fabricated balanced single-band filter. Fig. 15 shows the measured and simulated differential- and commonmode response of the fabricated balanced single-band bandpass filter. Again, the design parameters are identical to those

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TABLE I PERFORMANCE COMPARISON WITH PREVIOUS WORKS

VI. CONCLUSION

Fig. 15. Measurement and simulation results of the proposed balanced single-band bandpass filter with a wide stopband using the resistor-loaded SIRs. (a) Differential-mode response. (b) Common-mode response.

In this paper, a half-wavelength SIR loaded with a capacitor or a resistor has been studied. The capacitor or resistor can help reduce the common-mode response without affecting the differential-mode of the SIR. With such, the dual-band and wide-stopband balanced bandpass filters, which are designed by incorporating four proposed resonators, are proposed for improving the common-mode suppression. Meanwhile, high selectivity of the differential-mode frequency response is realized by providing two coupling paths. It has been observed that the balanced dual-band bandpass filter can easily realize two differential-mode passbands at 2.46 and 5.56 GHz with four transmission zeros, with the common-mode suppression level higher than 31 dB from 1 to 7 GHz. In addition to a good differential-mode passband, the wide-stopband balanced single-band bandpass filter has two transmission zeros. Both the differential- and common-mode stopbands (with a 35-dB suppression . level) are up to REFERENCES

in Section IV and . The differential-mode passband is loaded at 0.9 GHz with a 1-dB bandwidth of 110 MHz or 12.2%. The minimum insertion loss including SMA connectors is measured to be 2 dB. Two transmission zeros are realized at 0.72 and 1.13 GHz. The differential-mode stopband with a 35-dB suppression level stretches up . The common-mode supto 6.74 GHz, which is about pression level is higher than 35 dB from 0 to 6.35 GHz, which . The common-mode rejection ration inside the is about differential-mode passband is larger than 32.5 dB. The size of (20.1 mm 30 mm), which the filter is about is very compact. The difference between the differential-mode measurement and simulation bandwidth is also caused by the fabrication tolerance of the gap, which also makes the measurement and simulation common-mode suppression have about 3-dB difference. The balanced filter in Fig. 10 is summarized and compared with the previous works in Table I. The proposed wide-stopband single-band balanced filter shows a lower insertion loss and wider stopband with higher suppression. For design and tuning processes, the filter in Fig. 10 using the proposed resonators is simpler than the previous ones because the common-mode suppression can be made independent of the differential-mode response.

[1] A. Ziroff, M. Nalezinski, and W. Menzel, “A 40 GHz LTCC receiver module using a novel submerged balancing filter structure,” in Proc. Radio Wireless Conf., 2003, pp. 151–154. [2] Y.-S. Lin and C. H. Chen, “Novel balanced microstrip coupled-line bandpass filters,” in Proc. URSI Int. Electromagn. Theory Symp., 2004, pp. 567–569. [3] J. Shi, J.-X. Chen, and Q. Xue, “A novel differential bandpass filter based on double-sided parallel-strip line dual-mode resonator,” Microw. Opt. Technol. Lett., vol. 50, no. 7, pp. 1733–1735, Mar. 2008. [4] C.-H. Wu, C.-H. Wang, and C. H. Chen, “Novel balanced coupled-line bandpass filters with common-mode noise suppression,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 2, pp. 287–295, Feb. 2007. [5] C.-H. Wu, C.-H. Wang, and C. H. Chen, “Stopband-extended balanced bandpass filter using coupled stepped-impedance resonators,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, pp. 507–509, Jul. 2007. [6] C.-H. Wu, C.-H. Wang, and C. H. Chen, “Balanced coupled-resonator filters using multisection resonators for common-mode suppression and stopband extension,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 8, pp. 287–295, Aug. 2007. [7] B. L. Teck and Z. Lei, “A differential-mode wideband bandpass filter on microstrip line for UWB application,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 10, pp. 632–634, Oct. 2009. [8] C.-Y. Chen and C.-Y. Hsu, “A simple and effective method for microstrip dual-band filters design,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 3, pp. 246–248, May. 2006. [9] J. Shi, J.-X. Chen, and Q. Xue, “A quasi-elliptic function dual-band bandpass filter stacking spiral-shaped CPW defected ground structure and back-side coupled strip lines,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 6, pp. 430–432, Jun. 2007. [10] S. Sun and L. Zhu, “Compact dual-band microstrip bandpass filter without external feeds,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 644–646, Oct. 2005.

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[11] Y. P. Zhang and M. Sun, “Dual-band microstrip bandpass filter using stepped-impedance resonators with new coupling schemes,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3779–3885, Oct. 2006. [12] J.-T. Kuo, T.-H. Yeh, and C.-C. Yeh, “Design of microstrip bandpass filter with a dual-passband response,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1331–1337, Apr. 2005. [13] L. K. Yeung and K.-L. Wu, “A dual-band coupled-line balun filter,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 11, pp. 2406–2411, Nov. 2007. [14] J. Shi and Q. Xue, “Novel balanced dual-band bandpass filter using coupled stepped-impedance resonators,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. , pp. 19–21, Jan. 2010. [15] S. Morikazu, M. Mitsuo, and Y. Sadahiko, “Geometrical structures and fundamental characteristics of microwave stepped-impedance resonator,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1078–1085, Jul. 1997. [16] D. E. Bockelman and W. R. Eisenstant, “Combined differential and common-mode scattering parameters: Theory and simulation,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1530–1539, Jul. 1995. [17] W. R. Eisenstadt, B. Stengel, and B. M. Thompson, Microwave Differential Circuit Design Using Mixed-Mode S-Parameters. Boston, MA: Artech House, 2006. [18] J. S. Hong and M. J. Lancaster, Microwave Filter for RF/Microwave Application. New York: Wiley, 2001. [19] S.-C. Lin, P.-H. Deng, Y.-S. Lin, C.-H. Wang, and C. H. Chen, “Wide-stopband microstrip bandpass filters using dissimilar quarter-wavelength stepped-impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1011–1018, Mar. 2006. Jin Shi was born in Nantong, Jiangsu Province, China, in 1979. He received the B.S. degree from HuaiYin Teachers College, Huai’an City, Jiangsu Province, China, in 2001, the M.S. degree from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2004, and is currently working toward the Ph.D. degree at the City University of Hong Kong, Kowloon, Hong Kong. From 2004 to 2006, he was a Research Engineer with Comba, Hong Kong, working on RF repeater systems. Since 2007, he has been a Research Assistant with the City University of Hong Kong. His research interests are RF/microwave components and subsystems.

Quan Xue (M’02–SM’04) received the B.S., M.S., and Ph.D. degrees in electronic engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1988, 1990, and 1993, respectively. In 1993, he joined the UESTC, as a Lecturer. He became an Associate Professor in 1995 and a Professor in 1997. He became a Distinguished Academic Staff Member for his contribution to the development of millimeter-wave components and subsystems. From October 1997 to October 1998, he was a Research Associate and then a Research Fellow with the Chinese University of Hong Kong. In 1999, he joined the City University of Hong Kong, Kowloon, Hong Kong, where he is currently an Associate Professor and the Director of the Applied Electromagnetics Laboratory. Since May 2004, he has been the Principal Technological Specialist of State Integrated Circuit (IC) Design Base, Chengdu, Sichuan Province, China. He has authored or coauthored over 150 internationally referred papers His current research interests include antennas, smart antenna arrays, active integrated antennas, power amplifier linearization, microwave filters, millimeter-wave components and subsystems, and microwave monolithic integrated circuit (MMIC) RF integrated circuits (RFIC). Dr. Xue is the MTT-S Regional Coordinator of IEEE Region 10. He cosupervised two IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) Best Student Contest papers [third place (2003) and first place (2004)].

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Varactor-Tuned Dual-Mode Bandpass Filters Wenxing Tang, Student Member, IEEE, and Jia-Sheng Hong, Senior Member, IEEE

Abstract—This paper presents a new type of varactor-tuned dual-mode bandpass filter. Since the two operating modes (i.e., the odd and even modes) in a dual-mode microstrip open-loop resonator do not couple to each other, the tuning of the passband frequency becomes simple with a single dc-bias circuit while keeping nearly constant absolute bandwidth. Design equations and procedures are derived, and two two-pole tunable bandpass filters of this type are demonstrated experimentally. Index Terms—Dual-mode resonator filter, microstrip filters, tunable filter.

I. INTRODUCTION

E

LECTRONICALLY tunable microwave filters are attracting more attention for research and development because of their increasing importance in improving the capability of current and future wireless systems [1]–[11]. To develop compact tunable filters, dual-mode microstrip resonator tunable filters are attractive because each dual-mode resonator can be used as a doubly tuned resonant circuit, and, therefore, the number of resonators required for a given degree of filter is reduced by half, resulting in a compact filter configuration [12]–[18]. Tang et al. [12] introduced a new type of electronically tunable dualmode microstrip open-loop resonator bandpass filter, but only showed a tuning range of 12%, and not necessarily for the constant absolute bandwidth tuning. Being different from the conventional dual-mode filter, this new type of dual-mode resonator filter exhibits a distinct characteristic for which the dual modes do not couple [18]. This leads to a simple tuning scheme since tuning the passband frequency is accomplished by merely changing the two modal frequencies proportionally using a single dc bias. In this paper, a systematical method has been derived to design this kind of electronically tunable filter for a constant absolute bandwidth tuning, and moreover, for achieving a wide tuning range (41%). The proposed filter structure is shown in Fig. 1. It can be seen that a wideband transformer is applied as the input/output (I/O) coupling structure for the filter [5]. The wideband transformer may be considered as a type of interdigital coupled lines, which is known to have a dominant inductive coupling with wideband behavior. Therefore, the I/O coupling can be adjusted to achieve constant bandwidth tuning requirement over a wide tuning range. While for the transformer used in the previous designs [12], [13], capacitive coupling is dominant and it is difficult to control I/O coupling properly over a wide tuning range due to the lack of inductive coupling associated with the structure. In a conventional tunable dual-mode filter, I/O coupling is used Manuscript received December 02, 2009; revised March 05, 2010; accepted April 28, 2010. Date of publication July 12, 2010; date of current version August 13, 2010. This work was supported by the U.K. Engineering and Physical Science Research Council. The authors are with the Department of Electrical, Electronic and Computer Engineering, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, U.K. (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.2010.2052958

Fig. 1. Layout of the proposed two-pole tunable filter.

Fig. 2. Coupling scheme of the filter.

to compensate both the variation of a slope parameter x (or b) and the coupling between resonators due to the frequency change. However, for the proposed dual-mode tunable filter, I/O coupling is only used to compensate the variation of x (or b) since its even and odd modes do not couple to each other. II. THEORY AND DESIGN EQUATIONS A. Filter Operation For our investigation, Fig. 1 shows a layout of the proposed two-pole varactor-tuned dual-mode microstrip bandpass filter. Variable capacitances Cv are supposed to be loaded varactors. These three same varactors are to be used with a single dc-bias circuit, which makes both the implementation and tuning simple. The proposed dual-mode filter has a coupling scheme, as shown in Fig. 2, where S and L denote the input and output ports, respectively; node 1 denotes the odd mode and node 2 denotes the even mode. Since these two operating modes do not couple to each other, a simple tuning scheme can be obtained. How to achieve this ideally is discussed below. B. Ideal Requirement for Tunable Dual-Mode Filter To tune the center frequency of this type of dual-mode filter while keeping constant filter response shape and bandwidth, two factors ideally need to be considered. Firstly, the resonant frequencies of oddmode (f0o ) and even-mode (f0e ) need to be shifted proportionally. Secondly, the shape and bandwidth of odd- and even-mode frequency responses must keep constant over the entire tuning range, this would require the external quality factors for the odd mode (Qexo ) and even

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Q

mode ( exe ) vary directly with the tuning frequency [19]. These parameters may be represented by

f0e f0o = A e Qexe = 1ffe0 3 dB o f Qexo = 1f o0 3 dB j

A

0

(1)

j

(2) (3)

where denotes the separation between the even- and odd-mode resonant frequencies; 1 3o dB 1 3e dB are the 3-dB bandwidths of the oddand even-modes, respectively, [21]. Note that ideally 1 3e dB , 1 3o dB , and are to be constant over the entire tuning range for constant absolute bandwidth tuning.

; f

f

f

A

f

C. Odd- and Even-Mode Tuning Rates To shift the odd- and even-mode resonant frequencies proportionally by loading the same capacitance, the tuning rates of odd and even modes need to be characterized. The tuning rate indicates how much a modal frequency is shifted by varying capacitance v . Assume that when v varies from v1 to v2 , the odd-mode frequency shifts from o o e e 01 to 02 and the even-mode frequency shifts from 01 to 02 . Thus, the tuning rate can be defined as

f

C f

C f

C

C

f02o=e f01o=e Cv2 Cv1 0

Tuning rate =

o=e

j

0

j

f

=pF)

(GHz

(4)

where the superscript denotes the odd or even modes. By placing a short or open circuit at the symmetric plane of the circuit in Fig. 1, we obtain the circuit model for the odd or even modes without the I/O coupling, as shown in Fig. 3(a) and (c). To demonstrate how to control the modal tuning rate, the circuit models of Fis. 3(a) and (c) may be modified as that of Fig. 3(b) and (d), respectively. A reference port is added for deriving an input admittance, i.e., for the odd mode,

Yino = j !Cv

0

Yo tan o

(5)

and for the even mode,

o + !Cv Yine = j Yo YYoL + YYLo tan tan o

(6)

0

with

) + Ye tan e YL = Ye Y!e(Cv[!=2(C+vC=2stub + Cstub )] tan e 0

C

C 

where v is the loading capacitance; stub represents the bended short open-circuited stub, which may be estimated from stub = ( 1 tan 1 ) . o , e , 1 , o , e , and 1 are the admittances and electrical lengths for the transmission line sections shown in Fig. 3. The resonant frequencies of the odd and even mode can be found from (7)

C

Y

 =! Y Y Y   Y

Im[ ino ] = 0

Y

:

Im[ ine ] = 0

(7)

From (5), it can be seen that the resonant frequency of the odd mode depends on the parameters v , o , and o . Assume that the loading capacitance v varies from v1 (0.6 pF) to v2 (5 pF). For o = 80 at a nominal frequency of 1 GHz, the modal tuning rate can be calculated by (4) with different values o , and the results are plotted in Fig. 4(a). Similarly, for o = 0 02 S, the modal tuning rate can also be calculated against different values of o , and the results are present in Fig. 4(b). These two sets of results show that the smaller the admittance o or the shorter the electrical length o , the larger the modal tuning rate. With

C

Y

:

C Y C Y  



C



Y

Fig. 3. Circuit models. (a) and (b) Odd mode. (c) and (d) Even mode.

loading the same capacitance range, the tuning rate of the even mode depending on the parameters e and e is investigated. From (5) and (6), it is envisaged that the resonant frequency of the even mode can be shifted by varying e and e while the resonant frequency of the odd mode is not changed for the given o and o . This means that the tuning rate of the even mode may be controlled so as to match the tuning rate of the odd mode by varying e and e . For given separations between the odd- and even-mode resonant frequencies over a loading capacitance range, the values of e and e may be determined from (8) and (9) as follows:

Y

Y





Y

Y





Y

f01e f01o f02e f02o

A

j

0

j

j

0

j



A =B =

B

(8) (9)

where and are the separations between the odd- and even-mode resonant frequencies for the loading capacitances v1 and v2 , respectively. Ideally, they are also proportional to the bandwidth. For the constant absolute bandwidth tuning, the resonant frequencies of the odd and even modes need to be shifted proportionally; hence, should be equal to . To demonstrate how to achieve this, assume that  = 100 MHz; o = 0 02 S, o = 80 at 1 GHz, which is the nominal high frequency of a given tuning range, v1 = 0 6 pF, and v 2 = 5 0 pF. Also, for the demonstration, stub is chosen for three values: stub = 0 stub = 0 3 pF and stub = 1 3 pF to represent different cases of the short open-circuited stub. By applying (5) and (6) = as follows: and (8) and (9), e and e can be determined for  e = 0 018 S and e = 68 54 when stub = 0; e = 0 016 S and e = 62 17 when stub = 0 3 pF, and e = 0 006 S and  e = 23 43 when stub = 1 3 pF. After e and e are determined, the tuning rate of the even mode can then be calculated by (4), for all the three cases, as 9.2%, which is equal to that of the odd mode (see Fig. 4). The resonant frequencies of the even and odd modes against the loading capacitance v , varying from 0.6 to 5.0 pF, are plotted in Fig. 5 for a comparison. Note that, for the even mode, the curves for the three different values of stub coincide together, which implies that the effect of the short open-circuited stub can easily be compensated by e and e with negligible influence to the even-mode tuning rate. This, however, allows a more flexible design for the loading element inside the open loop. From the analysis above, it is clear that the tuning rate of the even mode is controllable to be equal to the tuning rate of the odd mode. As such, the resonant frequencies of the two operating modes can be controlled to shift proportionally. The separation between the modal frequencies, which is ideally proportional to the bandwidth, varies with the tuning range present in

A C

C

B

A Y

: C

Y :   :

:

:

 C

:



C

C

C

Y





;C

Y

C

:

:

C

:

C

C

Y

C

:

: B A Y : Y : 

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Fig. 6. Modal frequency separation varies against the tuning range. (Note that for the tuning range of 29.5%, Y : : ; for the tuning S and  : : ; for the tuning range of 55.5%, range of 45.1%, Y S and  : : .) S and  Y

= 0 0215

= 0 0185 = 68 44

= 0 0163 = 68 8

= 69 04

Fig. 7. Circuit model of: (a) odd mode and (b) even mode with I/O coupling. Fig. 4. Tuning rate of odd-mode varies with: (a) Y and (b)  (where C : pF). : pF; C

06

= 50

= D. External Coupling of the Filter With the I/O coupling, the circuit models for the odd and even modes are present in Fig. 7. The input reflection coefficient of the odd mode (S11o ) and even mode (S11e ) may be found by

S11o (f ) = S11 (f ) 0 S21 (f ) S11e (f ) = S11 (f ) + S21 (f )

(10) (11)

where S11 (f ) and S21 (f ) are the two-port scattering parameters of the filter [21], which can be extracted using a full-wave electromagnetic (EM) simulation tool [22]. The external quality factor of the odd mode (Qexo ) and even mode (Qexe ) of the proposed filter may be derived from the group delay of the input reflection coefficient of odd and even mode [21] as follows:

Qexo = Fig. 5. Resonant frequency of the even mode compared with that of the odd mode.

Fig. 6, obtained by using (5) and (6) and (8) and (9). It shows that the wider the tuning range, the larger the variation of the frequency separation. This somewhat limits the tuning range for a large constant absolute bandwidth.

Qexe =

1 S11o (f0o )

(12)

1

(13)

o 2f0 e 2f0

2 S11e (f0e ) 2

where

@'S11o (f ) (2 )@f @'S11e (f ) S11e (f ) = 0 (2 )@f

S11o (f ) = 0

(14) (15)

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Fig. 8. Typical group-delay response of input reflection coefficient of odd mode.

where S 11o (f ) and S 11e (f ) are the group delays of the odd and even modes, 'S 11o (f ) and 'S 11e (f ) denote the phase responses of the input reflection coefficient of odd and even mode. A typical group-delay response of the input reflection coefficient of the odd mode is shown in Fig. 8. To maintain the shape and bandwidth of the odd- and even-mode frequency responses over the entire tuning range, Qexo and Qexe of the proposed tunable filter must satisfy (2) and (3) and this may be achieved by properly choosing the transformer parameters w11 ,l1 , s, and Cs shown in Fig. 7. In general, for given l1 , which depends on the contour of the open loop, a stronger coupling can be obtained with a smaller s. In addition, the coupling turns to be more inductive and can be enhanced with a narrower w11 and larger Cs . E. Design Procedure Step 1: Determine the Requirements of the Ideal Tunable Dual-Mode Filter: The requirements of the ideal tunable filter may be derived from its fixed frequency response centring at the high-frequency edge of a given tuning range. The separation (A) of the odd- and even-mode resonant frequencies and the required Qexo and Qexe for the ideal tunable filter can then be determined from (1)–(3). Step 2: Design of Transformer Network: To maintain the shape and bandwidth of odd- and even-mode frequency responses over the tuning range, Qexo and Qexe of the proposed tunable filter given in (12) and (13) must satisfy (2) and (3) by properly choosing the transformer parameters w11 , l1 , s, and Cs . Step 3: Design of the Even-Mode Tuning Rate: The tuning rate of the even mode may be designed to match the tuning rate of the odd mode by using (8) and (9) with B = A for the constant absolute bandwidth tuning. As there are six degrees of freedom to determine the tuning rate of the even mode, i.e., w3 , l3 , w4 , l4 , w5 , and l5 , four parameters may be chosen first, and then the last two can be determined by (8) and (9). Since the purpose of using w4 , l4 , w5 , and l5 is to shorten the length of the even mode within the square loop of the odd mode (see Fig. 1), these parameters may be chosen first. To go through the design procedure, two examples are given in Section III. III. DESIGN EXAMPLES A. Tunable Filter With Finite-Frequency Transmission Zero Located at High Side of the Passband (Filter A) The proposed filter has been designed using microstrip lines with the following specifications: Tunable range: Fractional bandwidth (FBW) Number of poles:

0.6–1.07 GHz 2.9% at 1.07 GHz 2:

To derive the required design parameters for the tunable filter, a filter centring at 1.07 GHz has been designed firstly with a finite-frequency transmission zero located at 1.12 GHz, a fractional bandwidth of

Q

Q

Fig. 9. (a) External quality factor and . (b) Desired and extracted for the proposed tunable filter (filter A).

f 0f

2.9%, and passband return loss of 20 dB. The desired parameters may be derived from a target coupling matrix corresponding to the prescribed response [23], which are f0o = 1:053 GHz, f0e = 1:098 GHz, Qexo = 27:2, and Qexe = 105:2. With these design parameters, the desired values for the ideal tunable filter are derived according to (1)–(3) and plotted in Fig. 9 using the full line. The proposed tunable filter can be designed in light of full-wave EM simulation. By comparing the extracted Qexo , Qexe , f0o , and f0e from the EM simulation (see Section II-D) with the desired ones from the target coupling matrix, the dimensions of the dual-mode filter can be determined. For our example, a substrate with a relative dielectric constant of 10.2 and a thickness of 1.27 mm is used. The odd-mode resonator is formed using a 50- line, which has a tuning rate as that shown in Fig. 4. Although it is possible to use another characteristic impedance line, the resultant tuning rate will be different. The loading capacitance Cv is chosen to be 0.6 pF. The electrical length of the odd-mode resonator can then be determined initially for f0o = 1:053 GHz. To achieve desired values for Qexo and Qexe , obtained from (2) and (3), respectively, as shown (full line) in Fig. 9(a), the transformer parameters, i.e., w11 , l1 , s, and Cs are chosen so that the extracted external quality factors from (12) and (13) could best satisfy those desired over the tuning range. The resultant extracted Qexo and Qexe are also plotted (dotted line) in Fig. 9(a). The parameters for the even mode, i.e., w3 , l3 , w4 , l4 , w5 , and l5 are derived based on (8) and (9) with B = A so that the resonant frequencies of the odd and

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TABLE I PARAMETERS OF THE PROPOSED TUNABLE DUAL-MODE FILTER (FILTER A) (IN MILLIMETERS) (SEE FIG. 1)

Fig. 11. Photograph of the fabricated proposed filter (filter A).

Fig. 12. Layout of filter B: (a) with symmetrical structure and (b) with asymmetrical structure (dimensions refer to Fig. 1).

Fig. 10. Measured and EM simulated S -parameters of the proposed filter (filter A). (a) S and (b) S . The bias voltage is between 2.2–22 V. The 3-dB absolute bandwidth is 85 5 MHz from 0.6 to 1.03 GHz.

6

even modes can be shifted proportionally. Initially, w4 , l4 , w5 , and l5 were chosen as 4, 8, 1, and 1 mm, respectively. After that, w3 and l3 are determined from (8) and (9). Note that w4 , l4 , w5 , l5 may be readjusted so that the loading elements of the even mode are within the square loop of the odd mode. The plot of the separation of the odd-and even-mode resonant frequencies is displayed in Fig. 9(b). It is shown that the resonant frequencies of the odd and even modes are shifted nearly proportionally after the tuning rate of the even mode is modified. Note that the extracted result of Fig. 9(b) was obtained by EM simulation and with the I/O structure, whereas the result shown

in Fig. 5 was obtained from the theory without considering the I/O coupling. Nevertheless, they all showed that both modes can be tuned proportionally over a certain tuning range. The parameters for the final layout of the proposed tunable filter, as referring to Fig. 1, are given in Table I. Full-wave EM simulated responses of the filter are present in Fig. 10. It can be seen that the tunable filter exhibits a nearly constant 3-dB absolute bandwidth (80 6 5 MHz), and its center frequency is tuned from 0.6 to 1.03 GHz with the capacitance Cv of loading varactors changes from 5.0 to 0.6 pF. To demonstrate this type of tunable filter experimentally, the designed filter is fabricated as shown in Fig. 11. The matching capacitance Cs is realized by an AVX chip capacitor [24], while the variable capacitance Cv is implemented by a M/A COM MA46H202 varactor [25]. In this case, three varactors are used, which are applied with a single dc-bias circuit. The measured results for a dc bias ranging from 2.2 to 22.0 V are also plotted in Fig. 10, which are obtained using Agilent 8510B network analyzer. From Fig. 10, we can observe that the measured tunable characteristics are in good agreement with the simulated ones. The experimental varactor-tuned bandpass filter shows a high selectivity on the high side of the passband with less than 1.8-dB insertion loss and more than 10-dB return loss over a tuning range of 41% from 0.6 to 1.03 GHz. The measured 3-dB bandwidth is 85 6 5 MHz. A further discussion is made here on the design. From Fig. 9(a), it can be seen that the Qexe does not meet the ideal values after the even-mode tuning rate being modified. This is because the parameters of the even mode have the effects on both Qexe and the tuning rate. Therefore, there is a tradeoff in the design, and as a result, the designed bandwidth would be slightly different from the desired specification. However, the bandwidth can be readjusted by slightly varying the even-mode resonant frequency. Comparing Fig. 9(a) and (b), it can be seen that when the extracted Qexe is smaller (the coupling to the even mode being stronger), the separation between the odd- and even-mode frequencies is smaller, which thus minimizes the variation of the 3-dB bandwidth

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 58, NO. 8, AUGUST 2010

TABLE II PARAMETERS OF THE PROPOSED TUNABLE DUAL-MODE FILTER (FILTER B) (IN MILLIMETERS) (SEE FIG. 1)

Fig. 15. Photograph of the fabricated proposed filter (filter B).

Fig. 13. EM simulated performance of filter B with symmetrical structure and asymmetrical structure.

Fig. 14. Desired and extracted jf (filter B).

0

f

j

for the proposed tunable filter

at these frequencies within the tuning range. In addition, it is recommended to design the filter with a slightly larger bandwidth than the required specification. B. Tunable Filter with Finite-Frequency Transmission Zero Located at Low Side of the Passband (Filter B) Filter B can be designed following the same design procedure as filter A. The layout of filter B is present in Fig. 12. It should be noticed that the location of loaded varactor for the even mode is different from that of filter A. This is because the resonant frequency of the even mode of filter B is required to be lower than that of the odd mode in order to produce a finite-frequency transmission zero located at low side of the passband. By employing the similar design procedure, filter B with a symmetrical varactor loading structure can be designed. The

Fig. 16. Measured and EM simulated S -parameters of the proposed filter (filter B). (a) S and (b) S . The bias voltage is between 2.2–22 V. The 3-dB absolute-bandwidth is 91 6 6 MHz from 0.57 to 0.98 GHz.

filter parameters are listed in Table II. In order to reduce the number of varactors, filter B with an asymmetrical varactor loading structure can then be developed from the designed symmetrical one. A symmetrically loaded element for the even mode can have an equivalent asymmetrically loaded element as long as the loading point is symmetrical with respect to the I/O ports (see Fig. 12). Both designs result in the same filtering responses as shown in Fig. 13. In this case, the only difference between the two designs, i.e., symmetrical and asymmetrical, is the loading capacitance for the even mode, as indicated in Table II.

TANG AND HONG: VARACTOR-TUNED DUAL-MODE BANDPASS FILTERS

While the extracted external quality factors for both modes are similar to that of Fig. 9(a), the extracted even–odd-mode separation with respect to the loading capacitance Cv shows a different characteristic, as illustrated in Fig. 14. The 3-dB bandwidth of filter B has a larger variation than filter A, the reason is that when the coupling to even mode is stronger, the frequency separation is larger as well, resulting in a larger 3-dB bandwidth. The experimental filter for filter B, fabricated using the same substrate as that for filter A, is shown in Fig. 15. It is EM-simulated and measured responses are displayed in Fig. 16. From Fig. 16, it can be seen that the measured tunable characteristics are similar to the simulated ones. The experimental varactor-tuned bandpass filter shows a high selectivity on the low side of the passband with less than 2.2-dB insertion loss and more than 10-dB return loss over a tuning range of 41% from 0.57 to 0.98 GHz. The applied bias voltage is between 2.2–22.0 V. The measured 3-dB bandwidth is 91 6 6 MHz over the tuning range.

IV. CONCLUSION A new type of varactor-tuned dual-mode microstrip open-loop resonator bandpass filter has been investigated for the constant absolute bandwidth tuning. This type of filter can be tuned in a simple manner by controlling the resonant frequencies of the odd and even modes since these two operating modes do not couple to each other. Design equations and design procedures have been present. By applying the design procedures, two two-pole tunable bandpass filters with opposite asymmetrical frequency responses have been demonstrated with both simulated and experimental results. Good agreement between simulation and measurement is obtained.

REFERENCES [1] W. W. Peng and I. C. Hunter, “A new class of low-loss high-linearity electronically reconfigurable microwave filter,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 8, pp. 1945–1953, Aug. 2008. [2] R. Zhang and R. R. Mansour, “Novel digital and analogue tunable lowpass filters,” IET Microw., Antennas, Propag., vol. 1, no. 3, pp. 549–555, Jun. 2007. [3] Y. H. Chun, H. Shaman, and J.-S. Hong, “Switchable embedded notch structure for UWB bandpass filter,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 9, pp. 590–592, Sep. 2008. [4] M. Houssini, A. Pothier, A. Crunteanu, and P. Blondy, “A 2-pole digitally tunable filter using local one bit varactors,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2008, pp. 37–40. [5] P. S. June and G. M. Rebeiz, “Low-loss two-pole tunable filters with three different predefined bandwidth characteristics,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 5, pp. 1137–1148, May 2008. [6] Y. H. Chun, J.-S. Hong, P. Bao, T. Jackson, and M. J. Lancaster, “Tunable bandstop filters using BST varactor chips,” in Proc. 37th Eur. Microw. Conf., Oct. 2007, pp. 110–113. [7] K. Entesari, K. Obeidat, A. R. Brown, and G. M. Rebeiz, “A 25–75-MHz RF MEMS tunable filter,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 11, pp. 2399–2405, Nov. 2007. [8] B. W. Kim and S. W. Yun, “Varactor-tuned combline bandpass filter using step-impedance microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1279–1283, Apr. 2004. [9] M. S. Chung, I.-S. Kim, and S. W. Yun, “Varactor-tuned hairpin bandpass filter with an attenuation pole,” in Proc. Asia–Pacific Conf., Dec. 2005, vol. 4, 4 pps. [10] C.-K. Liao, C.-Y. Chang, and J. Lin, “A reconfigurable filter based on doublet configuration,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 1607–1610. [11] Y. H. Chun and J.-S. Hong, “Electronically reconfigurable dual-mode microstrip open-loop resonator filter,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 7, pp. 449–451, Jul. 2008.

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[12] W. Tang, J.-S. Hong, and Y. H. Chun, “Compact tunable microstrip bandpass filters with asymmetrical frequency response,” in Eur. Microw. Conf., Amsterdam, The Netherlands, 2008, pp. 599–602. [13] W. Tang and J.-S. Hong, “Tunable microstrip quasi-elliptic function bandpass filters,” in Eur. Microw. Conf., Rome, Italy, 2009, pp. 767–770. [14] I. Wolff, “Microstrip bandpass filter using degenerate modes of a microstrip ring resonator,” Electron. Lett., vol. 8, no. 12, pp. 302–303, Jun. 1972. [15] J.-S. Hong and M. J. Lancaster, “Bandpass characteristics of new dualmode microstrip square loop resonators,” Electron. Lett., vol. 31, no. 11, pp. 891–892, May 1995. [16] J.-S. Hong and M. J. Lancaster, “Microstrip bandpass filter using degenerate modes of a novel meander loop resonator,” IEEE Microw. Wireless Compon. Lett., vol. 5, no. 11, pp. 371–372, Nov. 1995. [17] M. Matsuo, H. Yabuki, and M. Makimoto, “Dual-mode steppedimpedance ring resonator for bandpass filter applications,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 7, pp. 1235–1240, Jul. 2001. [18] J.-S. Hong and S. Li, “Theory and experiment of dual-mode microstrip triangular patch resonators and filters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1237–1243, Apr. 2004. [19] J.-S. Hong, H. Shaman, and Y. H. Chun, “Dual-mode microstrip openloop resonators and filters,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 8, pp. 1764–1770, Aug. 2007. [20] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures. Norwood, MA: Artech House, 1980. [21] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [22] “EM User’s Manual, Version 11.54,” Sonnet Softw. Inc., North Syracuse, NY, 2007. [23] 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. [24] “AVX Accu-P data sheet,” AVX Corporation, Myrtle Beach, SC, 2006. [25] “M/A COM MA46H202 data sheet,” M/A COM, Lowell, MA, 2006. Wenxing Tang (S’08) was born in Guangxi, China. He received the Undergraduate Diploma from Beijing University of Posts and Telecommunications, Beijing, China, in 2002, the M.Sc. degree in electrical engineering from Heriot-Watt University, Edinburgh, U.K., in 2005, and is currently working toward the Ph.D. degree in electrical engineering at Heriot-Watt University. He is currently a Part-Time Research Associate with the Microwave Research Group, Heriot-Watt University. His research interests include microwave and millimeter-wave tunable/reconfigurable devices and multilayer liquid crystal polymer (LCP) technology.

Jia-Sheng Hong (M’94–SM’05) received the D.Phil. degree in engineering science from the University of Oxford, Oxford, U.K., in 1994. His doctoral dissertation concerned EM theory and applications. In 1994, he joined the University of Birmingham, Edgbaston, Birmingham, U.K., where he was involved with microwave applications of high-temperature superconductors, EM modeling, and circuit optimization. In 2001, he joined the Department of Electrical, Electronic and Computer Engineering, Heriot-Watt University, Edinburgh, U.K., as a faculty member leading a team for research into advanced RF/microwave device technologies. He has authored or coauthored over 190 journal and conference papers and also Microstrip Filters for RF/Microwave Applications (Wiley, 2001) and RF and Microwave Coupled-Line Circuits, Second Edition (Artech House, 2007). His current interests involve RF/microwave devices, such as antennas and filters for wireless communications and radar systems, as well as novel material and device technologies including RF microelectromechanical systems (MEMS), and ferroelectric and high-temperature superconducting devices.

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Accurate Synthesis and Design of Wideband and Inhomogeneous Inductive Waveguide Filters Pablo Soto, Member, IEEE, Eva Tarín, Vicente E. Boria, Senior Member, IEEE, Carlos Vicente, Member, IEEE, Jordi Gil, Member, IEEE, and Benito Gimeno, Member, IEEE

Abstract—In this paper, a new synthesis and design methodology is presented and applied for the fast and accurate design of inductive rectangular waveguide filters. By using this technique, the dimensions of both homogeneous and inhomogeneous filters can be successfully synthesized for almost any practical filter bandwidth, return loss, or filter order. This novel technique is based on a prototype with additional degrees of freedom, able to match the response of the different filter components in a wideband frequency range, and an elaborated design procedure that fully exploits this flexibility. During the design procedure, the prototype and the real structure are continuously aligned in order to have the same electromagnetic behavior and jointly evolve to obtain an equiripple response. Once the final prototype has been synthesized, excellent filter dimensions can be extracted that, in most cases, do not require further optimization. Examples will show the outstanding performance of the proposed design technique in terms of versatility, accuracy, and CPU time. Index Terms—Circuit synthesis, design automation, design methodology, distributed parameter circuits, inmittance inverters, losses, stopband performance, waveguide filters.

I. INTRODUCTION

NE OF the most important contributions in microwave filter history is the work of Cohn [1]. Although several significant advances have been obtained over the last decades [2], the design procedure of microwave filters (and, in general, any microwave passive device) is based on Cohn’s paper. This procedure can be summarized in the following steps. • Step S1: Synthesis of an equivalent-circuit prototype. The prototype must be as similar as possible to the real structure and, simultaneously, simple enough to be synthesized in an analytical form to obtain a suitable ideal response.

O

Manuscript received January 27, 2010; revised April 21, 2010; accepted May 16, 2010. Date of publication July 01, 2010; date of current version August 13, 2010. This work was supported by the Ministerio de Educación y Ciencia, Spanish Government, under Research Project TEC2007/67630-C03-01. P. Soto, E. Tarín, and V. E. Boria are with the Grupo de Aplicaciones de las Microondas (GAM), Instituto de Telecomunicaciones y Aplicaciones Multimedia (iTEAM), Universidad Politécnica de Valencia, E-46022 Valencia, Spain (e-mail: [email protected]; [email protected]; [email protected]). C. Vicente and J. Gil are with Aurora Software and Testing S.L., E-46022 Valencia, Spain (e-mail: [email protected]; [email protected]). B. Gimeno is with the Departamento de Física Aplicada y Electromagnetismo, Instituto de Ciencas de los Materiales, Universidad de Valencia, E-46100 Burjassot, Valencia, Spain (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.2010.2052668

Fig. 1. Inhomogeneous inductive rectangular waveguide filter and significant physical dimensions.

• Step S2: Carry out an equivalence between the prototype elements and the real structure parts, thus obtaining an initial value for the device physical dimensions. • Step S3: Perform an optimization or final refinement of the filter dimensions obtained in Step S2. In the past, Step S3 was accomplished by a cut-and-try procedure assisted with tuning screws. Over the last two decades, this final refinement is efficiently performed with computer-aided design (CAD) tools; the use of tuning screws is now required only to compensate for the manufacture tolerances in sensitive dimensions. The success of this design methodology is based on the similarities between the prototype and the structure. When the equivalent prototype behaves as the real structure, very good starting dimensions can be obtained after Step S2. A filter with excellent performance is then easily obtained by computer optimization. When the starting dimensions are poor, however, the optimization procedure is cumbersome, and Step S3 tends to fail or to provide a filter with unsatisfactory response. For narrowband filters, this design procedure works nicely, as the filter can be accurately represented by means of a lumped-element prototype. Moreover, the optimization in Step S3 can be based in the synthesis of the protoype that recovers the same defective response provided by the real structure. After comparing this prototype with the ideal prototype from Step S1, the structure dimensions are adjusted to obtain the ideal response. Excellent design techniques for narrowband filters have been proposed, such as [3]–[6] to cite a few. These design techniques are not, however, suitable for filters with wider bandwidths. Since the prototype must be simple

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Fig. 2. Generalized half-wave prototype for inductive rectangular waveguide filters, including its degrees of freedom.

enough to be synthesized analytically in Step S1, the frequency dependence of the real structure over a wide bandwidth cannot be represented. As a result, the starting point obtained is usually poor and the design procedure is liable to fail. This problem is further increased when the filter elements exhibit different frequency dependence. Inhomogeneous structures, namely, structures in which the ratios of the internal wavelengths and characteristic impedances at different positions along the direction of propagation change with frequency [7, p. 257], are very hard to design. Nonetheless, these structures are of interest to designers since their additional degrees of freedom can be exploited to improve the filter performances. This investigation approaches this problem with a new and general design methodology also suitable for wideband and inhomogeneous structures, which will be developed for the particular case of inductive rectangular waveguide filters. Although simple structures, inductive filters are widely used in several applications, due to their quality factor and powerhandling capability. The first design procedure was based on a lumped-element prototype [1]. This method provides good results for narrowband filters, but fails for higher bandwidths since the structure resonators are distributed in nature. For moderate bandwidth filters (i.e., relative bandwidth of 5%–15% in wavelength terms), better results are obtained with the distributed homogeneous prototype proposed in [8]. Its main inaccuracy is a frequency constant model of the inductive shunts elements. This problem was solved by Levy in [9] with an exact model for ideal shunt inductive obstacles, thus obtaining good results for narrow and moderate bandwidths. A similar method was proposed by Rhodes in [10], [11] with explicit formulas for the prototype elements. The technique in [9] was recently refined in [12] after applying the procedure [13] to model the real coupling window rather than an ideal shunt inductive obstacle. In spite of these efforts, the prototype is kept simple enough to perform an analytical synthesis and is not able to exactly represent the structure elements in a wide frequency range. As a result, the response obtained for wideband filters is flawed (i.e., reflection zeros are usually lost). Additionally, since the prototype is homogeneous, this technique is unable to deal with inhomogeneous filters with different resonator widths. A generalization of [10], [11] to the inhomogeneous case was reported in [14], but the resulting method is only valid in a very narrowband scope. The accurate representation of dispersive inductive obstacles [15] is crucial for wideband filters. In [16], a generalized and flexible prototype able to represent the wideband frequency behavior of a wide class of direct-coupled-cavity filters was presented. The design technique, however, was very simple and

only able to provide an impaired prototype response that can be exactly recovered with a real structure. The quality of the recovered response depends on the similarities between the real filter and the classic half-wave prototype, and is unsatisfactory for filter with demanding specs. In this paper, a novel and complete design methodology that overcomes the limitations of [16] is presented for the particular case of inline inductive iris rectangular waveguide filters. Using this methodology, the dimensions of any type of inline inductive iris filter can be accurately synthesized in negligible CPU time regardless of the filter bandwidth, order, return loss, inhomogeneity, or symmetry along the direction of propagation. An outstanding synthesized structure is obtained that, in most cases, does not even require the final optimization Step S3, thus speeding up the whole design procedure. II. WIDEBAND CIRCUIT PROTOTYPE A. Prototype Description To model a general inductive iris filter in rectangular waveguide (see Fig. 1), the prototype shown in Fig. 2 is proposed. This prototype is an improved version of the one described in [16]. A monomode prototype has been chosen since the higher order mode coupling between the structure elements is nearly negligible. To accurately represent the structure, the propagaof the th transtion constant and the guide wavelength mission line are chosen to be the same of the propagating mode in its corresponding filter waveguide, whereas the characteristic is defined as impedance (1) where is the intrinsic impedance of the homogeneous medium and is a prototype free parameter. The characteristic impedance of a TEM mode, which is constant with frequency, . To represent a TE and TM mode, the is obtained with parameter must be set to 1 and 1, respectively. The representation of a frequency-dependent impedance inverter is performed following [16]. As a result, the unnormalized impedance parameter of the th prototype inverter is given by (2) models the variation of the structure where the parameter with frequency. To simplify the representaparameter , is used for all the tion, a common reference frequency, . prototype impedance inverters so that

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Further, the filter elements implementing the impedance inverters introduce a frequency-dependent length correction in their adjacent resonators. To describe this effect in a wide bandwidth, the length of the th prototype transmission line incorporates a quadratic correction term

(3) and , whereas is the with polynomial coefficients guide wavelength of the th resonator at its resonant frequency. Note that, at this frequency, the transmission line length is equal to and a resonance occurs. In contrast to classic prototypes, each resonator can have a different resonant frequency. Although this point complicates to a great extent the synthesis of the proposed prototype, it allows an accurate representation of both homogeneous and inhomogeneous filters. It is worth recalling that inhomogeneous filters are asynchronously tuned devices. To obtain the ideal response, a synthesis variable in each resonator and impedance inverter is required. For this prototype, and the central frethe impedance inverter parameters of the resonators (where the transmission line length quency is a half guide wavelength) will be used. The remaining param, eters are, therefore, free. The prototype free parameters , and will be used to model the real inverter frequency dependence. The parameter , not included in [16], will be used to reduce the step between successive prototypes in the design procedure described in Section III. It can be seen that the prototype in Fig. 2 is a generalization of the distributed prototypes traditionally used for the design of inductive filters. In fact, the classic distributed half-wave prototype [8] is a synchronously tuned prototype with all the free to have a constant charparameters equal to 0, including acteristic impedance. The prototype proposed in [9] is obtained and the resonator parameters are once the parameters set to 1 to model an ideal shunt inductance between rectangular waveguides. From now on, the prototype in Fig. 2 will be called the generalized half-wave prototype. B. Prototype Synthesis Conventional prototypes can be synthesized in an analytical form since their response can be represented in terms of a polynomial function in a suitable wavelength related variable. However, the response of the generalized half-wave prototype described in Section II-A cannot be expressed by using a polynomial function in just one variable. In fact, for the inhomogeneous case, each resonator exhibits a different frequency behavior that cannot be summarized in a common wavelength variable. The prototype generalization cost is the inability to define an analytical synthesis procedure. Instead, an approximate synthesis method between a generalized half-wave prototype and a previously synthesized one will be used. To complete the synthesis, the prototype resulting from the approximate synthesis procedure must be refined with a fine circuit optimizer.

Fig. 3. (a) Frequency-dependent impedance inverter from Fig. 2. (b) Equivalent circuit where the frequency dependence is transferred to the input and output transformers.

Fig. 4. Equivalence of a transmission line between two transformers of the same turns ratio and connected in the opposite way.

The first step in the approximate synthesis procedure consists in transforming the prototype in Fig. 2 to a prototype with constant impedance inverters. To accomplish this task, the inverter frequency dependence must be transferred to the transmission lines by using the equivalence shown in Fig. 3, where and are free constant parameters and the functions and must fulfill (4)

To keep the transmission line symmetry after performing the equivalences, the same turns ratio will be used for both transformers connected in the opposite way to the th transmission line (5) so that the equivalence in Fig. 4 can be applied. Accordingly, for the th inverter, the turns ratios and are chosen th and th transmission lines, to be those related to the respectively,

(6)

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Fig. 5. Generalized half-wave prototype in the form used to perform the approximate synthesis procedure.

As a result, after applying the circuit equivalences compiled in Figs. 3 and 4 to the original generalized half-wave prototype, the prototype form shown in Fig. 5 is obtained. This form, with frequency-independent impedance inverters, will be used to perform the approximate synthesis procedure. In this equivalence, the transformers in the prototype input and output are removed by including their impedance multiplicative effect in the prototype access ports. Since condition (2) must be fulfilled when relation (4) is used to perform the equivalences, the parameters must satisfy (7) which sets all the parameters , except for . Taking into consideration that the prototype synthesis is easier when the transmission lines are more similar, and the central elements of all-pole Chebyshev prototypes are close to be periodic, the is set to make the center of the prototype as simparameter ilar as possible. If is even, the parameters of the resonators adjacent to the central coupling window are equated . From this condition, the value of can be deduced to be (8) whereas if is odd, the central resonator parameter is chosen to be half of the average value of its neighboring coupling window parameters (9) so that the parameter

given by

(10) must be used. For symmetric filter topologies, the prototype obtained after applying (7)–(10) is also symmetric. This fact simplifies the design procedure since only half of the prototype and the real structure must be considered. Once the generalized half-wave prototype with frequency-independent impedance inverters is obtained, an equivalence with a previously synthesized prototype, also in the form shown in Fig. 5, is performed. This equivalence is carried out on an element-by-element basis. Let us denote by prototype A the previously synthesized prototype, and by prototype B the goal prototype that is going to be synthesized from the prototype A.

First, the equivalence between transmission lines is accomplished. The matrix representation of a transmission line section of length is (11) with and representing the characteristic impedance and guide wavelength of the line, respectively. Close to the transresonance, and with an error lower than 3% mission line in a 15% wavelength bandwidth, the term can be approx. This result suggests the following mapping imated by between both lines to establish the equivalence: (12) where the tilde and normal terms represent the parameters of the transmission lines of prototypes A and B, respectively. In the case of two generalized half-wave prototypes in their synthesis form (see Fig. 5), the mapping (12) for the th transmission line can be stated as

(13) using (1), (3), and (5) for the transmission line characteristic impedance, length, and turns ratio, respectively. Observe that , and in both prototypes can be the parameters , , different. are free, two conditions Since only the parameters and can be enforced to establish the equivalence. To have a good behavior in the whole filter passband, (13) is imposed at the filter passband cutoff frequencies. After applying these conditions, a nonlinear system of two equations is obtained. This system and by using the Newton–Raphson is easily solved for method [17]. For a transmission line, (13) can be interpreted as a frequency mapping that relates each frequency in the passband of prototype A to a frequency where prototype B has an equivalent electrical behavior. This mapping coincides at the passband limits. Since each transmission line establishes a different mapping, the prototype A response is not exactly recovered with prototype B. The resonant frequency of the generalized half-wave prototype transmission lines will be different in a general case. This point is essential, however, for a method able to cope with inhomogeneous filters that have resonators of diverse characteristics. In homogeneous wideband filters, the coupling windows also exhibit different response variation with frequency. Since this variation influences the adjacent resonators, when an ideal transfer function (i.e., passband equiripple) is exactly achieved, the real filter resonators will also have slightly different resonant

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frequencies. Hence, a prototype with this enhanced flexibility is able to represent the behavior of real structures more accurately, even for the homogeneous case. With regard to the input and output prototype ports, their impedance must be made as similar as possible in the filter passband, namely,

(14) where only the parameter is available. Therefore, the equality can only be imposed at a particular frequency in the passband, and the equivalence degrades as the frequency moves away. The option chosen was to adjust the parameter in order to have the same balanced error at the passband cutoff frequencies. By forcing these conditions, a system of two equations in the variables and is obtained; this provides the parameter and also the maximum error in the passband. Finally, the prototype B impedance inverters are equated to those of the previously synthesized prototype A. For the th inverter, we have (15)

where all the terms, except for , are known once the equivalences between the transmission lines and prototype input and output ports have been performed. The accuracy of the approximated synthesis technique depends on the proximity between prototypes A and B, along with the differences between the resonator mappings. After performing the synthesis of a vast range of prototypes, the results reveal that, for the same structure, the resonator mappings are are similar (variations greater than 0.5% in the value of very rare even for wideband inhomogeneous filters). This fact guarantees that if both prototypes are not very different, a very good starting point will be obtained after applying the approximate synthesis procedure. To recover the goal response, only and a final optimization from the initial values of must be performed. In this paper, the equal ripple bandpass response will be considered as the goal function. Thus, an equiripple optimizer has been developed. This optimizer uses an enhanced version of the technique described in [18] with improved robustness due to its integration in a trust region strategy [19]. III. DESIGN PROCEDURE The design procedure of microwave components can be divided into the three separated steps enumerated in Section I. In this section, a fully automated design procedure that integrates the first two steps is proposed. By combining both steps, the quality of the extracted initial structure can be greatly improved. This excellent starting point for Step S3 guarantees the success of the design procedure after a simple and fast final optimization. The key point of this novel design procedure is the availability of a very flexible prototype, which has an analytically

synthesizable form, can be tuned to match the behavior of the real structure, and can be gradually moved between an already synthesized prototype and a goal prototype. The price is a complex synthesis procedure, which requires the combination of an approximate synthesis algorithm, an intermediate prototype strategy and a circuit optimizer. The overall design procedure starts with the exact synthesis of a classic distributed half-wave prototype, the particular case of a generalized half-wave prototype with all the parameters , , , and set to 0. Since this particular prototype must be homogeneous, the cutoff wavenumber of all the prototype is chosen to be lines (16) where is the fundamental mode cutoff wavenumber in the structure waveguide related to the prototype th transmission line. The resulting prototype is analytically synthesized using traditional techniques [8]. From the synthesized classic prototype, the synthesis of a prototype more similar to the real structure is attempted. mode behavior is For this goal prototype, the rigorously considered by setting , and by using in each prototype transmission its cutoff wavenumber line. Nevertheless, since a priori information of the real coupling windows frequency dependence is not available, a pure inductive behavior is assumed by taking the parameters and . Once the goal prototype is successfully synthesized, initial structure dimensions must be deduced. The electromagnetic (EM) simulation of the real inductive windows, with their input and output waveguides, will be performed. From this simulation, the length correction that must be applied to the access waveguides to have a pure impedance inverter behavior can be computed. More precisely, for the th real inductive window, these corrections are the shortest lengths in magnitude satisfying

(17) that provide an impedance inverter with parameter given by (18) and are the magwhere and are integers, input reflection coefficient, whereas nitude and phase of the is the phase of the transmission parameter. The terms and denote, respectively, the length correction at the waveguide related to the th prototype line due to their input and output real coupling windows. Based on (18), the physical dimensions of the coupling windows that provide the required prototype impedance parameter at can be determined. To perform this task, an

SOTO et al.: ACCURATE SYNTHESIS AND DESIGN OF WIDEBAND AND INHOMOGENEOUS INDUCTIVE WAVEGUIDE FILTERS

automated algorithm that uses the root-finding Newton–Rapshon and Brent’s methods, has been implemented [17]. Once the physical dimensions of the structure coupling windows have been extracted, the physical lengths of the resonator waveguide sections are set according to (19) was obtained during the prototype synthesis procewhere is the resonant frequency of the prototype th transdure, and ). mission line (where At this point, an initial structure has been obtained, and, in a traditional design procedure, the final optimization of the structure starts. In the novel design methodology proposed in this paper, however, the extracted coupling windows are simulated in the filter passband in order to extract their wideband behavior. From this analysis, a least square fit is used to represent the impedance inverter parameters in the form (2) and also the coracrection lengths using a parabolic expansion in terms of cording to (3). As a result, new values are set for the parameters , , and . By introducing this feedback in the design procedure, the prototype inherits the frequency dependence of the real structure. The resulting prototype is synthesized using the technique detailed in Section II-B from the last goal prototype; a new iteration then begins. The algorithm is repeated until the variation of the structure dimensions in two consecutive iterations is lower than a fixed tolerance. The convergence is usually achieved in very few iterations. Following this procedure, the prototype and the structure feedback to each other and jointly evolve to obtain an excellent extracted structure. Moreover, the computational burden is centered in the prototype synthesis (more precisely, in the prototype optimization) since the EM simulations are reduced to a minimum. Therefore, only circuit optimizations are required. The entire procedure is extremely fast for an outstanding extracted final structure. In fact, the overall design time is greatly reduced due to the time saved in the final optimization step that, in many cases, is not even required. To integrate the design procedure in a robust and fully automated CAD tool, the prototype synthesis must succeed in all the iterations by recovering the desired equiripple response. In difficult filters, mainly in the first two iterations, the differences between the generalized half-wave goal prototype and the previously synthesized one are important. The starting point provided by the approximate synthesis procedure is therefore not suitable to perform the optimization (for instance, a reflection zero is lost and the equal ripple optimizer cannot be initialized) or simply is not good enough to have an agile and fast circuit optimization. When these situations are detected, the prototype synthesis task is considered too difficult. Instead, an intermediate proto, , , and type is generated with the parameters , placed midway between the parameters of the synthesized prototype and the parameters of the goal prototype. Next, the intermediate prototype synthesis is attempted. Once the intermediate prototype has been successfully optimized, it is used as a reference to try to synthesize the original goal prototype from

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it. In addition, the error between the final values of the protoand ) of the intermediate type synthesis variables ( prototype and the starting values that were provided by the approximate synthesis procedure are stored. These values are then used to estimate and compensate for the error of the approximate initial guess in the next synthesis. This compensation dramatically improves the synthesis steps performed from intermediate prototypes. This procedure is recursive and an additional intermediate prototype can be generated to synthesize a previous one whose synthesis is troublesome. Since the differences between successive intermediate prototypes can be as small as required, the resulting algorithm proves to be extremely robust. The design of almost any practical filter can be successfully accomplished regardless of their bandwidth, order, and return loss. To illustrate these concepts, the design of a seventh-order 1-GHz bandwidth filter centered at 11 GHz and with 27-dB return loss will be described. The ports are WR-75 waveguides, whereas the resonator widths alternate between 20.55 and 17.05 mm. The window lengths are set to 2 mm. Since the generalized half-wave prototype to be synthesized in the first iteration is very different from the initial classic distributed half-wave prototype, the approximate synthesis technique is unable to provide a good starting point [see Fig. 6(b)]. Hence, the intermediate prototype 1 is created, but again the obtained starting point is not fair and another intermediate prototype is generated. For this second intermediate prototype, closer to the classic half-wave prototype, the approximate synthesis technique provides an initial point with all the reflection zeros, and the equiripple response can be recovered after the prototype optimization [see Fig. 6(a) and (b)]. From this prototype, the synthesis of the intermediate prototype 1 is accomplished. Finally, from the first intermediate prototype, the original goal prototype is successfully synthesized. As shown in Fig. 6(a), the two last steps are straightforward due to the error estimation algorithm. The synthesized goal prototype is used to extract the physical dimensions of an initial structure. Next, the free parameters of the generalized prototype are updated to represent the frequency dependence of the extracted structure elements. A new iteration then starts from the last iteration goal prototype. Four iterations are required to obtain a convergence in the structure physical dimensions. Both the structure and the prototype jointly evolve iteration after iteration towards the final solution. The summarized data compiled in Table I shows that the prototype synthesis parameters converge in the fourth iteraalso progress with the structure uption. The parameters dates to represent the frequency behavior of their coupling windows. Since the structure is inhomogeneous, different central are obtained for each prototype and filter resfrequencies onator. Note that the symmetry of the selected filter topology is preserved by the design procedure. The response of the extracted structure is compared in Fig. 7 with the response of the final synthesized prototype. As expected, the prototype shows a perfect equiripple response. The extracted structure response is excellent for the exigent set of specifications, although is not exactly equiripple. The main reason for the slight variations is the higher order modes

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Fig. 7. Responses of the extracted structure after the combined prototype-structure synthesis, the final generalized half-wave prototype, the structure after an equiripple optimization, and the extracted structures obtained using the methods in [14] and [16]. The dimensions of the extracted structure (and their differences with the post-synthesis optimized structure) l : mm m , l l : mm m , are l l l : mm m , l : mm m , w w : mm w : mm m , w m , w : mm m , and w w : mm m . w

= = 12 196 (+13 ) = = = 14 775 (0 ) = = 12 707 (+9 ) = = = 9 191 (+5 ) =

Fig. 6. (a) Schematic description of the first iteration of the combined prototype-structure synthesis, where the dashed and continuous lines denotes failed and successfull synthesis attempts. (b) Responses of the initial starting points provided by the approximated synthesis technique in the first three steps.

TABLE I EVOLUTION OF SELECTED PROTOTYPE PARAMETERS DURING THE COMBINED PROTOTYPE-STRUCTURE SYNTHESIS

effect, not considered in the monomode model described in Section II-A. The presence of these modes in the wider resonators introduces slight differences between the structure and the prototype model that cause minor degradations in the extracted dimensions (around 10 m) and the filter response. After performing an extremely fast optimization from this excellent starting point, the equiripple response of the final prototype is exactly recovered with the real structure.

= 17 967 (018 ) = 18 876 (08 ) = 10 144 (+12 ) = 8 968 (+6 )

Although the computational time required by all the synthesis methods used in Fig. 7 is negligible (below 1 s), their performances are quite different. The method reported in [14] uses approximations that neglect the diverse frequency variations of the filter resonators, thus being only valid in a narrowband scope for low sensitive structures. As a result, the deviations of the extracted dimensions from the optimized filter dimensions are in the range of 100–500 m for this particular filter. Better results are obtained using [16], as the average error in the extracted dimensions is reduced to about 80 m. The filter passband cutoff frequencies are successfully recovered, and only a serious impairment in the passband ripple is observed. This example, however, illustrates the method limitations. The method in [16] only performs the approximated synthesis procedure of Step 1 in Fig. 6(a) from the classic half-wave prototype to the goal prototype. The resulting prototype is accepted regardless of the obtained response and without any optimization (in the new methodology, this prototype is rejected and the intermediate prototype strategy is launched to perform an accurate synthesis). The structure dimensions are then extracted from this imperfect prototype. Next, the free parameters of the generalized prototype are updated to model the extracted filter elements. Finally, a new iteration starts with an approximated synthesis procedure from the same classic half-wave prototype to the new generalized prototype, which will provide a defective response that is closer to be recovered with a real structure. The similarities between the extracted response in Fig. 7 from [16] and the Step 1 response in Fig. 6(b) are evident. In contrast to the design methodology presented in this paper, the method in [16] is unable to progress. It is only able to provide an imperfect response that can be recovered with a real structure. The impairment grade of the extracted structure depends on the differences between the real filter and classic half-wave prototype.

SOTO et al.: ACCURATE SYNTHESIS AND DESIGN OF WIDEBAND AND INHOMOGENEOUS INDUCTIVE WAVEGUIDE FILTERS

Fig. 8. Comparison between the responses of the different synthesized structures for the WR75 homogeneous wideband filter. The dimensions of the extracted structure with the present method (and their differences with the postl : mm m , synthesis optimized equiripple filter) are l l : l : l mm m , l mm m , l l : mm m , l l : mm m , l : mm m , w w : mm m , w w mm m , w w : mm m , w w : : mm m , w w : mm m , and w w mm m . :

= = 12 409 9 920 8 719 8 571

= = 9 959 (+33 = 11 493 (06 ) = = 12 102 (07 = 12 312 (07 ) = = 12 389 (06 ) (06 ) = = 12 638 (+4 ) = (+17 ) = = 9 013 (+18 ) = (+19 ) = = 8 610 (+18 ) = (+18 )

) ) = = = =

As shown in Fig. 7, the design methodology proposed in this paper does not present these limitations. In fact, for the range of filters that can be accurately modeled by the generalized and flexible prototype (in this application, inline inductive waveguide filters), it is able to provide an outstanding extracted structure in a very reduced CPU time. This structure can be straightforwardly optimized, if required. The excellent starting point guarantees the fast success of the final optimization and the whole design procedure. IV. RESULTS An accurate CAD tool based on the design procedure described in this paper has been implemented. To perform the full-wave simulations, the efficient and accurate modal analysis techniques for inductive filters described in [20] and [21] have been used. The tool is completely automated and only requires the user contribution to introduce the input file with the filter specifications and topology considerations. All the CPU times reported in this section were obtained in a PC with a 2.66-GHz E6750 Intel Core 2 Duo processor. For the first example, a homogeneous wideband filter has been considered. The filter uses WR75 resonators coupled through 1.5-mm thickness inductive irises. The filter passband extends from 11.25 to 13.75 GHz, i.e., 20% relative bandwidth in frequency (33.2% in terms of guide wavelength). The filter order is 11 and the return loss is 25 dB. The structure has been synthesized with the procedure described in this paper, the recent technique [12], and the method due to Rhodes [10], [11]. In Fig. 8, the extracted structure responses are compared. The methods in [12] and [10], [11] provide good responses for this extremely wide filter. Both responses are very similar,

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with a 100-MHz reduction in the filter upper passband cutoff frequency, and with only seven of the 11 reflection zeros on the frequency axis (although the position of some of the remaining zeros can be guessed). For both extracted structures, the average error with the dimensions of the equiripple optimized structure are about 150 m with a maximum error close to 500 m. Since the structure is not sensitive, these differences are surmountable. In fact, using an appropriate optimization method with suitable goals and settings, and after an exhaustive optimization that took about 2 h, the authors were able to obtain the optimized structure with equal ripple response from both starting points. Instead, the structure obtained in negligible CPU time with the combined prototype-structure synthesis procedure is excellent, very close to be perfect equiripple, and does not require further optimization. Also, the structure can be optimized in only 15 s to match the equal ripple transfer function. To check the accuracy of the new CAD tool, the extracted structure was simulated using Ansoft’s High Frequency Structure Simulator (HFSS) [22]. The agreement between both responses is excellent, as shown in Fig. 8. In order to test the limits of the novel methodology, the filter specifications were changed. Filters with orders greater than 30, or with return loss over 35 dB, were also successfully designed even though these limits are beyond the specifications of practical filters. This example also reveals the importance of passband equal ripple responses. These responses optimize the filter skirt selectivity close to the passband, and therefore, reduce the filter order required to satisfy the stopband selectivity specifications. Fig. 8 shows that the rejection at 10.5 GHz of the extracted structure is greater than the rejection provided by the other synthesized structures whose responses are farther from the equiripple response, even though their passbands are narrower. This effect is also observed above the passband if the upper passband cutoff frequency shift is compensated. In fact, the same rejection of both eleventh-order nonequiripple structures can be obtained with an equivalent tenth-order equiripple filter with the same passband bandwidth. The method proposed in this paper can be applied to any inductive obstacle, provided that an appropriate EM simulator tool is available. In order to expand the technique range of application, it has been integrated inside the FEST3D CAD framework [23] that uses the boundary integral–resonant mode expansion (BI–RME) method [21] to analyze arbitrarily shaped inductive obstacles. For the next example, a tenth-order filter with inductive rounded posts as coupling elements has been designed [24]. The filter ports are standard WR-75 waveguides. The passband is centered at 10.5 GHz with a 250-MHz bandwidth and 23-dB return loss. The radii of the rounded posts were fixed, and the post offsets were the coupling design parameters. To improve the spurious free stopband extent, an inhomogeneous filter with different resonator widths was chosen. The use of nonidentical resonators to improve the stopband response was reported many years ago [25], and recently used for -plane filters in [26], [27] and for -plane filters in [28]. For the filter with rounded posts, the resonator widths were carefully chosen. A filter with four 24-mm-wide resonators followed by six WR-75 resonators was considered.

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Fig. 10. (a) Top view (drawn to scale) of the inhomogeneous filter with different ports. (b) Response comparison of the implementation alternatives.

Fig. 9. (a) Topology of the designed filter from FEST3D. (b) and (c) Simulated passband and stopband responses, respectively.

The filter was synthesized with the new methodology in less than 2 min. Note that the structure includes arbitrarily shaped obstacles, and the design of asymmetric filters is more challenging since the design variables are doubled for the same order. The filter layout, as well as the simulated passband and stopband responses, are depicted in Fig. 9. The passband response of the extracted structure is excellent and a post-synthesis optimization is not required. The agreement with the simulated results provided by HFSS is very good [see Fig. 9(b)]. Conversely, the response of the structure obtained by using [14] is not completely satisfactory. Although this response is good when the frequency is very close to

10.5 GHz, it degrades as the frequency moves from the passband center frequency. In fact, this filter cannot be considered narrowband since, in terms of the WR-75 guide wavelength, its relative bandwidth is close to 6%. The authors were not able to recover the equiripple response from this starting point with computer optimization. The simulated stopband performance is very good, as shown in Fig. 9(c). A rejection greater than 50 dB is obtained from 10.7 to 21.4 GHz (covering the second harmonic) that diminmode starts ishes to 40 dB up to 23.6 GHz, where the to propagate in the filter access ports. The improvement on an equivalent homogeneous structure is remarkable. From a practical point of view, manufacture tolerances can introduce asymmetries that excite unexpected modes and degrade the stopband filter response depicted in Fig. 9(c) performance. The shows that these asymmetries could introduce narrow spikes around 21.4 GHz. On the other hand, the spikes related to the mode can be avoided by reducing the filter height to keep this mode below cutoff in the stopband. Besides the out-of-band performance, the additional degrees of freedom of inhomogeneous structures can be used to reduce the filter length or tradeoff the response selectivity above and below the filter passband. Wider resonators provide shorter filters and improve the skirt selectivity above the passband. The authors have not observed, however, a noticeable insertion loss improvement related to the resonator width choice. Finally, inhomogeneous structures can be used to implement devices with different waveguide access ports. The last application example consists of a filtering structure with a passband covering the common frequency range between the WR-90 and WR-75 recommended operation band. The structure input port is a WR-90 standard waveguide, whereas the output port is a

SOTO et al.: ACCURATE SYNTHESIS AND DESIGN OF WIDEBAND AND INHOMOGENEOUS INDUCTIVE WAVEGUIDE FILTERS

19.05-mm-wide waveguide (the height is kept constant to have an inductive structure). This problem can be solved with a transformer followed by a filter with 19.05 10.16 mm cross-section resonators or a homogeneous filter in WR-90 waveguide followed by a transformer to the output port. In addition, an inhomogeneous filter whose resonator widths gradually reduce from the input to the output waveguide can be used [see Fig. 10(a)]. This filter can be synthesized using the novel methodology proposed in this paper. The three solutions were implemented using eighth-order filters. The filters were synthesized and optimized to obtain an equiripple response with the CAD tool developed. To perform the impedance matching in the first two solutions, a secondorder transformer was required. The simulated responses are compared in Fig. 10(b). The inhomogeneous filter selectivity is placed between those of the other two structures, according to a structure whose resonator widths vary between 19.05–22.86 mm. On the other hand, and due to the lack of a separate transformer, the proposed topology presents the advantage of reduced length (about 15%–20%, excluding the structure ports). It can be concluded that the inhomogeneous topology is a more compact solution that performs the filter and transformer function simultaneously, without penalty in the filter selectivity.

V. CONCLUSIONS In this paper, a novel methodology for the design of microwave structures is proposed. This methodology is based on merging the prototype synthesis and the structure extraction. A combined prototype and structure synthesis is then obtained, where both jointly evolve and feedback to each other to obtain an excellent starting structure in reduced CPU times. This structure requires, at most, a fast final optimization. The technique key points are the availability of a very general and flexible prototype, and the use of an elaborated synthesis methodology that fully exploits this flexibility. At each iteration, the synthesis algorithm lines up the prototype with the real structure and then succesfully performs a complex prototype synthesis to obtain the goal transfer function (combining the approximate synthesis procedure, the intermediate prototype strategy and a circuit optimizer). The extracted real structure, guided by the aligned prototype, progresses iteration after iteration to finally recover the goal response. This methodology has been presented for the particular case of inline inductive waveguide filters. As a result, almost any inductive filter with practical interest can be accurately designed regardless of its bandwidth, return loss, order, or structure symmetry and homogeneity. After the combined prototype-structure synthesis, an outstanding structure is obtained. This synthesized structure guarantees the very fast success of the whole design procedure since, at worst, only a final refinement is required. The results clearly prove the advantages of this new approach over traditional design methodologies. It is the authors’ intention to extend this novel and powerful methodology to a wider range of microwave structures.

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ACKNOWLEDGMENT The authors are grateful to the reviewers for their constructive suggestions that have been incorporated in this paper. REFERENCES [1] S. B. Cohn, “Direct-coupled-resonator filters,” Proc. IRE, vol. 45, no. 2, pp. 187–196, Feb. 1957. [2] R. J. Cameron, C. M. Kudsia, and R. R. Mansour, Microwave Filters for Communication Systems: Fudamentals, Design and Applications. Hoboken, NJ: Wiley, 2007. [3] S. Bila, D. Baillargeat, M. Aubourg, S. Verdeyme, P. Guillon, F. Seyfert, J. Grimm, L. Baratchart, C. Zanchi, and J. Sombrin, “Direct electromagnetic optimization of microwave filters,” IEEE Microw. Mag., vol. 2, no. 1, pp. 46–51, Mar. 2001. [4] S. F. Peik and R. R. Mansour, “A novel design approach for microwave planar filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, Jun. 2002, pp. 1109–1112. [5] P. Kozakowski and M. Mrozowski, “Automated CAD of coupled resonator filters,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 12, pp. 470–472, Dec. 2002. [6] A. Garcia-Lamperez, S. Llorente-Romano, M. Salazar-Palma, and T. K. Sarkar, “Efficient electromagnetic optimization of microwave filters and multiplexers using rational models,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 508–521, Feb. 2004. [7] G. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures. Norwood, MA: Artech House, 1980. [8] L. Young, “Direct-coupled cavity filters for wide and narrow bandwidths,” IEEE Trans. Microw. Theory Tech., vol. MTT-11, no. 3, pp. 162–178, May 1963. [9] R. Levy, “Theory of direct-coupled-cavity filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-15, no. 6, pp. 340–348, Jun. 1967. [10] J. D. Rhodes, “The generalized direct-coupled cavity linear phase filter,” IEEE Trans. Microw. Theory Tech., vol. MTT-18, no. 6, pp. 308–313, Feb. 1970. [11] J. D. Rhodes, Theory of Electrical Filters. New York: Wiley, 1976. [12] F. M. Vanin, D. Schmitt, and R. Levy, “Dimensional synthesis for wide-band waveguide filters and diplexers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2488–2495, Nov. 2004. [13] H. Y. Hwang and S.-W. Yun, “The design of bandpass filters considering frequency dependence of inverters,” Microw. J., vol. 45, no. 9, pp. 154–163, Sep. 2002. [14] R. Balasubramanian and P. Pramanick, “Computer-aided design of -plane tapered corrugated waveguide bandpass filters,” Int. J. RF Microw. Comput.-Aided Eng., vol. 9, no. 1, pp. 14–21, Jan. 1999. [15] S. Amari, J. Bornemann, W. Menzel, and F. Alessandri, “Diplexer design using pre-synthesized waveguide filters with strongly dispersive inverters,” in IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, May 2001, pp. 1627–1630. [16] P. Soto and V. E. Boria, “A versatile prototype for the accurate design of homogeneous and inhomogeneous wide bandwidth direct-coupledcavity filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 451–454. [17] W. H. Press, S. A. Teukolsky, W. S. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing. New York: Cambridge Univ. Press, 1999. [18] D. Budimir, Generalized Filter Design by Computer Optimization. Norwood, MA: Artech House, 1998. [19] J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlineal Equations. Englewood Cliffs, NJ: Prentice-Hall, 1987. [20] M. Guglielmi, G. Gheri, M. Calamia, and G. Pelosi, “Rigorous multimode network numerical representation of inductive step,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 2, pp. 317–326, Feb. 1994. [21] G. Conciauro, M. Guglielmi, and R. Sorrentino, Advanced Modal Analysis: CAD Techniques for Waveguide Components. New York: Wiley, 2000. [22] “HFSS Manual,” Ansoft Corporation, Pittsburgh, PA, 2008, rel. 11.1.3. [23] FEST3D 6.5.0 Aurora Software and Testing S. L., Valencia, Spain, 2010. [Online]. Available: www.fest3d.com, (on behalf of ESA/ESTEC) [24] S. Yin, T. Vasilyeva, and P. Pramanick, “Use of three-dimensional field simulators in the synthesis of waveguide round rod bandpass filters,” Int. J. RF Microw. Comput.-Aided Eng., vol. 8, no. 6, pp. 484–497, Dec. 1998.

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[25] H. J. Riblet, “Waveguide filter having nonidentical sections resonant at same fundamental frequency and different harmonic frequencies,” U.S. Patent 3 153 208, Oct. 13, 1964. [26] M. Guglielmi, “Simple CAD procedure for microwave filters and multiplexers,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 7, pp. 1347–1352, Jul. 1994. [27] M. Morelli, I. Hunter, R. Parry, and V. Postoyalko, “Stop-band improvement of rectangular waveguide filters using different width resonators: Selection of resonator widths,” in IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, May 2001, pp. 1623–1626. [28] R. Vahldieck, “Quasi-planar filters for millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 2, pp. 324–334, Feb. 1989. Pablo Soto (S’01–M’06) was born in Cartagena, Spain, in 1975. He received the M.S. degree in telecommunication engineering from the Universidad Politécnica de Valencia, Valencia, Spain, in 1999, and is currently working toward the Ph.D. degree at the Universidad Politécnica de Valencia. In 2000, he joined the Departamento de Comunicaciones, Universidad Politécnica de Valencia, where he has been a Lecturer since 2002. In 2000, he was a Fellow with the European Space Research and Technology Centre (ESTEC)–European Space Agency (ESA), Noordwijk, The Netherlands. His research interests comprise numerical methods for the analysis and automated design of passive waveguide components. Mr. Soto was the recipient of the 2000 COIT/AEIT National Award for the best master’s thesis on basic information and communication technologies.

Eva Tarín was born in Cheste, Spain, in 1983. She received the B.S. and M.S. degree in telecommunication engineering (with first-class honors) from the Universidad Politécnica de Valencia, Valencia, Spain, in 2005 and 2008, respectively. Her research interests concern numerical methods for the analysis and automated design of waveguide components. Ms. Tarín was the recipient of several regional and national awards for her outstanding academic record. She also held an IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Undergraduate/pre-Graduate Scholarship for the 2005–2006 academic period.

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 the 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 he has been a Full Professor since 2003. In 1995 and 1996, he 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 EM analysis and design of passive waveguide devices. He has authored or coauthored five chapters in technical textbooks, 60 papers in refereed international technical journals, and over 150 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 has been 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 member of the Technical Committees of the IEEE MTT-S International Microwave Symposium (IMS) and the European Microwave Conference. He was the recipient of the 2001 Social Council of the Universidad Politécnica de Valencia First Research Prize for his outstanding activity during 1995–2000.

Carlos Vicente (M’08) was born in Elche, Spain, in 1976. He received the Dipl. degree in physics from the Universidad de Valencia, Valencia, Spain, in 1999, and the Dr.-Ing degree in engineering from the Technical University of Darmstadt, Darmstadt, Germany, in 2005. From 1999 to the beginning of 2001, he was a Research Assistant with the Department of Theoretical Physics, Universidad de Valencia. From 2001 to 2005, he was a Professor Assistant with the Institute of Microwave Engineering, Technical University of Darmstadt. Since 2005, he has been with the Microwave Applications Group, Universidad Polit´ecnica de Valencia, Valencia, Spain. In 2006, he co-founded Aurora Software and Testing S. L., which is devoted to the telecommunications sector. His research concerns the analysis and design of passive components for communications satellites with a special emphasis on high-power practical aspects such as passive intermodulation, corona discharge, and multipaction.

Jordi Gil (M’08) was born in Valencia, Spain, on April 27, 1977. He received the Licenciado degree in physics from the Universidad de Valencia, Valencia, Spain, in 2000, the Master Philosophy Diploma in telecommunications engineering from the Universidad Polit´ecnica de Valencia, Valencia, Spain, in 2005, and is currently working toward the Ph.D. degree at the Universidad Polit´ecnica de Valencia. From 2001 to 2004, he was Researcher with the Aerospatiale Italian Company, Ingegneria Dei Sistemi-S.p.A., under the frame of the V European Framework Programme. From 2004 to 2006, he joined the Microwave Applications Group, Universidad Polit´ecnica de Valencia, in the frame of a European reintegration grant funded by the VI European Framework Programme. In 2006 he cofounded the company Aurora Software and Testing S.L., devoted to the space sector. He is currently continuing his research activities with Aurora Software and Testing S.L., as he works toward the Ph.D. degree under the frame of a Torres Quevedo Grant funded by the Spanish Government. His current research interests include numerical methods in computer-aided techniques for the analysis of microwave and millimeter passive components based on waveguide technology, and nonlinear phenomena appearing in power microwave subsystems for space applications.

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. Beginning in 1990, he was an Assistant Professor with the Departamento de Física Aplicada y Electromagnetismo, Universidad de Valencia, where, in 1997, he became an Associate Professor. During 1994 and 1995, he was a Research Fellow with the European Space Research and Technology Centre (ESTEC)–European Space Agency (ESA), Noordwijk, The Netherlands. In 2003, he received a fellowship from the Spanish Government for a short stay (three months) with the Università degli Studi di Pavia, Pavia, Italy, as a Visiting Scientific. His current research interests include the areas of computer-aided techniques for analysis of passive components for space applications, waveguides and cavities including dielectric objects, EM bandgap structures, frequency-selective surfaces, and nonlinear phenomena appearing in power microwave subsystems (multipactor and corona effects).

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Analysis and Measurement of a Time-Varying Matching Scheme for Pulse-Based Receivers With High-Q Sources Xin Wang, Linda P. B. Katehi, Fellow, IEEE, and Dimitrios Peroulis, Member, IEEE

Abstract—A time-varying matching scheme designed for pulsebased high-quality factor ( ) receiving systems is introduced and analyzed. The proposed technique takes advantage of the time-domain characteristics of the short-duration pulses in a way that the matching network is optimized for absorbing energy from the pulse within its duration, and is reconfigured after the pulse so that the captured energy is efficiently delivered to the load. Particularly, this paper extends the authors’ earlier work by providing a comprehensive analysis of the capability of the proposed design in achieving exceptional voltage gain and width compression. Measurements in the VHF range are provided to show the performance of the proposed design versus the conventional ones. The proposed technique is applied to the matching design for electrically small antennas, which achieves output signals with ten times narrower pulsewidths compared to original pulses, 3–4 times larger amplitudes, and 0.3–3-dB energy gains compared to conventional matching designs. New results also demonstrate the effects of timing/synchronization, which show that accurate synchronization (with typical error smaller than 10% of the original pulsewidth) is required to maintain large output amplitude (e.g., higher than 90% of the maximum achievable level). The proposed technique is particularly suitable for receiving systems using common modulation schemes including pulse amplitude modulation and on–off keying. Index Terms—High- circuits, matching network, pulse circuits, time-varying circuits.

I. INTRODUCTION

C

ONVENTIONAL impedance-matching techniques are usually designed in the frequency domain, which is most efficient in traditional transceiver systems that utilize continuous signals with limited frequency bandwidth to transmit information. In narrowband systems, these techniques lead to simple - or -type matching networks that can be readily designed using closed-form expressions [1]. For applications where wideband characteristics are required, a number of more

Manuscript received May 11, 2009; revised October 27, 2009 and April 15, 2010; accepted April 23, 2010. Date of publication July 12, 2010; date of current version August 13, 2010. This work was supported in part by the National Science Foundation (NSF) under CAREER Grant 0747766. X. Wang and D. Peroulis are with the School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]; [email protected]). L. P. B. Katehi is with the Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA 95616 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2052870

advanced design techniques have been developed for syntheladder matching networks, including sizing multistage analytic approximation approaches (e.g., Butterworth, Chebyshev, and elliptic approximations) and numerical methods such as the real frequency technique [2]. However, due to their linear time-invariant (LTI) nature, the resultant matching networks are limited by the gain-bandwidth restrictions derived by Bode [3] and Fano [4]. As the intended frequency bandwidth increases, the frequency-domain techniques usually become less efficient and more difficult to use. This is especially true in pulse-based systems such as impulse ultra-wideband communication and radar systems. Instead of using continuous waveforms, these systems employ short-duration pulses, which call for a very wide instantaneous bandwidth for successful transmission and reception. Nevertheless, the simple time-domain characteristics (e.g., short duration, known or predictable pulse shapes, etc.) of the pulse-based systems make time-domain techniques better-suited solutions [5], [6]. A time-domain technique of impedance matching specifically designed for pulse-based transmitting systems has recently been developed by the authors [7]–[9]. The proposed time-varying matching scheme achieves exceptional performance by exploiting the time-domain characteristics of the pulse signals of interest. Specifically, this time-varying design pre-stores energy in critical reactive components to setup proper initial conditions for each incoming pulse. As a result, the design is not limited by the gain-bandwidth restrictions, and nearly perfect match can be obtained from simple single-ladder networks for monocycle pulses with 10:1 10-dB bandwidth. This technique was successfully implemented for matching and designs between 50- source and different resistive, loads, with measured results showing a typical 5–8-dB increase/decrease in the transmitted/reflected energy compared to conventional matching designs. However, this scheme requires knowledge of the pulse shape, timing, and magnitudes. While this is acceptable in the transmitting mode, it cannot be practically employed in receivers, where the required initial conditions are unknown and the only information available is the pulse timing (synchronization). To address this problem, we proposed in [10] and [11] a different time-varying matching design that has the ability to efficiently capture and deliver the energy from a high- source to the load in the receiving mode. In this matching design, there is no preset voltage or current, and no information is used on the incoming pulse, except its timing. This technique is useful in many applications where wideband pulse signals need to be received

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Q source and a resistive load (after

Fig. 1. Impedance matching between a high[11]).

through high- components such as electrically small antennas, microelectromechanical systems or nanoelectromechanical systems (MEMS/NEMS) resonators or high- filters, as shown in Fig. 1. Compared to conventional matching designs that disperse the pulse shape, our time-varying matching scheme reconfigures the circuit topology in real time, and hence, achieves a much larger amplitude and shorter pulsewidth in the signal delivered to the load. One may notice that the schematic of our time-varying matching circuit may look, at first sight, analogous to typical signal-matched filters (e.g., [12]–[14]). This is because both schematics illustrate a switch sampling an output signal at appropriate instants. However, their functions are quite distinct. Signal-matched filters focus on maximizing the signal-to-noise ratio typically after the down-conversion and demodulation stage (unless an expensive direct RF-to-digital conversion is employed), and require a coherent receiving architecture. On the other hand, the proposed time-varying designs operate directly on the received RF signal, with a focus on maximizing the received energy with a given time window by compressing the received pulse in this window. In addition, coherent detection is not required. This paper builds on the authors’ work published in [11] for receiver systems with high- sources. With extensive numerical and experimental results, it expands the previous work in the following aspects. 1) A full analysis of the time-varying matching design is provided with a particular emphasis on designs aiming for maximum voltage gain and width compression. 2) Analysis and new results are provided demonstrating the effects of timing/synchronization errors in the proposed matching design. 3) It is demonstrated for the first time that the benefits of the time-varying design in the receiver mode are independent of the shape of the incoming pulse and can be found even for arbitrary waveforms. Notice that the work presented in this paper is a basic research toward understanding the design and working mechanism of the time-varying matching technique. These findings can be useful

Fig. 2. Energy flowchart when receiving a short pulse signal: (a) within the duration of a incoming pulse and (b) after the pulse ends.

to a variety of applications. We do not target a specific system level application in this paper. Instead, we focus on the circuit level design in demonstrating the performance of the proposed matching scheme. II. MOTIVATION AND ANALYSIS Definitions and Basic Idea As illustrated in Fig. 1, we consider the impedance-matching problem between a high- source and a resistive load (often 50 ). Traditionally, an LTI matching network is used to achieve the tradeoff between the bandwidth and gain of the system so that minimum reflection and maximum energy transmission are obtained. However, due to the large factor of the source, the achievable bandwidth will be very limited compared to that of the original pulse signals, which leads to a long ringing tail at the load. Fig. 2 schematically illustrates an interpretation of this limitation from the time-domain point of view. In the figure, the pulse source is replaced by its Thevenin equivalent circuit with and a source impedance an open circuit voltage . The resistive load is denoted by . The matching network is assumed to be lossless, with an input current denoted and an output voltage (across ) . With the source by voltage assumed to be a pulse starting at time 0 and ending at time , the transient response of the system can be divided into two stages: within the duration of the pulse and after the end of the pulse . Fig. 2(a) shows the energy flow within the pulse duration . In of the the end of the pulse at time , the available energy incoming pulse is converted into the following three different components.

WANG et al.: ANALYSIS AND MEASUREMENT OF TIME-VARYING MATCHING SCHEME

1)

: the energy stored in the reactive components in the source and the matching network at time . : the energy delivered to by the time . 2) 3) : the reflected energy by the time . The above energy components can be expressed as (1) (2) (3) (4) where and , respectively, represent the th capacitor and th inductor in the system. Correspondingly, the symbols and represent the voltage across and the current through , respectively. , as shown in Fig. 2(b), After the pulse ends is released and converted into the following. : the energy delivered to after the time . 1) 2) : the reflected energy after the time . These are expressed as (5) (6) The overall energy reflected to the source is (7) The overall energy delivered to the load is (8) Given the source and load impedances, conventional matching designs can usually be formulated as a problem of finding the optimal passive lossless network that maximizes the energy transmission efficiency, i.e., (9) can be calculated by finding the solution of in the conventional LTI designs, can be expressed as

, which,

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and values considered in this paper (e.g., For the high and ), the overall quality factor of the system with a lossless matching network will also be high. Notice that the quality factor of a system can be defined as Maximum energy stored Energy dissipated per cycle

(13)

can be much larger than for which suggests that comes from the dishigh- pulsed systems. Notice that charging of , which can also be much larger than . constitutes the major part of the overall transIn this case, . Therefore, we expect a good matching netmitted energy in Fig. 2(a) and 2) efwork to be able to: 1) maximize ficiently convert to in Fig. 2(b). When designed , conventional matching designs effectively to maximize achieve a tradeoff between the above two targets. On the one usually leads to a high- factor in the hand, maximizing system of Fig. 2(a). On the other hand, the high- factor leads to a low damping ratio in the system of Fig. 2(b), which causes substantial pulse distortion and additional energy loss. To address the above problem, we propose a time-varying matching scheme. Instead of employing a fixed matching network for all time, as in conventional matching designs, our timevarying matching network is able to reconfigure itself near the end of each incoming pulse. As a result, the matching network in the two stages of the transient response in Fig. 2(a) and (b) may have different topologies that can be optimized separately. Specifically, the real-time configurable matching network operates in the following two fundamental steps. Step 1) Within the duration of the incoming pulse, a reactive network is connected to the high- source to capture as much as possible of the available pulse energy. The resistive load is isolated from the source by switches so that the stored energy can be maximized. Step 2) During the intermission of two consecutive pulses, the circuit topology is reconfigured in a way that the energy stored in the high- system is efficiently delivered to the load. A. Illustrative Example We utilize the conventional matching network design besource with and a tween a series resistive load as an example to further illustrate the above idea. Specifically, the matching design and simulation results in Fig. 3 are used to provide a quantitative assessment of Fig. 2. is used as the As shown in Fig. 3(a), a series capacitor matching network for the sinusoidal monocycle pulse source

(10) is the impulse response function of the system. where To characterize the source and load impedances, we define and an external load quality factor the source quality factor at the center frequency of the pulse as (11) (12)

otherwise

(14)

is the center angular frequency of the where monocycle. We focus on this single capacitor matching network design for the sake of simplicity of discussion. A more complex ladder network may be used to provide more degrees of freedom in achieving a desired performance. However, in terms of maximum energy transfer of the wideband pulses, the improvement

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The impulse response function of the system found to be

can then be

(20) where (21) (22) Following the design ideas discussed in Section II-A, we only . need to find the optimal value that maximizes With the optimized matching design, Fig. 3(b) illustrates the , , and norcalculated energy components malized to as a function of for different . Due to and decrease the gain-bandwidth restriction, both . Nevertheless, we notice that both with increasing and are generally much larger than , especially for to high values. This means that the conversion of after the end of the incoming pulse constitutes the major part . For a fixed , higher of the overall transmitted energy leads to higher . However, it is seen that the conversion ratio from to decreases as increases. This is because the load is in series with the source, and hence, the conversion ratio, can be estimated as (23)

Q

Fig. 3. Conventional impedance-matching design between a high- inductive source and a resistive load. (a) Circuit schematic and (b) calculated energy components for different source and load quality factors.

will be very limited, whereas additional loss will arise from the components. nonideality of the With the circuit topology determined, the source quality facand can be explicitly written as tors

Therefore, the series capacitor matching design does not proto for large vide an efficient means of converting , which limits the overall performance of the matching. Nevertheless, this limitation can be mitigated by reconfiguring the discharges so that the stored matching network before energy can be delivered to the load in a more efficient way. This leads to a reconfigurable matching design, which is discussed in Section III.

(15) (16) The energy components

and

can be simplified as (17) (18)

where and are, respectively, the voltage across the and the current through the source inmatching capacitor ductor . The energy can be found using (8) with given by (10). To facilitate the analysis that follows, an auxiliary resonant frequency determined by the matching capacitor is defined as (19)

III. TIME-VARYING MATCHING DESIGN Based on the idea illustrated in Section II, the time-varying matching network can be designed in the following steps. Step 1) Select the matching network configuration that maxwithin the duration of the incoming imizes pulse. Step 2) Select the reconfiguration of the network of Step 1) that provides an efficient way of converting to . Step 3) Determine the timing of the network reconfiguraand minimizes the ringing tion that maximizes effect. As in Section II, we utilize the problem of matching an source to a resistive load for the monocycle pulse (14) as a vehicle to demonstrate the circuit design and the performance. Fig. 4 shows the schematic of the time-varying matching circuit. In the following, the design principle and simulation results are presented for this circuit.

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Q

Fig. 4. Proposed time-varying matching design between a high- inductive source and a resistive load. (a) Network configuration during the monocycle pulse. (b) Network configuration after the monocycle pulse. Fig. 5. Shape of the received pulse signal depending on the timing of the switching operation.

A. Network Design During the Incoming Pulse Fig. 4(a) shows the matching network configuration within the duration of the incoming monocycle pulse. As shown in the figure, the inductive source is loaded with a single capacitor , whereas the load is isolated by a switch. Since there is no load resistance in series with the capacitor, the circuit is equivalent to the conventional matching design of Fig. 3(a) with as far as is concerned. In other words, the effect of on is avoided. As a result, the maximum achievable stored energy [the far right ends of the curves in Fig. 3(b)] can be obtained. Specifically, we select the capacitance to be the one that maximizes . Here, can still be calculated using (18) and given by with (24)

(25) where and are still given by (21) and (22), but with replaced by . B. Network Design After the Incoming Pulse Fig. 4(b) shows the matching network configuration after the incoming pulse. With the switch closes at time , the stored energy starts discharging, which is a combined process of the . Notice that discharge of the capacitor and the inductor is connected to the source and matching capacitor the load in parallel. This prevents the system from oscillating as the is found to be load voltage signal

where (27) (28) For the high and values considered in this paper, both and are real. As will be shown later, when the switch closing time is appropriately selected, most of the stored energy can be delivered to the load. C. Timing of the Switch Operation According to (26), is determined by the initial conditions of the reactive elements at time . Before the switch is turned on, the stored energy bounces back and forth between the source inductor and matching capacitor. Therefore, the timing of the switch operation have effects on the performance of the matching network by determining the initial conditions and . Fig. 5 shows typical waveforms of , as well as the associated energies stored in the inductor and capacitor when the , , and switch is turned on at three different times: . Here, , , and , respectively, correspond to the moments when the stored energy is entirely located in the capacitor, evenly distributed between the inductor and capacitor, and entirely located in the inductor. , as shown 1) Matching Capacitor Discharging: For in Fig. 5(a), and comprises only the first two terms in (26). Notice that and

(29)

for the and values considered in this paper. The first term is negligibly small compared to the second term. Therefore, the can be approximated as expression of (26)

(30)

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The energy conversion ratio calculated is

(31) In other words, nearly all the stored energy is delivered to the load. This can be explained by the fact that is in parallel with in the discharge path of the capacitor. Since , the discharge current through is negligible , and therefore, most of the energy compared to that through . is dissipated in , as shown in 2) Source Inductor Discharging: For Fig. 5(c), and comprises only the last term in (26), i.e.,

Fig. 6. Effect of the switch synchronization error on the pulse magnitude of the load voltage in a typical time-varying matching network design of Fig. 4 and . The synchronization error is normalized to the with pulsewidth of the incoming pulse, as defined by (34). The load voltage is normalized to its maximum value that is obtained when there is no synchronization error.

Q = 100

Q =6

(32) The energy conversion ratio is found to be approximately

(33) which is approximately the same as (23) in the conventional matching design. This can be explained by the fact that is in series with the parallel combination of and in the is much smaller discharge path of the source inductor. As and than the impedance of , the discharge currents through are approximately the same. Therefore, the conversion ratio to decreases as increases. from , as shown in 3) Combined Discharging: For waveform is a combination of the above two Fig. 5(b), the discharging processes. The preceding narrow spike is due to the capacitor discharge and the wider pulse that follows is due to the inductor discharge. Based on the above observations, the capacitor discharge delivers the energy to the load much more efficiently than the inwaveforms in ductor discharge does. It is also seen from the Fig. 5 that the capacitor discharge produces a much larger pulse magnitude and shorter pulsewidth, as compared to the inductor expressions (30) discharge. This can be explained by the and (32). The load voltage caused by the capacitor discharge that is much (30) has an attenuation coefficient of in larger than the dominant attenuation coefficient the inductor discharge (32). Therefore, it is desirable to select the switch closing time to be when most of the stored energy is in the matching capacitor. In practical applications, the optimal time needs to be recovered during the synchronization process. Hence, the output pulse amplitude is sensitive to the synchronization error. Fig. 6 shows the calculated pulse amplitude degradation as a function of synchronization error for a typical matching problem with and . Here, the synchronization error relative to the width of the incoming pulse as Relative synchronization error

(34)

It is seen that in order to prevent the output pulse amplitude from decreasing to lower than 90% of its maximum achievable level, the sychronization error of the switch operation has to be smaller than 10% of the incoming pulsewidth. In practical applications, accurate timing is necessary to achieve the desired pulse shape (large amplitude and narrow pulsewidth) and avoid error in retrieving the information. Nevertheless, the sensitivity of the pulse shape to the timing may also be exploited for synchronization as its varying pattern is predictable. In the simulation and measurement results that follows, unless otherwise mentioned, we assume the switch is fully synchronized to the incoming pulse so that maximum output amplitude is achieved. D. Simulation Results and Discussion Fig. 7 shows the simulated waveforms of in the timevarying matching design compared to the conventional design and . As seen in the figure, a ringing when effect occurs in the conventional matching design, which constitutes the main part of the output pulse signal. According to the expression of the impulse response function (20), the ringing . Obviously, the width of tail follows the form of the received pulse at the load is determined by . We define to be the period the pulsewidth of the load voltage signal of its maximum magnitude. before the pulse decreases to in the conventional design then has a pulsewidth three times as large as that of in this particular case. Since begins norat time 0 and starts decaying exponentially at time , malized to can be estimated to be (35) In the time-varying matching design, with the switch closing time selected to be , can be estimated to be (36) As , the pulsewidth of in the time-varying matching design is much smaller than that in the conventional has a width matching design. In the particular case of Fig. 7, about 20 times narrower compared to that of the incoming pulse. Notice that the peak amplitude of the pulse in the time-varying

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Q = 100

Fig. 7. Simulation results of the two matching designs for and . (a) Given incoming monocycle signal and (b) load voltages in the conventional matching design of Fig. 3(a) and the time-varying matching design of Fig. 4.

Q =6

design is about seven times as large as that of the conventional design. In addition, the overall energy of the received pulse calculated using (8) is about 0.8 dB larger in the time-varying design compared to the conventional design. Fig. 8 compares the simulated performance of the two matching network designs for different source and load quality factors, including the energy transmitted to the load , the normalized peak amplitude of the load voltage , and the normalized pulsewidth of the load voltage . As shown in the figure, the proposed time-varying matching technique achieves a better energy transmission efficiency compared to the conventional matching design. For example, the middle two curves in Fig. 8(a) show the energy efficiency of the two matching designs as a function for . A minimum improvement of about 1.1 dB of by the time-varying matching is achieved around design compared to the conventional one. For the case of , the bottom two lines in Fig. 8(a) show a 0.6-dB . These are improvement in the energy efficiency at consistent with the observed 0.8-dB improvement in the simuand . For another lation results in Fig. 7 for example of and , a 4-dB improvement of energy efficiency is achieved by our time-varying matching, as indicated by the right ends of the top two curves in Fig. 8(a). of the time-varying matching design (dotted Notice that lines) is almost constant as increases. This is the consequence of the capability of the matching network to maximize independent of within the duration of the incoming to an efficiently. pulse, and after that, convert

Fig. 8. Performance of the proposed time-varying matching design of Fig. 4 compared to that of the conventional design of Fig. 3(a) in terms of the: (a) energy transfer efficiency, (b) peak voltage amplitude, and (c) received pulsewidth.

Besides the improved energy transmission efficiency, our time-varying design features a much larger output amplitude and narrower output pulsewidth compared to that of the conventional design, as illustrated in Fig. 8(b) and (c). The exceptional pulse compression effect is one of the major advantages of the proposed time-varying matching technique and provides an effective means of achieving a high pulse repetition rate that is not attainable in conventional high- systems. Specifically, the maximum achievable pulse repetition rate can be estimated in the time-varying matching design and as in the conventional matching design if one pulse is repeated immediately after the previous one ends. As illustrated in Fig. 8(c), in the conventional matching design is much larger in the time-varying matching design, especially for than high values. Consequently, a much higher pulse repetition rate can be achieved in the time-varying matching design. It is also worth demonstrating the performance of the proposed matching network in terms of the time- and frequency-

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Fig. 10. S -parameters of the two matching network designs. The S parameters are calculated as the ratio between the spectra of the incident and reflected currents, as shown in Fig. 9.

With and obtained, we can define and calculate parameter of the time-varying the energy-spectrum-based with an amplitude matching network

(40)

Q = 10

Fig. 9. Simulation results of the two matching designs for and . (a) Incident current signal, and (b) reflected current signal in the conventional matching design of Fig. 3(a) and the time-varying matching design of Fig. 4.

Q = 20

domain reflection. For the problem of matching between the source and resistive load, the incident and reflected current signal is defined as

(37) (38) and the energy of the reflected pulse signal can be calculated as (39) which is equivalent to the previous definitions (4), (6), and (7). Fig. 9 shows the simulated waveforms of the incident and reflected current signal in the two matching network designs when and . As seen in the figure, the reflected current in the conventional matching design has a ringing tail similar to that of the load voltage as a result of the discharge of the stored energy. However, the reflection in the time-varying matching design takes place within the duration of the incoming pulse. With the matching network reconfigured in the end of the pulse, the reflection becomes negligibly small. Obviously, the proposed matching network provides a better way of reducing the reflection, and as a result, minimizing the inter-symbol interferences caused by the reflected signal [15].

and are the Fourier transform of where and , respectively. Fig. 10 shows the -parameter of the time-varying matching network compared to that of the conventional one over the frequency range covered by the spectrum of the incident current signal. It is seen that a much wider bandwidth is achieved with our time-varying matching network. Although the conventional design achieves a better matching , its overall energy efficiency is about around the frequency 4 dB lower than that of the time-varying matching design due to the narrow bandwidth. As a result, the energy of the reflected signal is 2 dB lower in our time-varying matching design compared to the conventional matching design. IV. EXPERIMENTAL RESULTS This section provides an experimental validation of the proposed matching network design. We implemented the time-varying matching scheme for an application of receiving short pulses using an electrically small loop antenna. Electrically small antennas have many attractive features including compact size, omni-directivity, constant radiation pattern, and minimized dispersion that are desirable for many mobile applications. However, due to their high- nature, small antennas often result in large mismatch loss, which is even more pronounced in wideband pulses. For instance, a small loop antenna has a low radiation resistance and large inductive reactance, which can be seen as a high- source similar to the simple source we have analyzed in Section III. Therefore, the matching network design proposed in Fig. 4 is directly applied with the source replaced by a small half-loop antenna. We carried out the experiments in a low-frequency range (with center frequency of the pulse signal to be 20–40 MHz) due to the great need of electrically small antennas around

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Fig. 11. Schematic of the measurement setups of a half-loop receiving antenna formed by a semicircle wire with one end mounted on the ground plan and the other as the antenna output connected to the matching network. (a) Receiving design without any matching network. (b) With the conventional matching network. (c) With the proposed time-varying matching work.

Fig. 12. Equivalent-circuit model and the input impedance of the receiving half-loop antenna utilized in the experiments. (a) Schematic of the equivalent jX and the circuit (after Fig. 8 in [16]). (b) Simulated input impedance R quality factor as a function of the frequency.

this frequency range. Nevertheless, the design is frequency scalable [9], [10] . Scaling the center frequency of the pulse signal of interest to a higher or lower level does not change the relative bandwidth in terms of the highest-to-lowest frequency ratio. For example, the sinusoidal monocycle pulse has 10-dB bandwidth regardless of its a 10:1 center frequency. Application of our proposed technique to a higher frequency range can be achieved by scaling down the values of reactive components (inversely proportional to the frequency) and adopting faster switches with switching time reasonably shorter than the pulse duration (e.g., one-tenth of the pulsewidth). Fig. 11 shows the schematic of three different measurement setups of the receiver. The half-loop antenna has a diameter of 44 cm, thus the circumference is about 1–2 tenths of the wavelength at the center frequency of the pulse signals used in the experiments. The transmitted pulse signals were generated by an Agilent 33250 A arbitrary waveform generator and fed directly to the transmitting antenna of the same type as the receiving antenna. The switch is realized with a CMOS reflective singlepole single-throw (SPST) switch (ADG 902 with 3.6-ns ON and 5.8-ns OFF switching time) made by Analog Devices. The switch is controlled with another Agilent 33250 A signal generator that is synchronized to the one in the transmitter side. A Tektronix 1-GHz oscilloscope with 50- input impedance is used as the load. The matching capacitors were implemented with ZEEX @ 50 MHz, minZC836 varactor diodes (minimum imum tuning ratio 5:1), which can be easily tuned with the bias voltage to achieve optimal matching performance.

Fig. 12 illustrates the modeling and simulation results of the antenna input impedance. For the time-domain simulation, an equivalent-circuit model of the receiving antenna needs to be used to replace the source impedances in Figs. 3 and 4. Fig. 12(a) shows a typical circuit model that is derived in [16]. To determine the circuit elements, the half-loop antenna is simulated with the Ansoft High Frequency Structure (HFSS) software. The simulated input impedance , as well as the quality factor, are shown in Fig. 12(b). Fig. 13 shows the measured waveforms of the received signals when a 35-ns sinusoidal monocycle is transmitted. Compared to the case of no matching, the conventional matching design improves the transferred energy by 4.5 dB and the time-varying design improves the energy by 4.9 dB. In the conventional matching design, the ringing effect makes the received pulse about three times wider than the transmitted pulse. However, in the time-varying design, we measured a shortest pulsewidth of about 5 ns, which is seven times shorter than the original one. As expected, the compressed pulse in our time-varying design features a much larger peak amplitude that is about 3.2 times as large as that of the conventional design. Table I shows the measured performance of the time-varying matching scheme compared to the simulation results. Here, sinusoidal monocycles as the transmitted signals with three corresponding to three difdifferent pulsewidth values ferent and values are measured and simulated. Due to the difficulty in measuring the open circuit voltage signal, we verify the performance of the time-varying matching with the measurements of three parameters, and as listed in the table:

+

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Fig. 13. Measured voltage waveforms of the : (a) transmitted sinusoidal monocycle and (b) received pulse signals in different matching schemes (after [11]). TABLE I MEASURED AND SIMULATED PULSE MAGNITUDE AND PULSEWIDTH

Fig. 14. Measured and simulated pulse shape of the received signal for different timing of the switching operation.

1) the voltage gain in peak amplitude compared to the conventional design; 2) the width compression compared to the transmitted pulse; and 3) the received energy gain compared to the conventional matching design, i.e., Voltage Gain Width Compression Energy Gain

time varying matching conventional matching time varying matching transmitted pulsewidth time varying matching conventional matching

(41) (42) (43)

It is seen that the measured voltage gain is about 20% lower than simulation results, whereas the measured pulsewidth is wider than the simulation results. Correspondingly, the measured pulse energy is lower than the simulation prediction. This is mainly caused by the nonideal switches and parasitic inductances in the fabricated circuits. Particularly, the clock-feedthrough and other switching transients of the switch being turned on will affect the capacitor voltage, and as a result,

causing degradation of the pulse amplitude and energy. The on-resistance of the switch can lead to additional loss. The effect of the nonideal switch can be minimized by choosing higher performance switches and by compensation schemes [17], [18]. Fig. 14 illustrates the dependence of the received pulse on the switch closing time compared to the simulation. In the simulation, the coupling between the transmitting and receiving antennas has been matched to the measured results. The three switch closing times are selected the same way as in Fig. 5, leading to typical waveforms of the received pulse due, respectively, to: 1) the matching capacitor discharge only; 2) the combined discharge of the antenna and the matching capacitor; and 3) the antenna discharge only. It can be seen that, as the switch closing time changes, the variation of the pulse shape follows the same pattern as predicted by the simulation results. Similar measurement results were obtained for transmitting other pulses including a second derivative Gaussian pulse, as shown in Fig. 15, and a third derivative Gaussian pulse in Fig. 16, both of which have a center frequency around 30 MHz. Similar performances were achieved in both cases: an approximately three times larger voltage amplitude compared to the waveforms in conventional design and a pulsewidth of approximately 5 ns. In practical applications, the received signal may be a superposition of distorted pulses that arrive randomly from different

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Fig. 15. Measured voltage waveforms of: (a) the transmitted second derivative Gaussian pulse and (b) the received pulse signals in different matching schemes (after [11]).

Fig. 17. Measured voltage waveforms of the: (a) transmitted arbitrary pulse and (b) received pulse signals in different matching schemes.

Fig. 16. Measured voltage waveforms of: (a) the transmitted third derivative Gaussian pulse and (b) the received pulse signals in different matching schemes.

propagation paths. This is particularly the case in a multipath environment. As a result, the received pulse signal often appears

to be a random waveform [19], [20] . Fig. 17 shows the performances of different matching schemes when a random arbitrary pulse signal is received. We tuned the matching capacitance and the timing of the switching operation so that maximum voltage amplitude is achieved. The received pulse, as shown in the figure, has a pulsewidth of about 10 ns and 2.8 times voltage gain compared to the conventional design. It can be seen that the received pulse in the time-varying matching design maintains a large magnitude, a narrow width, and a pulse shape independent of the incoming pulse. This indicates that our time-varying matching technique provides an efficient way of collecting energy from pulses with random waveforms. One of the limiting factors of the pulse resolution or repetition rate is the inter-symbol interference caused by the pulse dispersion in a conventional band-limited system. The pulse compression effect achieved in our time-varying design also makes it easier to discriminate consecutive pulse signals that are closely aligned. Fig. 18 demonstrates the performance of the time-varying matching scheme compared to the conventional one when consecutive pulses are transmitted and received. As shown in Fig. 18(a), five sinusoidal monocycles representing signal symbols “1-0-0-1-1” with opposite polarities are transmitted. In the conventional matching design, as shown in Fig. 18(b), the received signal waveform exhibits a ringing effect, resulting in an inter-symbol interference, which adds to the difficulty of detecting the signals. However, as seen in Fig. 18(c), our time-varying matching design maintains a

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closely aligned pulses. These advantages make the time-varying matching technique promising for applications in ultra-wideband frequency systems and digital communications. REFERENCES [1] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2004. [2] H. J. Carlin and P. P. Civalleri, Wideband Circuit Design, 1st ed. Boca Raton, FL: CRC, 1997. [3] H. Bode, Network Analysis and Feedback Amplifier Design. New York: Van Nostrand, 1945. [4] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,” J. Franklin Inst., vol. 249, pp. 57–83, 1950. [5] C. Bennett and G. Ross, “Time-domain electromagnetics and its applications,” Proc. IEEE, vol. 66, no. 3, pp. 299–318, Mar. 1978. [6] H. Schantz, The Art and Science of Ultrawideband Antennas. Norwood, MA: Artech House, 2005. [7] X. Wang, L. P. Katehi, and D. Peroulis, “Time-domain impedance RC loads,” in IEEE adaptors for pulse-based systems with high MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 1741–1744. [8] X. Wang, L. P. Katehi, and D. Peroulis, “Time-varying matching network for antennas in pulse-based systems,” in IEEE Antennas Propag. Symp. Dig., Jun. 2007, pp. 89–92. [9] X. Wang, L. P. Katehi, and D. Peroulis, “Time-varying matching networks for signal-centric systems,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 12, pp. 2599–2613, Dec. 2007. [10] X. Wang, L. P. Katehi, and D. Peroulis, “Time-varying matching for receiving wideband pulse through electrically small antennas,” in IEEE Antennas Propag. Symp. Dig., Jul. 2008, pp. 1–4. [11] X. Wang, L. P. Katehi, and D. Peroulis, “A time-varying matching scheme for pulse-based high-Q receivers,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 837–840. [12] S. Sussman, “A matched filter communication system for multipath channels,” IRE Trans. Inform. Theory, vol. 6, no. 3, pp. 367–373, Jun. 1960. [13] P. Z. Peebles, Radar Principles. New York: Wiley, 2007. [14] M. Zahabi, V. Meghdadi, J. Cances, and A. Saemi, “A mixed-signal matched-filter design and simulation,” in 15th Int. Digital Signal Process. Conf., Jul. 2007, pp. 272–275. [15] S. B. A. Kashem, S. Raju, and M. I. Raza, “Modified physical configuration to compensate parasitic effects in high speed systems,” in Elect. Comput. Eng. Int. Conf., Dhaka, Bangladesh, 2008, pp. 741–744. [16] G. Streable and L. Pearson, “A numerical study on realizable broadband and equivalent admittances for dipole and loop antennas,” IEEE Trans. Antennas Propag., vol. AP-29, no. 5, pp. 707–717, Sep. 1981. [17] C. Eichenberger and W. Guggenbuhl, “Dummy transistor compensation of analog MOS switches,” IEEE J. Solid-State Circuits, vol. 24, no. 8, pp. 1143–1146, Aug. 1989. [18] S. Willingham and K. Martin, “Effective clock-feedthrough reduction in switched capacitor circuits,” in Proc. IEEE Int. Circuits Syst. Symp., May 1990, vol. 4, pp. 2821–2824. [19] J. H. Reed, An Introduction to Ultra Wideband Communication Systems, Illustrated ed. New York: Prentice-Hall, 2005. [20] S. Song and Q. Zhang, “Multidimensional detector for UWB ranging systems in dense multipath environments,” IEEE Trans. Wireless Commun., vol. 7, no. 1, pp. 175–183, Jan. 2008.

Q

Fig. 18. Measured voltage waveforms of: (a) consecutive transmitted pulse signals and received pulse signals with (b) conventional matching and (c) timevarying matching schemes (after [11]).

narrow pulse output, which is a desirable feature in applications where high data rate or high resolution is required. Since the receiver design is a linear system, the information is carried in the amplitude of the output pulse, which is proportional to that of the incident signal. This makes the time-varying matching design particularly suitable for signal detection of many common modulation schemes including pulse amplitude modulation (PAM) and on–off keying (OOK).

V. CONCLUSION A time-varying matching scheme for pulse-based high- receiving systems is introduced. With the ability to reconfigure itself in real time, the proposed design can efficiently capture the incoming pulse within its duration and thereafter, quickly deliver the stored energy to the load without dispersion. As a result, an exceptional pulse compression effect can be obtained without sacrificing the total energy transfer efficiency. The received pulse signal features a much larger amplitude, narrower duration, and a shape that is not dependent on that of the incoming pulse. Measurement results also show that the proposed matching scheme is capable of efficiently collecting energy from arbitrary pulse waveforms, as well as discriminating

Xin Wang received the B.S. and M.S. degrees in electronic engineering from Tsinghua University, Beijing, China, in 2000 and 2002, respectively, and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 2009. He is currently a Post-Doctoral Research Associate with Purdue University. His current research interests include miniaturized and reconfigurable RF circuits and antennas designs.

WANG et al.: ANALYSIS AND MEASUREMENT OF TIME-VARYING MATCHING SCHEME

Linda P. B. Katehi (S’81–M’84–SM’89–F’95) received the Bachelor’s degree in electrical engineering from the National Technical University of Athens, Athens, Greece, in 1977, and the Master’s and Doctoral degrees in electrical engineering from the University of California at Los Angeles (UCLA), in 1981 and 1984, respectively. On August 17, 2009, she became the sixth Chancellor of the University of California at Davis (UC Davis). As Chief Executive Officer, she oversees all aspects of the university’s teaching, research, and public service mission. She also holds UC Davis faculty appointments in electrical and computer engineering and in women and gender studies. She was Provost and Vice Chancellor for academic affairs with the University of Illinois at Urbana-Champaign, the John A. Edwardson Dean of Engineering and Professor of electrical and computer engineering at Purdue University, and Associate Dean for academic affairs and graduate education at the College of Engineering and Professor of electrical engineering and computer science at The University of Michigan at Ann Arbor. Since her early years as a faculty member, she has focused on expanding research opportunities for undergraduates and improving the education and professional experience of graduate students with an emphasis on underrepresented groups. She has mentored over 70 postdoctoral fellows and doctoral and master’s students in electrical and computer engineering. Twenty-one of the 42 doctoral students who graduated under her supervision have become faculty members in research universities in the U.S. and abroad. She has authored or coauthored ten book chapters and approximately 600 refereed publications in journals and symposia proceedings. She holds 16 U.S. patents with six U.S. patent applications pending. Dr. Katehi is a member of the National Academy of Engineering. Until 2010, she chaired the President’s Committee for the National Medal of Science and the Secretary of Commerce’s Committee for the National Medal of Technology and Innovation. She is a Fellow and board member of the American Association for the Advancement of Science. She is also a member of many other national boards and committees. She has been the recipient of numerous national and international awards both as a technical leader and educator for her work in electronic circuit design.

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Dimitrios Peroulis (S’98–M’03) received the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 2003. Since August 2003, he has been with Purdue University, West Lafayette, IN, where he currently leads a group of graduate students on a variety of research projects in the areas of RF microelectromechanical systems (MEMS), sensing, and power harvesting applications, as well as RF identification (RFID) sensors for the health monitoring of sensitive equipment. He has been a Principal Investigator (PI) or a co-PI in numerous projects funded by government agencies and industry in these areas. He is currently a key contributor in two Defense Advanced Projects Agency 1000) RF (DARPA) projects at Purdue focusing on: very high-quality ( tunable filters in mobile form factors (DARPA Analog Spectral Processing Program, Phases I, II, and III) and on developing comprehensive characterization methods and models for understanding the viscoelasticity/creep phenomena in high-power RF MEMS devices (DARPA M/NEMS S&T Fundamentals Program, Phases I and II). Furthermore, he leads the experimental program of the Center for the Prediction of Reliability, Integrity and Survivability of Microsystems (PRISM) funded by the National Nuclear Security Administration. In addition, he heads the development of MEMS technology in a U.S. Navy project (Marines) funded under the Technology Insertion Program for Savings (TIPS) program focused on harsh-environment wireless microsensors for the health monitoring of aircraft engines. He has authored or coauthored over 110 refereed journal and conference publications in the areas of microwave integrated circuits and antennas. Dr. Peroulis was the recipient of the 2008 National Science Foundation (NSF) CAREER Award. His students have received numerous Student Paper Awards and other student research-based scholarships. He has also been the recipient of eight teaching awards including the 2010 HKN C. Holmes MacDonald Outstanding Teaching Award and the 2010 Charles B. Murphy award, which is Purdue University’s highest undergraduate teaching honor.

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Design of Flip-Chip Interconnect Using Epoxy-Based Underfill Up to V -Band Frequencies With Excellent Reliability Li-Han Hsu, Student Member, IEEE, Wei-Cheng Wu, Edward Yi Chang, Senior Member, IEEE, Herbert Zirath, Senior Member, IEEE, Yin-Chu Hu, Chin-Te Wang, Yun-Chi Wu, and Szu-Ping Tsai

Abstract—This study demonstrates a flip-chip interconnect with and at 10 MHz) epoxy-based underfill ( for packaging applications up to -band frequencies. To achieve the best interconnect performance, both the matching designs on GaAs chip and Al2 O3 substrate were adopted with the underfill effects taken into consideration. The optimized flip-chip interconnect showed excellent performance from dc to 67 GHz with return loss below 20 dB and insertion loss less than 0.6 dB. Furthermore, the dielectric loss induced by the underfill was extracted from measurement and compared with the simulation results. % relative humidity test, The reliability tests including 85 C thermal cycling test, and shear force test were performed. For the first time, the -parameters measurement was performed to check the flip-chip reliability, and no performance decay was observed after 1000 thermal cycles. Moreover, the mechanical strength was improved about 12 times after the underfill was applied. The results show that the proposed flip-chip architecture has excellent reliability and can be applied for commercial applications.

= 35

tan = 0 02

85

Index Terms—Design, epoxy resin, flip-chip, interconnect, millimeter wave (MMW), packaging, reliability, underfill, -band.

I. INTRODUCTION

I

N RECENT years, with the demands for wireless communication systems increases rapidly, the operating frequency for the portable wireless is moving toward millimeter waves (MMWs). To meet the demands for commercial applications,

Manuscript received May 14, 2009; revised January 09, 2010; accepted March 22, 2010. Date of publication July 08, 2010; date of current version August 13, 2010. This work was supported by the National Science Council of Taiwan and the Ministry of Economic Affairs, Taiwan under Contract NSC 96-2752-E-009-001-PAE and Contract 95-EC-17-A-05-S1-020. L.-H. Hsu and W.-C. Wu are with the Department of Materials Science and Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, and also with the Microwave Electronics Laboratory, Department of Microtechnology and Nanoscience, MC2, Chalmers University of Technology, Göteborg SE-412 96, Sweden (e-mail: [email protected]; williamwu.mse90g@nctu. edu.tw; [email protected]). E. Y. Chang, C.-T. Wang, Y.-C. Wu, and S.-P. Tsai are with the Department of Materials Science and Engineering, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). H. Zirath is with the Microwave Electronics Laboratory, Department of Microtechnology and Nanoscience, MC2, Chalmers University of Technology, Göteborg SE-412 96, Sweden (e-mail: [email protected]). Y.-C. Hu was with the Department of Materials Science and Engineering, National Chiao Tung University, Hsinchu 300, Taiwan. He is now with Everlight Electronics, Taipei 236, Taiwan. 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.2010.2052960

package with low power consumption, low cost, small size, and light weight becomes indispensable. In this respect, the flip-chip interconnect has been regarded as a promising packaging technology for cost-effective module assembly in MMW systems due to its shorter interconnect length, higher throughput for production, and smaller package size [1]–[9]. The flip-chip reliability, however, needs to be carefully considered since it relies only on several metallic connections between chip and carrier. One promising solution is using underfill to improve the reliability by enhancing the joint strength, protecting the interconnect from mechanical shock, and helping the heat dissipation [10]–[13]. However, underfill may result in performance degraand loss tangent dation due to its higher dielectric constant compared to the air [14]–[16], which limits the usage of the flip-chip interconnect for high-frequency applications. This is one of the reasons why bond-wire is still the favorite packaging technology in microwave industry. The RF characteristics of the flip-chip assembly with epoxy-based underfill have been investigated up to 40 GHz [15], demonstrating an additional insertion loss of 0.5 dB at 30 GHz. Kusamitsu et al. reported the RF characteristics of the flip-chip assembled 30-, 60-, and 77-GHz low-noise amplifier (LNA) monolithic microwave integrated circuits (MMICs) [16]. Due to the underfill effect, the frequency response of the MMICs shifted to lower frequency bands. The 30-GHz LNA was shifted by 3 GHz; the 60- and 77-GHz LNAs were shifted by 9 GHz. In a flip-chip interconnect, the RF degradation due to underfill is induced by three major factors: chip impedance change, parasitic capacitance, and material dielectric loss. The high dielectric constant of the underfill ( typically) tends to reduce the chip line impedance, resulting in impedance mismatch and reflection at the transitions [14]. By using low- underfill or designing in advance with the underfill effects taken into account can ease this problem. Regarding to the parasitic effect, the flip-chip interconnect generally shows an overall capacitive effects [1], [2], which could degenerate the interconnect performance at MMW frequencies. Due to the underfill injection, the parasitic capacitance at the interconnect increases, indicating that a more inductive counterpart is needed for compensation. Furthermore, a dielectric loss is induced as an epoxy-based is applied. The loss can underfill be reduced by introducing low-loss materials such as benzocyclobutene (BCB) or other porous low- materials. Moreover, for most commercial applications, reliability investigations are generally required. The adhesion and coefficient

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HSU et al.: DESIGN OF FLIP-CHIP INTERCONNECT USING EPOXY-BASED UNDERFILL UP TO

Fig. 1. Schematic of the flip-chip interconnect structure with underfill in this study (without any matching structure adopted).

of thermal expansion (CTE) mismatch between chip and carrier are very important factors for the flip-chip reliability. For MMW flip-chips, the adhesion relies only on a few bump interconnections, which is deficient and very fragile at mechanical vibration. Besides, the CTE mismatch between chip and carrier could lead to joint fatigue during temperature variation. By using underfill as a buffer layer, the adhesion can be improved and the thermal stress can be reduced significantly. Some investigations about the MMW flip-chip reliability with epoxy underfill have been reported previously [15]–[17]. The fatigue life of the flip-chip assembly with underfill was investigated using finite-element analysis (FEA) simulation [15]. The simulation results indicated that the joint fatigue life improved significantly after underfill was applied. Schmückle et al. demonstrated that using an Al O carrier ppm/K for a ppm/K flipped-chip has negligible thermal GaAs effect due to their small CTE mismatch [17]. However, the adhesion between chip and carrier should be taken into account before commercial applications can be realized. In this study, the flip-chip assembly with epoxy-based underfill was designed, fabricated, and characterized up to 67 GHz. The reliability tests including a 85 C 85 relative humidity (RH) test, thermal cycling test, and shear force test were performed to evaluate the feasibility of such packages for practical applications. II. TEST STRUCTURE AND FABRICATION Fig. 1 shows the schematic of the flip-chip interconnect structure with underfill in this study (without any matching structure adopted). The GaAs chip and Al O substrate with the thickness of 100 and 254 m, respectively, were employed. The metallization was 3 m Au (gold). The characteristic impedances of the coplanar waveguide (CPW) transmission lines on the chip and substrate were both 50 . The total length of the back-to-back flip-chip interconnect structure was 3000 m, including 1000 m on the chip and 2000 m on the substrate. The dimensions of the bump were fixed due to the fabrication concern. The diameter of the bump was 50 m; the bump height was 20 m. The CPW transmission lines were first patterned and electroplated on the GaAs chip. The chip was then upside-down mounted on a sapphire carrier for thinning down to 100 m. After de-mounting from the sapphire carrier, the chip was diced into individual dies. The substrate with an Au CPW circuit and a bump was fabricated by an in-house bumping process, as reported in [18]. The Au-to-Au thermo-compression process was

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Fig. 2. SEM images of the: (a) fabricated flip-chip interconnect structure and (b) cross-sectional SEM image of the flip-chip interconnect structure at the bump transition region.

Fig. 3. Measured and simulated S -parameters of the flip-chip interconnect with and without underfill.

performed to bond the flip-chip interconnect structure. Fig. 2(a) shows the scanning electron microscope (SEM) image of the fabricated flip-chip interconnect structure. After flip-chip bonding, the epoxy-based underfill ( and at 10 MHz) was injected into the gap between the chip and substrate by a capillary underfill process and cured at 150 C for 2 h. Fig. 2(b) shows the cross-sectional SEM image at the Au bump region. As shown in the micrograph, the underfill was successfully filled into the gap without any voids. III. DESIGN AND OPTIMIZATION The fabricated flip-chip samples were measured using on-wafer probing measurement with a short-open-load-thru (SOLT) calibration technique. During the measurement, a at 26.5 GHz) was 10-mm-thick layer of Rohacell 31 ( placed between the sample and the metal chuck of the probe station to avoid the grounded backside under the substrate. Fig. 3 shows the measured and simulated -parameters of the flip-chip interconnect with and without underfill from dc to ) and insertion 40 GHz. It is shown that both the return loss ( ) became worse after the underfill was applied. The loss ( was increased about 5 10 dB from dc to 40 GHz and the resonance frequency with lowest reflection shifted from 33 to 28 GHz. The reason is that the effective dielectric constant changed due to the insertion of the underfill. This effect will be further discussed in the following Section III-A. Furthermore, the underfill also induced significant dielectric loss

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Fig. 5. Measured and simulated and without Z matching.

Fig. 4. (a) Simulation structures and CPW parameters for Case I (bare CPW line on GaAs chip), Case II (CPW + Al O ), Case III (CPW + Al O + Underfill), and Case IV (CPW + Al O + Underfill with Z matching). (b) Simulated S -parameters. (The CPW line length is 1000 m; the gap between the GaAs chip and Al O substrate is 20 m; the thickness of the GaAs is 100 m; the thickness of the Al O is 254 m.)

to the overall loss. The degraded about 0.2 0.4 dB from dc to 40 GHz. In Sections III-A–C, both matching designs on the GaAs chip and Al O substrate were performed, targeting on a broadband interconnect performance with low return loss and low insertion loss. Moreover, the dielectric loss induced by underfill was extracted from the measurement and simulation for comparison. A. Matching Design on GaAs Chip acts to lower the of the transmisAn underfill with sion line, leading to the performance deviation of the MMIC chip [14]. To solve this issue, the most effective method is to design in advance by taking the underfill effects into account. Fig. 4 shows the simulated CPW parameters and -parameters of a 50- CPW transmission line on GaAs chip. As can be seen, showed a simulated of a 50- CPW line below 30 dB from dc to 100 GHz (Case I). After the flip-chip, an Al O substrate is present under the CPW circuit (Case II).

S -parameters of the flip-chip assembly with

slightly increased to 7.10 and became 45 due to the degraded about 10 dB in avflip-chip detuning effect. The erage as compared to Case I. When an epoxy-based underfill was applied (Case III), the increased to 8.77 with and the minimum shifted to lower frequency (from 57 to was further lowered to 41 and decayed up 51 GHz). was 50 after modifying G (the to 16 dB. In Case IV, gap between the signal and ground conductor) to 84 m. was 6.52 and the simulated was below 25 dB from dc to 100 GHz. Based on the simulation, the flip-chip assembly with the impedance matching design on the GaAs chip was fabricated and measured. Fig. 5 shows measurement and simulation and results. After the matching design was adopted, improved below 25 GHz, but became worse beyond 25 GHz. At higher frequencies, the parasitic capacitance becomes significant, especially when the underfill is injected. The parasitic capacitance can be reduced and compensated with proper matching design on the Al O substrate. B. Matching Design on Al O Substrate Generally speaking, a flip-chip interconnect shows an overall capacitive effect, which would result in impedance mismatch and reflection at MMW frequencies [1], [2]. To improve the interconnect performance, reducing and compensating the parasitic capacitance by employing proper matching design on the packaging carrier is essential. In previous reports, the MMW flip-chip interconnect with a matching design on the packaging carrier has been studied and investigated [1]–[9]. It has been demonstrated that reducing the metal pad overlap, increasing the bump height, and employing the inductive compensations can improve the reflection at the transitions [1]–[9]. Fig. 6 shows the optimized interconnect structure and performance after both matching designs on the GaAs chip and Al O substrate were adopted. The matching designs on the Al O substrate included m , a high impedance line a small metal pad overlap m , and a ground pad shrinking m . The bump height was fixed to 20 m due to the fabrication

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Fig. 6. Optimized interconnect structure and performance.

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,L , L Al O , and L Fig. 8. L sembly as extracted from the EM simulation; the L was extracted from the measurement for comparison.

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of the flip-chip as-

dielectric loss, and radiation loss. For a flip-chip assembly, the radiation loss is very small so that it can be neglected. The dielectric loss means an overall loss in carriers (chip and substrate) , unand underfill. To specifically identify the carrier loss derfill loss , and metal loss , the following calculations were employed. Equation (1) is the definition of the loss factor [19], the real loss (1)

Fig. 7. De-embedded S 11 of the optimized interconnect structure from 1 to 67 GHz (EM simulation).

concern. As can be seen, the flip-chip interconnect showed good broadband performance up to 67 GHz. From dc to 40 GHz, was less than 20 dB; from 40 to 67 GHz, was less than 25 dB. was within 0.6 dB from dc to 67 GHz, demonstrating excellent performance for flip-chip assembly with unfrom the derfill material. Fig. 7 shows the de-embedded electromagnetic (EM) simulation. The inductive and capacitive behaviors versus frequency bands indicate that the matching designs were well adopted with a good counterbalance between the inductance and capacitance at the transition. C. Dielectric Loss of Underfill Material The loss induced by the underfill is not only due to the mismatch loss (reflection), but also due to the real loss (attenuation). In general, the real loss consists of three components: metal loss,

To get the real loss induced by underfill with , one can subtract the real loss of the flip-chip assembly with from the real loss of the flip-chip assembly with . Equation (2) gives the real loss induced by underfill. A similar approach can be applied for calculating the chip loss and substrate loss Al O . Equation (3) gives the real loss of GaAs and induced by carriers. In the simulation, the Al O were set to be 0.006 and 0.0002, respectively, (2) (3) On the other hand, to get the real loss induced by metal (gold), one can subtract the real loss of the flip-chip assembly with a perfect electrical conductor (PEC) from the real loss of the flipchip assembly with a gold conductor. Equation (4) gives the real loss induced by metal as follows: (4) Fig. 8 shows the underfill loss , chip loss , substrate loss , and metal loss Al O of the flip-chip assembly as extracted from the EM simulation. The underfill loss was also extracted from the for comparison. In measurement of the underfill was 0.02 at 10 MHz. As can this study,

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% RH test results of the flip-chip assembly.

Fig. 9. 85 C=85

be seen, at lower frequencies, the measurement and simulation showed fair agreement, below 20 dB from dc to 35 GHz. Beyond 35 GHz, the underfill loss decayed up to 13 dB, which is almost the same level as the metal loss, the dominant loss at MMW frequency. The underfill loss can be minimized . Further by using low-loss underfill material with to less than 0.0002 is not necessary since the reducing induced loss is too small and can be neglected.

Fig. 10. Thermal cycling test results of the flip-chip assembly.

IV. RELIABILITY AND MECHANICAL STRENGTH Reliability investigation is always essential for commercial applications. In Sections IV-A–C, three types of reliability tests, i.e., 85 C 85 RH test, thermal cycling test, and shear force test, were performed to test the interconnect reliability of the flip-chip assembly. A. 85 C 85

RH Test

The water absorption is an important issue, which would compromise the reliability during a long-term operation, especially for a package with polymer materials. To test the water absorption of the flip-chip assembly, the samples were stored in an testing environment of 85 C and 85% RH for 96 h. Fig. 9 shows the testing results. The weight increased 4.3% (0.6 mg) for the sample without underfill and 6.8% (0.9 mg) for the sample with underfill after 96-h testing. The underfill contributed the water absorption of around 2.5% weight to the tested samples. Furthermore, no electrical failure was observed after the test. B. Thermal Cycling Test During temperature variation, the CTE mismatch between the chip and packaging carrier could lead to joint fatigue and consequent failure. To test the interconnect reliability, the thermal cycling test, i.e., temperature range from 55 C to 125 C with 15-min dwell time (specification of the Joint Electron Device Engineering Council (JEDEC) standard) was employed. The contact resistance measurement and -parameters measurement were used to check the testing results. In Fig. 10, it indicates that no sample failed and the contact resistance showed negligible change during the test no matter with or without underfill. Fig. 11 shows the comparison of the measured -parameters (with underfill) before and after 1000 thermal cycles. The

Fig. 11. Measured S -parameters of the flip-chip assembly (with underfill) before and after thermal cycling test.

RF performance did not decay after the test, showing excellent thermal mechanical stability of the flip-chip assembly. To the best of our knowledge, this was the first time the -parameters measurement was employed to check the flip-chip reliability in open literature. Table I shows the material properties of some commonly used chip and substrate materials. As can be seen, GaAs, Si, and Al O have similar CTE and small CTE mismatch. Hence, using the Al O substrate for GaAs or Si flipped-chips has a negligible thermal effect in temperature variation [17]. This point is also supported by the testing results above. However, if one wants to further reduce cost by introducing organic substrates, the CTE mismatch becomes an important issue, and hence, the underfill is essential to improve the reliability. C. Shear Force Test Adhesion between the chip and substrate is very critical to the flip-chip reliability since it relies only on a few metallic connections, which is deficient and very fragile at mechanical vibration. The shear force test was performed to investigate the adhesion of the flip-chip assembly. Fig. 12 shows the testing results. Four samples were tested for each condition to obtain the average shear force. The shear force of the samples without

HSU et al.: DESIGN OF FLIP-CHIP INTERCONNECT USING EPOXY-BASED UNDERFILL UP TO

Tan , "

TABLE I , COST AND CTE OF COMMON CHIP AND SUBSTRATE MATERIALS

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ACKNOWLEDGMENT The authors would like to thank to C.-H. Huang, National Chiao Tung University, Hsinchu, Taiwan, C.-W. Oh, and W.-C. Lim, National Chiao Tung University, for the discussion and the help of the experimental work. REFERENCES

Fig. 12. Shear force test results of the flip-chip assembly.

underfill was 173 g. With the underfill application, the shear force was improved to 2052 g, which is about 12 times improvement, as compared to the samples without underfill. After 1000 thermal cycles, the shear force of the samples were 1584 g (with underfill) and 19 g (without underfill), respectively. The shear strength decayed after the thermal cycling test, especially for the samples without underfill. It is shown from these testing results that the application of the underfill has significantly improved the interconnect reliability of the flip-chip assembly. V. CONCLUSION In this study, the use of the epoxy-based underfill in the flip-chip interconnect is evaluated for applications from dc to 67 GHz. The matching designs on both the GaAs chip and Al O substrate were employed with the underfill effects taken into account. The optimized structure showed excellent below 20 dB and less than 0.6 dB performance of from dc to 67 GHz. For the dielectric loss induced by the underfill, the extracted results indicated that the loss can be further improved by using other lower loss underfill materials. The reliability tests including the 85 C 85 RH test, thermal cycling test, and shear force test were performed. The testing results revealed that with underfill, the flip-chip assembly had low water absorption, sustainable joint fatigue life, and robust joint strength, showing its potential for commercial MMW packaging applications.

[1] A. Jentzsch and W. Heinrich, “Theory and measurements of flip-chip interconnects for frequencies up to 100 GHz,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 5, pp. 871–878, May 2001. [2] C. L. Wang and R. B. Wu, “Modeling and design for electrical performance of wideband flip-chip transition,” IEEE Trans. Adv. Packag., vol. 26, no. 4, pp. 385–391, Nov. 2003. [3] D. Staiculescu, J. Laskar, and E. M. Tentzeris, “Design rule development for microwave flip-chip applications,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1476–1481, Sep. 2000. [4] C. Karnfelt, H. Zirath, J. P. Starski, and J. Rudnicki, “Flip chip assembly of a 40–60 GHz GaAs microstrip amplifier,” in 34th Eur. Microw. Conf., Amsterdam, The Netherlands, Oct. 11–15, 2004, pp. 89–92. [5] W.-C. Wu, L.-H. Hsu, E. Y. Chang, C. Karnfelt, H. Zirath, J. P. Starski, and Y.-C. Wu, “60 GHz broadband MS-to-CPW hot-via flip chip interconnects,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 11, pp. 784–786, Nov. 2007. [6] W. Wu, E. Y. Chang, C. H. Huang, L. H. Hsu, J. P. Starski, and H. Zirath, “Coaxial transitions for CPW-to-CPW flip chip interconnects,” Electron. Lett., vol. 43, pp. 929–930, Aug. 2007. [7] W.-C. Wu, E. Y. Chang, R.-B. Hwang, L.-H. Hsu, C.-H. Huang, C. Karnfelt, and H. Zirath, “Design, fabrication, and characterization of novel vertical coaxial transitions for flip-chip interconnects,” IEEE Trans. Adv. Packag., vol. 32, no. 2, pp. 362–371, May 2009. [8] W. C. Wu, L. H. Hsu, E. Y. Chang, J. P. Starski, and H. Zirath, “60 GHz broadband 0/1-level RF-via interconnect for RF-MEMS packaging,” Electron. Lett., vol. 43, no. 22, Oct. 2007. [9] L.-H. Hsu, W.-C. Wu, E. Y. Chang, H. Zirath, Y.-C. Wu, C.-T. Wang, and C.-T. Lee, “Design and fabrication of 0/1-level RF-via interconnect for RF-MEMS packaging applications,” IEEE Trans. Adv. Packag., vol. 33, no. 1, pp. 30–36, Feb. 2009. [10] K. Chai and L. Wu, “The underfill processing technologies for flip chip packaging,” in 1st Int. IEEE Polymers and Adhesives in Microelectron. Photon. Conf. , Potsdam, Germany, Oct. 2001, pp. 119–123. [11] L. Nguyen and H. Nguyen, “Effect of underfill fillet configuration on flip chip package reliability,” in 27th Annu. IEEE/SEMI Int. Electron. Manuf. Technol. Symp., San Jose, CA, Jul. 2002, pp. 291–303. [12] K. H. Teo, “Reliability assessment of flip chip on board connections,” in Proc. 2nd Electron. Packag. Technol. Conf., Singapore, Dec. 1998, pp. 269–273. [13] J. Sun, H. Fatima, A. Koudymov, A. Chitnis, X. Hu, H.-M. Wang, J. Zhang, G. Simin, J. Yang, and M. A. Khan, “Thermal management of AlGaN–GaN HFETs on sapphire using flip-chip bonding with epoxy underfill,” IEEE Electron Device Lett., vol. 24, no. 6, pp. 375–377, Jun. 2003. [14] G. Baumann, E. Muller, F. Buchali, D. Ferling, H. Richter, and W. Heinrich, “Evaluation of glob top and underfill encapsulated active and passive structures for millimeter wave applications,” in 27th Eur. Microw. Conf., Jerusalem, Israel, Oct. 1997, vol. 1, pp. 26–31. [15] Z. Feng, W. Zhang, B. Su, K. C. Gupta, and Y. C. Lee, “RF and mechanical characterization of flip-chip interconnects in CPW circuits with underfill,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2269–2275, Dec. 1999. [16] H. Kusamitsu, Y. Morishita, K. Maruhasi, M. Ito, and K. Ohata, “The flip-chip bump interconnection for millimeter-wave GaAs MMIC,” IEEE Trans. Electron. Packag. Manuf., vol. 22, no. 1, pp. 23–28, Jan. 1999. [17] F. J. Schmuckle, F. Lenk, M. Hutter, M. Klein, H. Oppermann, G. Engelmann, M. Topper, K. Riepe, and W. Heinrich, “W -band flip-chip VCO in thin-film environment,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 12–17, 2005, pp. 1–4. [18] W. C. Wu, H. T. Hsu, E. Y. Chang, C. S. Lee, C. H. Huang, Y. C. Hu, L. H. Hsu, and Y. C. Lien, “Flip-chip packaged In Al As/InGaAs metamorphic HEMT device for millimeter wave application,” in Proc. CS-MAX, Compound Semiconduct. Manuf. Expo, Palm Spring, CA, Nov. 2005, pp. 94–97. [19] J. Capwell, T. W. D. Markell, and L. Dunleavy, “Automation and real-time verification of passive component S -parameter measurements using loss factor calculations,” Microw. J., vol. 47, no. 3, pp. 82–90, Mar. 2004.

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Li-Han Hsu (S’08) was born in Tainan, Taiwan, in 1981. He received the B.S. and M.S. degrees in materials science and engineering from the National Chiao Tung University, Hsinchu, Taiwan, in 2003 and 2005, respectively, and is currently working toward the dual Ph.D. degrees in materials science and engineering and microtechnology and nanoscience from National Chiao Tung University, Hsinchu, Taiwan and the Chalmers University of Technology, Göteborg, Sweden. His main research interest is millimeter-wave packaging technology including flip-chip interconnects, hot-via interconnects, and integration of V -/E -band multichip module (MCM) transceiver modules.

Wei-Cheng Wu was born in Hsinchu, Taiwan, in 1979. He received the B.S. degree from in materials science and engineering from National Chiao Tung University, Hsinchu, Taiwan, in 2001, and is currently working toward the dual Ph.D. degrees in materials science and engineering and microtechnology and nanoscience from National Chiao Tung University, Hsinchu, Taiwan and the Chalmers University of Technology, Göteborg, Sweden. His research interests include fabrication, characterization, and packaging technologies of compound semiconductor devices and integrated circuits (ICs) for high-frequency applications, especially flip-chip interconnects and transition design.

Edward Yi Chang (S’85–M’85–SM’04) received the B.S. degree in materials science and engineering from National Tsing Hua University, Hsinchu, Taiwan, in 1977, and the Ph.D. degree in materials science and engineering from the University of Minnesota at Minneapolis–St. Paul, in 1985. From 1985 to 1988, he was with the GaAs Component Group, Unisys Corporation, Eagan, MN. From 1988 to 1992, he was with the Microelectronic Group, Comsat Laboratories. In 1992, he was involved with the GaAs monolithic microwave integrated circuit (MMIC) programs for both groups. In 1992, he was with National Chiao Tung University (NCTU), Hsinchu, Taiwan. In 1994, he helped set up the first GaAs MMIC production line in Taiwan, and in 1995, he became the President of Hexawave Inc., Hsinchu, Taiwan. In 1999, he returned to NCTU, where he is currently a Professor with the Department of Materials Science and Engineering. His research interests include new device and process technologies for compound semiconductor RF integrated circuits (RFICs) for wireless communication. Dr. Chang is a Senior Member and Distinguished Lecturer of the IEEE Electronic Devices Society.

Herbert Zirath (S’84–M’86) was born in Göteborg, Sweden, on March 20, 1955. He received the M.Sc. and Ph.D. degrees from the Chalmers University, Göteborg, Sweden, in 1980 and 1986, respectively. He is currently a Professor of high-speed electronics with the Department of Microtechnology and Nanoscience, Chalmers University. In 2001, he became the Head of the Microwave Electronics Laboratory. He currently leads a group of approximately 30 researchers in the area of high-frequency semiconductor devices and circuits. His main research interests include InP-HEMT devices and circuits, SiC- and GaN-based transistors for high-power applications, device modeling including noise and large-signal models for field-effect transistor (FET) and bipolar devices, and foundry-related monolithic microwave ICs for millimeter-wave applications based on both III–V and silicon devices. He also works part-time with Ericsson AB, Mölndal, Sweden, as a Microwave Circuit Expert. He has authored or coauthored over 220 papers in international journals and conference proceedings and one book. He holds four patents.

Yin-Chu Hu was born in Taipei, Taiwan, in 1984. She received the B.S. and M.S. degrees in materials science and engineering from National Chiao Tung University, Hsinchu, Taiwan, in 2005 and 2007, respectively. She is currently with Everlight Electronics, Taipei, Taiwan. Her major research interest is using various dielectric materials as underfill in flip-chip interconnect for millimeter-wave applications.

Chin-Te Wang was born in Taipei, Taiwan, on November 6, 1983. He received the B.S. degree in materials science and engineering from National Chung Hsing University (NCHU), Taichung, Taiwan, in 2006, and is currently working toward the M.S. and Ph.D. degrees at National Chiao Tung University (NCTU), Hsinchu, Taiwan. While with NCHU, he was interested in electric materials. He then joined the Compound Semiconductor Device Laboratory, NCTU.

Yun-Chi Wu was born in Chiayi, Taiwan. He received the B.S. degree in materials science and engineering from Tatung University, Taipei, Taiwan, in 2001, the M.S. degree in materials science and engineering from National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 2003, and is currently working toward the Ph.D. degree in materials science and engineering at NCTU. He is currently with the Compound Semiconductor Device Laboratory, Department of Materials Science and Engineering, NCTU. His research is focused on HEMT device and process technologies for wireless communication applications.

Szu-Ping Tsai was born in Pingtung, Taiwan on February 2, 1985. She received the B.S. degree in materials science and engineering from National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 2007, and is currently working toward the M.S. degree at NCTU. In 2007, she joined the Compound Semiconductor Device Laboratory, NCTU, where she has been involved in the area of the flip-chip technology with a focus on aspects of finite-element thermomechanical modeling and simulation.

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An Intrinsic Circuit Model for Multiple Vias in an Irregular Plate Pair Through Rigorous Electromagnetic Analysis Yao-Jiang Zhang, Member, IEEE, and Jun Fan, Senior Member, IEEE

Abstract—An irregular plate pair with multiple vias is analyzed by the segmentation method that divides the plate pair into a plate domain and via domains. In the via domains, all the parallel-plate modes are considered, while in the plate domain, only the propagating modes are included to account for the coupling among vias and the reflection from plate edges. Boundary conditions at both vias and plate edges are enforced and all parasitic components of via circuit are expressed analytically in terms of parallel-plate modes. The work presented in this paper indicates that a previous physics-based via circuit model from intuition is a low-frequency approximation. Analytical and numerical simulations, as well as measurements, have been used to validate the intrinsic via circuit model. Index Terms—Intrinsic via circuit model, parallel-plate modes, physics-based via circuit model, segmentation technique, signal and power integrity.

I. INTRODUCTION IAS ARE widely used in multilayer high-speed printed circuit boards (PCBs) or packages to connect signal traces on different layers or connect devices to power and ground planes [1]–[4]. As discontinuities, vias may cause mismatch, crosstalk, mode conversion, and other signal integrity issues in a signal link path. Moreover, vias passing through a parallel power/ground plate pair could effectively pick up the power distribution network noise, resulting in degraded signal quality. Similarly, high-speed transient currents flowing along vertical vias could also excite the parallel plates they penetrate, causing serious voltage fluctuations in power distribution network or electromagnetic interference problems due to strong edge radiation. Therefore, modeling the electromagnetic behavior of vias in parallel plates plays a critical role in analyses of signal integrity, power integrity, and electromagnetic interference for multilayer PCBs and packages. Via-plate interactions have been extensively studied. For vias crossing a single plate, either full-wave methods or quasi-static

V

Manuscript received October 15, 2009; revised April 13, 2010; accepted May 01, 2010. Date of publication July 15, 2010; date of current version August 13, 2010. The authors are with the Department of Electrical and Computer Engineering, Electromagnetic Compatibility Laboratory, Missouri University of Science and Technology, Rolla, MO 65409 USA (e-mail: [email protected]; jfan@mst. edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2052956

approaches are effective due to the localized field distribution near the vias [5]–[12]. For vias crossing a plate pair, however, quasi-static approximation and simple lumped circuit modeling are not suitable any more as the electromagnetic fields are not localized near vias. Although the higher order evanescent parallel-plate modes are localized in proximity to the vias as energy stored in the electric and magnetic fields, the propagating modes spread over the entire plate pair. This makes it more difficult to model a via crossing a plate pair than a via crossing a single plate. An analytical method for the via–plate–pair interaction was first reported in [13] for a single via, and later extended to the coupling of two vias [14]. A magnetic frill current was assumed in a via-hole in a plate (the gap region between a via and a plate), and cylindrical waves were used to describe the parallel-plate modes. Boundary conditions at the vias were explicitly enforced. However, the analytical method cannot handle a plate pair with more than two vias. Moreover, the method is restricted to an infinitely large plate pair, and no edge reflection was considered. Recently, an algorithm denoted by the Foldy–Lax multiple scattering method, was proposed to extend the analytical method to multiple vias in a plate pair [15]–[19]. The dyadic Green’s function of an infinitely large plate pair or a finite circular plate pair, as well as the addition theorem of cylindrical harmonics, was used to analyze the multiple scattering among vias. All the via-holes in plates were regarded as ports of a multiport microwave network whose admittance matrix was obtained from current distributions in the ports. The multiple scattering method can be regarded as an efficient semianalytical method. However, the method is still restricted to either an infinite or a finite circular plate pair since an analytical dyadic Green’s function is not available for an irregular finite plate pair. Different from the rigorous multiple scattering method, an alternative solution denoted a physics-based via circuit model was proposed in [20]–[24]. In this model, the via–plate–pair interaction was represented simply by a -type circuit. Each via barrel was viewed as a simple short circuit and the displacement currents between the via and the top/bottom plates were represented by two shunt capacitors. The impedance of the plate pair, widely studied in [25]–[32], was used as the return path for the signal current along the via barrel. The combination of the lumped via circuits and the full-wave plate pair impedance correctly reflects the fact that vias are usually electrically small, while the plate pair is comparable to the wavelength of interest. Moreover, the

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model is suitable for any irregular plate pair since its impedance matrix can be easily obtained from the cavity model and the segmentation technique [28], [30]. Despite its flexibility in handling edge boundaries, the physics-based via circuit model is based on physical intuition. The boundary conditions at the vias are not satisfied by the lumped via circuit. In addition, the coupling among vias is described by the parallel-plate impedance matrix (the modes) only. Therefore, when vias are closely spaced and the coupling due to the higher order modes cannot be neglected, the accuracy of the physics-based via circuit model will start to deteriorate. The purpose of this paper is to establish a theoretical foundation for the physics-based via circuit model for multiple vias in an irregular plate pair. An intrinsic via circuit model is derived from a rigorous electromagnetic analysis where boundary conditions at both the vias and the plate edges are explicitly enforced. The term “physics-based via circuit model” is used herein to refer to the model proposed in [20]–[24] based on physical intuition, while the term “intrinsic via circuit model” refers to the model presented in this paper based on rigorous electromagnetic analysis. Later, the physics-based via circuit model will be demonstrated to be an approximation of the intrinsic via circuit model at low frequencies. Virtual circular boundaries can be introduced to divide the entire plate pair into via domains and a plate domain, as shown in Fig. 1, where a via domain associated with Via is enlarged for illustration. The virtual circular boundaries are located where the higher order parallel-plate modes excited by scattering at each via are confined within the corresponding via domain, and, are negligibly small in the plate domain. The segmentation technique is then used to enforce the continuity of fields along these virtual circular boundaries. This paper focuses on the via domain modeling, and simply uses an impedance matrix to represent the plate domain behaviors. The details of the plate domain modeling will be addressed in a future publication. To construct an intrinsic via circuit model in a via domain, the electric and magnetic components excited by a magnetic frill current in the via domain are obtained first using the same method discussed in [24]. The via domain is then regarded as a two-port network whose admittance matrix is expressed analytically in terms of the parallel-plate modes. An intrinsic via circuit model is obtained through a -type equivalent-circuit representation of the two-port network. The equivalent circuit associated with both the evanescent and propagating modes are derived separately. There are two main contributions of this paper, which are: 1) it establishes a rigorous theoretical foundation for the equivalent-circuit modeling of the via-plate-pair interaction and 2) it provides an empirical expression to determine the virtual circular boundary, within which the higher order modes are well confined. Further, the virtual circular boundary can be used to quantify the limitation of the equivalent-circuit modeling in practical engineering designs. When the spacing between vias is larger than two times the radius of the virtual circular boundary, which is usually the case in most practical PCB and package designs, the coupling between the vias is dominated

Fig. 1. Domain segmentation of an irregular plate pair with multiple vias. (a) Top view. (b) Side view.

by the zeroth-order waves. Thus, the equivalent-circuit modeling approach presented in this paper can accurately describe the via–plate–pair interaction, which, compared with other modeling approaches, is often preferred for its fast simulation speed, easy interface with other circuit elements, as well as clear physical meaning for analysis and design. II. SEGMENTATION OF VIAS AND A PLATE PAIR The top view of an irregular plate pair loaded with vias is shown in Fig. 1(a). Bounded by the edge , the region of the entire via-plate structure is denoted that is decomposed into separate regions: via domains , and one plate domain , i.e., . , is enlarged in As an example, the via-domain for Via , Fig. 1(a) and (b), showing the top and side views. The barrel, pad, and via-hole radii of the via are denoted , , and , respectively. The via height or the separation of the plates is , and a dielectric layer is between the plates with a relative permittivity of . A local cylindrical coordinate system is set up with the origin at the center of the bottom plate surface in , as shown in Fig. 1. The virtual via boundary between the via domain and . the plate domain is located at The vertical ( -directional) electric field near the th via can be expressed as (1)

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where and are the expansion coefficients for the inparallel-plate modes ward and outward and , respectively, which are expressed as

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At Port 3 of Via , the port voltage and current can be defined waves as in terms of the (4) (5)

(2) (3) are the th-order Bessel and Hankel (the where is the cylindrical second kind) functions, respectively, coordinates of the via, and is the parallel-plate modes are radial wavenumber. Only the modes are negligible due considered in this paper. The to the specific excitations typical in a via geometry (a circular magnetic current in a via-hole, or a vertical electric current along a via barrel). In most practical designs, the following assumptions are satisfied. • Only the TEM mode is considered in the via-holes. In other words, magnetic frill currents are assumed to excite the parallel-plate pair, which have been widely used in [13]–[19], [24], [33], and [34]. This implies that sources modes in in the via-holes only excite the the region between the two parallel plates, assuming vias and their corresponding via-holes are concentric. • The height is electrically small so that only the cylindrical waves can propagate. All the higher order decay rapidly along the radial modes direction. Therefore, there is a virtual via boundary (in local coordinates) for each via beyond which all the higher order modes can be neglected. In other words, only modes are considered in the the plate domain outside the via domains. • The virtual via boundary for each via is electrically small so that the azimuthal variation of fields due to the asymmetry of the via-plate structure is negligible. This means modes need to be considered for only the the via domain modeling. These assumptions indicate that each via domain is an electrically small region bounded by a virtual circular boundary. Thus, voltages and currents are well defined and circuit ports can be specified. As shown in Fig. 1, Ports 1 and 2 are defined as coaxial ports on the inner surface of the plates, between the vias and the plates, and across the via-holes. Port 3 is defined as a radial port at the virtual circular boundary between the two parallel plates. Furthermore, these assumptions determine that only the waves need to be considered in the plate domain. Therefore, properly setting the virtual circular boundaries between the via domains and the plate domain is critical for the validity of the via circuit model derived later on. In this paper, it is also assumed that the virtual circular boundaries do not overlap, as shown in Fig. 1. This guarantees that the higher order modes of one via do not illuminate another via. This assumption limits the approach presented herein to the cases where vias are not placed very closely to each other. Section IV will provide an empirical formula to determine the virtual via boundary, or in other words, the minimum spacing between two adjacent vias for the validity of the approach.

The plate domain in Fig. 1(a) can then be modeled as a -port network to describe the coupling among the vias using the modes. The impedance matrix of this -port network is different to the conventional impedance matrix of a rectangular plate pair widely studied in [25]–[32]. The new impedance definition and calculations will be introduced in a future publication. The via, inside its via domain, can be viewed as a three-port network, as shown in Fig. 1. Ports 1 and 2 can connect other layers in a multilayer PCB or package, and Port 3 connects to the network that describes the plate domain to ensure the continuity . of the tangential fields at the virtual via boundary at This segmentation approach enforces the boundary conditions at both the via barrels and the plate edges, as explained later in detail. III. INTRINSIC THREE-PORT VIA CIRCUIT MODEL A via domain can be viewed as a via located at the center of . The field distria circular plate pair with a dimension of butions excited by a magnetic frill current can then be derived using the same method as in [24]. A. Field Distributions Excited by a Magnetic Frill Current Using the equivalence principle of electromagnetics, the TEM mode at Port 1 or Port 2 of a via, shown in Fig. 1(b) as an example, can be regarded as an equivalent magnetic frill current

(6) where is the voltage across the via-hole at or . The magnetic field distribution due to (6) has been derived as [24]

(7) for

, and

for

(8) , and the two auxiliary functions are defined as

(9)

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(10) and are the reflection coefficients for the th where cylindrical waves from the via barrel and the outer , respectively, and given as radial boundary

Fig. 2. Illustration of a via domain used for extracting an equivalent two-port network.

(11) Irregular (12)

where denote the zeroth (first)-order Bessel and Hankel function of the second kind, respectively, and is the radial wavenumber for the modes as (13) has four choices depending As shown in (12), the values of . The superscript in the on the boundary conditions at and indicates one of the choices. auxiliary functions and denote the perfect electric conductor, perfect magnetic conductor, and perfectly matched layer conditions, respectively. represents According to the discussion in Section II, the case where all the higher order modes are well confined in the via domain. Therefore, in this case, only waves, , is nonzero. the reflection coefficient of the The other boundary conditions are introduced here for the dein the following termination of the virtual via boundary sections. The electrical-field components can be obtained from (7) and (8) from the magnetic field as

(14)

(15) for

(17) for . The following identity can be used to obtain the explicit expression of (18) where functions.

is the first-order (zeroth-order) Bessel or Hankel

B. Admittance Matrix of a Two-Port Network A via domain, shown in Fig. 1(b) as an example, can be regarded as a two-port microwave network with the radial port (Port 3) terminated with the impedance of the plate domain, which is related to the reflection coefficient as [27] (19) Therefore, an admittance matrix of the two-port network can be defined as (20) where and are the voltage and current pair of Ports 1 and 2, respectively. In practice, the dielectric material between the plate pair is normally reciprocal. As a result, the . Let two-port via network satisfies reciprocity, i.e., and . and can then be calculated through the port currents, as shown in Fig. 2, with the top hole of the via closed by a perfect electric conductor (PEC) boundary and an equivalent magnetic frill current located at the bottom via-hole. in (8) leads to a description The magnetic field for of the current distribution on the via barrel and pad as [15]

and

(21) Substituting (8) into (21) using (16)

and

yields (22)

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Fig. 3.  -type equivalent circuit for a reciprocal two-port network from its admittance matrix. The circuit is rotated by 90 , to be consistent to the ports illustrated in Fig. 1.

As shown in Fig. 2, the port currents can be obtained by letting and . The self-admittance and can then be obtained from (22) as mutual admittance (23) (24) A reciprocal two-port network can be represented by a -type equivalent circuit, as shown in Fig. 3 [35]. Using (23) and (24), the following shunt and series elements for the -type equivalent circuit model can then be expressed as (25)

Fig. 4. Via circuit model due to the higher order modes with two shunt and one series parasitic capacitances.

In the case of a via without a pad, i.e., the pad radius is equal , the pad-plate capacitance vanto the barrel radius ishes, and the barrel-plate capacitance in (30) is reduced to the analytical expression of the barrel-plate capacitance given in [24]. represents Similarly, the shunt capacitance from the capacitive coupling between the via and top plate. Its value if the pad and via-hole dimensions are the is the same as same in both the top and bottom plates. , 2) Series Capacitance: The series admittance in (26), is expressed in terms of both the zeroth-order propagating mode and the higher order evanescent modes . It can be separated into two parts

(26) (32) where the admittance order mode

C. Via Circuit Components Due to Higher Order Modes has a 1) Shunt Capacitances: The shunt admittance clear physical meaning that it represents the capacitive coupling between the via (the barrel and pad) and the bottom plate. Thus, a shunt capacitance can be defined as (27)

represents the part due to the zeroth-

(33) and the series capacitance expressed as

due to the higher order modes is

(34)

Therefore, substituting (25) into (27) yields (28) where is the mode number used to truncate the infinite summation in practical calculations. can be divided into two parts The shunt capacitance (29) and the pad-plate capacwhere the barrel-plate capacitance itance are expressed, respectively, as (30)

(31)

Note that the series capacitance is not included in the previous physics-based via circuit model in [20]–[22], [24], as it cannot be attributed to a static capacitance between two sepain (15) and (17) lead to the rate conductors. While the fields in capacitive element , the higher order parts of the fields (14) and (16) are responsible for the parasitic capacitance . Equations (27) and (32) convert the general -circuit shown in Fig. 3 into the via circuit model shown in Fig. 4. The parasitic and characterize the energy stored in the capacitances electric field near the via due to the higher order parallel-plate modes. D. Via Circuit Components Due to Zeroth-Order Waves In the two-port via circuit model shown in Fig. 4, both and are related to the higher order modes, which are well confined in the via domain according to the assumptions in

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is the impedance looking at Port 3 outward into the where plate domain by definition, and has the same form as (19) for a circular plate pair. For an irregular plate pair, (40) is still valid and a generalized (19) will be derived in a future publication. Substituting (40) and (37) into (36), as well as using (35), yields (41) flowing into the rim The zeroth-order current component and is a combination of the effects of the via pad at of both and . It can be obtained as

Fig. 5. Alternative derivation of Y by the segmentation technique.

Section II. Only the series admittance in (33), the component due to the zeroth-order waves, is then related to both the via is provided and plate domains. An alternative derivation of herein based on the segmentation of the via domain and the plate domain, as shown in Fig. 5. The derivation will show clearly can be decomposed into an intrinsic part due to the via how domain and an impedance due to the plate domain. As discussed in Section II, the via domain I and the plate , as illustrated in Fig. 5. The domain II are separated at boundary condition on the interface requires

(42) The first term of the right-hand side of (42) is derived from the with a zeroth-order current component of (33) caused by PMC boundary at , and (43) reflects the magnetic field at The transform coefficient excited by the current at as

(35) (36) and denote the where left- and right-side port currents and voltages at the segmentation interface. These voltages and currents can be well defined similarly as in (4) and (5) because of the assumption that only the zeroth-order waves can reach the boundary between the via and plate domains. According to the equivalence principle, a perfect magnetic conductor (PMC) boundary can be assumed for the interface at if the equivalent currents and are impressed along can then be obtained as both sides. In region I, the voltage

(44)

Substituting (41) into (42), the series admittance rived as

is then de-

(45) When all the via geometrical dimensions are electrically behaves like a capacitance, i.e., small, the admittance , and the impedance acts as an inductance, i.e., . From (43) and (38), it can be derived that (46)

(37) where

and (47)

(38) relating to a part and the voltage transform coefficient , which only results from the zeroth-order waves, is of

The series admittance

in (45) can then be rewritten as (48)

(39) where the ideal transformation ratio Equation (39) is obtained from the zeroth-order component of with a PMC boundary condition. (14) at In region II, when there is only one via in the plate pair, the is related to as voltage (40)

is defined as (49)

Formula (48) implies an equivalent circuit shown in Fig. 6. A similar equivalent -network has been proposed for the radial-line/coaxial-line junction in [36] and [37] with no circuit extraction.

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is an auxiliary function defined as a determinant and of the Bessel and Hankel functions as (54) and are the th Bessel and th second-order Hankel functions, respectively. Similarly, the parameters due to the zeroth-order waves can be expressed in their concise forms as

Fig. 6. Via circuit elements due to the zeroth-order mode.

(55) (56) (57) where

(58) (59) Fig. 7. Complete three-port via circuit model.

E. Complete Intrinsic Three-Port via Circuit Model By combining the via circuits due to the higher order modes in Fig. 4 and the zeroth-order waves in Fig. 6, a complete three-port via circuit model and its connection to the impedance matrix of can be obtained, as shown in Fig. 7. the plate domain , , , , and To demonstrate that all the parasitics are intrinsic to a via domain, and also to facilitate the usage of the derived intrinsic via circuit model, their expressions are given here again explicitly as

provided It can be shown from the properties of in the Appendix that all the parasitic parameters from (50)–(59) are real values despite that the imaginary unit may be included in some expressions. Moreover, at the low frequencies where , using the small argument approximations and given in the Appendix, the static of approximations of the parasitic parameters caused by the zerothorder mode can be obtained as (60) (61)

(50) (62) (51) where is the per-unit-length capacitance of a coaxial geometry with inner and outer radii of and , can be expressed in respectively. The auxiliary function a concise form as

(52) where

for

,

for

,

(53)

Equations (50)–(59) provide analytical expressions for all the components in the intrinsic three-port via circuit model shown in Fig. 7. Clearly, all of them are only related to the via structure and only. The impact of itself, i.e., are functions of the plate edges is described by the impedance matrix of the plate domain only, which will be addressed in a future publication. It is worth giving a brief explanation on the physical meaning comes of each parasitic component. The shunt capacitance components in (15) and (17), the series from the higher order capacitance is due to the higher order components in (14) reflects the effects of the zerothand (16), the capacitance component between the two radial plates and order in Fig. 5, the inductance is due to the radial loop , and, the transformer ratio is to relate the vertical voltage to the horizontal excitation .

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Fig. 8. Convergence of the via capacitances versus the mode number N (r = 9. Convergence of the via capacitances versus the radius of the virtual 0:1, a = 0:4, b = 0:8, h = 0:8, R = 40 (unit: millimeters); " = 3:84; Fig. boundary with different boundary conditions (r = 0:1, a = 0:4, b = 0:8, frequency = 1:0 GHz). h = 0:8 (unit: millimeters); " = 3:84; mode number N = 31; frequency = 1:0 GHz).

F. Properties of the Via Parasitic Components have been extensively studied in [24] The properties of for a via without a pad. Therefore, only convergence with the number of the parallel-plate modes, as well as the selection of , is disthe boundary condition at the virtual boundary at and . cussed for both An example of the convergence of the parasitic capacitances in (50) and (51) with the increase of , the number of the higher order modes used in the calculations, is shown in Fig. 8. Here is selected to be 40 mm, far enough the outer boundary away from the via. It can be seen that only tens of the higher order modes are required to obtain the converged values. An is neginteresting observation is that the series capacitance ative, and its physical meaning is yet to be revealed. The convergence of (50) and (51) with the increase of is demonstrated in Fig. 9. The shunt and series capacitances and converge to a constant value when is approximately 1.6 times of the via-hole radius in this example, independent of the . This is because the higher boundary condition applied at order modes decay very quickly from the via due to the small plate separation . This demonstrates the previous assertion that to separate the via and plate there is a virtual via boundary domains. The plate size, shape, or other vias outside of the via domain have a negligible impact on the parasitic capacitances and . In addition, the result shown in Fig. 9 implies that the difor values with different boundary conditions ference in can be used to determine where the virtual boundary at should be located in order to separate the via and plate domains. Based on this idea, an empirical formula will be developed in Section IV to determine the virtual boundary from geometrical dimensions. From the discussion of the virtual boundary, it is clear that both the intrinsic and physics-based via circuit models are only valid when the distance between any two vias is larger than . Otherwise, the multiple higher order scattering among vias cannot be neglected. This is the limitation of this category of via circuit models. The frequency-dependent properties of the parasitics due to the zeroth-order mode from (55)–(59) are shown in Fig. 10, is used in the calculations. At the frequencies lower where

than 10 GHz, these parasitic parameters remain approximately constant in this example, which are consistent with the predictions of (60)–(62). However, beyond 10 GHz, the frequency dependence is significant, and a larger via-hole radius results in a faster increase of their values with frequency. IV. DETERMINATION OF THE VIRTUAL VIA BOUNDARY The validity of the intrinsic via circuit model depends on the assumptions discussed in Section II. One of the most important ones is that all the circular via domains do not overlap. Therefore, the virtual circular boundary can be used to quantify the limitation of the intrinsic via circuit model in practical applications. reflects the energy stored in the The via-plate capacitance higher order electric fields adjacent to a via. Thus, it is used herein to quantify the size of a via domain. will reFrom (50), different boundary conditions at is far away sult in different values of . However, when values for different boundary confrom the via barrel, the ditions converge to the one obtained using the PML boundary, as illustrated in Fig. 9. This implies that the higher order modes are negligible at the boundary and the boundary condition is not important any more. In other words, the virtual via boundary values using can be determined by examining whether the different boundary conditions are close enough. To do so, a relative difference is introduced to define the difference in the values with the PMC and PML boundary conditions at as (63) where and are obtained by (52) and (53). For in higher order modes at low frequencies, the wavenumber (13) can be approximated as a purely imaginary number, i.e., . Therefore, is only related to , , and , independent of the dielectric constant and frequency. Note that all the geometrical parameters are normalized to the . via-hole radius since must be larger than or equal to and values for the Some examples with different relative difference are shown in Fig. 11. It can be seen that

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Fig. 11. Examples of the relative difference versus the radius of the virtual : ; mode : ,b : (unit: millimeters); " : ,a boundary (r number N ; frequency : GHz).

= 01 = 04 = 08 = 31 =10

= 3 84

Fig. 12. Validation of the curve-fitting formula for the relative difference [lines without symbols: from (63); lines with symbols: from (64)].

Fig. 10. Frequency-dependent properties of the parasitics due to the zerotha : mm, h : mm, order waves with different via-hole radii (r " : ;l b is used here).

=42 =

= = 02

= 08

in logarithmic scale decays approximately linearly with Therefore, can be approximated as

In most practical designs, and are suitable choices for estimating and . The curve-fitting formula (64) and the accurate expression (63) of the relative difference for several vias with different barrel-to-pad radius are compared in Fig. 12. Note that although this is a relatively extreme case with and values, good agreement is achieved between large (63) and (64). can be Using (64), the virtual via domain boundary at determined by specifying a small tolerance . This yields

.

(64) where, for a specific via structure, and are two constant parameters that can be calculated by selecting two different values in (63) as (65) (66)

(67) For signal integrity analyses, is an acceptable choice. The smaller the relative difference is required, the larger the value of the virtual via boundary .

V. COMPLETE MODEL FOR MULTIPLE VIAS IN AN IRREGULAR PLATE PAIR The intrinsic via circuit model for a single via, as shown in Fig. 7, can be easily extended to the cases involving multiple

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Fig. 15. Solution domain for the shunt capacitance C and the boundary conditions when the finite-difference method is applied. Fig. 13.

ith

via circuit model when P vias are present in a parallel-plate pair.

Fig. 14. Schematic of circuit model for pair.

P

vias in an irregular parallel-plate

vias. With vias in a plate pair, the voltage and current relationship of (40) is replaced by

(68) is used to describe port voltages An impedance matrix as functions of port currents in the plate domain, as shown in Fig. 1(a). Using (68) instead of (40), and following the same procedure in the derivation of the via circuit due to the zeroth-order waves, an equivalent-circuit model for the th via can be obtained, as shown in Fig. 13, using the ideal transformer theory [38]. For any arbitrarily shaped plate pair with vias, Fig. 13 is valid for each via, which leads to a complete circuit model shown in Fig. 14 for multiple vias in an irregular plate pair. The impedance of a plate pair used in (40) or (68) is different from the conventional one defined in [26] where rectangular ports and area integration of the Green’s function are used to analytically evaluate the impedance matrix in a rectangular plate pair. In this application, ports in a plate pair should be circular, should be the impedance as already pointed out in [32], and matrix for the plate pair with multiple PMC holes (with the via will be addressed in detail in domains excluded). The new a future publication.

Fig. 16. Comparisons of the barrel-plate, pad-plate, and shunt capacitances : , using analytical formulas and the finite-difference method (FDM) (r b : ,h : ,R : (unit: millimeters); mil mm; : frequency of 1.0 GHz is chosen in capacitance evaluations using analytical formula).

= 0 4318 = 0 2286 = 1 27

1

= 0 1016 = 0 0254

VI. VALIDATION AND MEASUREMENTS A. Evaluation of , , and Finite-Difference Method

Using

, In the intrinsic via circuit model, the shunt capacitance including and , can be validated using quasi-static approximations. Due to the radial symmetry, the potential satisfies the 2-D Laplace equation (69) The finite-difference method is used to solve (69), and the solution domain is defined in a rectangular region, as shown in Fig. 15, where the Neumann boundary condition is specified , as discussed in [24]. on the plane of The details of the finite-difference method implementation have been given in [39]. The shunt capacitances , , and obtained by analytical formulas (28), (30), and (31), as well as by the finite-difference method are compared in Fig. 16. When other parameters are fixed, with the increase of the pad radius

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, increases and decreases steadily; the combined cadecreases initially and then pacitance increases rapidly. The analytical evaluations and the numerical finite-difference simulations agree well. The meshing grid in the finite-difference method was chosen to be either 1.27 10 mm (0.05 mil) or 0.635 10 mm (0.025 mil), and convergence with mesh density was ensured. B. Input Impedance of a Probe in a Circular Plate Pair A probe can be viewed as a special via with its top coaxial port closed by a PEC, as shown in Fig. 2. Consequently, an equivalent probe circuit model can be obtained by shorting the top coaxial port (Port 1) in the three-port via circuit model given in Fig. 7. The input admittance of a centrally located probe in a circular plate pair can then be obtained as

Fig. 17. Comparisons of the input impedance results of a probe in an infinite : ,b : ,h : , unit: mm). plate pair (r a

= = 0 254 = 0 8382 = 1 4732

(70) Another expression for the input admittance of a probe in a circular plate pair seen from Fig. 4 is (71) where is obtained by (33). Equations (70) and (71) can be in (23), which can regarded as two different expressions for be further proven to be consistent with the formulas for a probe in an infinite plate pair derived in [15], [33], and [34] when is selected for calculations. Therefore, in (23) can be viewed as a general analytical formula for the input admittance of a probe located in a circular plate pair. The input admittance can also be obtained from the physicsbased via circuit model as (72)

Fig. 18. Comparisons of the input impedance results of a probe in a finite circular plate pair. Via dimensions are the same as those in Fig. 17, and the radius of the plate pair is 10.2 cm.

where is calculated from (19) by replacing with , the via barrel radius [20]–[22]. This indicates for the physics-based via circuit model, the radial port (Port 3) should be defined at the , , barrel radius to minimize the parasitic effects due to , and instead of the virtual boundary of for the intrinsic via circuit model. By numerically comparing the input admittances of a probe using (70) of the equivalent probe circuit, (71) of the analytical formula, and (72) of the physics-based circuit, the equivalence of (70) and (71) can be verified, and also the limitation of the physics-based via circuit model can be observed. The input impedance results of a probe in an infinite plate pair using the three different expressions are compared in Fig. 17. The radii of the probe barrel and the via-hole are 0.254 mm (10 mil) and 0.8382 mm (33 mil), respectively. The separation of the plates is 1.4732 mm (58 mil). A Debye dielectric model, , is used for the permittivity of the dielectric material between the two plates. The relative are set to be static permittivity and the optical permittivity 4.3 and 4.1, respectively; and, the relaxation time is 3.1831 10 . The input impedance calculated from (70) of the equivalent probe circuit agrees very well with (71) of the analytical formula

for both the real and imaginary parts. The input impedance calculated from (72) of the physics-based via circuit model, however, matches (70) and (71) at low frequencies only. Moreover, the resonant frequency predicted from (72) is 15 GHz, as opposed to 20 GHz calculated from (70) and (71). The three different formulas to calculate the input impedance of a probe centered in a finite circular plate pair with a radius of 10.2 cm and a PMC edge boundary condition are compared in Fig. 18. Calculations from (70) and (71) match each other in the entire frequency band. Further, (72) agrees well with both (70) and (71) at the first several resonant frequencies, but deviates from them as frequency increases. The comparisons in these two figures indicate that: 1) the new expression for in (48), which is key to derive the intrinsic three-port via circuit model, agrees very well with that in (33) and 2) the physics-based via circuit model can be viewed as an approximation of the intrinsic three-port via circuit model at low frequencies. The difference between these two models becomes larger with the increase of frequency and plate-pair sizes. In practical applications, the physics-based circuit model is usually acceptable due to several reasons. First, only lowest resonant frequencies of a plate pair are usually critical for signal/

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Fig. 19. (a) Test board geometry (courtesy of the authors of [21]) (1 mil = 0:0254 mm). (b) Equivalent circuit using the intrinsic three-port via model for the test board geometry shown in (a).

Fig. 20. Comparisons of the S -parameters obtained by the intrinsic via circuit model, the physics-based via circuit model, and measurements without a local shorting via.

power integrity analysis depending on the specific data rate and rise time. Secondly, a plate pair in practical PCBs is usually not large enough so that the approximation of using the physicsbased equivalent circuit can still result in an acceptable accuracy. Thirdly, to provide a current return path, shorting vias or decoupling capacitors are often used in a plate pair, which in most cases reduces the impact of the plate-pair sizes. Therefore, the rigorous intrinsic via circuit model derived here justifies the engineering applications of the physics-based via circuit introduced in [20] and [21] at low frequencies in the cases with a relatively small plate pair, or a plate pair with dense shorting vias/decoupling capacitors.

approximate PEC cavity). An additional shorting via is located at (4.064, 5.080) mm or (160, 200) mil to provide an adjacent return path for the signal via. The relative permittivity and loss tangent of the dielectric layers between the plates are 3.5 and 0.014, respectively. The transmission property between Ports 1 and 2 of the test board geometry is simulated by the equivalent circuit shown in Fig. 19(b). Note that each stripline is split into two microstrip lines referenced to the top and bottom plates, as proposed in [21], in order to be connected to the intrinsic via circuit model. The length of the microstrip lines is assumed to be 6.35 mm (250 mil) in the simulation. The intrinsic via circuit models are connected to the impedance matrix of each plate pair, which is . Two cases were studied: terminated by a load impedance with and without the local shorting via at (4.064, 5.080) mm. is used for the case with the local shorting via while is used for the case without the local shorting via. The -parameters calculated from the rigorous intrinsic via circuit model, the approximate physics-based via circuit model, and measurements for the cases with and without the local shorting via are compared in Figs. 20 and 21, respectively. The circuit models were simulated using the Advanced Design System (ADS), a circuit simulator from Agilent Technologies, Santa Clara, CA. The -parameters of the test geometry were measured using an Agilent vector network analyzer (VNA)

C. Measurements Fig. 19(a) illustrates a test board geometry measured to validate the intrinsic via circuit model. It contains seven plate pair with two striplines located in the top and bottom ones, respectively. The separations of the plate pair are 0.3048 mm (12 mil) or 0.2032 mm (8 mil). A signal via located at (4.064, 4.064) mm or (160, 160) mil, as shown in Fig. 19(a), connects these two striplines. The radii of the via barrel and via hole are 0.127 and 0.381 mm, respectively. For each plate pair, the fields are restricted inside a 9.144 9.144 mm or 360 360 mil cavity constructed by dense stitching vias connecting all the plates (an

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Fig. 22. Phase comparison of S obtained by the intrinsic via circuit model and measurements without a local shorting via.

Fig. 21. Comparisons of the S -parameters obtained by the intrinsic via circuit model, the physics-based via circuit model and measurements when a local shorting via is present.

E8364A in the frequency band from 45 MHz to 40 GHz. Microwave probes, model 40A G-S and S-G with a 225- m pitch, from GGB Industries Inc., Naples, FL, were used to minimize the effects of test fixtures. The short-open-load-thru (SOLT) calibration was conducted using a CS-14 calibration substrate, also from GGB Industries Inc. There are only slight differences between the results obtained from the rigorous intrinsic via circuit model and the approximate physics-based via circuit model. These two models match the measured results very well up to 15 GHz. At higher frequencies, these two models can also predict the same trends of the -parameters as the measurements, but with larger discrepancies. A possible reason for the discrepancies is that the trace-to-via discontinuities are neglected in the simulations. In addition, Fig. 22 also shows good agreement up to 30 GHz obtained by the intrinsic via circuit between the phase of model and the measurement. , as illustrated in The impedance of each plate pair, Fig. 19(b), is a critical parameter to understand the simulation . A smaller will result in a better transmission results of property between the two ports. The impedance magnitudes of the second plate pair obtained from the intrinsic and physics-based via circuit models are compared in Fig. 23. These two models provide almost the same series impedances. As mentioned earlier for the input impedance

Fig. 23. Magnitudes of the input impedance calculated from the intrinsic via circuit and the physics-based via circuit models.

of a probe, these two via circuit models result in quite similar results for a small plate pair at the first several resonant frequencies. The test board geometry in this example is very small in dimension. Therefore, at the frequencies of interest (less than 40 GHz), the physics-based via circuit is still a good approximation. This explains why these two models provide almost the same -parameters in Figs. 20 and 21. The resonant peaks in Fig. 23 lead to the minimum values in both Figs. 20(b) and 21(b). This is because is the impedance of the return current path of the signal via. At low frequencies, the surrounding stitching vias are an effective low-impedance return current path. Thus, the signal can easily pass through several layers of the plate pair. Another observation from Fig. 23 is that the local shorting via shifts the first resonant frequency from 12.35 to 14.58 GHz, enhancing the bandwidth of the signal channel. VII. CONCLUSIONS An intrinsic three-port via circuit model and its connection to the impedance matrix of a plate pair is derived rigorously through electromagnetic analysis. Both boundary conditions at vias and plate edges are satisfied explicitly. This provides a theoretical foundation for the physics-based via circuit model that

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was developed ad hoc from physical intuition. Analytical formulas for the input impedance of a probe in a circular plate pair are used to verify the derived intrinsic via circuit model. It shows that the physics-based via circuit model is an acceptable approximation to the rigorous one derived in this paper for a relatively small plate pair or for a plate pair with dense shorting vias/decoupling capacitors. Furthermore, test boards with a signal via and a seven-plate pair with and without a local shorting via are employed to validate the intrinsic via circuit model with good agreement. APPENDIX PROPERTIES OF THE AUXILIARY FUNCTION • For real values of

and ,

is imaginary as (73)

is a Neumann function or th-order Bessel where function of the second kind. can be expressed • For real values of and , as (74) where and are the th-order modified Bessel functions of the first and second kind, respectively. • Using small argument approximations of the Bessel and and ), it can be shown that Hankel functions ( (75) (76) ACKNOWLEDGMENT The authors would like to thank Dr. C. Schuster, Hamburg University of Technology, Hamburg, Germany, Dr. G. Selli, Amkor Technology, Phoenix, AZ, and Dr. Y. Kwark and Dr. M. Ritter, both with the IBM T. J. Watson Research Center, Yorktown Heights, NY, for their kind support with measurements and/or constructive discussions. REFERENCES [1] H. W. Johnson and M. Graham, High-Speed Digital Design: A Handbook of Black Magic. New York: Prentice-Hall, 1993, ch. 7. [2] S. H. Hall, G. W. Hall, and J. A. McCall, High-Speed Digital System Design—A Handbook of Interconnect Theory and Design Practices. New York: Wiley, 2000, ch. 5. [3] E. laermans, J. Geest, D. Zutter, F. Olyslager, S. Sercu, and D. Morlion, “Modeling complex via hole structure,” IEEE Trans. Adv. Packag., vol. 25, no. 2, pp. 206–214, May 2002. [4] R. Abhari, G. V. Eleftheriades, and E. V. Deventer-Perkins, “Physics-based CAD models for the anaysis of vias in parallel-plate environments,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 10, pp. 1697–1707, Oct. 2001. [5] S. Maeda, T. Kashiwa, and I. Fukai, “Full wave analysis of propagation characteristics of a through hole using the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 12, pp. 2154–2159, Dec. 1991. [6] T. Wang, R. F. harington, and J. R. Mautz, “Quasi-static analysis of a microstrip via through a hole in a ground plane,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 6, pp. 1008–1013, Jun. 1988.

[7] P. Kok and D. D. Zutter, “Capacitance of a circular symmetric model of a via hole including finite ground plane thickness,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1229–1234, Jul. 1991. [8] P. A. Kok and D. D. Zutter, “Prediction of the excess capacitance of a via-hole through a multilayered board including the effect of connecting microstrips or striplines,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2270–2276, Dec. 1994. [9] K. S. Oh, J. E. Schutt-Aine, R. Mittra, and W. Bu, “Computation of the equivalent capacitance of a via in a multilayered board using the closed-form Green’s function,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 2, pp. 347–349, Feb. 1996. [10] A. W. Mathis, A. F. Peterson, and C. M. Butler, “Rigorous and simplified models for the capacitance of a circularly symmetric via,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 10, pp. 1875–1878, Oct. 1997. [11] F. Tefiku and E. Yamashita, “Efficient method for the capacitance calculation of circularly symmetric via in multilayered media,” IEEE Microw. Guided Wave Lett., vol. 5, no. 9, pp. 305–307, Sep. 1995. [12] S.-G. Hsu and R.-B. Wu, “Full-wave characterization of a through hole via in multilayered packaging,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 5, pp. 1073–1081, May 1995. [13] Q. Gu, Y. E. Yang, and M. A. Tassoudji, “Modeling and analysis of vias in multilayered integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 2, pp. 206–214, Feb. 1993. [14] Q. Gu, A. Tassoudji, S. Y. Poh, R. T. Shin, and J. A. Kong, “Coupled noise analysis for adjacent vias in multilayered digital circuits,” IEEE Trans. Circuit Syst., vol. 41, no. 12, pp. 796–804, Dec. 1994. [15] H. Chen, Q. Lin, L. Tsang, C.-C. Huang, and V. Jandhyala, “Analysis of a large number of vias and differential signaling in multilayered structures,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 818–829, Mar. 2003. [16] L. Tsang and D. Miller, “Coupling of vias in elctronic packaging and printed circuit board structures with finite ground plane,” IEEE Trans. Adv. Packag., vol. 26, no. 4, pp. 375–384, Nov. 2003. [17] C. C. Huang, L. Tsang, C. H. Chan, and K. H. Ding, “Multiple scattering among vias in planar waveguides using preconditioned SMCG method,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 20–28, Jan. 2004. [18] C.-J. Ong, D. Miller, L. Tsang, B. Wu, and C.-C. Huang, “Application of the Foldy–Lax multiple scattering method to the analysis of vias in ball grid arrays and interior layers of printed circuit boards,” Microw. Optical Technol, Lett., vol. 49, no. 1, pp. 225–31, Jan. 2007. [19] X. Gu and M. B. Ritter, “Application of Foldy–Lax multiple scattering method to via analysis in multi-layered printed circuit board,” in DesignCon 2008, Santa Clara, CA, Feb. 4–7, 2008, pp. 1–18. [20] C. Schuster, Y. Kwark, G. Selli, and P. Muthana, “Developing a ‘Physical’ model for vias,” in Proc. IEC DesignCon Conf., Santa Clara, CA, Feb. 6–9 2006, pp. 1–24. [21] G. Selli, C. Schuster, Y. H. Kwark, M. B. Ritter, and J. L. Drewniak, “Developing a physical via model for vias—Part II: Coupled and ground return vias,” in Proc. IEC DesignCon Conf., Santa Clara, CA, Jan. 29 –Feb. 1, 2007, pp. 1–22. [22] G. Selli, C. Schuster, and Y. Kwark, “Model-to-hardware correlation of physics based via models with the parallel-plate impedance included,” in Proc. IEEE Electromagn. Compat. Symp., Portland, OR, Aug. 14–18, 2006, pp. 781–785. [23] M. Pajovic, J. Xu, and D. Milojkovic, “Analysis of via capacitance in arbitrary multilayer PCBs,” IEEE Trans. Electromagn. Compat., vol. 49, no. 3, pp. 722–726, Aug. 2007. [24] Y. Zhang, J. Fan, G. Selli, M. Cocchini, and F. D. Paulis, “Analytical evaluation of via-plate capacitance for multilayer printed circuit boards and packages,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 9, pp. 2118–2128, Sep. 2008. [25] Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and experiment on microstrip antennas,” IEEE Trans. Antennas Propag., vol. AP-27, no. 2, pp. 137–145, Mar. 1979. [26] G.-T. Lei, R. W. Techentin, P. R. Hayes, D. J. Schwab, and B. K. Gilbert, “Wave model solution to the ground/power plane noise problem,” IEEE Trans. Instrum. Meas., vol. 44, no. 2, pp. 300–303, Apr. 1995. [27] M. Xu and T. H. Hubing, “The development of a closed-form expression for the input impedance of power-return plane structures,” IEEE Trans. Electromagn. Compat., vol. 45, no. 3, pp. 478–485, Aug. 2008. [28] Z. L. Wang, O. Wada, Y. Toyota, and R. Koga, “Convergence acceleration and accuracy improvement in power bus impedance calculation with a fast algorithm using cavity modes,” IEEE Trans. Electromagn. Compat., vol. 47, no. 1, pp. 2–9, Feb. 2005.

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[29] Y. Joeong, A. C. Lu, L. L. Wai, W. Fan, B. K. Lok, H. Park, and J. Kim, “Hybrid analytical modeling method for split power bus in multilayered package,” IEEE Trans. Electromagn. Compat., vol. 48, no. 1, pp. 82–94, Feb. 2006. [30] C. Wang, J. Mao, G. Selli, S. Luan, L. Zhang, J. Fan, D. J. Pommerenke, R. E. DuBroff, and J. L. Drewniak, “An efficient approach for power delivery network design with closed-form expressions for parasitic interconnect inductances,” IEEE Trans. Adv. Packag., vol. 29, no. 2, pp. 320–334, May 2006. [31] J. Trinkle and A. Cantoni, “Impedance expressions for unloaded and loaded power ground planes,” IEEE Trans. Electromagn. Compat., vol. 50, no. 2, pp. 390–398, May 2008. [32] K.-B. Wu, G.-H. Shiue, W.-D. Guo, C.-M. Lin, and R.-B. Wu, “Delaunay–Voronoi modeling of power-ground planes with source correction,” IEEE Trans. Adv. Packag., vol. 31, no. 2, pp. 303–310, May 2008. [33] J.-X. Zheng and D. C. Chang, “End-correction network of a coaxial probe for microstrip patch antennas,” IEEE Trans. Antenna Propag., vol. 39, no. 1, pp. 115–118, Jan. 1991. [34] H. Xu, D. R. Jackson, and J. T. Williams, “Comparison of models for the probe inductance for a parallel plate waveguide and a microstrip patch,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3229–3235, Oct. 2005. [35] D. M. Pozar, Microwave Engineering. New York: Wiley, 2005. [36] A. G. Williamson, “Equivalent circuit for radial-line/coaxial-line junction,” Electron. Lett., vol. 17, no. 8, pp. 300–301, Nov. 1987. [37] A. G. Williamson, “Radial-line/coaxial-line step junction,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 1, pp. 56–59, Jan. 1985. [38] C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 2nd ed. New York: McGraw-Hill, 2002, pp. 573–577. [39] Y. Zhang, E. Li, A. R. Chada, and J. Fan, “Calculation of the via-plate capacitance of a via with pad using finite difference method for signal/ power integrity analysis,” in Int. Electromagn. Compat. Symp., Kyoto, Japan, Jul. 20–24, 2009.

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Yao-Jiang Zhang (S’97–M’01) received the B.E. and M.E. degrees in electrical engineering from the University of Science and Technology of China, Hefei, Anhui, China, in 1991 and 1994, respectively, and the Ph.D. degree in physical electronics from Peking University, Beijing, China, in 1999. From 1999 to 2001, he was with Tsinghua University, as a Post-Doctoral Research Fellow. From August 2001 to August 2006, he was a Senior Research Engineer with the Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A STAR), Singapore. From September 2006 to September 2008, he was with the Electromagnetic Compatibility (EMC) Laboratory, Missouri University of Science and Technology (M S&T) (formerly the University of Missouri–Rolla). From September 2008 to April 2010, he was a Research Scientist with the IHPC. He is currently an Associate Research Professor with the EMC Laboratory, M S&T. His research interests include computational electromagnetics, parallel computing techniques, signal integrity, and power integrity issues in high-speed electronic packages or PCBs.

Jun Fan (S’97–M’00–SM’06) received the B.S. and M.S. degrees in electrical engineering from Tsinghua University, Beijing, China, in 1994 and 1997, respectively, and the Ph.D. degree in electrical engineering from the University of Missouri–Rolla in 2000. From 2000 to 2007, he was with the NCR Corporation, San Diego, CA, as a Consultant Engineer. In July 2007, he joined the Missouri University of Science and Technology (formerly University of Missouri–Rolla), and is currently an Assistant Professor with the Missouri Science and Technology (S&T) Electromagnetic Compatibility (EMC) Laboratory. His research interests include signal integrity and electromagnetic interference (EMI) designs in high-speed digital systems, dc power-bus modeling, intra-system EMI and RF interference, PCB noise reduction, differential signaling, and cable/connector designs. Dr. Fan was the chair of the IEEE EMC Society TC-9 Computational Electromagnetics Committee (2006–2008). He was a Distinguished Lecturer with the IEEE EMC Society (2007 and 2008). He is currently the vice chair of the Technical Advisory Committee of the IEEE EMC Society. He was the recipient of the IEEE EMC Society Technical Achievement Award in 2009.

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Miniaturized Coupled-Line Couplers Using Uniplanar Synthesized Coplanar Waveguides Chen-Cheng Wang, Chi-Hui Lai, Student Member, IEEE, and Tzyh-Ghuang Ma, Member, IEEE

Abstract—A novel slow-wave synthesized coplanar waveguide (CPW), namely, the uniplanar synthesized CPW, is investigated in this paper. Quasi-lumped CPW inductors and capacitors are used in synthesizing the new slow-wave structure. The synthesis method, lumped equivalent-circuit model, and simulated and experimental results are discussed. The synthesized line, featuring excellent miniaturization capabilities, has a moderate quality factor. The slow wave factor is 7.5, while the unloaded quality factor is 30–60. By utilizing the synthesized CPW, three novel miniaturized coupled-line directional couplers are proposed and experimentally verified. With the help of even/odd-mode analysis, design charts are summarized for understanding of the developed miniaturized couplers. When compared with previous designs, the miniaturized couplers show comparable performance, but significantly reduced sizes. Additionally, they have quasi-square appearances, which are suitable for circuit integration in cascade connection. Index Terms—Backward-wave coupler, circuit miniaturization, coupled lines, forward-wave coupler, synthesized coplanar waveguide (CPW).

I. INTRODUCTION

P

ASSIVE microwave components, principally developed by transmission lines, are large in size and have become the major bottleneck in realizing highly integrated systems. To tackle this problem, slow-wave artificial transmission lines have drawn a great deal of attention in the microwave community because of their capability of miniaturization and cost reduction. In the literature, the artificial transmission line is also known as the synthesized transmission line. Investigations on the miniaturizations of transmission lines, based on slow wave structures, have been found in [1]–[8]. When compared with other size-reduction techniques [9]–[11], the slow-wave artificial transmission line provides a simple and systematic way to synthesize miniaturized components without the need of surface mounted devices, via-holes, or photonic-bandgap structures.

Manuscript received February 19, 2010; revised May 04, 2010; accepted May 31, 2010. Date of publication July 01, 2010; date of current version August 13, 2010. This work was supported by the National Science Council, R.O.C., under Grant 97-2221-E-011-019-MY2 and Grant 98-2221-E-011-029. C.-C. Wang was with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 10607, Taiwan. He is now with the Advantech Company Ltd., Taipei 11466, Taiwan. C.-H. Lai and T.-G. Ma are with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 10607, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2052667

Meanwhile, the coupled-line couplers, an important building block in microwave systems, have been investigated intensively over past decades [12]. Advanced directional couplers in coplanar waveguide (CPW) form have been reported in both edge- and broadside-coupled configurations. Edge-coupled CPW couplers with interdigital capacitor loading, floating finite-extent backed conductor, and composite right/left-handed lines have been reported in [7], [13], and [14], respectively. Broadside-coupled CPW couplers with a defected ground structure, dielectric overlay, and hexagonal slot were investigated in [15]–[17]. These advanced designs are capable of improving the coupling strength, reducing the occupied size, or widening the operating bandwidth. Nevertheless, except for [7], the physical lengths of the previously reported couplers are all one-quarter-wavelength long. This is unacceptable in modern applications and further miniaturization is required. We proposed a novel slow-wave synthesized CPW, namely, the uniplanar synthesized CPW, in [8]. The new design is composed of quasi-lumped CPW meander line inductors and interdigital capacitors. The synthesized line is only one-sixth the length of a conventional CPW with identical characteristic impedance and electrical length. The preliminary results, including the circuit layout and equivalent lumped circuit model, have been introduced in [8]. A miniaturized rat-race coupler, which is merely 7.2% the size of a conventional one, was developed as a demonstration of the miniaturization capability of the new synthesized line. In the first part of this paper, i.e., Section II, the electrical characteristics of the new synthesized line are investigated in greater detail. The explicit synthesis procedure, which was not discussed in [8], is given along with the extraction method for each individual component in the equivalent-circuit model. The attenuation constants, slow wave factors, and quality factors of the synthesized CPWs are compared to those of a conventional CPW. Synthesized CPWs, with alternative layouts, but identical in-band responses, are discussed in terms of the transmission coefficients as well. By utilizing the uniplanar synthesized CPW, in the second half of this paper, i.e., Section III, three novel miniaturized coupled-line couplers, including a 10-dB backward-wave edge-coupled CPW coupler, a 6-dB backward-wave broadside-coupled CPW coupler, and a 3-dB forward-wave CPW coupler, are proposed and experimentally verified. The even/odd-mode analysis is applied to achieve the design charts with respect to the geometric parameters of the synthesized coupled lines. For a specific coupling coefficient, the dimensions of the miniaturized couplers can be determined in accordance with the design charts. The miniaturized backward-wave couplers are approx-

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connection lines of the interdigital capacitors. The short conventional 50- CPWs on each side of the unit cell are used for phase adjustment between the actual phase of the unit cell and the targeted value. The lumped circuit model in Fig. 2 is too difficult to analyze directly due to the complicated interconnections. To simplify the analysis, the circuit model can be approximately rearranged as a simple -section with an equivalent per-unit-cell series inand per-unit-cell shunt-to-ground capacitance , ductance where (1) and (2) With (1) and (2), the characteristic impedance and guided of the unit cell can be estimated by [4] wavenumber (3) and Fig. 1. Circuit layout of the uniplanar synthesized CPW. (a) 3-D view. (b) Top view.

Fig. 2. Lumped equivalent-circuit model.

imately one-third the size of their corresponding conventional designs. The responses, on the other hand, remain almost unaltered. The design methodology, even/odd-mode analysis, design charts, and experimental results will be investigated thoroughly. II. UNIPLANAR SYNTHESIZED CPW A. Circuit Configuration and Synthesis Procedure The circuit layout and lumped equivalent-circuit model of the uniplanar synthesized CPW are shown in Figs. 1 and 2, respectively. Here, a 50- 90 synthesized line, which will be referred to as a unit cell in the following discussion, is demonstrated as an example. As shown in Fig. 1, the meander line inductors for replacing the series inductance in a conventional CPW are arranged symmetrically with respect to the center of the unit cell. In the firstorder approximation, each of the meander line inductors can be represented by a network consisting of a series lumped inand four parasitic shunt capacitors , as indicated ductor in Fig. 2. The interdigital capacitors on each side of the me. ander line inductors can be summed up as a shunt capacitor Two series lumped inductors are then added to the model to account for the additional current paths flowing through the

(4) where is the physical length of the unit cell. As and in (1) and (2) rise proportionally, the guided wavenumber in (4) can be dramatically increased, whereas the characteristic impedance in (3) remains invariant. The physical length of the unit cell, with a specific electrical length, is hence significantly reduced. The approximate expressions (1)–(4) provide very accurate results as long as the unit cell is electrically small when compared with the guided wavelength. The synthesis procedure for a uniplanar synthesized CPW with an arbitrary electrical length and characteristic impedance is straightforward. First, the lumped equivalents of each individual quasi-lumped component should be extracted. As indicated in Fig. 2, each quasi-lumped component can be treated as a two-port network; the associate port definitions ( and ) are shown in the figure. The -parameters of each two-port network can be simulated by a full-wave electromagnetic (EM) simulator, e.g., Ansoft High Frequency Structure Simulator (HFSS). For the interdigital capacitors, the simulated -parameters are de-embedded to the reference planes and , as indicated in Fig. 2. Using the simulated -parameters, the corresponding or matrix can be retrieved by standard matrix operations; its associated - or T-equivalent circuit network can be obtained at the same time. The lumped equivalents of each quasi-lumped component can be determined by the or T-equivalent circuit parameters. After determining all element values of the equivalent-circuit model, the characteristic impedance and propagation constant of the unit cell can be calculated using (1)–(4). The phase delay of the synthesized line is (5) By repeating the extraction procedure, a set of design curves versus the dimensions of the meander line inductors and interdigital capacitors could be retrieved. This, in turn, helps determine the dimensions of a unit cell with specific electrical properties. A post-integration tuning process might be required to ac-

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TABLE I GEOMETRIES OF THE 35-, 50-, AND 70- 90 UNIT CELLS

count for the parasitic coupling effects of the closely arranged quasi-lumped elements. These design curves are easy to build and similar to Fig. 2 in [4]. They will not be repeated for brevity. B. Experimental Results and Discussion The 50- 90 unit cell was developed on a 0.508-mm RO4003C substrate. The relative dielectric constant is 3.55 and the loss tangent is 0.0027. Air bridges are used to suppress the unwanted slotline mode. For easy mounting of subminiature A (SMA) adapters, two additional 5-mm CPWs are connected to each side of the unit cell. The additional lines are de-embedded using the thru-reflect-line (TRL) calibration technique in the measurement. The dimensions of the unit cell are summarized in Table I for easy reference. The subscript “u” represents the parameters associated with the unit cells. At a center frequency of 915 MHz, the occupied size of the unit cell, including the finite-extent ground plane, is merely 10.6 mm 12.75 mm, or 0.05 . It is only one-sixth the length equivalently, 0.041 of a conventional CPW with the same electric length. Here, is the guided wavelength of a 50- CPW on the same substrate at 915 MHz. The finite-extent ground plane, whose width is set equal to that of the signal trace of a 50- CPW, accounts for the return current path on the ground plane. The simulated, measured, and calculated -parameters of the unit cell are shown in Fig. 3(a). The full-wave simulation was carried out by Ansoft HFSS, while the measurement was taken by an Agilent E8363B network analyzer. The calculation, on the other side, was completed by the Agilent ADS, using the equivalent-circuit model in Fig. 2. The extracted element values nH, pF, pF, and are nH. By utilizing the -parameters, Fig. 3(b) shows the characteristic impedance of the 50- unit cell from 0.5 to 1.3 GHz using [18] (6) In (6), the -parameters can be derived from full-wave simulation, measurement, or model calculation. Fig. 3(b) also shows the phase delay of the synthesized line. Excellent agreement between the results is observed. The measured characteristic impedance and phase delay are 51.8 and 90.5 , respectively, at the center frequency. Within a bandwidth of 110 MHz, the variation of the characteristic impedance is less than 5%.

Fig. 3. Simulated, measured, and calculated: (a) S -parameters and (b) characteristic impedances and phase delays of the 50- uniplanar synthesized CPW.

To synthesize a 50- 90 CPW, the layout in Fig. 1 is not the only approach. As illustrated in Fig. 4(a), by cascaded connection of two 45 unit cells, three 30 unit cells, or four 22.5 unit cells, a 50- 90 synthesized line can be realized, as well. At the center frequency, all synthesized lines have the same electrical properties. However, their out-of-band transmission responses are dramatically different, as shown in Fig. 4(b). The unit cell is naturally a low-pass filter whose cutoff frequency is inversely proportional to the square root of the product and per-unit-cell capacitance of the per-unit-cell inductance . As the per-unit-cell inductance and capacitance become larger, the cutoff frequency moves toward a lower frequency range. Accordingly, the 90 line formed by a single unit cell has the lowest cutoff frequency among the designs. The synthesized line, using four cascaded unit cells, on the other hand, features the steepest fall-off rate in addition to the highest cutoff frequency; this is because the four cascaded 22.5 unit cells can be approximated by an eighth-order low-pass filter, inherently having good frequency selectivity in the transition region. Two additional 90 unit cells were obtained using the same synthesis procedure. At 915 MHz, the characteristic impedances of the additional cells are 35.4 and 70.7 . The geometric parameters are also summarized in Table I. The sizes of the 35- and 0.06 and 0.043 0.043 , 70- unit cells are 0.042

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Fig. 5. Comparisons of the normalized attenuation constants and slow wave factors of the 35-, 50- and 70- unit cells and a conventional CPW.

its promising miniaturization capability is simultaneously taken show into consideration. The unloaded quality factors similar results. At the center frequency, the unloaded quality factors of the 35-, 50-, and 70- unit cells are 66, 31, and 27, respectively; this is good enough for components developed for circuit miniaturization. III. MINIATURIZED COUPLED-LINE COUPLERS

Fig. 4. (a) 50- 90 uniplanar synthesized CPWs with two 45, three 30, and four 22.5 unit cells. (b) Comparisons of the transmission coefficients (jS j) of the 90 synthesized lines in (a).

respectively. For the 35- synthesized line, the meander line inductors are replaced by straight line inductors, as the required series inductance is relatively low. and guided wavenumbers of The attenuation constants the 35-, 50-, and 70- unit cells can be determined by [18]

(7) where (8) In Fig. 5, the normalized attenuation constants (dB/ ) and norof the unit cells are commalized guided wavenumbers pared to those of a conventional 50- CPW. The results are simulated by Ansoft HFSS. The normalized guided wavenumber is known as the slow wave factor of a transmission line. At the center frequency, the slow wave factors of the synthesized lines are 7.5, five times higher than that of a conventional CPW. The attenuation constant in Fig. 5, on the other side, is normalized by the guided wavelength. This represents the power dissipated by a transmission line in one guided wavelength. The power dissipation of the synthesized line, though higher than that of a conventional CPW, is still in an acceptable range if

In this section, the uniplanar synthesized CPWs are used to develop novel miniaturized CPW couplers with a high degree of miniaturization and well-behaved circuit responses. With the help of even/odd-mode analysis, a 10-dB backward-wave edgecoupled directional coupler, a 6-dB backward-wave broadsidecoupled directional coupler, and a 3-dB forward-wave edgecoupled coupler are designed and investigated consecutively in the following Sections III-A–C. All designs were developed on a 0.508-mm RO4003C substrate with a center frequency of 915 MHz. A. Backward-Wave Edge-Coupled Coupler The circuit layout of the proposed miniaturized edge-coupled directional coupler is shown in Fig. 6(a). The coupler consists of two mutually coupled 90 uniplanar synthesized CPWs. To widen the operating frequency range, each of the coupled lines is formed by a cascaded connection of two 45 unit cells. Two interdigital capacitors, each with a finger length , are inserted in between the synthesized lines to activate the mutual coupling. The shunt-to-ground interdigital capacitors of each individual line, on the other side, are arranged only on one side of the synthesized line with a finger length . Rigorously speaking, the analysis of the synthesized edge coupler is involved since this coupler lacks a plane of symmetry due to the interdigitated fingers. It makes the even/odd-mode analysis, the most straightforward way to extract the electrical parameters, failed to be applied directly. However, as indicated in [19], the mutual capacitors can be assumed to be approximately symmetrical with respect to the center line T–T’ without loss of generality. With this assumption, the proposed miniaturized coupler can be approximately analyzed using the even/odd-mode analysis, thus significantly reducing the design complexity.

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Fig. 7. Excitation arrangements for even/odd-mode analysis.

In (13), the subscript or represents the even- or odd-mode , the even/oddtwo-port -parameters. With the impedance mode characteristic impedances can be evaluated by Fig. 6. Miniaturized edge-coupled directional coupler using uniplanar synthesized CPWs. (a) Circuit layout. (b) Lumped equivalent-circuit model.

(14) and

The equivalent-circuit model for the synthesized edge coupler is depicted in Fig. 6(b). The per-unit-cell even- and odd-mode inductances and capacitances are given by the following. Even mode (9) (10) Odd mode (11) (12) To determine the even/odd-mode electrical parameters, including the characteristic impedances, effective dielectric constants, coupling coefficient, and lumped equivalents, the synthesized edge coupler is first treated as a full four-port network in a full-wave simulator. The simulated four-port -parameters are then imported into the Agilent Advanced Design System (ADS). With the excitation arrangements shown in Fig. 7, the even/odd-mode analysis is completed, and the four-port -parameters are converted into a pair of two-port -parameters representing the even- and odd-mode half circuits. By substituting can be the two-port -parameters into (6), the impedances derived by (13)

(15) Using (14) and (15), the coupling coefficient sized edge coupler is

of the synthe-

(16) Similarly, the complex propagation constants associated with the even- and odd-mode half circuits can be extracted from the two-port -parameters, as given by (17), shown at the bottom of this page. The even/odd-mode effective dielectric constants are shown in the following: (18)

Design charts for the synthesized edge coupler are derived and summarized in Fig. 8 using the even/odd-mode analysis and and , are studied to (13)–(18). Two geometric parameters, unveil their influence on the even/odd-mode electrical paramdetermines the shunt-to-ground caeters. The finger length pacitor . It affects both even- and odd-mode characteristic impedances and effective dielectric constants. In contrast, the only affects the mutual capacitor . This, in finger length turn, controls the odd-mode characteristic impedance and effective dielectric constant. The effective dielectric constants shown in Fig. 8(b) are relatively high, which indicates the outstanding miniaturization capability of the proposed design. In addition, for the case of

(17)

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Additionally, the lumped values in Fig. 6(b) can be determined using the even/odd-mode characteristic impedances and effective dielectric constants with the following equations: (19) (20) (21) (22)

Fig. 8. (a) Even/odd-mode characteristic impedances. (b) Even/odd-mode effective dielectric constants. (c) Coupling coefficient of the miniaturized edge coupler versus l .

mm and mm, the even- and odd-mode effective dielectric constants are nearly the same. This suggests that the phase velocities of both modes are equal, thus resulting in a directional coupler with high directivity. With the knowledge of even/odd-mode characteristic impedances, the coupling coefficient of the miniaturized coupler can be evaluated using (16). The results are shown in Fig. 8(c). The maximum attainable coupling coefficient is around 8.5 dB. For tighter coupling, one may use broadside-coupled configuration or some advanced schemes, such as the tandem coupler and Lange coupler [20].

A miniaturized 10-dB edge coupler was developed as a demonstration. The design begins with two uncoupled 50- synthesized lines. From the design charts, the initial lengths of the interdigital fingers are mm and mm, which corresponds to a coupling coefficient of 9.75 dB. A post-integration iterative process is required to minimize the parasitic coupling between the connecting lines, to optimize the impedance matching, as well as to adjust the overall electrical length. The final dimensions are mm and mm. The dimensions of the unit cells are mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, mm, and mm. The geometric parameters associated with the interconnections mm, mm, and . The are subscript “ ” represents the parameters associated with the edge coupler. The calculated even- and odd-mode characteristic impedances are 73 and 37.1 , while the even- and odd-mode guided wavenumbers are 144.6 and 158 rad/m, respectively, at the center frequency. The extracted lumped equivalents nH, pF, using (19)–(22) are nH, and pF. The occupied size of the synthesized edge coupler, including the finite-extent ground plane, is 11 mm 19.2 mm, or equiva0.075 . Instead of a long and narrow aplently, 0.043 pearance, like the design in [7], the proposed coupler has a quasi-square appearance, which makes the cascade-connected circuit components be able to be integrated in a more compact way. When compared with a conventional design, the miniaturized coupler shows a size reduction of 68%; moreover, it has an 83% decrease in the coupling length. Additionally, the proposed design is 30% the length and 90% the size of the coupler in [7]. The simulated and measured -parameters are shown in Fig. 9(a) and (b) from 0.5 to 1.3 GHz. The simulated and measured phase differences between the output ports are depicted in Fig. 9(c). The agreement between the simulated and measured results is good. The discrepancy can be attributed to the fabrication tolerance, the parasitic effects of the bond wires, and the ignored finite conductor thickness in the simulation. At are 915 MHz, the measured return loss and isolation and are 0.7 and 22.1 and 23.0 dB. The measured 10.3 dB, respectively. At this frequency, the measured phase and is 90.8 . The phase imbalance differences between is less than 2 over the entire band. From 720 to 1140 MHz, the miniaturized edge coupler simultaneously satisfies the following specifications: the variation of coupling coefficient

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TABLE II COMPARISONS OF THE MINIATURIZED EDGE COUPLER AND THE COUPLER IN [7]

Fig. 10. Circuit layout of the miniaturized broadside-coupled directional coupler using uniplanar synthesized CPWs. (a) Top view. (b) Bottom view.

Fig. 9. Simulated and measured: (a) S , S , (b) differences of the miniaturized 10-dB edge coupler.

S

,

S

, and (c) phase

dB, the phase imbalance 2 , and the return loss and isolation 20 dB. This corresponds to a factional bandwidth of 46 %. Table II compares the electrical performances of the proposed miniaturized edge coupler with those of the design in [7]. The data are estimated from the figures of [7]. The proposed design features a wider fractional bandwidth, lower power dissipation, and reduced size. B. Backward-Wave Broadside-Coupled Coupler The second design for demonstrating the minimization capability of the uniplanar synthesized CPWs is a backward-wave broadside-coupled directional coupler. The circuit layout is

shown in Fig. 10. The coupled synthesized lines are arranged on the top and bottom layers of the substrate, which facilitates tight coupling between the lines. Four conductor-backed CPWs are used to connect the coupled synthesized lines to SMA adapters. Additional bond wires can suppress the higher order modes. The vias around the coupler keep the ground planes in equal potential. With the same extraction procedure, the design charts for the synthesized broadside coupler are shown in Fig. 11. The couand pled lines are investigated in terms of two variables . The length determines the mutual capacitances between the synthesized lines on the top and bottom layers. This, in turn, controls the odd-mode characteristic impedance and effective dielectric constant. On the other hand, the even-mode parameters are adjusted by the length , which determines the shunt-to-ground capacitances of the lines. Differing from the

WANG et al.: MINIATURIZED COUPLED-LINE COUPLERS USING UNIPLANAR SYNTHESIZED CPWs

Fig. 11. (a) Even/odd-mode characteristic impedances. (b) Even/odd-mode effective dielectric constants. (c) Coupling coefficient of the miniaturized broadside coupler versus l .

edge-coupled coupler, here also affects the even-mode effective dielectric constant, whereas has a negligible effect on the odd-mode parameters. Based on the even/odd-mode characteristic impedances, Fig. 11(c) shows the calculated coupling coefficient of the synthesized broadside coupler. The maximum attainable coupling is around 3.2 dB, significantly larger than that achieved by the edge coupler. The maximum achievable coupling coefficient is principally limited by the degraded directivity when the difference between the even- and odd-mode phase velocities

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mm and becomes large. This can be observed when mm. For tighter coupling, e.g., a coupling coefficient greater than 3 dB, the substrate could be replaced by one with thinner thickness or higher dielectric constant. Additionally, since the broadside coupler is fully symmetric with respect to the middle plane of the substrate, the design charts in Fig. 11 are expected to have better accuracy than those in Fig. 8. For demonstrative purpose, a miniaturized 6-dB broadside coupler was developed and experimentally verified. The initial dimensions of the interdigital fingers from the design charts are mm and mm. To compensate for the parasitic coupling and to optimize the impedance matching, the is fine tuned as 3.8 mm. The dimensions of the unit length mm, mm, mm, cells are mm, mm, mm, mm, mm, mm, mm, mm, mm, and mm. The geometric parammm, eters associated with the interconnections are mm, mm, mm, and . The subscript “ ” represents the parameters associated with the broadside coupler. At the center frequency, the calculated evenand odd-mode characteristic impedances are 85.2 and 25.8 , while the even- and odd-mode guided wavenumbers are 134.2 and 167.8 rad/m, respectively. The extracted lumped equivanH, pF, lents using (19)–(22) are nH, and pF. The size of the miniaturized broadside coupler, including the finite-extent ground plane, is 11 mm 13 mm, or equivalently, 0.051 . When compared with a conventional 0.043 broadside-coupled CPW coupler, it features a size reduction of 65%; in addition, it also has an 83% decrease in the coupling length. The miniaturized broadside coupler is merely 6% the size of the coupler in [15]. The simulated and measured -parameters are shown in Fig. 12(a) and (b), while the simulated and measured phase differences between the output ports are depicted in Fig. 12(c). The agreement between the results is good. The measured center frequency shifts slightly to the lower frequency side. At are 900 MHz, the measured return loss and isolation 21.8 and 16.9 dB, respectively. The measured and are 1.6 and 6.1 dB at the center frequency. The measured and at 900 MHz is 90.7 . phase differences between According to Fig. 12(b), the phase imbalance is less than 2 up to 1.1 GHz. From 580 to 1065 MHz, the broadside coupler meets the following specifications: the variation of coupling dB, the phase imbalance 2 , and the coefficient return loss and isolation 15 dB. This is equivalent to a factional bandwidth of 53%, comparable to the bandwidth of a conventional coupled-line coupler [21, Ch. 6]. The comparison of the performances of the miniaturized broadside coupler and the design in [15] is shown in Table III. Although the coupler in [15] shows a very wide operation bandwidth and negligible in-band loss, it is 5.8 times the size of a conventional CPW design. C. 3-dB Forward-Wave Edge-Coupled Coupler The last example is a 3-dB forward-wave edge-coupled directional coupler. A conventional forward-wave coupler generally

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TABLE III COMPARISONS OF THE MINIATURIZED BROADSIDE COUPLER AND THE COUPLER IN [15]

Fig. 13. Circuit layout of the miniaturized 3-dB forward-wave directional coupler using uniplanar synthesized CPWs.

at ports 1 and 3 at the center frequency. Meanwhile, at ports 2 and 4, the input power is equally split and the phases are in quadrature (25) (26) This alternate forward-wave coupler is inherently a narrowband design. Fig. 13 shows the circuit layout of the miniaturized 3-dB forward-wave directional coupler using uniplanar synthesized CPWs. The proposed design adopts an edge-coupled configuration, as well. To satisfy the criterion (23), the even-mode characteristic impedance Fig. 12. Simulated and measured: (a) S , S , (b) S , and differences of the miniaturized 6-dB broadside coupler.

S

. (c) Phase

requires a coupling length of several wavelengths. An alternative approach, resulting in compact size, was presented in [19] with microstrip technology. In this approach, the requirement of for a 3-dB forward-wave directional coupler is satisfied with (23) and (24) where is the coupling length of the coupler. With (23) and (24), the coupler is perfectly matched and isolated, respectively,

(27) should be lowered down by replacing the meander line inductors by straight line ones to decrease the series inductance . must be lengthened to In the meantime, the finger length of the unit cells. Theoretincrease the shunt capacitance should be chosen as close to 50 ically speaking, as possible. However, to avoid parasitic resonances, the length should not violate the quasi-lumped assumption, which requires the finger length to be shorter than one-tenth the guided wavelength. This sets a lower bound of the realizable even-mode characteristic impedance. In order to meet the condition of (24), the odd-mode guided wavenumber (28)

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TABLE IV COMPARISONS OF THE MINIATURIZED FORWARD-WAVE COUPLER AND THE COUPLER IN [19]

mm, mm, and mm. The subscript “ ” represents the parameters associated with the forward-wave coupler. The size of the miniaturized forwardwave directional coupler is 11 mm 31.85 mm, or equivalently, 0.125 . This is only 60% the size and 32% the cou0.043 pling length of the design in [19]. The simulated and measured -parameters are shown in Fig. 14(a) and (b), while the phase differences between the output ports are depicted in Fig. 14(c). The agreement between the results is reasonably good. The ignored finite conductor thickness in the simulation is the most significant factor contributing to the discrepancy since the interdigital fingers are extraordinarily long. At the center frequency, i.e., 915 MHz, are 15.9 and the measured return loss and isolation 15.6 dB, respectively. The measured and are both 3.6 dB. The phase difference between the output ports is 90.1 . The additional loss at the output ports can also be attributed to the extra long fingers. The forward-wave coupler is narrowband, as expected. From 909 to 942 MHz, i.e., a fractional bandwidth of 3.6%, the miniaturized coupler meets the following specifications: the variation of coupling coefficient dB, the phase imbalance 2 , and the return loss and isolation 15 dB. Table IV compares the performances of the miniaturized forward-wave coupler with those of the design in [19]. The proposed coupler features a more compact size at the expanse of a narrower bandwidth and slightly higher in-band power dissipation. Fig. 14. Simulated and measured: (a) S , S , (b) S , and S . (c) Phase differences of the miniaturized 3-dB forward-wave directional coupler.

should be raised by lengthening the length to increase the . An iterative process is required, as mutual capacitance simultaneously affects the odd-mode guided wavenumber. The extraction procedure in Section III-A should be included to determine the even/odd-mode electrical parameters during the iteration. In the current design, the optimized finger lengths are mm and mm, which correspond to an even-mode characteristic impedance of 60.8 , and a ratio of 1.95 between the odd- and even-mode guided wavenum. The geometric parameters are mm, bers mm, mm, mm,

IV. CONCLUSION A novel slow-wave synthesized CPW, namely, the uniplanar synthesized CPW, has been proposed and comprehensively discussed in this paper. The unit cell is realized by direct integration of quasi-lumped CPW components. It features outstanding size reduction capability with a moderate quality factor. The physical length of a 90 synthesized line is less than 1/20 of a guided wavelength. By utilizing the uniplanar synthesized CPWs, three miniaturized CPW directional couplers with coupling coefficients of 3, 6, and 10 dB have been designed and experimentally verified. Benefitting from the synthesized CPWs, the proposed directional couplers feature remarkable reductions in both sizes and coupling lengths. The proposed backward-wave couplers are only one-third the size of the conventional designs. To the

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authors’ knowledge, the miniaturized couplers are the smallest designs ever reported using printed circuit board technology. The electrical properties, on the other hand, are comparable to those in literature. The proposed miniaturized couplers are especially suitable for circuit integration, owing to their quasi-square appearances. The uniplanar synthesized CPW is believed to be suitable for monolithic microwave integrated circuit (MMIC) applications as well. With better resolution, including thinner linewidth and narrower line spacing, the proposed designs can be developed with even more compact sizes. Exploiting the harmonic suppression capability, by introducing transmission zeros to the synthesized CPW, will be an interesting topic worthy of study in the future. REFERENCES [1] K. W. Eccleston and S. H. M. Ong, “Compact planar microstrip line branch-line and rat-race couplers,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 10, pp. 2119–2125, Oct. 2003. [2] P. Kangaslahti, P. Alinikula, and V. Porra, “Miniaturized artificial-transmission-line monolithic millimeter-wave frequency doubler,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 4, pp. 510–518, Apr. 2000. [3] 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. [4] C. W. Wang, T. G. Ma, and C. F. Yang, “A new planar artificial transmission line and its applications to a miniaturized Butler matrix,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 12, pp. 2792–2801, Dec. 2007. [5] T. Fujii, I. Ohta, T. Kawai, and Y. Kokubo, “Miniature broad-band CPW 3-dB branch-line couplers in slow-wave structure,” IEICE Trans. Electron., vol. E90-C, no. 12, pp. 2245–2253, Dec. 2007. [6] K. Hettak, G. A. Morin, and M. G. Stubbs, “Compact MMIC CPW and asymmetric CPS branch-line couplers and Wilkinson dividers using shunt and series stub loading,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1624–1635, May 2005. [7] L. Li, F. Xu, K. Wu, J. Ho, and M. Chaker, “Slow-wave line coupler with interdigital capacitor loading,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 11, pp. 2427–2433, Nov. 2007. [8] C.-C. Wang, C.-H. Lai, and T.-G. Ma, “Novel uniplanar synthesized coplanar waveguide and the application to miniaturized rat-race coupler,” presented at the IEEE MTT-S Int. Microw. Symp, 2010. [9] C. T. Lin, C. L. Liao, and C. H. Chen, “Finite-ground coplanar waveguide branch-line couplers,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 3, pp. 127–129, Mar. 2001. [10] Y.-C. Chiang and C.-Y. Chen, “Design of a wide-band lumped-element 3-dB quadrature coupler,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 476–479, Mar. 2001. [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. 798–804, Mar. 2004. [12] R. K. Mongia, I. J. Bahl, P. Bhartia, and J. S. Hong, RF and Microwave Coupled-line Circuits, 2nd ed. Norwood, MA: Artech House, 2007. [13] C.-L. Liao and C. H. Chen, “A novel coplanar-waveguide directional coupler with finite-extent backed conductor,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 200–206, Jan. 2003. [14] S.-G. Mao and M.-S. Wu, “A novel 3-dB directional coupler with broad bandwidth and compact size using composite right/left-handed coplanar waveguide,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 5, pp. 331–333, May 2007.

[15] D.-J. Kim, Y. Jeong, J.-H. Kang, J.-H. Kim, C.-S. Kim, J.-S. Lim, and A. Ahn, “A novel design of high directivity CPW directional coupler design by using DGS,” in IEEE-MTT Int. Microw. Symp. Dig., Long Beach, CA, Jun. 1995, pp. 1239–1243. [16] F. Tefiku, E. Yamashita, and J. Funada, “Novel directional couplers using broadside-coupled coplanar waveguides for double-sided printed antennas,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 2, pp. 275–282, Feb. 1996. [17] M. Nedil and T. A. Denidni, “Quasi-static analysis of a new wideband directional coupler using CPW multilayer technology,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 2006, pp. 1133–1136. [18] 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. [19] Z. Liu and R. M. Weikle, “A compact quadrature coupler based on coupled artificial transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 12, pp. 889–891, Dec. 2005. [20] J. Lange, “Interdigital stripline quadrature hybrid,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 12, pp. 1150–1151, Dec. 1969. [21] R. K. Mongia, I. J. Bahl, P. Bhartia, and J. S. Hong, RF and Microwave Coupled-line Circuits, 2nd ed. Norwood, MA: Artech House, 2007.

Chen-Cheng Wang was born in Keelung, Taiwan, in 1985. He received the B.S. and M.S. degrees in electrical engineering from the National Taiwan University of Science and Technology, Taipei, Taiwan, in 2007 and 2009, respectively. In 2009, he joined the Advantech Company Ltd., Taipei, Taiwan, where he is involved in research and development of substitute military services. His research interests include microwave passive circuit designs, RF identification (RFID), and mobile antenna designs.

Chi-Hui Lai (S’10) was born in Taichung, Taiwan, in 1985. He received the B.S. degree in communication engineering from Feng Chia University, Taichung, Taiwan, in 2007, the M.S. degree in electrical engineering from the National Taiwan University of Science and Technology, Taipei, Taiwan, in 2009, and is currently working toward the Ph.D. degree at the National Taiwan University of Science and Technology. His research interests include miniaturized microwave circuit design and its system applications.

Tzyh-Ghuang Ma (S’00–M’06) was born in Taipei, Taiwan, in 1973. He received the B.S. and M.S. degrees in electrical engineering and Ph.D. degree in communication engineering from National Taiwan University, Taipei, Taiwan, in 1995, 1997, and 2005, respectively. In 2005, he joined the faculty of the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, where he is currently an Associate Professor. His research interests include miniaturized microwave circuit designs, ultra-wideband antennas, antenna arrays, and RF identification (RFID). Dr. Ma was the recipient of the Poster Presentation Award presented at the 2008 International Workshop on Antenna Technology (iWAT), Chiba, Japan.

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A Novel Miniaturized Forward-Wave Directional Coupler With Periodical Mushroom-Shaped Ground Plane Sen-Kuei Hsu, Student Member, IEEE, Chung-Hao Tsai, Student Member, IEEE, and Tzong-Lin Wu, Senior Member, IEEE

Abstract—A novel miniaturized forward-wave directional coupler with periodical mushroom-shaped ground plane is proposed. The coupler can be designed with identical characteristic impedances for even and odd mode and enhanced the difference of propagation constants between the even and odd mode. These distinct propagation characteristics can be predicted by the equivalent-circuit model of a unit-cell using the Bloch-Floquet theorem. A tested sample is designed and fabricated on the FR4 substrate. We de, and 1.0-dB signed a 0-dB coupler with the length about 1 28 coupling is measured at 2.9 GHz due to the loss. Compared with previously studies, the proposed coupler can be implemented to attain the highest coupling level with a smaller size. Index Terms—Forward-wave directional coupler.

I. INTRODUCTION

C

OUPLED-LINE directional couplers are used in microwave and millimeter-wave circuits design extensively. Commonly, there are two kinds of coupling mechanisms to design a coupled-line directional coupler. The first one is based on the backward coupling. The coupling strength is decided by the difference of the characteristic impedances of the even and odd mode under the assumption of equal phase velocities. Typically, the coupler is designed about a quarter-wave length at which the coupling level is maximum [1], [2]. The other coupling mechanism is based on forward coupling. The coupling level is mainly decided by the accumulated phase difference between the even and odd mode along the coupler. Ideally, 0-dB coupling can be achieved for the symmetrical couplers with the identical characteristic impedances of the even and odd mode. There are two main issues in implementing the forward-wave directional couplers by using the microstrip structures. The first one is small difference of even- and odd-mode propagation constants. This issue causes a very long coupling length and a small gapwidth in practical design that is not suitable for lower

Manuscript received January 12, 2010; revised May 14, 2010; accepted May 19, 2010. Date of publication July 08, 2010; date of current version August 13, 2010. This work was supported by the National Science Council, Taiwan, R.O.C., under Grant NSC 97-2221-E-002-060-MY3 and in part by National Taiwan University under Grant 98R0062-3. The authors are with the Department of Electrical of Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 10617 (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2052869

frequency applications [3]. The second one is the nonidentical characteristic impedances of the even and odd mode for the coupler. It will degrade the coupling level and directivity performance. The tapered lines are thus required to do the impedance matching and avoid the backward-wave reflection. Several different methods have been proposed to solve aforementioned problems [4]. The difference of the even- and odd-mode propagation constants could be increased by using the nonuniform transmission line such as the wiggly or serpentine configurations [5]–[7], the periodic shunt capacitive stubs along the coupler [8], the patterned ground planes for the microstrip lines [9], or the composite right/left-handed (CRLH) transmission lines [10]–[12]. They all contribute for size reduction, but most of them need the tapered lines for impedance matching and the total size of the coupler would be enlarged. In this paper, a novel miniaturized forward-wave directional coupler is proposed with the identical characteristic impedances of the even and odd mode and enhanced difference of propagation constants between the two modes. The proposed coupler consists of a pair of microstrip coupled-line and periodical mushroom-shaped ground plane. The periodically perturbed ground plane significantly increases the propagation constant of the even mode due to the slow-wave effect, but has very small effects for the odd mode. Furthermore, the characteristic impedances of the two modes can be designed to be equal and matched to the port impedance of 50 . Based on the proposed structure, a high coupling level (close to 0 dB) forward-wave coupler can be achieved with a compact size. II. STRUCTURE AND MODELING THEORY A. Forward Directional-Coupler Structure Fig. 1(a) shows the structure of the proposed forward-wave directional coupler. It is composed of a pair of a microstrip coupled line on the top layer with periodical mushroom-shaped structures on the ground plane. For each cell, a via connects a patch of mushroom on the middle layer to ground plane. The top and side views of the unit-cell and their corresponding geometrical notations are shown in Fig. 1(b) and (c), respectively. The total length of the unit-cell is and the width and the space and , respectively. The width and of the coupled-line are and , respeclength of the mushroom are designed with and radius tively. The metal via is designed with the height . The gap between the adjacent patches on the second layer is . The total thickness of the substrate is .

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Fig. 1. Proposed structure in: (a) forward-wave directional coupler with mushroom-shaped ground plane and the configurations of the unit-cell in the: (b) top view and (c) side view.

Slow-wave transmission line with mushroom-shaped ground plane has been well investigated and applied on the circuit design for several years [13], [14]. Our group also applies the mushroom-shaped ground plane to design the broadband common-mode noise filter [15]. The common-mode noise on the high-speed differential signals is rejected by using the property of the frequency stopband of the common mode. In this work, unlike the common-mode filter, the forward-wave coupler is designed by employing the special propagation characteristics of the even and odd mode within the passband. The problems of impedance mismatching and large size for the conventional forward-wave coupler will be alleviated and controlled based on the proposed idea. B. Equivalent Models and Propagation Characteristics The sectioning of the unit-cell and the corresponding equivalent-circuit model are shown in Fig. 2(a) and (b), respectively. As shown in Fig. 2(a) and (b), the equivalent model consists of the three sections, i.e., A, B, and A (from left to right). The lump , and in section A represent a pair of elements , , coupled lines with the reference plane on the bottom layer. Section B covers the coupled microstrip with the mushroom-shaped , , and deground plane. The lump elements , scribe the coupled microstrip with the reference plane on the patch (second layer). The mushroom structure is modeled by a

Fig. 2. (a) Sectioning of the unit-cell. (b) Equivalent-circuit model of proposed forward coupler. (c) Odd-mode equivalent-circuit model. (d) Even-mode equivalent-circuit model.

simple parallel resonator and , which correspond to the capacitance between the patch and ground plane and the inductance of a via, respectively. Due to the electrical length, the unit-cell is kept smaller than one-tenth of the wavelength, and the lumped model shown in Fig. 2(b) for the unit-cell can well predict the propagation characteristics of the proposed coupler. Fig. 2(c) and (d) show the equivalent circuits for the odd and even mode, respectively. They correspond to the structure with a perfect electrical wall and perfect magnetic wall at the center of the coupled line, respectively. The total inductance and total and capacitance are shown as , respectively. The total mutual inductance and capacitance and are represented as , respectively. The dispersion relations of these two models can be obtained by applying the periodic boundary conditions related with the Bloch–Floquet theorem [16]. For the odd and even mode, the dispersion relations can be expressed as (1a)

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TABLE I GEOMETRICAL PARAMETERS, CORRESPONDING INDUCTANCE AND CAPACITANCE

(1b) and are the propagation conrespectively, where stants for Bloch waves of the odd and even mode. As shown in Fig. 2(c) and (d), the series impedance and shunt admittance for the odd and even modes are defined as (2a) (2b) (2c) (2d) and ; therewhere fore the cutoff frequency and resonant frequency of the mushroom structure can be defined as (3a) and (3b) The Bloch impedances (or characteristic impedances) for these two modes are defined as (4a) and (4b)

Under the assumption of , the propagation constants of the two modes can be derived from (1a) and (1b) using Taylor’s expansion as

Fig. 3. Simulation results for: (a) odd- and even-mode characteristic impedances and (b) odd- and even-mode phase constants.

(5a)

and (5b). Full-wave simulations based on Ansoft’s High Frequency Structure Simulator (HFSS) are also demonstrated in Fig. 3 to validate the accuracy of the equivalent model. The characteristic impedances and propagation constants are extracted, respectively, from full-wave -parameters simulation as [17] (6a)

(5b) In order to see the frequency dependence of the propagation characteristics of the odd and even mode, the equivalent-circuit model of the tested sample are extracted. The geometrical divalues are mensions of the unit-cell and the corresponding listed in Table I. Fig. 3(a) and (b) shows the characteristic impedances and propagation constants (or dispersion curves) of the odd and even mode, respectively, based on (4a), (4b), (5a),

(6b) where even or odd, and is the complex propagation constant of mode . As shown in Fig. 3(a) and (b), the agreement between the equivalent model and the full-wave simulated results are reasonably good.

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Fig. 4. Dispersion curves of coupled microstrip and proposed coupled line.

Several interesting phenomena could be seen in Fig. 3(a) and (b). As shown in Fig. 3(a), odd-mode impedance is almost frequency independent and kept at 50 from 1 to 4 GHz, as predicted by (4a). However, even-mode characteristic impedance gradually decreases from 60 to 0 . The zero impedance ocGHz, as indicated curs at the cutoff frequency of in (4b). A crossing point of can GHz. At this frequency, the proposed forbe obtained at ward-wave coupler can achieve a high coupling level with perfect matching conditions. It is noted that the even-mode characteristic impedance becomes imaginary at frequency higher than cutoff frequency, and is shown as zero in Fig. 3(a). The crossing point frequency can be derived by (4a) and (4b) as

Fig. 5. S -parameter comparison of the simulation and equivalent model for: (a) S and S and (b) S and S .

(7a) The parameter in (7a) is determined by the coupled microstrip, and is shown as (7b) This formulation based on the equivalent model is helpful for designing the operation frequency. As shown in Fig. 3(b), the propagation constant of the odd mode is almost linearly dependent on the frequency below 4 GHz. It implied that the perturbed mushroom ground structure has very little effect on odd-mode propagation. However, the even-mode propagation constant is dramatically increased as the frequency is increased due to the slow-wave effect. As will approach infinity as the frequency indicated by (5b), is close to the cutoff frequency . The dispersive behavior of the even mode significantly enhances the difference of propagation constants between two modes, and thus can be used for miniaturization of the forward-wave coupler. In general, the structure will be designed at the frequency , where the coupler could have both identical characteristic impedance of those two coupled modes and reasonably large phase constant difference between two modes, as shown in Fig. 3(b). In the

previous work, the difference of propagation constants between the two modes is enhanced by the CRLH transmission line in [10]–[12] and the difference can also be enlarged by right-handed transmission line in this work. Fig. 4 shows a comparison of dispersion curves between the proposed structure and the conventional microstrip coupled line. The conventional structure is designed with odd-mode characteristic impedance of 50 on a two-layer substrate with a thickness of 0.6 mm, which is the same thickness between the top and second layer used in the proposed structure. If the coupler is deGHz, it is clear that the phase difference is only signed about 2 for the microstrip structure, but it can reach about 15 for the proposed structure. The scattering parameters is calculated from the equivalent models, and shown in Fig. 5. Fig. 5(a) shows the comparison of and between full-wave simulation and the modeling results. The agreement is reasonably good. The proposed coupler has an asymmetric frequency response around the center frequency of 2.9 GHz. This is due to the fact that the difference of the propagation constants between the even and odd mode is slowly increased below 2.9 GHz and has significant variations above 2.9 GHz. The other two-port performance ( and ) is also well predicted by the equivalent model and is shown in Fig. 5(b).

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Fig. 6. Four-port general symmetrical coupler.

Fig. 7. Photograph of 12-cell forward coupler.

C. Coupling Model of Forward-Wave Coupler Fig. 6 shows a general symmetrical coupler. Under the assumption of , the four-port -parameters of an ideal forward-wave directional coupler can be derived as [4]

(8a)

(8b) . and are two-port -paand rameters for the half circuit models with a magnetic wall (for even) and an electric wall (for odd) on the coupled line. The length of the coupler is defined as . The fractional powers coupled from port 1 to port 2 and from port 1 to port 3 are given, respectively, by

Fig. 8. S -parameter for the proposed forward coupler: (a) (b) S and S . (c) In-band response of S and S .

S

and

S

and

(9a) (9b) It is evident that the coupler can achieve 0-dB forward-wave coupling (or complete power transferring) if the length of the coupler is chosen as (10) where . For the conventional microstrip forward-wave coupler, there is still a tradeoff between the strong coupling level and coupling length. If we wish to achieve a full

power transfer, two microstrip lines would be designed in very . In this situation, weak coupling with the coupler length would be very long due to a small difference of the propagation constants between those two modes. On the other hand, if we wish to reduce the coupler length by increasing the mutual coupling, the impedance matching condition would be lost and backward coupling will take place. However, based on the proposed forward-wave coupler structure, the impedance-matching condition could be easily designed at in (7a) and (7b). More importhe crossing point frequency tantly, the propagation constant difference of the two modes

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TABLE II CHARACTERISTIC OF CONVENTIONAL FORWARD COUPLER

near this frequency is inherently large. It will be helpful to design a forward-wave coupler with both high coupling level and a compact size. III. MEASUREMENT AND DISCUSSIONS A forward-wave directional coupler with 12 cells is fabricated on an FR4 substrate based on the parameters described in Table I. The dielectric constant and loss tangent of the substrate are 4 and 0.01, respectively. The total length of the coupler is 79.2 mm and a photograph is shown in Fig. 7. Fig. 8(a) and (b) shows the four-port -parameters and , respectively. Both measurement results and full-wave simulation results (by HFSS) are demonstrated and compared. Good agreement is seen. As shown in Fig. 8(a) and (b), the performance at the directional port (port 2) is close to 0 dB at 1 GHz and decreased gradually as frequency is increased. degrade gradually and attain to the lowest The behavior of level 30 dB at 2.9 GHz. However, the behavior at coupled port (port 3) is increased gradually, and the coupling level of 1.0 dB can be obtained at 2.9 GHz. Fig. 8(c) shows the detailed and from 2.1 to 3.3 GHz. The fractional behavior of bandwidth (FBW) of the proposed structure is about 19% for amplitude variance of 1 dB. The small insertion loss at 2.9 GHz could be a result of the dielectric loss of the FR4 substrate. As and are less than 20 dB at shown in Fig. 8(b), the operation frequency. It implies that most of the power is forward coupled to port 3 within the short coupled length 79.2 . On average, and are mm, which is about less than 15 dB from 1 to 3.3 GHz, but significantly increased to 5 dB for a frequency higher than the cutoff frequency GHz. This phenomenon is caused because the even mode characteristic impedance becomes imaginary above the cutoff frequency. The phase difference of port 2 and port 3 is presented in Fig. 9. Both measurement and simulation results are shown for comparison. Good consistency is seen. It is seen that the phase difference between port 2 and port 3 is about 90 at the designed frequency 2.9 GHz, which is very consistent with (8a) and (8b). Table II shows comparisons of conventional forward-wave couplers published in related literatures. Compared to the couplers with the coupling level higher than 3 dB, it is clearly found that the proposed structure has the smallest size. The size of the forward-wave coupler using the microstrip coupled-line [3], wiggly and serpentine microstrip lines [5], [6], and the photonic-

Fig. 9. Difference of phase angle between S

and S .

bandgap structure [9] and the metamaterial transmission line , , , , and , re[10] is about , spectively. The length of the proposed structure is only which is much smaller than the others. Compared to the microstrip coupler, the size reduction is about 80%. Furthermore, the measured coupling level reaches 1.0 dB with almost total power transfer. Compared to previously reported data, the proposed structure has the highest coupling power, as shown in Table II. Two smaller couplers are shown in Table II. The first one in [8] is the forward-wave coupler designed by the shunt without constub transmission line of a length of about sidering the length of impedance transformers. The size of the transformers are much larger than the coupler itself, therefore the total dimension of the coupler will become very large. The other one in [12] is the forward-wave coupler implemented by , but the CRLH transmission lines of the length about coupling level is lower than 10 dB. The FBW of the proposed structure is 19% for 1.0-dB amplitude variations. Most of forward couplers in Table II are smaller than 19%, except two couplers in [5] and [12]. The first one is the forward coupler designed by using the wiggly coupled-line, which has broad band. The width, but it requires longer coupling length other is the forward-wave coupler based on the CRLH transmission line. Although it has 43% FBW, the coupling level is relatively low (about 10 dB). The directivity of the proposed coupler is also compared to previous works, and is shown in Table II. The directivity is higher than other works because of the perfect impedance matching for the even and odd mode.

HSU et al.: NOVEL MINIATURIZED FORWARD-WAVE DIRECTIONAL COUPLER

IV. CONCLUSION A novel miniaturized forward-wave directional coupler with a high coupling level has been proposed. The characteristic impedances and dispersion relations based on the equivalent-circuit models are derived from the Bloch–Floquet theorem. The models explain that the impedances of the even and odd mode can be designed at an equal value of 50 and the phase constant difference of the two modes is inherently large at the operation frequency due to the slow-wave effect. A forward-wave coupler is designed and fabricated on an FR4 substrate based on the proposed structure. The 1.0-dB coupling level can be measured, and the size reduction compared to the conventional microstrip forward-wave coupler is 80%. REFERENCES [1] R. Levy, “Directional couplers,” in Advances in Microwaves, L. Young, Ed. New York: Academic, 1966, vol. 1, pp. 115–209. [2] S. Uysal, Nonuniform Line Microstrip Directional Couplers and Filters. Boston, MA: Artech House, 1993. [3] P. K. Ikäläinen and G. L. Matthaei, “Wide-band forward-coupling microstrip hybrids with high directivity,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 8, pp. 719–725, Aug. 1987. [4] R. Mongia, I. Bahl, P. Bhartia, and J. Hong, RF and Microwave Coupled-Line Circuits. Boston, MA: Artech House, 2007. [5] S. Uysal, C. W. Turner, and J. Watkins, “Nonuniform transmission line codirectional couplers for hybrid MIMIC and superconductive applications,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 3, pp. 407–414, Mar. 1994. [6] S. Uysal and J. Watkins, “Novel microstrip multifunction directional couplers and filters for microwave and millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 6, pp. 977–985, Jun. 1991. [7] S. Uysal, J. Watkins, and C. W. Turner, “Sum-difference circuits using 0 dB and 3 dB co-directional couplers for hybrid microwave and mimic circuit applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1992, pp. 937–940. [8] T. Fujii and I. Ohta, “Size-reduction of coupled-microstrip 3 dB forward couplers by loading with periodic shunt capacitive stubs,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1235–1238. [9] C. C. Chang, Y. Qian, and T. Itoh, “Enhanced forward coupling phenomena between microstrip lines on periodically patterned ground plane,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, pp. 2039–2042. [10] L. Liu, C. Caloz, C.-C. Chang, and T. Itoh, “Forward coupling phenomena between artificial left-handed (LH) transmission lines,” J. Appl. Phys., vol. 92, no. 9, pp. 5560–5565, Nov. 2002. [11] A. Hirota, Y. Tahara, and N. Yoneda, “A compact coupled-line forward coupler using composite right/left-handed transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 601–604. [12] A. Hirota, Y. Tahara, and N. Yoneda, “A compact forward coupler using coupled composite right/left-handed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 12, pp. 3127–3113, Dec. 2009. [13] C. Zhou and H. Y. D. Yang, “Design considerations of miniaturized least dispersive periodic slow-wave structures,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 2, pp. 467–474, Feb. 2008. [14] F. Elek and G. V. Eleftheriades, “Dispersion analysis of the shielded sievenpiper structure using multiconductor transmission-line theory,” IEEE Microwave Wireless Compon. Lett., vol. 14, no. 5, pp. 219–221, May 2004. [15] C.-H. Tsai and T.-L. Wu, “A broadband and miniaturized commonmode filter for gigahertz differential signals based on negative permittivity metamaterials,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 1, pp. 195–202, Jan. 2010.

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[16] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [17] C.-K. Wu, H.-S. Wu, and C.-K. C. Tzuang, “Electric-magnetic-electric slow-wave microstrip line and bandpass filter of compressed size,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 8, pp. 1996–2004, Aug. 2002. Sen-Kuei Hsu (S’08) received the B.S.E.E. degree from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 2005, the M.S. degree in communication engineering from National Tsing-Hua University, Hsinchu, Taiwan, in 2007, and is currently working toward the Ph.D. degree at the Graduate Institute of Communication Engineering, National Taiwan University. His current research interests are microwave circuits and high-speed digital circuits.

Chung-Hao Tsai (S’08) was born in Changhua, Taiwan, in 1984. He received the B.S. degree in electrical engineering from National Sun Yat-sen University, Kaohsiung, Taiwan, in 2006, and is currently working toward the Ph.D. degree in communication engineering at National Taiwan University, Taipei, Taiwan. His current research interests include metamaterials in electromagnetic compatibility (EMC) applications and signal integrity for high-speed digital circuits.

Tzong-Lin Wu (S’93–M’98–SM’04) received the B.S.E.E. and Ph.D. degrees from National Taiwan University (NTU), Taipei, Taiwan, in 1991 and 1995, respectively. From 1995 to 1996, he was a Senior Engineer with Microelectronics Technology Inc., Hsinchu, Taiwan. From 1996 to 1998, he was with the Central Research Institute, Tatung Company, Taipei, Taiwan, where he was involved with the analysis and measurement of EMC/electromagnetic interference (EMI) problems of high-speed digital systems. From 1998 to 2005, he was with the Electrical Engineering Department, National Sun Yat-Sen University. He is currently a Professor with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, NTU, Taiwan. In Summer 2008, he was a Visiting Professor with the Electrical Engineering Department, University of California at Los Angeles (UCLA). He has been an Associate Editor of the International Journal of Electrical Engineering (IJEE) since 2006. His research interests include EMC/EMI and signal/power integrity design for high-speed digital/optical systems. Dr. Wu is a member of Institute of Electronics, Information and Communication Engineers (IEICE), Japan, and the Chinese Institute of Electrical Engineers (CIEE). He is the chair of the Taipei Section, IEICE (2007–2011). He was the treasurer of the IEEE Taipei Section (2007–2008). He serves on the Board of Directors (BoD) of the IEEE Taipei Section (2009 to 2010). He was a Distinguished Lecturer of IEEE EMC Society (2008–2009). He actively participates in IEEE activity. He was the co-chair of the 2007 IEEE EDAPS Workshop and the chair of the 2008 International Workshop on EMC. He was the recipient of the 2000 Excellent Research Award and 2003 Excellent Advisor Award presented by NSYSU, the 2002 Outstanding Young Engineers Award presented by the CIEE, the 2005 Wu Ta-You Memorial Award presented by the National Science Council (NSC), and the 2009 Technical Achievement Award presented by the IEEE EMC Society.

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Extraction of Intrinsic and Extrinsic Parameters in Electroabsorption Modulators Mauricio Yañez, Student Member, IEEE, and John C. Cartledge, Fellow, IEEE

Abstract—A new method is presented for extracting the intrinsic and extrinsic parameters of electroabsorption modulators (EAMs) from measurements of their electrical to optical and optical to electrical responses over frequency. The method is analytically described and experimentally verified with the use of an electrical vector network analyzer. It is shown that the use of EAMs as both modulators and receivers allows the extraction of the parameters, which govern their intrinsic frequency response while removing the effect of the interconnect/package used and the need for an accurately calibrated optical modulator and receiver pair used to do the measurements. A set of -parameters accounting for the effect of the interconnect/package is also calculated and used along with the intrinsic parameters to reproduce the measured frequency response of the EAM. Good agreement is obtained between calculated and measured results, thus confirming the validity of the proposed technique. Index Terms—Electroabsorption modulator (EAM), frequency response, parameter extraction.

I. INTRODUCTION

D

URING THE past two decades, electroabsoption modulators (EAMs) have become fundamental components in optical communication systems. Their compactness, high speed of operation, and small driving voltage have contributed to their wide use as modulators in digital fiber-optic systems and have ignited interest on their use for radio-over-fiber systems [1], [2]. While mostly used for the generation of 10-Gb/s signals in on–off keying digital syshigh-speed tems, their use has also been investigated for the generation of phase-modulated optical signals [3]. Monolithic integration with other opto-electronic components ranges from the single electroabsorption modulated laser [4] to more recent photonic integrated circuits containing several EAMs and capable of providing aggregate capacities of 100 Gb/s on a single chip [5]. The performance of EAMs is determined by both their optical (absorption transfer function, coupling losses, etc.) and electronic (intrinsic capacitance, escape time of the photogenerated carriers, etc.) properties. Moreover, these two sets of properties mutually affect each other and influence the frequency response of the EAM [6]–[8]. The optical and electronic properties of EAMs depend, in turn, on the applied bias voltage and input Manuscript received April 08, 2010; revised May 06, 2010; accepted May 06, 2010. Date of publication July 08, 2010; date of current version August 13, 2010. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON, Canada K7L 3N6 (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2052868

optical power [11]. Finally, the frequency response of the interconnect used to drive the EAM and that of the package used to enclose it also contribute to the overall performance [12]. The combined effect of all these factors has been used to obtain dynamic absorption curves, which provide useful qualitative insight into the expected performance of the EAM [13]. In general, the linear effects introduced by the package and interconnect need to be decoupled from the nonlinear response of the active region in the EAM. For an arbitrary voltage driving signal, the carrier rate equation must be solved in the time domain and the dependence of the output optical power on the photo-generated carriers and intrinsic applied voltage needs to be carefully considered [7], [8]. Several techniques have been proposed for the purpose of extracting the intrinsic and extrinsic parameters of opto-electronic components such as p-i-n photodiodes and laser diodes from their measured -parameters [14]–[17]. These techniques, however, are not directly applicable to EAMs since they rely only on either optical to electrical (OE) or electrical to optical (EO) measurements, respectively. Techniques aimed at directly extracting the intrinsic parameters of EAMs either ignore the effect of the interconnect/package or require the use of sub-picosecond optical pulse sources [18], [19]. In general, extrinsic effects like those introduced by the frequency response of the interconnect/package and intrinsic ones like the bias voltage and input optical power dependence of the electronic properties of the active section within the EAM must be taken into account during the parameter-extraction process. In this paper, we present a method for extracting the intrinsic and extrinsic parameters of EAMs. The technique makes use of both EO and OE measurements over frequency for several bias voltages at a fixed input optical power. The formalism of optoelectronic scattering matrices described by Hale and Williams [20] is used along with the carrier rate equation to derive an expression that represents the response of the EAM as a modulator and receiver at different bias voltages. This expression is only a function of the intrinsic parameters of the EAM. Numerical fitting between measured EO and OE data and the derived expression is used to obtain a set of physically reasonable intrinsic parameters for the EAM. Those parameters are then used to extract -parameters representing the combined effect of the interconnect/package. It is shown that, in the absence of explicit information about the physical dimensions and composition of the active section of the EAM, different sets of numerically calculated intrinsic and extrinsic parameters can be used to reproduce with good accuracy the measured EO and OE frequency responses of the EAM. A notable feature of the proposed technique is that it relies on multiple ratios of the EO and OE fre-

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quency response of the EAM. As a result of this, the photocurrent conversion efficiency, optical coupling loss, and intrinsic optical loss coefficient need not be known in order to calculate the parameters that determine the intrinsic frequency response of the EAM. II. PARAMETER EXTRACTION AT DC In order to determine the extrinsic parameters, which govern the frequency response of the EAM, it is first necessary to determine the optical coupling loss between a light source and the EAM, the photocurrent conversion efficiency, and the intrinsic optical loss coefficient. Techniques for obtaining these parameters were proposed in [9]–[11]. In this section, we calculate the parameters in the unsaturated optical absorption regime of the EAM. Such a regime neglects the longitudinal dependence of the optical absorption within the EAM and is applicable for medium and low input optical powers. We begin by representing the optical power transmission and dc photocurrent as functions given of the applied bias voltage and input optical power by

(1)

(2) accounts for the optical coupling loss and is assumed to be the represent the same for both facets of the EAM. , , , and optical confinement factor inside the EAM, the electron charge, the length of the EAM, and the incident photon energy, respecconsists of the nontively. The total intrinsic optical loss photocurrent generating optical losses, , and those that lead to photocurrent generation through inter-band absorption between . The total intrinsic opthe conduction and valence bands, tical loss is given by [11] (3) represents the photocurrent conversion effiFinally, ciency of the EAM defined as the number of electron–hole pairs contributing to the photocurrent per the number of electron–hole pairs generated by photons [9]. Equations (1)–(3) can be combined to express the dc photocurrent as

(4) Equation (4) contains a number of unknown parameters, which need to be determined by numerical fitting with respect to measured data. This will be illustrated with measurements obtained from a packaged commercially available EAM from the Centre for Integrated Photonics, Ipswich, U.K. The EAM (40G-PSEAM-1550) is rated for operation up to 40 GHz. We use meaand numerically fit , , and to surements of

Fig. 1. Photocurrent conversion efficiency.

the measured photocurrent over a small to medium range of input optical powers to prevent optical power saturation thus obtained are then used to effects. The values of and find from measurements of the optical power transmission and photocurrent over a range of bias voltages and at a fixed input optical power of 4.6 dBm. The fitting procedure yielded dB with an optical coupling loss per facet and a photocurrent conversion efficiency shown in Fig. 1 along contributes to with a polynomial fit. The fact that creating residual optical absorption. On the other hand, is desirable if the current resulting from optical absorption is to be used. For the 40G-PS-EAM-1550, Fig. 1 shows that has reached 90% for less than half of the maximum rated bias voltage of 4 V. The resulting intrinsic optical loss coefficient defined as (5) is shown in Fig. 2, which shows that for the 40G-PS-EAM-1550, most of the transmission occurs for values of the bias voltage V V range. These calculated pawithin the rameters will be used in Section V during the extraction of the package/interconnect -parameters. III. INTRINSIC PARAMETER CALCULATION Electroabsorption modulators were primarily conceived for the purpose of externally modulating a continuous-wave (CW) optical signal. The photocurrent generated as a result of the electroabsorption process has, however, allowed them to be used as modulators/photodetectors in transceiver and opto-electronic clock recovery applications [2], [21]. Here we exploit the modulator/photodetector properties of an EAM to separate and estimate the intrinsic and extrinsic parameters that govern the frequency response. The multiple-quantum well nature of the intrinsic region in EAMs causes their frequency responses as photodiodes to depend strongly on the applied reverse-bias voltage. This effect is largely due to the junction voltage dependence of the carrier escape times for the electrons and holes generated through

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in Fig. 3 as a it is necessary to represent the photocurrent small-signal impedance around a bias voltage . Substituting and to first order (8) and (9) into (6) and (7), expanding around a bias voltage and solving the resulting equations with , gives [after using (10)] the as a function of the applied total small-signal photocurrent . From this, we have that the small-signal junction voltage small-signal impedance resulting from photoabsorption is given by (11) where (12) (13)

Fig. 2. Intrinsic optical loss coefficient.

(14)

Fig. 3. Electroabsorption modulator RF model.

photo-absorption. Along with the escape times of the photogenerated carriers, the intrinsic capacitance of the EAM contributes to the frequency response. These two elements are commonly used in circuit model representations of the intrinsic region of EAMs [6], [7], [22]. Fig. 3 shows such a representation where the total photogenerated current as a function of the junction is obtained from the solution of the carrier-rate voltage equations for electrons and holes and from the definition of electron and hole photocurrents given by (6) (7) (8) (9) (10) In (6) and (7), and represent the electron and hole densiand account for the escape times of the photies, while togenerated electrons and holes, respectively. The incident op. In Fig. 3, is tical power into the EAM is denoted by the real reference impedance equal to 50 , and the frequency response of everything but the intrinsic part of the EAM has . The external been lumped into the -parameter matrix (intrinsic) incident and reflected electrical waves are denoted and ( and ), respectively. In order to calculate by , the intrinsic reflection coefficient of the EAM

where and are the derivatives of the photocurrent conversion efficiency and intrinsic optical loss coefficient with respect to the junction voltage and evaluated at a bias voltage . It is worth noticing here that and . As a result of this and of being in parallel with the intrinsic capacitance , is in intrinsic circuit often simply approximated as representations of EAMs. We thus have that the total intrinsic , which impedance of the EAM is is necessary for the calculation of the intrinsic reflection coeffi. Using standard microwave theory [23], the cient complex response of the EAM as a modulator [20] can be obtained as (15) represents the small-signal output optical power where from the EAM. In order to obtain an expression for the complex response of the EAM as a receiver, it is necessary to find the smallsignal photocurrent resulting from an incident modulated optical signal. We therefore substitute (8) and (9) into (6) and (7) and solve the resulting equations with and . After using (10), the total small-signal photocuras a function of is obtained. From standard mirent crowave theory [23], it follows that the complex response of the EAM as a receiver [20] is given by (16) We point out here that we have intentionally kept the CW opfor both the EAM tical power going into the EAM equal to as modulator and the EAM as receiver scenarios. This is important since, in general, the intrinsic properties of the EAM depend on its input optical power through optical saturation effects [11]. To account for this dependence, EO and OE data is collected at . For this reason and to sima fixed CW input optical power plify our notation, we henceforth drop the explicit dependence in the intrinsic parameters described in this section. on

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For the receiver case, (18) while for both cases,

(19)

Fig. 4. Schematics for: (a) EO and (b) OE data collection including a polarization controller (PC).

If (17) and (18) correspond to the same value of bias voltage their ratio can be expressed as

,

(20) This last expression still remains a function of the -parameters of the interconnect/package and of the calibration coefficients and . In order to remove that dependence, ratios can further be taken for different values of the bias voltage in (20) with respect ). We can thus define to a single fixed value of (denoted as (21) which depends only on the intrinsic parameters of the EAM. and for EAMs in the unsatSince urated optical power regime, the following approximation can . Using this be made in (15): and substituting (15) and (16) into (21), we obtain

Fig. 5. Signal flow graph for the: (a) modulator and (b) receiver.

Fig. 4 shows the schematic for data collection in the proposed method. In Fig. 4(a), the EAM with input optical power is used as a modulator for several values of its bias voltage and EO data is collected. In Fig. 4(b), on the other hand, the EAM is used as a receiver where the CW input optical power is again ( to overcome the loss of the calibrated optical modulator used). OE data is collected at the same set of bias voltages used to take the EO measurements in Fig. 4(a). In order to obtain a set of usable EO and OE measurements, these are taken over a range of bias voltages within the transmission window of V V, as shown in Fig. 2). The polarthe EAM ( ization controller shown in Fig. 4 must be used when the EAM exhibits polarization-dependent effects within the range of bias voltages used to take the EO and OE measurements. Following the formalism of opto-electronic scattering matrices described by Hale and Williams [20], a signal flow graph representation of the schematic of Fig. 4 is shown in Fig. 5. The frequency dependence of the variables has been omitted ( and ) represent the comfor brevity, while and plex response and reflection coefficient of the calibrated optical receiver (modulator) used and can be calculated as described in [20]. Using the laws of signal-flow graph or applying the corresponding set of cascade matrices [23], the following equations can be obtained from Fig. 5. For the modulator case, (17)

(22) where (23) . ThereFrom (13) and (22), we have that fore, normalizing (22) with respect to its dc value and taking its magnitude, we have

(24) where subscripts are used on the right-hand side to denote bias voltage dependence. Equation (24) is only a function of the parameters that determine the intrinsic frequency response of the EAM. The set of dc parameters calculated in Section II is not necessary here since the combination of those parameters found is accounted for by the measured value of . in and nonlinear In Section V, measured values of least square numerical fitting are used to estimate the values of the intrinsic parameters in (24).

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IV. EXTRINSIC PARAMETER CALCULATION Instead of assuming a particular circuit topology for the interconnect network used to drive the EAM, as it is commonly done for laser diodes [14], [16], we use the intrinsic parameters in Fig. 3. We found as described in Section III to estimate choose to lump the response of the interconnect and package into a single set of -parameters for two reasons. First, the electrical interconnect of an EAM might contain an impedancematching network to improve the performance within some frequency range. An adequate circuit model for such a matching network can be difficult to determine from measurements of the overall EO and OE frequency response of the EAM only. Second, the possible resonances exhibited by the package contribute to the overall frequency response of the EAM [12] and would be difficult to separate from the effect of the interconnect and/or to model with discrete components. In order to find the , we begin by rewriting (18) as elements of (25) which is only a function of and when the intrinsic parameters and modulator calibration coefficient are known. For a series of bias voltage values, (25) can be solved in a least . is then square sense for those two parameters of . Finally, is calculated from (19). used in (20) to obtain The -parameters thus obtained along with the intrinsic parameters calculated as described in Section III will be used to reproduce the overall EO and OE frequency response of an EAM in Section V. V. EXPERIMENTAL RESULTS The parameter-extraction process described above was applied to the 40G-PS-EAM-1550 used in Section II. The vector network analyzer (VNA) and calibrated photodetector used were the 37397C and MN4765A both from the Anritsu Corporation, Atsugi, Japan, and both rated for operation up to 65 GHz. The calibration process described in [20] was used on another EAM, the OM5642W-30B from OKI Ltd., Tokyo, Japan, which was then used as the calibrated modulator. Fig. 6 shows the measured EO response, OE response, and reflection coefficient of 40G-PS-EAM-1550 for a series of bias voltages. The CW optical power into the EAM was dBm (as used in Section II) for both the EO and OE cases. Fig. 7(a) shows the normalized measured [as given in (21)] and numerically fitted [by using (24)] combination of the EO and OE responses of the EAM. For convenience, we have used the highest value of the bias V as our reference value of in (21). voltage The values of the carrier escape times and intrinsic capacitance that yield the fitted traces in Fig. 7(a) are shown in Fig. 7(b). As expected, the escape times of photogenerated carriers decrease with increasing reverse-bias voltage, while the calculated value of the intrinsic capacitance is physically reasonable for an EAM rated for operation up to 40 GHz. The set of initial values of , , and needed for the fitting procedure was obtained by initially neglecting the variation over of the intrinsic capacitance and assuming

Fig. 6. Measured EAM S -parameters. (a) EO, (b) OE, and (c) reflection coef0:3 V, 1V = 0:1 V. ficient within 1:0 V V

0



0

0

the set of bias voltages used. These assumptions reduce the fitting between (24) and the measured data to a linear least square problem the solution of which renders a set of initial values for the carrier escape times. These values were, in turn, used to find an initial estimate for the value of the intrinsic capacitance within the range of bias voltages used. Finally, a nonlinear least square fitting procedure was used to arrive at the set of values shown in Fig. 7. This set of intrinsic parameters was used as described in Section V to find the -parameters shown in Fig. 8, which represent the interconnect/package. The intrinsic and extrinsic parameters shown in the two previous figures were used to calculate the OE and EO frequency response of the EAM. The results, along with the measured data, are shown in Fig. 9 for a subset of the bias voltages used. The slight misfit exhibited at

YAÑEZ AND CARTLEDGE: EXTRACTION OF INTRINSIC AND EXTRINSIC PARAMETERS IN EAMs

Fig. 7. (a) Measured and fitted combination of EO and OE responses. (b) Intrinsic calculated EAM parameters.

Fig. 9. (a) Measured and calculated OE responses. (b) EO responses within 0:9 V V 0:3 V and 1V = 0:2 V using the intrinsic parameters from Fig. 7.

0

Fig. 8. Calculated S -parameters for the interconnect/package.

some bias voltages in the calculated EO response is likely due to a less than optimum value of the calculated intrinsic capacitance, which affects the modulator response through . VI. DISCUSSION ON LIMITATIONS The methodology described in Section III lead us to (24), which contains only the parameters governing the frequency response of the intrinsic EAM. Using (24) to fit the measured data yields two carrier escape times per bias voltage. It is not possible, however, to know which value should be assigned to the escape time of electrons and which should be assigned to that

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0

0

of the holes. Since the mobility of holes is much lower than that of electrons in the semiconductor materials used in EAMs, we have assigned, at each bias voltage, the lower value of the escape time to and the higher to . An adequately selected set of parameters will produce a fit between (24) and the corresponding measured values for all the bias voltages used and over the entire frequency range of operation of the EAM. The parameters cannot be independently optimized since (24) always contains the parameters of the ref. At each bias voltage, the values of , erence bias voltage , and must be found, the dimensionality of the fitting space can therefore increase rapidly (e.g., to 24 variables in the previous section). As such, one is faced with the possibility of having multiple values for the set of optimization parameters that yield almost identically good fits with respect to the measured data. In order to avoid obtaining values that are not physically meaningful, we have used a constrained nonlinear least square fitting procedure [24] in Section V. The fitting procedure relies on a primal-dual interior-point optimization method for semidefinite and second-order cone programming. A software package implementing such an optimization method [25]

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Fig. 10. Intrinsic parameters obtained by increasing the initial value of the intrinsic capacitance by 25% during data fitting.

was used along with an analytic expression for the Jacobian representing a first-order expansion of (24) around a given set of intrinsic parameters. A discussion about which numerical technique is better suited to solve the fitting problem or how to obtain the best set of initial values for the optimization parameters is beyond the scope of this paper. When detailed information is available about the physical dimensions and composition of the active region within the EAM, it is possible to calculate parameters such as the value of the intrinsic capacitance or to use that information in a more detailed model to predict the behavior of the EAM [26]. When detailed information is not available, all the parameters must be obtained though external measurements. The proposed method allows us to find a set of physically reasonable values for the parameters of the EAM solely from external measurements. The parameters thus found, however, are only accurate insofar as they can reproduce the EO and OE frequency response of the EAM. They are not, in general, the same as those obtained from a physics based model in which the composition of the quantum wells, dimensions of the active section and details of the interconnect and package used for the EAM are known. A somewhat different set of intrinsic and extrinsic parameters (from those found in Section V) could be obtained, for example, by starting the numerical fitting procedure from a different set of initial parameter values. To illustrate this, we have increased by 25% the initial estimate of the intrinsic capacitance and used again our constrained nonlinear least square fitting procedure. The intrinsic values thus obtained are shown in Fig. 10. Although somewhat different, the new found values for the intrinsic parameters still remain physically reasonable. The measured and calculated OE and EO responses using those values are shown in Fig. 11. The corresponding set of calculated -parameters for the interconnect/package are similar (but not identical) to those shown in Fig. 9 and are omitted for brevity. We can thus see that different sets of physically reasonable parameters can be used to reproduce the EO and OE response of the EAM when physical details about it are not available.

Fig. 11. (a) Measured and calculated OE responses. (b) EO responses within 0:9 V V 0:3 V and 1V = 0:2 V using the intrinsic parameters from Fig. 10.

0



0

0

VII. CONCLUSION We have proposed a parameter-extraction technique for EAMs, which exploits their use as both modulators and receivers. Opto-electronic scattering matrices were used to describe both modes of operation. By combining the EO and OE frequency responses of the EAM, it is possible to calculate the parameters that determine the intrinsic frequency response of the EAM while removing the effect of the interconnect/package and the need for an accurately calibrated optical transmitter and receiver pair. The intrinsic parameters can, in turn, be used to calculate a set of -parameters representing the effect of the interconnect/package, thus providing a complete characterization of the EAM. Input optical power dependence of the EAM properties is easily accounted for by using the same CW input optical power to collect the EO and OE data. With the possible exception of an additional polarization controller, the proposed method can be readily implemented and automated using a commercial lightwave component analyzer. Through numerical fitting, it was shown that in the absence of any detailed knowledge about the internal structure of the EAM, different sets of physically reasonable parameters obtained from our method

YAÑEZ AND CARTLEDGE: EXTRACTION OF INTRINSIC AND EXTRINSIC PARAMETERS IN EAMs

can be used to reproduce its OE and EO frequency responses with good accuracy. ACKNOWLEDGMENT The authors would like to thank the Advanced Photonic Systems Laboratory, Queen’s University, Kingston, ON, Canada, for providing the necessary equipment to perform the experimental verification of the parameter extraction technique. The authors especially wish to thank Dr. C. E. Saavedra, Queen’s University, for reviewing the first draft of this paper’s manuscript. REFERENCES [1] H. Fukano, T. Yamanaka, M. Tamura, and Y. Kondo, “Very-lowdriving-voltage electroabsorption modulators operating at 40 Gb/s,” J. Lightw. Technol., vol. 24, no. 5, pp. 2219–2224, May 2006. [2] S. Yaakob et al., “Adopting electroabsorption modulator for the WLAN 802.11a radio over fiber system,” in Proc. IEEE Int. Semiconduct. Electron. Conf., 2006, pp. 871–875. [3] C. Doerr et al., “Compact high-speed InP DQPSK modulator,” IEEE Photon. Technol. Lett., vol. 19, no. 8, pp. 1184–1186, Aug. 2007. [4] M. Aoki et al., “InGaAs/InGaAsP MQW electroabsorption-modulator integrated with a DFB laser fabricated by band-gap energy control selective area MOCVD,” IEEE J. Quantum Electron., vol. 29, no. 6, pp. 2088–2096, Jun. 1993. [5] “III–V photonic integrated circuits and their impact on optical network architectures,” in Optical Fiber Telecommunications, C. Joyner, D. Lambert, P. Evans, M. Raburn, V. A. I. P. Kaminow, T. Li, and A. E. Willner, Eds. New York: Academic, 2008, ch. 10. [6] G. L. Li, C. K. Sun, S. A. Pappert, W. X. Chen, and P. K. L. Yu, “Ultrahigh-speed traveling wave electroabsorption modulator—Design and analysis,” IEEE Trans. Microw, Theory Tech., vol. 47, no. 7, pp. 1177–1183, Jul. 1999. [7] F. Capelluti and G. Ghione, “Self-consistent time-domain large signal model of high-speed traveling-wave electroabsorption modulators,” IEEE Trans. Microw, Theory Tech., vol. 51, no. 4, pp. 1096–1104, Apr. 2003. [8] N. Chen and J. C. Cartledge, “Measurement-based model for MQW electroabsorption modulators,” J. Lightw. Technol., vol. 23, no. 12, pp. 4265–4269, Dec. 2005. [9] T. H. Wood, “Direct measurement of the electric-field-dependent absorption coefficient in GaAs/AlGaAs multiple quantum wells,” Appl. Phys. Lett., vol. 48, no. 21, pp. 1413–1415, May 1986. [10] M. K. Chin, “A simple method using photocurrent and power transmission for measuring the absorption coefficient in electroabsorption modulators,” IEEE Photon. Technol. Lett., vol. 4, pp. 866–869, Aug. 1992. [11] J. Shim, B. Liu, and J. Bowers, “Dependence of transmission curves on input optical power in an electrabsorption modulator,” IEEE J. Quantum Electron., vol. 40, no. 11, pp. 1622–1628, Nov. 2004. [12] S. Kaneko, H. Itamoto, T. Miyahara, and T. Hatta, “Impact of package resonance on eye diagram in high-speed optical modules,” in Proc. 27th Eur. Opt. Commun. Conf., Oct. 4, 2001, vol. 3, pp. 402–403. [13] N. H. Zhu, Q. Q. G. Hasen, H. G. Zhang, J. M. Wen, and L. Xie, “Dynamic P –I and P –V curves for semiconductor lasers and modulators,” J. Lightw. Technol., vol. 26, no. 19, pp. 3369–3375, Oct. 2008. [14] M. L. Majewski and D. Novak, “Method for characterization of intrinsic and extrinsic components of semiconductor laser diode circuit model,” IEEE Microw. Guided Wave Lett., vol. 4, no. 2, pp. 133–135, Feb. 1992. [15] J. C. Cartledge and R. C. Srinivsan, “Extraction of DFB laser rate equation parameters for system simulation purposes,” J. Lightw. Technol., vol. 15, no. 5, pp. 852–860, May 1997. [16] N. H. Zhu, C. Chen, E. Y. B. Pun, and P. S. Chung, “Extraction of intrinsic response from S -parameters of laser diodes,” IEEE Photon. Technol. Lett., vol. 17, no. 4, pp. 744–746, Apr. 2005. [17] H. P. Huang, N. H. Zhu, and J. Liu, “Extraction of intrinsic frequency response of p-i-n photodiodes,” IEEE Photon. Technol. Lett., vol. 17, no. 10, pp. 2155–2157, Oct. 2005.

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[18] T. Ido, H. Sano, S. Tanaka, and H. Inoue, “Frequency-domain measurement of carrier escape time in MQW electro-absorption optical modulators,” IEEE Photon. Technol. Lett., vol. 9, no. 12, pp. 1421–1423, Dec. 1995. [19] D. B. Malins, A. Gomez-Iglesias, S. J. White, W. Sibbett, and A. Miller, “Ultrafast electroabsorption dynamics in an inas quantum dot saturable absorber at 1.3 m,” Appl. Phys. Lett., vol. 89, pp. 17 111–17 113, 2006. [20] P. D. Hale and D. F. Williams, “Calibrated measurement of optoelectronic frequency response,” IEEE Trans. Microw, Theory Tech., vol. 51, no. 4, pp. 1422–1429, Apr. 2003. [21] Z. Hu, H. Chou, J. E. Bowers, and D. J. Blumenthal, “40-Gb/s optical clock recovery using a traveling-wave electroabsorption modulator-based ring oscillator,” IEEE Photon. Technol. Lett., vol. 16, no. 5, pp. 1376–1378, May 2004. [22] G. L. Li, P. K. L. Yu, W. S. C. Chang, K. K. Loi, C. K. Sun, and S. A. Pappert, “Concise RF equivalent circuit model for electroabsorption modulators,” Electron. Lett., vol. 36, no. 9, pp. 818–820, Apr. 2000. [23] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1997. [24] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2006. [25] J. F. Strum, “Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,” Optim. Methods Softw., vol. 11–12, pp. 625–653, 1999. [26] A. Vukovic, E. V. Bekker, P. Sewell, and T. M. Benson, “Efficient time domain modeling of rib waveguide RF modulators,” J. Lightw. Technol., vol. 24, no. 12, pp. 5044–5053, Dec. 2006. Mauricio Yañez (S’98) received the B.Sc. degree in electrical engineering from the Universidad de Guanajuato, Guanajuato, Mexico, in 1997, the MSc. degree in electrical engineering from the University of New Brunswick, Fredericton, NB, Canada, in 2000, and is currently working toward the Ph.D. degree at Queen’s University, Kingston, ON, Canada. From 2000 to 2002, he was a Member of the Technical Staff with the Advanced Fiber Optic Products Group, JDS Uniphase Corporation, Ottawa ON, Canada. He . His research interests include the development of high-speed circuits for communications applications, opto-electronic components for fiber-optic networks, and high-speed optical communication systems.

John C. Cartledge (S’74–M’79–SM’96–F’09) received the B.Sc. degree in mathematics and engineering, M.Sc. degree in mathematics, and Ph.D. degree in mathematics from Queen’s University, Kingston, ON, Canada, in 1974, 1976, and 1979, respectively. From 1979 to 1982, he was a Member of the Scientific Staff with Bell-Northern Research, Ottawa, ON, Canada, where his research involved fiber-optic systems for the exchange access network and high-capacity digital radio systems. Since 1982 he has been with the Department of Electrical and Computer Engineering, Queen’s University. In 2002, he was appointed a Queen’s University Research Chair. He has spent sabbatical leaves with Bellcore (1988–1989), Tele Danmark Research (1995–1996), and Corning (2009–2010). His current research interests include optical modulators, optical signal processing (wavelength converters and optical regenerators), electronic signal processing for optical waveform generation, and digital coherent systems. Dr. Cartledge is a Registered Professional Engineer (P.Eng.) and a Fellow of the Optical Society of America (OSA). He is an IEEE Photonics Society Distinguished Lecturer (2008–2010). His conference organization activities include serving on the Technical Program Committees for the Conference on Optical Fiber Communication (1994–1997 and 2005–2006), the IEEE Lasers and Electro-Optics Society (LEOS) Annual Meeting (2002–2008), and the European Conference on Optical Communication (1997 and 2006–2010). He was a program chair and general chair for the Conference on Optical Fiber Communication (2008 and 2010), and a program chair for the IEEE Photonics Society Summer Topical Meeting (2009). He is currently an associate editor for the IEEE PHOTONICS TECHNOLOGY LETTERS. He was a recipient of the IEEE Canada Outstanding Engineering Educator Award (2009).

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A Self-Started Laser Diode Pulsation Based Synthesizer-Free Optical Return-to-Zero On–Off-Keying Data Generator Yu-Chieh Chi and Gong-Ru Lin, Senior Member, IEEE

Abstract—A synthesizer-free return-to-zero on–off-keying (RZ-OOK) data generator is demonstrated by gain-switching a Fabry–Perot laser diode (FPLD) and driving a data-stream generator with a self-started opto-electronic oscillator (OEO). By self-feedback triggering the FPLD with the OEO, the FPLD with a modulation bandwidth of only 4 GHz can be self-pulsating to generate clock signal and return-to-zero (RZ) carrier at a repetition frequency of 10 GHz. The self-pulsated FPLD exhibits a pulsewidth of 19 ps associated with a pulse on/off extinction ratio of 7.7 dB and a timing jitter as low as 424 fs. Under self-started pulsation, the spectral linewidth enhancement factor of such a gain-switching FPLD is 12.8, which induces a dynamic frequency chirp ranging from 6 to 5 GHz within 30-ps duration. The corresponding chirp parameter of 0.22 is reported for such an OEO driven FPLD based RZ-OOK data generator. The lowest phase noise of 131.8 dBc/Hz at 10-MHz offset from 10-GHz carrier, and the maximum output power of 23 dBm is obtained after amplification. The generated RZ pulsed carrier exhibits a carrier-to-noise ratio of 53 dB and a harmonic suppression ratio up to 52 dB. Such a self-started OEO triggered FPLD is a new approach to the synthesizer-free optical clock and RZ-OOK data generation.

+

Index Terms—Chirp and jitter, Fabry–Perot laser diode (FPLD), on–off-keying (OOK), opto-electronic oscillator (OEO), return-tozero (RZ), self-started pulsation, synthesizer free.

I. INTRODUCTION

T

HE stabilized optical pulse clock at a high-repetition rate with ultra-low timing jitter is mandatory to enhance the transmission performance and to promote the optical network flexibility, which essentially helps to realize the high bit-rate and error-free transmission in optical-time-division-multiplexing (OTDM) networks. In addition to the commercially available synthesizers, the self-starting opto-electronic oscillator (OEO) is an alternative to create the high-purity signals and high repetition-rate optical pulse clock without using any signal generator for versatile applications [1]–[4]. The concept of the

Manuscript received October 14, 2009; revised February 26, 2010; accepted April 29, 2010. Date of publication July 12, 2010; date of current version August 13, 2010. This work was supported in part by the National Science Council, Taiwan, and the Excellent Research Projects of National Taiwan University, Taiwan, under Grant NSC98-2221-E-002-023-MY3, Grant NSC98-2623-E-002-002-ET, Grant NSC 98-2622-E-002-023-CC3, and Grant NTU98R0062-07. The authors are with the Graduate Institute of Photonics and Optoelectronics, and Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2052957

OEO was originally proposed by Kersten [5] in 1978. The first experimental demonstrations were reported by Schlaak et al. [6] using a waveguide modulator and by Damen et al. [7] using a laser diode in 1980. Thereafter, numerous investigations focus on developing versatile OEO architectures by using a electrical comb generator [8], a dual RF drive Mach–Zehnder intensity modulator (MZM) [9], an electroabsorption modulator [10], a broadband polymer electrooptic modulator [11], an LiNbO phase modulator [12], and an injection-locked dual loop [13]. Recently, Nakazawa et al. also proposed an OEO ring cavity consisting of a gain-switched vertical-cavity surface-emitting laser (VCSEL) at 850 nm and a single-mode photonic crystal fiber, which generated a 10-GHz optical pulse-train with timing jitter as low as 1.2 ps [14]. In comparison with the conventional nonreturn-to-zero (NRZ) data format, the pulsed return-to-zero on–off-keying (RZ-OOK) is a preferable data format with immunity to linear and nonlinear dispersion during high-speed and long-haul transmission. Traditionally, the RZ-OOK stream was demonstrated by modulating the optical pulse-train with the pseudorandom binary sequence (PRBS) data generator driven with additional microwave clock at punishments of increasing cost and complexity [15], [16]. The use of a self-feedback OEO for generating an RZ-OOK data stream has never been reported previously. Without using any external microwave frequency source in this work, we combine a self-feedback OEO configuration with a 1550-nm fiber-pigtailed Fabry–Perot laser diode (FPLD) packaged in a 5.6-mm transistor outline can (TO-56 can) [17], where the FPLD incorporated OEO forces its gain-switched pulsation self-started at 10 GHz to demonstrate the synthesizer-free RZ-OOK data generator at 10 Gbit/s. The self-pulsating technique effectively overcomes inherent limitation on direct-modulation bandwidth of the FPLD packaged with a TO-56 can, such that the self-started gain-switching FPLD can easily exceed the intrinsic bandwidth of 4 GHz set by the bonding wires of the TO-can module. We discuss the linewidth enhancement factor and chirp parameter of the OEO self-started gain-switching FPLD and also investigate the self-starting threshold at versatile feedback conditions, which facilitates to promote the sub-picosecond timing jitter and to enhance the on/off extinction ratio of gain-switched pulse. The shortening on OEO driven gain-switching FPLD pulsewidth at extremely low-power amplified output is demonstrated. Moreover, the single-sideband (SSB) phase noise, the carrier-to-noise ratio (CNR), and the second-order harmonics suppression ratio are also elucidated.

0018-9480/$26.00 © 2010 IEEE

CHI AND LIN: SELF-STARTED LASER DIODE PULSATION BASED SYNTHESIZER-FREE OPTICAL RZ-OOK DATA GENERATOR

Fig. 1. Schematic diagram of the self-starting OEO with gain-switching FPLD based synthesizer-free RZ-OOK data generator. Bandpass filter: BPF. Lownoise amplifier: LNA.

II. EXPERIMENTAL SETUP Fig. 1 illustrates the experimental setup of an OEO constructed using a 1550-nm FPLD without microwave signal generator for synthesizer-free RZ-OOK data generation. The gain-switching FPLD can only be self-started by the opto-electronic feedback from the FPLD, which is directly modulated at nearly or above threshold current condition. To assist the pick-up of 10-GHz feedback from FPLD noise, a -band erbium-doped fiber amplifier (EDFA) is used to amplify the weak output of FPLD before gain switching. There is no additional loss occurred on the OEO driven FPLD except the 50% coupler used for self-feedback triggering the OEO itself. The EDFA amplified in the OEO loop is for amplifying the FPLD output to initiate self-oscillation process, but not for compensating the insertion loss. An ultrafast photodetector (New focus 1434 with a cutoff frequency of 25 GHz) with associated pre-amplifier (New focus 1422), two sets of low-noise amplifiers, and microwave bandpass filters (Filtronic W5258) at a central frequency of 9.953 GHz is configured to extract the 10-GHz feedback, and a 35-dB gain power amplifier is employed to drive the FPLD for gain-switching operation via a bias-tee circuit. When the bias current of the FPLD is below threshold, even a weak gain-modulation cannot be triggered by the OEO, as no clock signal is filtered for the optical pulsation. By increasing the FPLD bias beyond threshold, the continuous-wave output of the FPLD transfers into the small-signal oscillation due to a sufficient gain-modulation via the feedback from OEO. The weak 10-GHz signal in the self-feedback OEO loop turns to gain-switch the FPLD itself eventually. Two sets of the microwave bandpass filters and amplifiers are mandatory to build up the feedback loop for self-starting and re-modulating the FPLD. The stabilized gain-switching FPLD pulse-train is obtained when the gain of the 10-GHz clock in the feedback OEO loop is sufficiently large and is coincident with the relaxation oscillation frequency of the FPLD. By using the microwave clock extracted from the OEO before driving the gain-switched FPLD, the PRBS data generator can be trigged to deliver the electrical NRZ data stream with a pattern length of 2 1. The optical NRZ data stream is simulated by externally encoding the optical pulse-train from the

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Fig. 2. Frequency response and power-current curve of the FPLD at 1550 nm.

OEO driven gain-switching FPLD with an amplified PRBS pattern at 10 Gbit/s through an MZM. Fig. 2 shows the frequency response of FPLD with bias currents of 9.2 and 15 mA, and the modulation bandwidth of about 4 and 6 GHz is originating from the FPLD bandwidth, the cutoff frequency set by the TO-56 can package, and the other parasitic components. However, a visible relaxation oscillation is still observable at around 10 GHz. The inset of Fig. 2 depicts that the power-current response of the FPLD with threshold current is nearly 9.2 mA. III. RESULTS AND DISCUSSIONS The OEO is self-started by increasing the bias current of FPLD beyond 12 mA, such that the gain-switching operation is automatically triggered to deliver the return-to-zero (RZ) pulsed carrier. The PRBS data generator is concurrently trigged by the microwave clock extracted from the OEO to deliver the electrical NRZ data stream at 10 Gbit/s. Fig. 3 illustrates the RZ-OOK data trace output from the PRBS driven MZM with a specific pattern of 10101110. Such an RZ-OOK data generator is completely synthesizer free; however, there is an intrinsic limitation on switching the gain characteristics of the FPLD. Typically, such an RZ-OOK generation cannot be automatically turned on when the bias current of the FPLD is close to the threshold condition. In the experiment, the gain-switching FPLD fails to self-start at a bias current less than 12 mA since there is no 10-GHz feedback signal picking up from the FPLD noise spectrum by the OEO circuitry. Although an inline EDFA has been inserted to enlarge the output power of the FPLD, the OEO driven gain-switching FPLD must be operated beyond threshold to trigger a sufficiently short RZ pulsed clock. Apparently, such an operation is contrary to the optimized condition for gain switching a free-running FPLD at slightly below threshold current. Nevertheless, the OEO driven gain-switching FPLD can respond to the incoming “on” bits consecutively, thus providing the RZ-OOK data with a greater modulation depth and a sharper bit-shape. Some parameters of the generated RZ clock due to such an extraordinary operation is found to strongly correlate with the bias current, which are discussed below. First of all, the red-shifted spectrum of the OEO driven gainswitching FPLD self-started is observed by detuning the inline

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Fig. 3. Specific 10101110 RZ-OOK data stream obtained after the PRBS driven MZM.

Fig. 4. Optical spectrum and mode spacing of OEO-driven gain-switching FPLD.

EDFA gain and the FPLD bias, which is mainly attributed to the dependence between the transient carrier concentration and the temporally changed refraction index. The correlation between the wavelength variation and the instantaneous change on the , where refractive index is described as and are the refractive indices at time and 0, respectively. In a gain-switched FPLD, the carrier-induced refractive index change and the wavelength variation can generally be correlated by [18] -

-

(1)

Typically, the is 2 3 10 for a general InGaAsP FPLD [21]. As predicted by (1) with a carrier density variation of and the negative , the FPLD spectrum red shifts toward longer wavelength after gain-switching, as shown in Fig. 4. In comparison with a freerunning FPLD, the OEO-driven gain-switching FPLD exhibits a red-shifted wavelength up to 3.2 nm, and the linewidth enhancement factor can be rewritten as

(2) With (2), the linewidth enhancement factor of the OEO driven gain-switching FPLD is calculated by assuming the cavity length as 600 m, the width , and the depth of the active region in FPLD as 1.5 and 0.2 m, respectively, the differential gain as 2.5 10 m , the electron charge as 1.6 10 , and the carrier lifetime as 2 ns, respectively. The linewidth enhancement factor of OEO driven gain-switching FPLD at a bias current of 15 mA is determined as 12.8, which is obtained under a transient current of , where is the bias current of FPLD

Fig. 5. Pulsewidth and output waveform of the OEO driven gain-switching FPLD at 10 GHz.

and is the modulated current at an angular frequency of . Due to the band-filling effect, the OEO driven gain-switching FPLD reveals a optical spectrum with a 3-dB linewidth broadening by at least 12.5 times when changing the operation condition from a free-running (36 pm) to gain-switching (0.46 nm) mode. The pulsewidth of the OEO driven gain-switching FPLD shown in Fig. 5 is obtained by the digital sampling oscilloscope (Agilent 86100 A 86116 A with a detecting bandwidth of 63 GHz). When enlarging the gain-switching FPLD output power up to 1.5 dBm by setting the inline EDFA gain as 11 dB, the OEO driven gain-switching pulsewidth is greatly shortened from 24 to 19 ps as the bias current increases from 12 to 15 mA. The inset of Fig. 5 depicts the oscilloscope trace for the perfect OEO-driven gain-switching FPLD pulse-train self-triggered at 10 GHz. In the following, we further discuss the relationship between pulsewidth and bias current of the OEO driven gain-switching FPLD. In general, the gain-switching

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FPLD output can be carried out by a set of nonlinear rate equations in (3) [18]

(3) where and denote the carrier and photon densities in the gain-switching FPLD cavity , the transparency carrier density , the differential gain , the spontaneous coupling factor , the electron charge , the active layer volume , the transient bias current , the group velocity , the optical confineand , the carrier and photon lifetimes, ment factor, and respectively. For the gain-switching FPLD, the current density is described by an electric pulse-train, and the photon lifeis given by , where and time denote the mirror and internal losses, respectively. In principle, the pulsewidth of the OEO driven gain-switching FPLD can be roughly estimated by using two combined expoand on rising and nential curves with time constants of , respectively. Theoretically, the rise time trailing edges of can be obtained by deriving the following equation:

Fig. 6. Calculated timing jitter from gain-switching FPLD (15-mA bias) at different nodes in the OEO feedback loop. The A–G checking points are defined as the different nodes shown in Fig. 1.

such that the rise time can be described as . In general, is approximately and the OEO driven gainswitching FPLD pulsewidth is given as (4) (4) Only when increasing the FPLD bias current from 9.2 to 12 mA, the gain-switching FPLD can be self-started to deliver an optical pulse-train at 10 GHz. A further reduction on pulsewidth relies strictly on increasing the bias current, and the minimum requested bias current is 1.2 times the threshold current. To function as a synthesizer-free optical clock, it is mandatory to characterize the spectral purity of the 10-GHz microwave clock extracted from the OEO driven gain-switching FPLD. After amplification, the maximum output power at the FPLD bias current of 15 mA is 23.2 dBm. Under a spectral analysis with a resolution bandwidth of 10 Hz, the CNR and second-order harmonic suppression ratio of the extracted 10-GHz microwave clock are 53 and 52 dB, respectively, which shows an SSB phase noise of 131.8 dBc/Hz at 10-MHz offset from the carrier. The CNR can be enhanced up to 60–65 dB by setting the resolution bandwidth as 1 Hz. The SSB phase noises measured at different nodes in the OEO feedback loop were shown in Fig. 6 to understand the noise-suppressing performance of each OEO element. In addition, the related timing jitter is calculated from the integral of the SSB phase-noise spectrum versus frequency measured at a specified node using the following equation [20]:

(5)

Fig. 7. Measured timing jitter of optical pulse-train at different driving currents and optical injection powers.

where is the repetition rate, is the SSB phase noise of the th harmonic frequency component of the OEO triggered gain-switching FPLD, is the harmonic number of the SSB and are the lower and higher limits of inphase noise, and tegration. With an integrating range from 10 Hz to 100 MHz, the minimized timing jitter of 352 fs can be obtained. The inset of Fig. 6 also shows the bandpass filtered spectrum with a quality , where the passband factor of of 7.5 MHz is still too wide to restrict the noise in the OEO feedback loop and inevitably degrades the phase-noise performance. Such a shortcoming can be improved if a high- filter and a high-gain and low thermal-noise amplifier are available. The measured timing jitter of the self-started gain-switching pulse is plotted as a function of the injection current and is shown in Fig. 7. The OEO feedback circulating loop plays an important role on suppressing the phase noise and corresponding jitter. Therefore, the timing jitter is greatly reduced with increasing bias current of the OEO driven gain-switching FPLD. If the output power by tuning the EDFA gain is enlarged, the timing jitter further indicate a decreasing trend as a larger 10-GHz feedback signal filtered from noise can be provided by increasing the EDFA gain, such that the self-starting oscillation

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Fig. 8. Pulse on/off extinction ratio in OEO dependence on the bias current and optical injection power.

is easier to be established. Nonetheless, the EDFA is also an active device with considerable noise figure up to 5–6 dB, which somewhat degrades the noise performance of the self-started gain-switching FPLD even with high-gain amplification. A minimized timing jitter of 424 fs is achieved at the FPLD bias current of 15 mA and at the EDFA output power of 1.5 dBm. Obviously, the cavity gain in the self-started OEO feedback loop must be appropriately enlarged to minimize the timing jitter of the RZ pulsed carrier for a further RZ-OOK data-stream generation. On the other hand, the pulse on/off extinction ratio (defined as the ratio of pulse amplitude to dc offset level) of the optical pulse-train generated from such a self-started OEO is also a key parameter for estimating the intensity noise performance of the RZ-OOK data stream. For obtaining a high on/off extinction ratio in our case, the precise control on the EDFA gain is mandatory to avoid additional spontaneous emission noise accompanied with the generated RZ pulsed carrier. However, such an issue on the optimization of the pulse on/off extinction ratio was seldom discussed in previous reports. The pulse on/off extinction ratio at versatile operating conditions shown in Fig. 8 reveals an increasing trend with enlarging FPLD bias, which is attributed to the larger microwave signal feedback into the OEO loop for clock extraction. By selecting the EDFA gain as 7, 9, and 11 dB, the pulse on/off extinction ratio increases from 5.5 to 7.7, 5.4 to 7.1, and 5.3 to 6.7 dB, respectively. The increasing EDFA gain essentially enhances the optoelectronic conversion to reduce the self-staring threshold current effectively. However, the spontaneous emission of the EDFA dominates to distort the gain-switching pulse, in which the induced dc level inevitably suppresses the pulse on/off extinction ratio by 1.4 dB at least. Subsequently, the EDFA gain is appropriately decreased and the FPLD bias is increased accordingly to promote the amplification on the gain-switched pulse-train. As the EDFA gain decreases from 11 to 7 dB, the pulse on/off extinction ratio is greatly enhanced from 6.8 to 7.7 dB. We conclude that the optimization on EDFA gain is more pronounced than the bias for adjusting better pulse quality. As mentioned above, we have increased the bias current to suppress the timing jitter. Assuming that the optical confinement factor is 0.5, the internal loss is 1000 m , and the facet reflective R is 30%, the calculated modulation bandwidth and output power [21] from the self-starting

Fig. 9. Calculated modulation bandwidths of the free-running and self-pulsated gain-switching FPLD, and their corresponding output powers and timing jitters as a function of the FPLD bias current.

Fig. 10. Optical pulses (blue line in online version) and corresponding peak-topeak chirps (red line in online version) of the pulse-train at bias current of 12 (dashed line) and 15 mA (solid line).

OEO under gain-switching operation as a function of the FPLD bias and the timing jitter are shown in Fig. 9. The simulated laser modulation bandwidth is increased from 13.9 to 16 GHz, whereas the timing jitter concurrently decreases due to the enhanced carrier accumulation in the gain-switching FPLD. Fig. 10 illustrates the optical pulse-train shapes and corresponding frequency chirps at bias current of 12 and 15 mA. The chirped frequency deviation of the OEO driven gain-switching FPLD pulse-train is positive at the rising edge and is negative at the falling edge , thus leading to a negative chirp parameter . Assuming the optical field of the gain-switching FPLD, at exhibits a linearly chirped Gaussian shape as given by [21]

(6)

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TABLE I PARAMETRIC COMPARISON ON THE OPTICAL PULSE-TRAIN AND THE MICROWAVE CLOCK GENERATED USING DIFFERENT METHODS

(7) where is the phase of and represents the gainswitching FPLD pulsewidth. The frequency change is related to the phase derivative given by

(8) With a chirp parameter analyzer, the maximum deviation of the chirped frequency (defined as ) are 10.7 and 11.8 GHz within the pulse durations of 24 and 19 ps at bias currents of 12 and 15 mA, which corresponds to an increment on the negative chirp parameter by 10%. In principle, the frequency chirp of the FPLD is [22], where is the phase of the output electric field and is the output power of the gain-switching FPLD, respectively. Since the pulse on/off extinction ratio is positively proportional to the bias current of the gain-switching FPLD, the dynamic chirp of the OEO-driven gain-switching pulse therefore shows a similar trend. Nonetheless, the linewidth enhancement factor concurrently decreases with increasing bias, which plays a less pronounced role than the pulse on/off extinction ratio for chirp degradation. Under a constant gain-switching FPLD bias of 15 mA, the chirp can be reduced by increasing the output power of the EDFA in the OEO feedback loop from 4.9 to 1.5 dBm. In brief, both the pulsewidth and timing jitter are decreased by increasing the bias current and by the remaining constant EDFA gain, whereas the pulse on/off extinction ratio and chirp parameter arise accordingly. In view of previous reports, most research was focused on either the optical pulse-train or the microwave signal generated from such kinds of self-started OEOs using different methods. There are only two groups simultaneously discussing the parametric performances of optical pulse-train and microwave signals [10], [14], but neither one discussed the possibility of the RZ-OOK data generation in reports. To compare, Table I lists the parameters regarding the performance of the optical pulse-train and the microwave clock generated using different methods. At the same repetition rate of 10 GHz, our proposed approach shows a shorter pulsewidth (19 ps), a lower timing jitter (0.424 ps), a higher CNR (53 dB), and a better SSB phase noise ( 131.8 dBc/Hz) than previous studies. Moreover, the on/off extinction ratio of the RZ pulsed carrier generated from such a self-started OEO driven gain-switching FPLD is preliminarily reported. Up to now, there is no demonstration on the RZ-OOK data-stream generation by using the synthesizer-free OEO driven gain-switching FPLD beyond 10 GHz. In contrast to the FPLD based all-optical NRZ-to-RZ data format conversion technology reported previously [23], [24], the proposed optical RZ-OOK data generator is not only a completely synthesizer-free and self-pulsating architecture, but also an electrical-to-optical OOK transmitter without the need of addition injection from the dropped optical NRZ data stream.

No matter the data format generator or convertor requires a microwave clock for driving the PRBS data generator, which is bulky and cost ineffective. The proposed RZ-OOK scheme is exempt from the use of a synthesizer, electrical RZ generator, and external optical injection in current optical NRZ-to-RZ converters. Future development on such kinds of self-started OEO triggered FPLDs will offer a new approach to the synthesizer-free optical clock and RZ-OOK data generation. IV. CONCLUSION Without using any external signal generators, we have demonstrated the self-started microwave clock and RZ pulsed carrier from a gain-switching FPLD at 1550 nm, which is achieved by using a self-feedback configured OEO to function as a synthesizer-free RZ-OOK data generator at 10 Gbit/s. Such a self-feedback gain-switching operation essentially overcomes the electrical modulation bandwidth of the commercial FPLD below 4 GHz. With continuous feedback and large gain in the feedback loop, the OEO driven gain-switching FPLD pulse-train repeated at 10 GHz is generated with shortened pulsewidth, timing jitter, and pulse on/off extinction ratio of 19 ps, 424 fs, and 7.7 dB, respectively. The chirp parameter and linewidth enhancement factor of the OEO driven gain-switching FPLD pulse are 0.22 and 12.8, respectively. The timing jitter in terms of laser modulation bandwidth in the OEO feedback loop is analyzed, and the self-started pulse on/off extinction ratio can be promoted by decreasing the EDFA gain and increasing the FPLD bias. For the OEO driven gain-switching FPLD, the measured output power can be as high as 23 dBm, and the SSB phase noise at 10-MHz frequency offset is as low as 131.8 dBc/Hz. The related CNR and the harmonics suppression ratio are determined as 60–65 and 52 dB, respectively. The OEO triggered gain-switching FPLD can generate RZ pulsed carrier with the lowest phase noise and jitter comparable to a commercial synthesizer in which the pulse on/off extinction ratio is greatly enhanced and the chirp can be minimized concurrently. REFERENCES [1] J. Lasri, P. Devgan, R. Tang, and P. Kumar, “Self-starting optoelectronic oscillator for generating ultra-low-jitter high-rate (10 GHz or higher) optical pulses,” Opt. Exp., vol. 11, no. 12, pp. 1430–1435, June 2003. [2] M. F. Lewis, “Novel RF oscillator using optical components,” Electron. Lett., vol. 28, no. 1, pp. 31–32, Jan. 1992. [3] X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 3, no. 8, pp. 1725–1735, Aug. 1996.

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[4] Y. Ji, X. S. Yao, and L. Maleki, “Compact optoelectronic oscillator with ultralow phase noise performance,” Electron. Lett., vol. 35, no. 18, pp. 1554–1555, Sep. 1999. [5] R. Th. Kersten, “Ein optisches Nachrichtensystem mit Bauelementen der integrierten Optik für die Übertragung hoher Bitraten,” Arch. Elektrotech., vol. 60, no. 6, pp. 353–359, Sep. 1978. [6] H. F. Schlaak, A. Neyer, and W. Sohler, “Electrooptical oscillator using an integrated cutoff modulator,” Opt. Commun., vol. 32, no. 1, pp. 72–74, Oct. 1980. [7] T. C. Damen and M. A. Duguay, “Optoelectronic regenerative pulser,” Electron. Lett., vol. 16, no. 5, pp. 166–167, Feb. 1980. [8] C. Lin, P. L. Liu, T. C. Damen, D. J. Eilenberger, and R. L. Hartman, “Simple picosecond pulse generation scheme for injection lasers,” Electron. Lett., vol. 16, no. 15, pp. 600–602, Jul. 1980. [9] X. S. Yao and L. Maleki, “Multiloop optoelectronic oscillator,” IEEE J. Quantum Electron., vol. 36, no. 1, pp. 79–84, Jan. 2000. [10] J. Lasri, P. Devgan, R. Tang, and P. Kumar, “Ultralow timing jitter 40-Gb/s clock recovery using a self-starting optoelectronic oscillator,” IEEE Photon. Technol. Lett., vol. 16, no. 1, pp. 263–265, Jan. 2004. [11] D. H. Chang, H. R. Fetterman, H. Erlig, H. Zhang, M. C. Oh, C. Zhang, and W. H. Steier, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photon. Technol. Lett., vol. 14, no. 2, pp. 191–193, Feb. 2002. [12] T. Sakamoto, T. Kawanishi, and M. Izutsu, “Optoelectronic oscillator using a LiNbO phase modulator for self-oscillating frequency comb generation,” Opt. Exp., vol. 31, no. 3, pp. 811–813, Mar. 2006. [13] W. Zhou and G. Blasche, “Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultra-low spurious level,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 929–933, Mar. 2005. [14] H. Hasegawa, Y. Oikawa, and M. Nakazawa, “A 10-GHz optoelectronic oscillator at 850 nm using a single-mode VCSEL and a photonic crystal fiber,” IEEE Photon. Technol. Lett., vol. 19, no. 19, pp. 1451–1453, Oct. 2007. [15] S. Bigo, E. Desurvire, S. Gauchard, and E. Brun, “Bit-rate enhancement through optical NRZ-to-RZ conversion and passive time-division multiplexing for soliton transmission systems,” Electron. Lett., vol. 30, no. 12, pp. 984–985, Jun. 1994. [16] D. Norte and A. E. Willner, “Demonstration of an all-optical data format transparent WDM-to-TDM network node with extinction ratio enhancement for reconfigurable WDM networks,” IEEE Photon. Technol. Lett., vol. 8, no. 5, pp. 715–717, May 1996. [17] Y. C. Keh and M. K. Park, “High speed TO-CAN based optical module,” U.S. Patent 200401266A1, Jun. 6, 2006. [18] P. P. Vasil’ev, Ultrafast Diode Lasers: Fundamentals and Applications. Norwood, MA: Artech House, 1995, ch. 3. [19] M. Osinski and M. Adams, “Picosecond pulse analysis of gain-switched 1.55 m InGaAsP laser,” IEEE J. Quantum Electron., vol. 21, no. 12, pp. 1929–1936, Dec. 1985.

[20] G. P. Agrawal, Fiber-Optic Communication Systems. New York: Wiley, 1992, ch. 3. [21] K. K. Gupta, D. Novak, and H. F. Liu, “Noise characterization of a regeneratively mode-locked fiber ring laser,” IEEE J. Quantum Electron., vol. 36, no. 1, pp. 70–78, Jan. 2000. [22] P. A. Yazaki, K. Komori, S. Arai, A. Endo, and Y. Suematsu, “Chirping compensation using a two-section semiconductor laser amplifier,” J. Lightw. Technol., vol. 10, no. 9, pp. 1247–1255, Sep. 1992. [23] C. C. Lin, H. C. Kuo, P. C. Peng, and G.-R. Lin, “Chirp and error rate analyses of an optical-injection gain-switching VCSEL based all-optical NRZ-to-PRZ converter,” Opt. Exp., vol. 16, no. 7, pp. 4838–4847, Mar. 2008. [24] Y. C. Chang, Y. H. Lin, J. H. Chen, and G.-R. Lin, “All-optical NRZ-to-PRZ format transformer with an injection-locked Fabry-Perot laser diode at unlasing condition,” Opt. Exp., vol. 12, no. 19, pp. 4449–4456, Sep. 2004. Yu-Chieh Chi was born in Taipei, Taiwan, in 1983. He received the B.S. degree in electrical engineering and M.S. degree in electro-optical engineering from National Taipei University of Technology (NTUT), Taiwan, in 2005 and 2007, and is currently working toward the Ph.D. degree at the Graduate Institute of Photonics and Optoelectronics, National Taiwan University (NTU), Taipei, Taiwan.

Gong-Ru Lin (S’93–M’96–SM’04) received the Ph.D. degree in electro-optical engineering from National Chiao Tung University, Hsinchu, Taiwan, in 1996. He currently directs the Laboratory of Fiber Laser Communications and Si Nano-Photonics, Graduate Institute of Photonics and Optoelectronics, Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan. His research concerns a broadband researching spectrum covering femtosecond fiber lasers, all-optical data processing, nanocrystallite Si photonics, and millimeter-wave photonic phase-locked loops. Dr. Lin is a member of the Optical Society of America (OSA). He has been elected as a Fellow of SPIE, and since 2008, a Fellow of the Institution of Engineering and Technology (IET) since 2009, and a Fellow of the Institute of Physics (IOP) since 2010. He is currently the chair of the IEEE Photonics Society Taipei Chapter.

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Microwave Human Vocal Vibration Signal Detection Based on Doppler Radar Technology Chien-San Lin, Sheng-Fuh Chang, Senior Member, IEEE, Chia-Chan Chang, Member, IEEE, and Chun-Chi Lin, Student Member, IEEE

Abstract—A speech radar system is presented for extracting speech information from the vocal vibration signal of a human subject. Due to the tiny glottis motion of several millimeters, a coherent homodyne demodulator with high sensitivity is developed to detect reflected radio signal, phase modulated by the vibrating vocal cords. The signal detection quality and system circuit design are described. Measurements of vowels and words, both with the speech radar system and the conventional acoustic microphone system, were conducted and compared. The essential speech information can be reliably obtained from the proposed speech radar, making it more appealing for speech applications in high background acoustic noise environment. Index Terms—Coherent demodulation, Doppler radar, glottis, homodyne demodulation, vocal cords.

I. INTRODUCTION T HAS been acknowledged that speech is a most effective method of human communication. Since the 1960s, scientists and engineers have made significant progress in speech signal processing, such as speech coding and synthesizing, speech recognition, speaker verification and identification, etc. Most techniques are based on the data of spoken voices. However, the recorded acoustic signals easily compose the surrounding background noise, which will considerably degrade the signal quality. In physiology, instruments including the electroglottograph [1] and high-speed video [2] have been adopted to study the motion of human vocal cords. From the microwave approaches, it has been reported that the motions of vocal cords, providing essential information associated with phonation, can be detected by electromagnetic pulsed radar [3] and continuous-wave radars [4], [5]. The first electromagnetic pulsed radar system, called the glottal electromagnetic micropower sensor (GEMS), was presented by a research team at the Lawrence Livermore National Laboratory [3]. In [6], a thorough description of a speech phonation mechanism and vocal tract excitation model are given. Numerous extended applications were reported based on GEMS

I

Manuscript received February 12, 2010; accepted April 20, 2010. Date of publication July 15, 2010; date of current version August 13, 2010. This work was supported in part by the National Science Council, Taiwan. C.-S. Lin was with the Department of Electrical Engineering, National Chung Cheng University, Chiayi 62102, Taiwan. He is now with MIPRO Electronics, Chiayi 600, Taiwan (e-mail: [email protected]; [email protected]). S.-F. Chang and C.-C. Chang are with the Department of Electrical Engineering and the Department of Communications Engineering, Center for Telecommunication Research, National Chung Cheng University, Chiayi 62102, Taiwan (e-mail: [email protected]; [email protected]). C.-C. Lin is with the Department of Electrical Engineering, National Chung Cheng University, Chiayi 62102, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2052968

signals, including speaker recognition [7], [8], speech enhancement [9], speech recognition [10], [11], and speech coding [12]. Using a different detection mechanism, the continuous-wave coherent speech radar was developed in [4] and [5] with special emphasis on low-cost fabrication. Similar continuous-wave coherent radars have also been seen in the detection of the human cardiopulmonary signal, where the heartbeat rate is to be precisely detected from a remote distance [13]–[18]. The difference is that the speech radars concern the waveform quality of the vocal vibration signal, while the heartbeat radars focus on precise heartbeat rate detection. The works of [3]–[5] achieved successful vocal vibration signal detection, but the detailed theoretical analysis of signal detection quality and the associated system circuit design are not given. With envisioning the great potential of vocal vibration detection in speech applications, this paper is aimed to a complete description of signal detection and system circuit design of the speech radar. The detection principle and the distortion of signal demodulation are described in Section II. The system circuit design is given in Section III. The experimental validation of developed speech radar circuitry, and experiments on vocal vibration detection of phonating vowels and words, are presented in Section IV. Finally, a conclusion is drawn in Section V. II. DETECTION OF VOCAL VIBRATION SIGNAL A. Coherent Homodyne Demodulation The human vocal vibration, as well as the cardiopulmonary movement, is in a periodic motion, which can be detected based on the Doppler radar principle. The cardiopulmonary signal detection focuses on the accurate detection of heartbeat and respiration rates [15]–[18], while the vocal vibration detection puts emphasis on the recovery of the original vocal excitation signal with minimum distortion. The single-path homodyne demodulation is adopted for the vocal vibration signal detection. The periodic opening and closing of vocal cords induces a phase modulation on the impinging electromagnetic wave. To correctly recover the phase-modulating signal, the coherent homodyne denotes transceiver is used, as illustrated in Fig. 1, where the vocal vibration displacement and is the averaged distance between the antenna and the vocal cords. The transmitted signal is generated from a phase-locked local oscillator, which is amplified and radiated by the antenna toward the throat of is fed into the receiver to cohuman subject. A portion of . herently demodulate the received signal The received signal , intercepted by the receiver antenna, contains three components: the desired phase-modulated signal

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Fig. 1. Block diagram of the speech radar system. Fig. 2. Calculated harmonics of the demodulated signal for the single-tone vocal vibration excitation x(t) = 1L sin (! t) for the case of  = 36 .

, the undesired interference , and the leakage component from the transmitter. The is the backscattered signal from the vibrating vocal cords, which can , where is be written as the angular frequency of the transmitted electromagnetic wave, is the wavelength inside the human body, denotes the represents the round-trip received signal amplitude, and phase difference between the received signal and the downrepresents the reconverting local oscillation signal. flected signals from surrounding stationary organs and denotes the leakage from the transmitter due to finite circuit isoand lation. Thus, . Since the same local oscillation signal is used for transmission and reception, the phase noise superimposed in the transmitted signal is carried over into the received components. Therefore, the down-converted baseband signal can be expressed as

denotes the angular vibrating frequency of signal and vocal cords. Equation (2) then becomes

(3) can be decomposed By applying the Fourier expansion, into a summation of dc term and harmonics as follows:

(4)

(1) denotes the total receiver gain. Since the same local In (1), oscillator is used for transmission and reception, the residual , , and ) are dramatically suppressed phase noises ( [15]. The residual phase noise can then be neglected in the following distortion analysis. By the rejection from the bandpass filter in baseband, the near-dc components of down-converted and become negligible. Consequently, the normalized baseband signal becomes (2)

B. Distortion of Extracted Vocal Vibration Signal 1) Single-Tone Voiced Speech Excitation: Assuming that a single-tone voiced speech is excited, i.e., , where is the amplitude of vocal vibration

is the th-order Bessel function of the first kind. where is From (4), it is seen that the original excitation signal maximally recovered when is odd multiples of , but equals to even multiples of . Therefore, vanished when a phase shifter is inserted into the receiver local oscillator path to obtain maximal recovered signal intensity in the system implementation. is tuned to an odd multiple of , the Even though . For human body motion can cause a phase tolerance to example, an unintended random body motion of 1 mm induces a 2.4 tolerance at 1 GHz and an intended 15-mm body motion causes a 36 phase tolerance. This phenomenon gets worse when higher operation frequency is adopted. At 15 GHz, even the unintended random motion of 1 mm will cause a dramatic 36 tolerance. Therefore, in the following calculation, the phase is included in the optimal . The first five tolerance harmonics of the demodulated baseband signal for the case of are shown in Fig. 2. The desired signal intensity increases with respect to the normalized vocal cords displacement and peaks around . However, the higher order harmonics increase as well. Therefore, the total harmonic

LIN et al.: MICROWAVE HUMAN VOCAL VIBRATION SIGNAL DETECTION BASED ON DOPPLER RADAR TECHNOLOGY

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Fig. 3. THD versus the normalized displacement L= for a single-tone voice excitation with different random-motion phase tolerances.

distortion (THD) is calculated for the signal quality assessment, defined in (5) at the bottom of this page. Fig. 3 shows the calculated THD with respect to the normalfor , , , and , reized displacement and also with the spectively. The THD increases with . Fig. 3 gives a guidebody random-motion phase tolerance line for selecting the operation frequency of the speech radar system. For a design example, the vocal cords displacement is around 0.5 mm, the random body motion is 1 mm, and the relative dielectric constant of the interior neck is 40 [19]. If the specified THD is less than 0.15%, the operation frequency must be selected at less than 1 GHz. 2) Multitone Voiced Speech Excitation: In general, the vocal is nonsinusoidal and is more appropricords displacement ately expressed as a summation of sinusoidal tones (6) where is the th harmonic of the vocal vibration displacement . Therefore, the demodulated baseband signal becomes (7), shown at the bottom of this page.

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Fig. 4. Simulated results of the demodulated baseband spectra for a three-tone excitation.

It can be seen from (7) that the nonlinearity of the coherent vocal signal demodulation causes vast intermodulation harmonics. Consider a voiced speech model, which has three in-phase tones with normalized amplitudes of 0 dB at 100 Hz, 0 dB at 200 Hz, and 6 dB at 300 Hz. The calculated spectra is defined of baseband signal are illustrated in Fig. 4, where as . When the normalized displacement of is 0.01 and , the demodulated signal has three primary tones at 100, 200, and 300 Hz, respectively, and all other intermodulated harmonics are smaller than 50 dB. However, when is increased to 0.1, the intermodulation and even worse distortion is degraded to 24 dB for . to 14 dB for III. SYSTEM CIRCUIT DESIGN A. System Description The speech radar was realized with the circuit block diagrams in Fig. 5. A phase-locked-loop frequency synthesizer generates a stable continuous-wave signal, where part of it is fed back through a phase shifter to the receiver and the other part, filtered by a low-pass filter for harmonic suppression, is radiated by the transmitter antenna. As the transmitted wave impinges the vibrating glottis, a phase-modulated signal is back-scattered

root-mean-squared summantion of harmonics root-mean-squared value of fundamental harmonic (5)

(7)

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Fig. 5. Block diagram of the speech radar system.

Fig. 6. Schematic circuit diagrams of the key building blocks.

and intercepted by the receiver antenna. The reflections from other organs can be picked up as well, depending on the antenna pattern. The received signals, including the desired phase-modulated signal and other unwanted interferences, are filtered by a bandpass filter to eliminate out-of-band interferences. The filtered signal is then amplified by the low-noise amplifier and coherently down-converted into the baseband. In the baseband domain, a low-frequency amplifier, followed by an antialiasing filter, optimizes the signal-to-noise ratio for driving the followed data acquisition (DAQ) module. The DAQ samples the baseband signal for executing data computation in the digital domain. The DAQ module also controls the phase shifter to meet requirement for obtaining a maximal basethe band signal intensity, as discussed in Section II-B. In addition to the speech radar system, an acoustic microphone system is

also incorporated to receive the acoustic speech, which is used for comparison with the electromagnetic vocal vibration signal. B. Circuit Design The schematic circuit diagrams of key building blocks are shown in Fig. 6. According to the analysis in Section II, the operating frequency is selected at 925 MHz. The circuit details are described as follows. 1) Frequency synthesizer: A phase-locked-loop integratedcircuit MB15E03 and microcontrol unit PIC12F629 are integrated with a 925-MHz Colpitts voltage-controlled oscillator to realize the frequency synthesizer. The synthesized frequency range is from 905 to 945 MHz in a 25-kHz step. The synthesized microwave signal has a 2-mW output power with phase noise of 120 dBc/Hz at 100-kHz offset from the carrier.

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Fig. 8. Measured and calculated THD for a 100-Hz single-tone excitation.

Fig. 7. Photograph of the developed speech radar system. (a) Antenna. (b) Transceiver circuits.

2) Power splitter: The power splitter was realized by a lumped T topology, which has a measured insertion loss smaller than 1 dB from 905 to 945 MHz. The amplitude imbalance and phase imbalance are within 0.5 dB and 5 , respectively. The output isolation is greater than 20 dB. 3) Transmitter low-pass filter: The transmitter low-pass filter was implemented by a third-order Butterworth topology. The circuit has an insertion loss less than 1 dB in 905–945 MHz and a stopband rejection greater than 20 dB in 1800–1900 MHz. 4) Receiver bandpass filter: A surface-acoustic-wave filter, centered at 925 MHz, was used as the receiver filter. The 1-dB passband is 6 MHz and the insertion loss is 4.2 dB, with stopband rejection better than 36 dB at 40 MHz away from the passband center. 5) Low-noise amplifier and mixer: The low-noise amplifier was designed with a common-emitter bipolar transistor, which has a measured gain of 15 dB, noise figure of of 2 dBm. The SA601 was used 1.6 dB, and input as a coherent down-mixer, which contains a pre-amplifier and a wide dynamic-range mixer. The pre-amplifier has an 11-dB gain, 1.6-dB noise figure, and 0-dBm input third-order intercept point (IP3). The mixer has a conversion gain of 6.5 dB, noise figure of 9.5 dB, and input IP3 of 2 dBm. 6) Phase shifter: A reflection-type phase shifter was designed for the phase tuning of local oscillation signal [20]. It includes a lumped-element branch-line coupler, terminated with two series-resonant surface-mounted silicon hyperabrupt-junction varactor diodes. The phase tuning range is larger than 190 with a 2-dB insertion loss. 7) Low-frequency amplifier and anti-aliasing lowpass filter: Operational amplifiers are used to realize the low-frequency gain amplifier and active low-pass filter. The

Fig. 9. Measured results of the demodulated baseband spectra for a three-tone excitation.

Fig. 10. Experimental setup of the detection of vocal vibration signal and acoustic signal. (a) Targeted person with microwave antennas and acoustic microphone. (b) Speech radar circuit.

low-pass filter has a cutoff frequency of 4 kHz to reject unwanted interference, which acts as the antialiasing filter for the followed data-acquisition module. 8) DAQ module: The National Instrument DAQ module USB9233 was used for baseband vocal vibration signal acquisition. It has four simultaneous process channels with 24-bit resolution, 102-dB dynamic range, and a maximum sampling rate of 50 kS/s. 9) Antennas: The beverage (wire) antenna at 925 MHz was designed as a near-field induction antenna. The photograph of the developed speech radar system is illustrated in Fig. 7.

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Fig. 12. Measurement waveforms of the word “Jason” by the acoustic microphone system (acoustic signal) and the speech radar system (vocal vibration signal). (a) Time-domain waveforms. (b) Frequency-domain spectra.

Fig. 11. Measurement waveforms of the vowel /a/ by the acoustic microphone system (acoustic signal) and the speech radar system (vocal vibration signal). (a) Time-domain waveforms. (b) Frequency-domain spectra. (c) Detailed spectral waveform of the fundamental and second harmonics.

ensures the modulator distortion is kept well below the demodulation distortion. First, a 925-MHz signal, phase modulated by a 100-Hz sinusoidal waveform, was generated from the above phase modulator and fed into the implemented speech receiver. The THD of the output demodulation signal is illustrated in Fig. 8. The meaand 2.6%-23.5% sured THD is within 1.1%–6.0% for for when is in the range of 0.01–0.1. The measured results agree very well with the calculation from (5). Second, another 925-MHz signal is modulated by a three-tone waveform, containing 100-, 200-, and 300-Hz sinusoidal frequency with normalized amplitudes of 0, 0, and 6 dB, respectively. The measured spectra are plotted in Fig. 9. As expected, for serious intermodulation products appear at . This result is consistent with the simuthe case of lation in Fig. 4. These experimental results validate the implemented speech radar circuits.

IV. EXPERIMENTAL RESULTS B. Vocal Vibration Signal Detection A. Experimental Validation of Implemented Coherent Speech Receiver Before conducting the voiced speech detection of human subjects, the implemented speech receiver was validated experimentally. An analog voltage-controlled phase shifter [20] was used as a phase modulator to produce the desired phase-modulated electromagnetic signal. The modulator distortion was measured and illustrated in Fig. 8, which indicates that the modup to 0.1. This ulation distortion is less than 1.3% for

The experiment setup for the detection of vocal vibration signal and acoustic signal is illustrated in Fig. 10. For the speech radar system, antennas are lightly placed on the larynx area of the human subject. Meanwhile, the acoustic signal is also recorded simultaneously by the microphone for data comparison. A 30-year-old male participant phonates in a variety of vowels and words. The experiment on the vowel of /a/ was conducted. The time-domain waveforms of the speech radar signal and acoustic

LIN et al.: MICROWAVE HUMAN VOCAL VIBRATION SIGNAL DETECTION BASED ON DOPPLER RADAR TECHNOLOGY

signal are displayed in Fig. 11(a), which indicate the same fundamental period, but are quite different on the waveform shape. Fig. 11(b) and (c) further illuminate their similarities and differences in the frequency domain. The fundamental and second harmonics detected in the speech radar signal are consistent with those in the acoustic signal, while extra higher harmonics exist in the acoustic signal. This result can be understood from the voiced speech articulation. When generating voiced speech, the vocal cords are pushed open by the air pressure from the lungs and then close because of the folds’ natural elasticity [21]. Therefore the vocal cords vibrate at the fundamental frequency with few harmonics, from which the waveform is detected by the speech radar system. Instantly, with the compressed air flow to the super-glottis, it is modulated by the vocal tract, tongue, lips, and jaw, generating rich harmonics in the spoken acoustic waveform. A further experiment was conducted on word detection. The word “Jason” was pronounced and the detected waveforms are shown in Fig. 12. The time-domain waveforms from the speech radar system have a consistent segment pattern when compared with those from the acoustic microphone system. From the aspect of the frequency domain, the fundamental and second harmonics of the speech radar signal are at the same frequencies with those from the acoustic signal, but they have different relative amplitudes due to the modulation of the vocal tract, tongue, lips, and jaw. V. CONCLUSION In this paper, the thorough analysis and system design of the speech radar is presented. To be able to detect the small phase fluctuation due to the tiny vocal vibration displacement, typically in millimeters, a coherent homodyne transceiver is used,where the phase noise can be significantly suppressed by the phase coherence of the transmitted and received signals. The THD is derived to assess the detected vocal vibration signal quality, where the single-tone and multitone models for the vocal cords waveform are considered. The effect of intended and unintended body motion is also included. From the derived result, the operation frequency of speech radar can be properly determined. Experiments on the detection of vowel and word phonations were conducted, while the conventional acoustic microphone detection was performed, as well for the purpose of comparison. The results show that the measured speech radar signals have excellent consistency with the acoustic signals, which validates the speech detection capability of the proposed radar system. The vocal vibration waveforms were found to have power content distribution dominantly over the first few harmonics, which is useful to the estimation of the sound pitch. In contrast, the acoustic waveforms contain richer harmonics, generated from the modulation by the vocal organs such as tract, lips, and jaw. The speech radar is essentially immune to background acoustic noise, making it more appealing for applications in high background acoustic-noise environments where acoustic signals are inaccessible or blocked. It shows a variety of potential applications, including the background acoustic noise

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removal, speaker verification, and identification, as well as medical uses.

REFERENCES [1] D. G. Childers and J. N. Larar, “Electroglottography for Laryngeal function assessment and speech analysis,” IEEE Trans. Biomed. Eng., vol. BME-31, no. 12, pp. 807–817, Dec. 1984. [2] T. Wittenberg, P. Mergell, M. Tigges, and U. Eysholdt, “Quantitative characterization of functional voice disorders using motion analysis of high-speed video and modeling, acoustics, speech, and signal processing,” in IEEE Int. Acous., Speech, Signal Process. Conf., Apr. 1997, vol. . 3, pp. 1663–1666. [3] J. F. Holzrichter, G. C. Burnett, L. C. Ng, and W. A. Lea, “Speech articulator measurements using low power EM wave sensor,” J. Acoust. Soc. Amer., vol. 103, no. 1, pp. 622–622, 1998. [4] C. Liang, “Novel microwave and millimeter-wave radar technologies and applications,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Univ. California at Davis, Davis, CA, 2001. [5] Y. Yang, “Radar based speech analysis system,” Master’s thesis, Dept. Elect. Comput. Eng., Univ. California at Davis, Davis, CA, 2000. [6] G. C. Burnett, “The physiological basis of glottal electromagnetic micropower sensor (GEMS) and their use in defining an excitation function for the human vocal tract,” Ph.D. dissertation, Dept. Appl. Sci., Univ. California at Davis, Davis, CA, 1999. [7] T. J. Gable, “Speaker verification using acoustic and glottal electromagnetic micropower sensor (GEMS) data,” Ph.D. dissertation, Dept. Appl. Sci., Univ. California at Davis, Davis, CA, 2000. [8] W. M. Campbell, T. F. Quatieri, J. P. Campbell, and C. J. Weinstein, “Multimodal speaker authentication using nonacoustic sensors,” in Proc. Multimodal User Authenatication Workshop, 2003, pp. 215–222. [9] L. C. Ng, G. C. Burnett, J. F. Holzrichter, and T. J. Gable, “Denoising of human speech using combined acoustic and EM sensor signal processing,” Proc. Int. Acoust., Speech, Signal Process. Conf., vol. 1, pp. 229–232, 2000. [10] B. Raj and R. Singh, “Feature compensation with secondary sensor measurements for robust speech recognition,” in Proc. EUSIPCO, 2005, 4 pp. [11] C. Demiroglu and D. V. Anderson, “Broad phoneme recognition in noisy environments using the GEMS device,” in Proc. Asilomar Signals, Syst. Comput. Conf., 2004, vol. 2, pp. 1805–1808. [12] T. F. Quatieri, K. Brady, D. Messing, J. P. Campbell, W. M. Campbell, M. S. Brandstein, C. J. Clifford, J. D. Tardelli, and P. D. Gatewood, “Exploiting nonacoustic sensors for speech encoding,” IEEE Trans. Audio Speech Language Process., vol. 14, no. 2, pp. 533–544, Mar. 2006. [13] K. M. Chen, D. Misra, H. Wang, H. R. Chuang, and E. Postow, “An -band microwave life-detection system,” IEEE Trans. Biomed. Eng., vol. BME-33, no. 7, pp. 697–702, Jul. 1986. [14] K. M. Chen, Y. Huang, J. Zhang, and A. Norman, “Microwave life-detection systems for searching human subjects under earthquake rubble and behind barrier,” IEEE Trans. Biomed. Eng., vol. 47, no. 1, pp. 105–114, Jan. 2000. [15] A. D. Droitcour, O. Boric-Lubecke, V. M. Lubecke, J. Lin, and G. T. A. Kovac, “Range correlation and I/Q performance benefits in single-chip silicon Doppler radars for noncontact cardiopulmonary monitoring,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 838–848, Mar. 2004. [16] Y. Xiao, J. Lin, O. Boric-Lubecke, and V. M. Lubecke, “Frequency tuning technique for remote detection of heartbeat and respiration using -band,” IEEE Trans. low-power double-sideband transmission in Microw. Theory Tech., vol. 54, no. 5, pp. 2023–2032, May 2006. [17] B. Park, O. Boric-Lubecke, and V. M. Lubecke, “Arctangent demodulation with DC offset compensation in quadrature Doppler radar receiver systems,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 5, pp. 1073–1079, May 2007. [18] C. Li and J. Lin, “Random body movement cancellation in Doppler radar vital sign detection,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 3143–3152, Dec. 2008. [19] T. Yilmaz, T. Karacolak, and E. Topsakal, “Characterization and test of a skin mimicking material for implantable antennas operating at ISM band (2.4 GHz–2.48 GHz),” IEEE Antennas Wireless Propag. Lett., vol. 7, no. 11, pp. 418–420, Nov. 2008.

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[20] C. S. Lin, S. F. Chang, C. C. Chang, and Y. H. Shu, “Design of a reflection-type phase shifter with wide relative phase shift and constant insertion loss,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 9, pp. 1862–1868, Sep. 2007. [21] C. Rowden, Speech Processing. New York: McGraw-Hill, 1992.

HBT, and pseudomorphic HEMT (pHEMT) technologies, multifunctional RF transceivers, smart-antenna RF system, and high-power microwave sources. Prof. Chang is a member of Phi Tau Phi and Sigma Xi.

Chien-San Lin was born in Chiayi, Taiwan, in 1976. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from National Chung Cheng University, Taiwan, in 1996, 2000, and 2010, respectively. In 2003, he joined MIPRO Electronics, Chiayi, Taiwan, where he is currently a Research and Development Engineer engaged in the development of professional microphone systems. His research interests include RF circuit designs in wireless communication systems.

Chia-Chan Chang (S’99–M’04) was born in Tainan, Taiwan, in 1973. She received the B.S. degree in communication engineering from National Chiao-Tung University, Hsinchu, Taiwan, in 1995, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of California at Davis (UCD), in 2001 and 2003, respectively. From 1995 to 1997, she was a Full-Time Teaching Assistant with the Department of Electronics Engineering, National Chiao-Tung University. Since February 2004, she has become an Assistant Professor with the Department of Electrical Engineering, National Chung-Cheng University, Chiayi, Taiwan. She also holds a joint-appointment with the Department of Communications Engineering, National Chung-Cheng University. Her research currently focuses on phased antenna array technology developments, microwave/millimeter-wave circuit designs, and the application of radar systems.

Sheng-Fuh Chang (S’83–M’84–SM’07) received the B.S. and M.S. degrees in communications engineering from National Chiao-Tung University, Taiwan, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Wisconsin–Madison, in 1991. He was involved with high-power microwave and millimeter-wave sources such as free-electron lasers and Cerenkov masers with the Center for Plasma Theory and Computation, University of Wisconsin–Madison. In 1992, he joined the Hyton Technology Corporation, where he was responsible for C - and Ku-band satellite low-noise down-converter and multichannel multipoint distribution system (MMDS) transceivers. In 1994, he joined the Department of Electrical Engineering, National Chung Cheng University, Taiwan, where he is currently a Full Professor with the Department of Electrical Engineering and Vice Director of the Center for Telecommunication Research. His research interests include microwave and millimeter-wave integrated circuits (ICs) with CMOS,

Chun-Chi Lin (S’06) was born in Chia-Yi, Taiwan, in 1979. He received the B.S. degree in electrical engineering from Yuan Ze University, Taoyuan, Taiwan, in 2002, the M.S. degree in electrical engineering from Chung-Cheng University, , Chiayi, Taiwan, in 2006, and is currently working toward the Ph.D. degree in electrical engineering at Chung-Cheng University. His primary research interests are 3-D human electromagnetic (EM) models, coated antennas, chip antennas, and chip antenna measurement systems. His additional research interests include antenna arrays and RF/microwave circuits for wireless applications.

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Exploring Joint Tissues With Microwave Imaging Sara M. Salvador, Elise C. Fear, Member, IEEE, Michal Okoniewski, and John R. Matyas

Abstract—In this paper, a feasibility study of the application of the tissue sensing adaptive radar technique for imaging of the knee is presented. Selected tissues of the knee joints were characterized, from the dielectric properties point of view, through measurements on bovine tissue samples using a high precision dielectric probe. Based on the acquired data, numerical studies were conducted on a realistic finite-difference time-domain model of a meniscus, as well as on a model of a patellar tendon to investigate the possibility of detecting lesions. To confirm our observations, measurements on healthy and experimentally torn menisci and tendons from bovines were performed. Overall, the results suggest that microwave imaging has potential for knee imaging. Index Terms—Dielectric spectroscopy, knee joint, ligaments, meniscus, microwave imaging.

I. INTRODUCTION

T

HE KNEE is one of the most frequently injured joints in the body [1]. The type of injury or disease is related to the age and the activity of the patient: younger people are more likely to suffer from traumatic injuries including meniscal and ligamentous pathologies, while older patients tend to be subject to degenerative joint diseases. Pathologies of the knee often involve tendons, ligaments, and menisci, and are common in humans and animals, including cattle and horses. A variety of imaging techniques are used to diagnose knee pathology in humans, including radiography, arthrography, computed tomography, magnetic resonance, and ultrasound [2]. For the diagnosis of knee diseases of large mammals, however, these methods are typically too expensive or complicated to be applied due to the need to place a sensor in direct contact with the joint of the animal. Microwave imaging has been studied as a complementary technique for detection of early stage breast tumors [3]–[7] and has also been proposed for the analysis of cardiac tissues, soft tissues, and bones [8]–[10]. Microwave methods are based on the analysis of the scattering of a wave propagating in the structure under examination: scattering originates from tissue discontinuities, specifically changes in the dielectric properties. The high conductivity of the tissues involved and the complex Manuscript received April 26, 2010; revised April 27, 2010; accepted May 20, 2010. Date of publication July 12, 2010; date of current version August 13, 2010. S. M. Salvador is with the Antenna and Electromagnetic Compatibility (EMC) Laboratory, Politecnico di Torino, Turin 10129, Italy (e-mail: sara. [email protected]). E. C. Fear and M. Okoniewski are with the Schulich School of Engineering, University of Calgary, Calgary, AB Canada T2N 1N4. J. R. Matyas is with the Faculty of Veterinary Medicine, University of Calgary, Calgary, AB, Canada T2N 1N4. 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.2010.2052662

geometry of the knee joint represent a challenge for the application of microwave techniques. Nevertheless, the dimensions of the structures and their accessibility are characteristics that make the knee a suitable target for testing and applying microwave imaging. Microwave imaging has the potential to become a complementary or alternative diagnostic imaging tool to X-ray, magnetic resonance, and ultrasound due to its low cost and capability of working without direct contact between the sensor and the joint under examination. Moreover, it is based on the use of nonionizing radiation, and at the frequencies used for imaging does not pose any known risk for the patient. In this paper, we report a preliminary study on the application of radar-based microwave imaging to knee anatomy. We adapt the previously introduced tissue sensing adaptive radar techniques and algorithms (first introduced in [11]) to the knee. In particular, we seek to detect lesions of clinically relevant sizes and shapes in the menisci, ligaments, and tendons. Further preliminary results, both numerical and experimental, that reinforce our initial findings with this technique ([12]) are presented here. In Section II, we report dielectric properties for the meniscus and ligaments at microwave frequencies. Together with previously published data for other tissues, these results provide the information required to assess the feasibility of microwave imaging of the limbs. The feasibility of detecting tears in the meniscus, and the supporting simulations and experiments, are discussed in Section III. In Section IV, the possibility of detecting lesions of the ligaments and tendons is analyzed. Finally, Section V is dedicated to discussion and conclusions.

II. ELECTRICAL PROPERTIES OF MENISCI AND LIGAMENTS The main components of the knee joint are bones (femur, tibia, fibula, and patella), articular cartilage, ligaments (anterior cruciate, posterior cruciate, lateral collateral, medial collateral, and patellar) and the lateral and medial menisci. While the microwave frequency properties of both bones and cartilage have already been characterized [13], to the best of the authors’ knowledge, no information is available for ligaments and menisci. Menisci and ligaments, as well as cartilage, are composed of collagen fibers and water. Due to the differences in composition and the water content (80% for cartilage and 70% for menisci and ligaments [14], [15]), we expect the dielectric constant of menisci and ligaments to be similar to, but lower than, measurements reported for cartilage. A technique developed earlier in our laboratory and described in [16] was used to measure the dielectric parameters of various components of the knee with an open-ended coaxial probe. Relative permittivity and conductivity of excised bovine menisci and cruciate ligaments (purchased as fresh intact knees from

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Fig. 1. Dielectric measurements: (a) on a whole bovine meniscus and (b) on a sample of meniscus.

a meat supplier) were measured over a frequency range extending from 50 MHz to 13.51 GHz. Samples were kept refrigerated for short-term storage, but were equilibrated to room temperature during measurement, and were kept moist throughout the process. Measurements were performed within one week after the death of the animal. A large number of measurements were performed with different samples and with different approaches. For selected samples, the measurements were taken both on the external and internal (cut) surface of the tissue, i.e., the sample was cut and the probe was pressed against the cut section. Fig. 1(a) and (b) shows, respectively, measurements on a whole meniscus and on a section of the meniscus. No appreciable differences were observed between the data obtained using these two approaches (i.e., external versus internal) for either the meniscus or ligaments. All the data are therefore presented together in the following graphs, where each line represents a different measurement. In total, five different menisci were used for measurements; for each meniscus, five or more sets of data (relative permittivity and conductivity) were collected. Eight cruciate ligaments were excised and multiple measurements taken on each sample for a total of 62 measurements. An initial analysis of the data indicated several clear outliers: in particular, all the data with a standard deviation (evaluated on permittivity data) higher than 6 with respect to the average of all the measurements were removed. All the data identified as outliers were characterized by very low relative permittivity and conductivity. We assumed, based on our previous experience with this system, such low dielectric properties resulted from incomplete contact between the probe and sample. In Fig. 2, the remaining measurements are presented. From Fig. 2, it is evident that there is variability in the dielectric properties of the considered tissues, both within and among specimens. The variability of response within the meniscus is not surprising, given the complex inhomogeneous histological structure of the meniscus and the spatial variation in meniscal composition, ranging from a stiff fibrous connective tissue, rich in nerves endings and blood vessels at the periphery, to an avascular fibrocartilage and hyaline cartilage interior [14]. Some inhomogeneities in the structure of ligaments exist, particularly in the bundling of collagen fibers, and, presumably, the variability in dielectric properties reflects these architectural inhomogeneities of ligament. To perform realistic numerical simulations, a one-pole Debye model was constructed to fit the average properties of the meniscus and the cruciate ligaments, as shown in Fig. 3. In Fig. 3(b), the dielectric properties of tendons and cartilage from

Fig. 2. (a) Measured dielectric properties of bovine meniscus (31 measurements are shown) and (b) of bovine cruciate ligaments (53 measurements are shown).

[13] are also shown. They are both very similar in composition to ligaments and it may be noted that the results from [13] are comparable to the measurements reported here, especially if considering the aforementioned variability. On the other hand, differences in methods and materials (conservation of samples, species of animal from which the tissue is extracted) may cause further variability in the measured dielectric properties. III. DETECTION OF MENISCAL TEARS With knowledge of the dielectric properties of the meniscus, we chose to investigate the detection of meniscal tears. Simulations indicating the potential for difference imaging are presented. This observation is also verified experimentally by imaging several menisci with and without tears. A. Simulations To define whether the detection of meniscal tears is feasible with a radar-based microwave imaging technique, 3-D numerical simulations of a meniscus illuminated with a short-time pulse of microwaves were performed. The finite-difference time-domain (FDTD) method was used (SEMCAD, SPEAG, Zurich, Switzerland). The antenna employed was the balanced antipodal Vivaldi antenna developed for breast cancer detection [17]. The meniscus model was built from the segmentation of an image of a bovine meniscus. To simplify the model, thickness was assumed uniform and the dielectric properties were assigned according to the Debye model mentioned in Section II. Both the meniscus

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Fig. 5. Example of a differential image of a lesioned meniscus obtained with numerical data. In the upper right corner, the model of the meniscus employed is shown. In Figs. 5, 10, and 11, the white lines represent the actual geometry of the imaged object, the red lines (in online version) represent the tear, the magenta stars (in online version) represent the positions into which the antenna is moved, and the colormap (in online version) represents the value of intensity associated with each pixel of the image by the focusing algorithm (high values of intensity on the reconstructed map correspond to scattering points in the imaged object).

Fig. 3. Average dielectric properties of: (a) meniscus and (b) cruciate ligaments given by measurements (continuous line) and by the Debye equation (dashed line). For comparison, in (b), the tendon’s and cartilage’s properties measured by Gabriel et al. [13] are also reported.

Fig. 4. Numerical model with meniscus and antenna.

and antenna were immersed in canola oil, characterized by a relative permittivity of 2.5 and a conductivity of 0.04 S/m. In Fig. 4, the models of the meniscus and the antenna are shown. The system is excited with a Gaussian derivative pulse [6] with a maximum frequency content near 3.6 GHz and full-width half-maximum (FWHM) bandwidth from 1.16 to 6.92 GHz. Reflections are observed at the port where the excitation is applied. Data are collected with the antenna scanned to 16 different positions around the center of the meniscus in a single plane, with an average distance between the antenna and the surface of 24.3 mm. This permits reconstruction of a 2-D image of the cross section of the meniscus. To create the image, data are processed using selected components of an algorithm developed for microwave breast cancer detection [18]. In particular, calibration is performed to remove the contributions from the antenna

and the external environment, surface recognition is applied to identify the position and shape of the imaged object, and the signals are focused to combine the information collected from different antenna’s positions to highlight scattering regions. More details on the different steps of the algorithm are contained in [18]. Simulations were conducted as follows. First, a complete acquisition (i.e., collection of signals from the 16 positions of the antenna) was performed with a “healthy” meniscus. Secondly, a tear was modeled as a cut of selected dimensions, extending through the whole thickness of the meniscus, and a new acquisition was performed. Data collected with the healthy meniscus and with the torn one were then processed and compared. Detailed numerical results are not shown here for brevity; however, a summary of our observations follows. First, there is a difference between the amplitude of signals collected with a healthy meniscus and with a lesioned one. When the antenna is located in close proximity to the tear, this difference can reach 35%. As the antenna is scanned away from the tear, this difference decreases to less than 10%. Nevertheless, the average energy of the difference signals is about 0.2% of the average energy of the collected signals so it is challenging to directly observe the difference between the two images. This is confirmed by the reconstructed image: by performing the focusing with the signals collected with the torn meniscus, the tear is not readily visible. However, when the difference between the two images (the one obtained with the healthy meniscus and the one obtained with the torn one) is computed, a peak of intensity is clearly present in the location corresponding to the tear. Fig. 5 shows clear detection of a 1-cm-long 2-mm-wide tear, suggesting the potential of difference imaging for monitoring the meniscus. B. Experimental Measurements To verify the observations made via simulations, experimental measurements are performed using a prototype scanner designed for breast cancer detection and the Vivaldi antenna. The experimental setup is shown in Fig. 6 and consists of a tank of canola oil in which the antenna and the meniscus are positioned. The tank rotates in order to scan the antenna around

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Fig. 6. Experimental setup. Fig. 8. (a) Examples of two corresponding signals collected before and after the movement of the meniscus. The two circles highlight the minima, which will be realigned. (b) Surface estimation obtained with the signals collected with a healthy meniscus (blue continuous line in online version), and with the torn one before (red dotted–dashed line in online version) and after (dashed green line in online version) realignment.

2



Fig. 7. (a) Actual ( marks) and estimated ( marks) surface location. Stars indicate the positions in which the antenna is moved. (b) Relative error in the estimation of surface location. marks represent the computed error.

2

the meniscus. The antenna is fixed to a Plexiglas arm in the tank, and connected to a vector network analyzer (VNA) via a coaxial cable. The bovine menisci used here have shape and size close to the numerical model (which was obtained by a segmentation of an image of a bovine meniscus) and were managed as described in Section II. The electric properties are expected to be similar to the ones assumed for simulations, as the Debye model employed for the numerical study was directly derived from the dielectric measurements performed on menisci purchased and managed in the same way (see Section II). was moved to 36 different positions around The antenna the meniscus, and the reflection coefficient ( ) was collected over 801 points in the frequency range of 50 MHz–13.51 GHz. Frequency data were then, in a post-processing phase, converted into the time domain using an inverse chirp z-transform to obtain the response of the system to the Gaussian derivative pulse used in simulations [18]. For every analyzed sample, an acquisition was first performed with the healthy meniscus. The meniscus was then removed, cut, and repositioned inside the scanner and a new acquisition followed. With measured data, the surface location estimation algorithm behaves relatively well with average error of about 6%, as shown in Fig. 7. Analogous to simulated data, the error is considerably higher in locations corresponding to the positions in which the antenna is facing the more irregular part of the meniscus. This is likely to be due to the nonconvex shape of the meniscus in the internal region. The surface location plays a fundamental role in an experimental scenario, as the two images have to be realigned before their difference is calculated. The realignment is performed on

Fig. 9. Difference between signals collected with and without a tear.

the collected signals rather than on the image itself. The collected signals are dominated by the reflections from the surface. Once the second set of data is collected, the signals are automatically realigned one-by-one to the corresponding signals of the first set, based on the position of the maximum (or minimum), as shown in Fig. 8(a). Note that this realignment procedure is based in the collected data rather than the actual position of the antennas. In this way, it is possible to compensate for rotations or translations of the imaged object to allow a correct comparison of the two images. Fig. 8(b) compares the surface estimation obtained with the signals collected with a healthy meniscus (blue line, in online version, with marks), and with the torn one before (red line, in online version, with stars) and after (green line, in online version, with marks) realignment. The considered , lesion is a 8.5 mm 12 mm radial tear ( is the wavelength, in canola oil, at the frequency with where maximum content, i.e., 3.6 GHz). It is clear that the position of the meniscus after it was removed, cut and put back in the tank was different from the initial one. The realignment operation allows for compensation of such a misalignment. Fig. 9 shows selected difference signals (i.e., difference between signals collected with and without a tear in the meniscus, after realignment) collected from different positions of the antenna. The tear in this case is an 8.5 mm 12 mm radial cut located in the internal part of the meniscus. For brevity, only

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be the monitoring of the progression of degenerative diseases after injury, by imaging the same meniscus over time. For both applications, identifying and aligning the meniscus is critical, as demonstrated by the experimental results. IV. DETECTION OF LESIONS OF THE LIGAMENTS Similar to the meniscus study, the feasibility of detecting lesions of the ligaments of the knee is investigated through numerical simulations and experiments. A. Numerical Simulations

Fig. 10. Image obtained by the difference between the signals collected with and without a tear.

Fig. 11. Differential image obtained when two tears are present.

four representative signals are shown. The signals are partially processed (time gating and calibration are performed) and the cases of healthy and torn menisci are compared. Antenna positions 23 and 24 face the tear, while positions 2 and 3 are located on the opposite side. Similar to simulations, the experimental results also demonstrated an increase in the amplitude of the reflected signal for the positions of the antenna facing the tear. To detect the small differences in signals, the realignment operation is fundamental. As in the simulated case, the tear is not directly observable in the reconstructed images, but the differential image clearly identifies its position, as shown in Fig. 10 and predicted in simulations. Good results are obtained even when two tears are present inside the meniscus. Fig. 11 shows a 1.3 mm 8 mm tear and a 4.3 mm 6.3 mm tear separated by 9 mm . This implies that the resolution (i.e., minimum distance between two tears to be detected as separate lesions) is of about 9 mm. This section demonstrates the potential for difference imaging of the meniscus in order to detect tears. In practice, the difference images can be reasonably obtained for example by acquiring images from both left and right knees, or by comparing images of the two menisci located in the same knee. Another promising application for this technique could

A model of the whole knee (including bones, cartilage ligaments, tendons and the menisci) is designed. The dielectric properties of the tissues are assigned according to the Debye models obtained from experimental data (see Section II and [20]). The model is described in [12]. Lesions occur for different reasons and are expected to have corresponding variability in their characteristics. To represent this variation, lesions of the tendon are modeled as regions of both increased and decreased dielectric properties with respect to the healthy tissue. In both cases, a total rupture of the medial collateral ligament (grade III lesion) is considered and modeled as a 4-mm cut extending for the whole width (26 mm) and thickness (5 mm) of the ligament. As with the meniscus study, numerical simulations are performed using the FDTD method. The Vivaldi antenna and the knee model are immersed in a material simulating canola oil. Data are collected by scanning the antenna to nine positions along the major axis of the knee in 1-cm steps. The system is excited with the same Gaussian derivative pulse described in Section III [6], and the data are processed using the calibration and the synthetic focusing steps of an algorithm developed for microwave breast cancer detection [18]. For every “clinical case,” simulations are run with the whole healthy knee first, and then analogous data are acquired after that a lesion is modeled in the ligament (while the rest of the knee model remains the same). A 2-D map of the imaged object is then obtained in both cases, and the difference map is calculated. Results, already reported in [12], show that regions of increased intensity are present in correspondence with the position of the lesion. B. Experimental Results To confirm the simulation results, experimental tests are conducted with the same experimental apparatus described in Section II and with bovine patellar tendons. The antenna is in this case moved to 19 positions along the major axis of the specimen with 5-mm steps. Data are collected over 1601 points in the frequency range of 50 MHz–15 GHz and converted into the time domain. The lesions are simulated by including small objects of selected size and dielectric properties in the tendon. A plastic cylindrical plug diameter mm length mm filled with water is used to represent lesions with permittivity higher than the tendon. A scar-tissue defect is simulated by shaping material developed to represent tumours [19] into a 17 10 8 mm volume; the relative permittivity of this material varies from 48 to 27 in the frequency range of 50 MHz–13.51 GHz, while

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Fig. 12. Patellar tendon in which an object (red plastic plug in online version) simulating an acute (edematous) tear is inserted. The plastic nail used to fix the specimen to the support is visible near the top, while the cut tibia is visible at the bottom.

Fig. 13. Differential maps obtained for two different positions of the lesion shown in Fig. 12.

the conductivity increases from 0.6 to 15 S/m. A sphere of made of nylon ( , low losses) diameter 1 cm represents lesions of lower permittivity than the tendon. In [12], some preliminary results were shown, obtained by simulating the tear with the low-permittivity material (nylon). In Fig. 12, a sample with a tear simulated by the plastic plug is shown. For all the three kinds of lesions considered, the difference maps show regions of increased intensity in the correspondence with the position of the inclusion. As with simulations, a low spatial resolution is observed. Two examples are presented in Fig. 13, where, in the upper right corner of the two maps, the specimen, including the lesion-simulant object, is shown. Other results are presented in [12]. When the lesion is very close to the bone, significant artifacts are present in the image, which almost mask the presence of the lesion. In this section, the feasibility of detecting variations in tendons is illustrated. Similar to the meniscus, challenges include acquisition of a baseline image for creating the difference scan, as well as alignment of the two scans. V. DISCUSSION AND CONCLUSIONS This paper presents numerical and experimental results exploring the potential of a radar-based microwave approach to imaging the knee joint, and more specifically for the detection of meniscal tears and lesions of the ligaments and tendons. Numerical and experimental tests have shown that meniscal tears as small as 1.3 mm 8 mm can be detected. In the experiments, the tear was located in the internal part of the meniscus where most of the tears occur. Furthermore, multiple tears can

be imaged and the system is able to distinguish tears positioned as closely as 9 mm apart, with an error in the estimation of their distance lower than 20%. Closer tears are imaged as a single lesion. The use of a frequency range including frequencies higher than 13.51 GHz could increase the resolution of the system: further studies are needed in this sense. Simulations have also shown that total ruptures of the medial collateral ligament (modeled as 4-mm gaps) can be imaged, and the results are partially confirmed by experimental measurements performed on real bovine patellar tendons. The capability of detecting the lesion seems to be influenced by the position of the lesion itself inside the tendon, and in particular, the proximity to the bones seems to remarkably reduce the quality of the image. This is observed in our experiments, especially when the lesion is modeled with a low-permittivity object (nylon). With lesions characterized by higher dielectric properties, such as scar tissue, the disturbance due to the presence of the bone is reduced. Depending on lesion’s properties, the scattering and reflections from other structures (bones in this case) can dominate the signals, partially or totally hiding the contribution from the lesion. Further studies are required in this sense, especially to develop algorithms able to remove the unwanted contributions from the signals while highlighting the useful ones. In both cases, the detection is based on differential imaging. For this reason, the more promising clinical application of this technique could be the monitoring of the development or healing of a lesion rather than the detection of the lesion itself. Furthermore, the practical application of microwave imaging of joint tissues might be in veterinary medicine, where injuries of large joint tissues are common, and where conventional imaging techniques are often limited by the ability to position a detector in close proximity to the tissue of interest. REFERENCES [1] I. S. Smillie, Injuries of the Knee Joint, 5th ed. New York: Churchill Livingstone, 1978. [2] A. M. Davies, Imaging of the Knee. Techniques and Applications. Berlin, Germany: Springer, 2002. [3] J. E. Bridges, “Noninvasive system for breast cancer detection,” U.S. Patent 5 704 355, Jan. 6, 1998. [4] S. C. Hagness, A. Taflove, and J. E. Bridges, “Two-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixed focus and antenna-array sensors,” IEEE Trans. Biomed. Eng., vol. 45, no. 12, pp. 1470–1479, Dec. 1998. [5] P. M. Meaney, M. W. Fanning, D. Li, S. P. Poplack, and K. D. Paulsen, “A clinical prototype for active microwave imaging of the breast,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 1841–1853, Nov. 2000. [6] E. C. Fear, J. Sill, and M. A. Stuchly, “Experimental feasibility study of confocal microwave imaging for breast tumor detection,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 887–892, Mar. 2003. [7] X. Li, S. K. Davis, S. C. Hagness, D. W. Van, d. Weide, and B. D. Van Veen, “Microwave imaging via space-time beamforming: Experimental investigation of tumor detection in multilayer breast phantoms,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1856–65, Aug. 2004. [8] S. Y. Semenov, R. H. Svenson, A. E. Boulyshev, A. E. Souvorov, V. Y. Borisov, Y. Sizov, A. N. Starostin, K. R. Dezern, G. P. Tatsis, and V. Y. Baranov, “Microwave tomography: Two-dimensional system for biological imaging,” IEEE Trans. Biomed. Eng., vol. 43, no. 9, pp. 869–877, Sep. 1996. [9] S. Semenov, J. Kellam, P. Althausen, T. Williams, A. Abubakar, A. Bulyshev, and Y. Sizov, “Microwave tomography for functional imaging of extremity soft tissues: Feasibility assessment,” Phys. Med. Biol., vol. 52, no. 18, pp. 5705–5719, Sep. 2007.

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[10] G. Bindu, V. Thomas, A. Lonappan, A. V. Praveen Kumar, V. Hamsakkutty, C. K. Aanandan, and K. T. Mathew, “Two-dimensional microwave tomographic imaging of low-water-content tissues,” Microw. Opt. Technol. Lett., vol. 46, no. 6, pp. 599–601, Jul. 2005. [11] E. C. Fear and J. M. Sill, “Preliminary investigations of tissue sensing adaptive radar for breast tumor detection,” in , Proc. 25th Annu. IEEE Int. Eng. Med. Biol. Soc. Conf., 2003, vol. 4, pp. 3787–3790. [12] S. M. Salvador, E. C. Fear, M. Okoniewski, and J. R. Matyas, “Microwave imaging of the knee: Application to ligaments and tendons,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, 2009, pp. 1437–1440. [13] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: III—Parametric models for the dielectric spectrum of tissues,” Phys. Med. Biol., vol. 41, no. 11, pp. 2271–93, Nov. 1996. [14] R. P. Jakob and H. U. Staubli, The Knee and the Cruciate Ligaments. Berlin, Germany: Springer-Verlag, 1992. [15] H. Van der Bracht, R. Verdonk, G. Verbruggen, D. Elewaut, and P. Verdonk, N. Ashammakhi and R. Reis, Eds., E. Chiellini, Ed., “Cellbased meniscus tissue engineering,” Top. Tissue Eng., vol. 3, pp. 1–13, 2007. [16] D. Popovic, L. McCartney, C. Beasley, M. Lazebnik, M. Okoniewski, S. C. Hagness, and J. H. Booske, “Precision open-ended coaxial probes for in vivo and ex vivo dielectric spectroscopy of biological tissues at microwave frequencies,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1713–1722, May 2005. [17] J. Bourqui, M. Okoniewski, and E. C. Fear, “Balanced antipodal vivaldi antenna for breast cancer detection,” in EuCAP, 2007, pp. 1–5. [18] J. M. Sill and E. C. Fear, “Tissue sensing adaptive radar for breast cancer detection—Experimental investigation of simple tumor models,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3312–3318, Nov. 2005. [19] J. Croteau, J. Sill, T. Williams, and E. Fear, “Phantoms for testing radar-based microwave breast imaging,” in ANTEM, 2009, 4 pp. [20] C. Gabriel, S. Gabriel, and E. Corthout, “The dielectric properties of biological tissues: I—Literature survey,” Phys. Med. Biol., vol. 41, no. 11, pp. 2231–2249, Apr. 1996. Sara M. Salvador received the B.S. and Master degree in biomedical engineering (both cum laude) from the Politecnico di Torino, Turin, Italy, in 2003 and 2005, respectively, and the Ph.D. (Dottorato di Ricerca) degree in electronic and communication engineering from the Politecnico di Torino, Turin, Italy (partially carried out at the University of Calgary, Calgary, AB, Canada), in 2009. Her main research activity concerns microwave imaging of biological structures and radar systems developed on digital signal processing (DSP) platforms.

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Elise C. Fear (S’98–M’02) received the Ph.D. degree in electrical engineering from the University of Victoria, Victoria, BC, Canada, in 2001. From 2001 to 2002, she was a Natural Sciences and Engineering Research Council of Canada (NSERC) Postdoctoral Fellow in electrical engineering with the University of Calgary, Calgary, AB, Canada. She is currently an Associate Professor with the Department of Electrical Engineering, University of Calgary. Her research interests include microwave breast cancer detection. Dr. Fear is currently an associate editor for the IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. She was the recipient of the 2007 Outstanding Paper Award of the IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING.

Michal Okoniewski is currently a Professor with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada, where he holds the Libin/Ingenuity Chair in biomedical-engineering and Canada Research Chair in applied electromagnetics. In 2004, he cofounded the Acceleware Corporation. His research interests range from computational electrodynamics, to tunable reflectarrays, RF microelectromechanical systems (MEMS) and RF micromachined devices, as well as hardware acceleration of computational methods. He is also involved in bioelectromagnetics and is involved with tissue spectroscopy and microimaging. Dr Okoniewski is a Registered Professional Engineer (P.Eng.).

John R. Matyas received the B.A. degree in biology and M.Sc. degree in medical sciences from Cornell University, Ithaca, NY, and the Ph.D. degree from the University of Calgary, Calgary, AB, Canada. He has been with The John Hopkins Hospital, where he was involved with bone marrow transplantation and with Squibb Pharmaceuticals, where he was involved with radiopharmaceutical imaging. He is currently a Full Professor with the Faculty of Veterinary Medicine, University of Calgary, where he is also a faculty member with the Biomedical Engineering Graduate Program. He currently holds research funding from the National Institutes of Health (NIH), Canadian Institutes of Health Research, and the Canadian Arthritis Network.

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Letters Corrections to “Analytical Extraction of Extrinsic and Intrinsic FET Parameters”

page. This change does not affect the results described in the above paper. The authors would like to apologize for any inconvenience to the readers.

B. L. Ooi, Z. Zhong, and M.-S. Leong The following corrections should be made to the above paper [1]. Equation (13g) should be corrected as shown at the bottom of this Manuscript received May 17, 2010; accepted May 24, 2010. Date of publication June 28, 2010; date of current version August 13, 2010. B. L. Ooi, deceased, was with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260. Z. Zhong and M.-S. Leong are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260 (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2052407

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[1] B. L. Ooi, Z. Zhong, and M.-S. Leong, “Analytical extraction of extrinsic and intrinsic FET parameters,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 2, pp. 254–261, Feb. 2009.

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P. Aaen A. Abbaspour-Tamijani A. Abbosh D. Abbott A. Abdipour M. Abe M. Abegaonkar R. Abhari A. Abramowicz M. Acar L. Accatino R. Achar E. Ackerman J. Adam K. Agawa M. Ahmad H.-R. Ahn B. Ai M. Aikawa J. Aikio C. Aitchison M. Akaike T. Akin S. Aksoy I. Aksun A. Akyurtlu G. Ala L. Albasha A. Alexanian W. Ali-Ahmad F. Alimenti R. Allam K. Allen A. Alphones A. Alu A. Álvarez-Melcon A. Al-Zayed S. Amari H. Amasuga R. Amaya H. An D. Anagnostou M. Andersen K. Andersson M. Ando Y. Ando P. Andreani M. Andrés W. Andress K. Ang C. Angell I. Angelov Y. Antar G. Antonini H. Aoki V. Aparin F. Apollonio R. Araneo J. Archer F. Ares F. Ariaei T. Arima M. Armendariz L. Arnaut F. Arndt E. Artal H. Arthaber F. Aryanfar U. Arz M. Asai Y. Asano A. Asensio-Lopez K. Ashby H. Ashoka A. Atalar A. Atia S. Auster I. Awai A. Aydiner M. Ayza K. Azadet R. Azaro A. Babakhani P. Baccarelli M. Baginski I. Bahl S. Bajpai J. Baker-Jarvis B. Bakkaloglu M. Bakr A. Baladin C. Balanis S. Balasubramaniam J. Balbastre J. Ball P. Balsara Q. Balzano A. Banai S. Banba R. Bansal D. Barataud A. Barbosa F. Bardati I. Bardi J. Bardin A. Barel S. Barker F. Barnes J. Barr G. Bartolucci R. Bashirullan S. Bastioli A. Basu B. Bates R. Baxley Y. Bayram J.-B. Bégueret N. Behdad F. Belgacem H. Bell D. Belot J. Benedikt T. Berceli C. Berland M. Berroth G. Bertin E. Bertran A. Bessemoulin M. Beurden A. Bevilacqua A. Beyer M. Bialkowski

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Digital Object Identifier 10.1109/TMTT.2010.2064950

S. Islam M. Ito K. Itoh T. Itoh Y. Itoh A. Ittipiboon F. Ivanek D. Iverson M. Iwamoto D. Jablonski D. Jachowski C. Jackson D. Jackson R. Jackson A. Jacob K. Jacobs S. Jacobsen D. Jaeger J. Jaeger S. Jagannathan N. Jain G. James M. Janezic S. Jang M. Jankovic D. Jansen L. Jansson H. Jantunen H. Jardon-Aguilar J. Jargon N. Jarosik B. Jarry P. Jarry A. Jastrzebski B. Jemison W. Jemison S. Jeng A. Jenkins S. Jeon D. Jeong J. Jeong Y. Jeong A. Jerng T. Jerse T. Jiang X. Jiang G. Jianjun D. Jiao J. Jin J. M. Jin J. Joe T. Johnson B. Jokanovic U. Jordan K. Joshin J. Joubert S. Jung T. Kaho S. Kanamaluru K. Kanaya S. Kang P. Kangaslahti B. Kapilevich I. Karanasiou M. Karim T. Kataoka A. Katz R. Kaul R. Kaunisto T. Kawai S. Kawasaki M. Kazimierczuk L. Kempel P. Kenington P. Kennedy A. Kerr D. Kettle A. Khalil W. Khalil S. Khang A. Khanifar A. Khanna R. Khazaka J. Khoja S. Kiaei J. Kiang B. Kim C. Kim D. Kim H. Kim I. Kim J. Kim S. Kim T. Kim W. Kim N. Kinayman R. King N. Kinzie S. Kirchoefer A. Kirilenko M. Kishihara T. Kitazawa J. Kitchen T. Klapwijk E. Klumperink D. Klymyshyn L. Knockaert R. Knoechel M. Koch K. Koh N. Kolias J. Komiak A. Komijani G. Kompa A. Konanur A. Konczykowska H. Kondoh B. Kopp B. Kormanyos J. Korvink P. Kosmas Y. Kotsuka S. Koziel A. Kozyrev V. Krishnamurthy H. Krishnaswamy C. Krowne J. Krupka D. Kryger H. Ku H. Kubo A. Kucar

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R. Mansour D. Manstretta J. Mao S. Mao F. Maradei A. Margomenos D. Markovic E. Márquez-Segura J. Martens F. Martin E. Martini K. Maruhashi J. Marzo D. Masotti A. Massa G. Massa F. Mastri J. Mateu A. Matsushima M. Mattes G. Matthaei K. Mayaram M. Mayer U. Mayer W. Mayer J. Mazeau S. Mazumder A. Mazzanti G. Mazzarella K. McCarthy G. McDonald I. McGregor M. McKinley J. McLean D. McQuiddy A. Mediano F. Medina M. Megahed I. Mehdi K. Mehrany A. Melcon R. Melville F. Mena D. Mencarelli C. Meng R. Menozzi W. Menzel P. Mercier B. Merkl F. Mesa R. Metaxas A. Metzger P. Meyer P. Mezzanotte E. Michielsen A. Mickelson D. Miller P. Millot J. Mingo F. Miranda D. Mirshekar A. Mirzaei S. Mitilineos R. Miyamoto K. Mizuno J. Modelski W. Moer M. Moghaddam A. Mohammadi S. Mohammadi A. Mohammadian P. Mohseni E. Moldovan M. Mollazadeh M. Mongiardo P. Monteiro J. Montejo-Garai G. Montoro J. Monzó-Cabrera J. Morente T. Morf D. Morgan M. Morgan A. Morini A. Morris J. Morsey A. Mortazawi M. Moussa M. Mrozowski Q. Mu J.-E. Mueller J. Muldavin K. Murata S.-S. Myoung M. Myslinski B. Nabet V. Nair K. Naishadham Y. Nakasha M. Nakatsugawa M. Nakhla J.-C. Nallatamby I. Nam S. Nam J. Nanzer T. Narhi A. Nashashibi A. Natarajan J. Nath A. Navarrini J. Navarro J. Nebus R. Negra J. Neilson B. Nelson P. Nepa A. Neri H. Newman G. Ng D. Ngo E. Ngoya C. Nguyen E. Nicol A. Nicolet S. Nicolson E. Niehenke M. Nielsen K. Nikita P. Nikitin N. Nikolova M. Nisenoff K. Nishikawa T. Nishino

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