IEEE MTT-V054-I09 (2006-09) [54, 9 ed.]

Citation preview

SEPTEMBER 2006

VOLUME 54

NUMBER 9

IETMAB

(ISSN 0018-9480)

CONTRIBUTED PAPERS

Active Circuits, Semiconductor Devices, and ICs Size Reduction of a MMIC Direct Up-Converter at 44 GHz in Multilayer CPW Technology Using Thin-Film Microstrip Stubs Loading . ......... ........ ......... ......... ........ ......... ......... ...... K. Hettak, G. A. Morin, and M. G. Stubbs

3453

Signal Generation, Frequency Conversion, and Control A Wide Tuning-Range CMOS VCO With a Differential Tunable Active Inductor ....... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... L.-H. Lu, H.-H. Hsieh, and Y.-T. Liao

3462

Wireless Communication Systems Piecewise Pre-Equalized Linearization of the Wireless Transmitter With a Doherty Amplifier .. ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ... W.-J. Kim, K.-J. Cho, S. P. Stapleton, and J.-H. Kim

3469

Field Analysis and Guided Waves Propagation Characteristics for Periodic Waveguide Based on Generalized Conservation of Complex Power Technique .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ........ H. K. Liu and T. L. Dong

3479

CAD Algorithms and Numerical Techniques Monostatic Reflectivity Measurement of Radar Absorbing Materials at 310 GHz ....... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ...... A. Lönnqvist, A. Tamminen, J. Mallat, and A. V. Räisänen

3486

Filters and Multiplexers Distortion Mechanisms in Varactor Diode-Tuned Microwave Filters .... ....... .. ....... B. E. Carey-Smith and P. A. Warr Design of Compensated Coupled-Stripline 3-dB Directional Couplers, Phase Shifters, and Magic-T’s—Part II: Broadband Coupled-Line Circuits . ........ ......... ......... ........ ......... ......... ..... S. Gruszczynski, K. Wincza, and K. Sachse

3492 3501

(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Compact Planar Microstrip Branch-Line Couplers Using the Quasi-Lumped Elements Approach With Nonsymmetrical and Symmetrical T-Shaped Structure .. ......... ........ ......... ......... ........ ......... ........ S.-S. Liao and J.-T. Peng A New Look at the Practical Design of Compact Diplexers ...... ...... A. Morini, T. Rozzi, M. Farina, and G. Venanzoni A 3-dB Quadrature Coupler Suitable for PCB Circuit Design ... ......... ........ .. J.-C. Chiu, C.-M. Lin, and Y.-H. Wang Novel Dual-Mode Bandpass Filters Using Hexagonal Loop Resonators . ........ ..... ..... ....... R.-J. Mao and X.-H. Tang Simple Analysis and Design of a New Leaky-Wave Directional Coupler in Hybrid Dielectric-Waveguide Printed-Circuit Technology ..... ......... ........ ......... ......... ...... J. L. Gómez-Tornero, S. Martínez-López, and A. Álvarez-Melcón Design of Composite Right/Left-Handed Coplanar-Waveguide Bandpass and Dual-Passband Filters .. ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .. S.-G. Mao, M.-S. Wu, and Y.-Z. Chueh Design of Dual- and Triple-Passband Filters Using Alternately Cascaded Multiband Resonators ....... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ . C.-F. Chen, T.-Y. Huang, and R.-B. Wu Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements Tunable Bandstop Defected Ground Structure Resonator Using Reconfigurable Dumbbell-Shaped Coplanar Waveguide ..... ......... ........ ......... . ......... ........ ......... .. A. M. E. Safwat, F. Podevin, P. Ferrari, and A. Vilcot Adaptive Nonuniform-Grid (NG) Algorithm for Fast Capacitance Extraction .. ......... ......... A. Boag and B. Livshitz Inductively Compensated Parallel Coupled Microstrip Lines and Their Applications ... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... .... R. Phromloungsri, M. Chongcheawchamnan, and I. D. Robertson

3508 3515 3521 3526 3534 3543 3550

3559 3565 3571

Instrumentation and Measurement Techniques Broadband Space Conservative On-Wafer Network Analyzer Calibrations With More Complex Load and Thru Models .. .. ........ ......... ......... ........ ......... .... S. Padmanabhan, L. Dunleavy, J. E. Daniel, A. Rodríguez, and P. L. Kirby

3583

MEMS and Acoustic Wave Components Scale-Changing Technique for the Electromagnetic Modeling of MEMS-Controlled Planar Phase Shifters .... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ...... E. Perret, H. Aubert, and H. Legay

3594

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

3602

IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY The Microwave Theory and Techniques Society is an organization, within the framework of the IEEE, of members with principal professional interests in the field of microwave theory and techniques. All members of the IEEE are eligible for membership in the Society upon payment of the annual Society membership fee of $14.00, plus an annual subscription fee of $16.00 per year for electronic media only or $32.00 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 K. VARIAN, President S. M. EL-GHAZALY J. HAUSNER K. ITOH M. HARRIS D. HARVEY

J. S. KENNEY, President Elect L. KATEHI N. KOLIAS J. S. KENNEY T. LEE

W. H. CANTRELL, Secretary J. LIN V. J. NAIR J. MODELSKI B. PERLMAN A. MORTAZAWI

Honorary Life Members T. ITOH A. A. OLINER

T. S. SAAD P. STAECKER

D. RUTLEDGE W. SHIROMA

N. KOLIAS, Treasurer R. WEIGEL R. SNYDER K. WU K. VARIAN R. YORK

Distinguished Lecturers K. TOMIYASU L. YOUNG

G. BOECK W. HOEFER T. ITOH

B. KIM J. LASKAR V. LUBECKE

J. C. RAUTIO D. ROOT D. RYTTING

Past Presidents M. SHUR P. SIEGEL A. SUAREZ

K. C. GUPTA (2005) R. J. TREW (2004) F. SCHINDLER (2003)

MTT-S Chapter Chairs Albuquerque: S. BIGELOW Atlanta: D. LEATHERWOOD Austria: R. WEIGEL Baltimore: A. D. BROWN Beijing: Z. FENG Beijing, Nanjing: W. X. ZHANG Belarus: A. GUSINSKY Benelux: D. V.-JANVIER Brasilia: A. KLAUTAU, JR. Buenaventura: C. SEABURY Buffalo: E. M. BALSER Bulgaria: K. ASPARUHOVA Cedar Rapids/Central Iowa: D. JOHNSON Central New England: K. ALAVI Central & South Italy: S. MACI Central No. Carolina: T. IVANOV Chicago: Z. LUBIN Cleveland: G. PONCHAK Columbus: F. TEIXEIRA Connecticut: C. BLAIR/R. ZEITLER Croatia: Z. SIPUS Czech/Slovakia: P. HAZDRA Dallas: R. EYE Dayton: A. TERZOUOLI, JR. Denver: M. JANEZIC Eastern No. Carolina: D. PALMER Egypt: I. A. SALEM Finland: T. KARTTAAVI Florida West Coast: K. O’CONNOR

Foothill: C. ANTONIAK France: P. EUDELINE Germany: W. HEINRICH Greece: I. XANTHAKIS Hong Kong: W. Y. TAM Houston: J. T. WILLIAMS Houston, College Station: C. MADSEN Hungary: T. BERCELI Huntsville: H. G. SCHANTZ India/Calcutta: P. K. SAHA India Council: K. S. CHARI Israel: S. AUSTER Japan Council: Y. TAKAYAMA Kitchener-Waterloo: R. R. MANSOUR Lithuania: V. URBANAVICIUS Long Island/New York: J. COLOTTI Los Angeles, Coastal: A. SHARMA Los Angeles, Metro: J. WEILER Malaysia: Z. AWANG Melbourne: J. KRALOVEC Milwaukee: S. G. JOSHI Mohawk Valley: P. RATAZZI Monterrey Mexico: R. M. RODRIGUEZ-DAGNINO Montreal: K. WU New Hampshire: T. PERKINS New Jersey Coast: D. REYNOLDS New South Wales: G. TOWN New Zealand: J. MAZIERSKA North Italy: G. GHIONE

North Jersey: K. DIXIT North Queensland: A. TSAKISSIRIS Northern Nevada: B. S. RAWAT Norway: S. E. WHEATLEY Orange County: H. J. DE LOS SANTOS Oregon: T. RUTTAN Orlando: P. WAHID Ottawa: Q. YE Philadelphia: J. NACHAMKIN Phoenix: C. WEITZEL Poland: M. P. MROZOWSKI Portugal: C. A. CARDOSO FERNANDES Princeton/Central Jersey: W. CURTICE/A. KATZ Queensland: A. ROBINSON Rio de Janeiro: J. R. BERGMANN Rochester: S. M. CICCARELLLI/J. VENKATARAMAN Romania: I. SIMA Russia, Nizhny-Novgorod: Y. BELOV Russia, Saint Petersburg: M. SITNIKOVA Russia, Moscow: V. KALOSHIN Russia, Saratov-Penza: N. RYSKIN Saint Louis: D. MACKE San Diego: J. TWOMEY Santa Clara Valley/San Francisco: J. J. SOWERS Seattle: K. POULSON Seoul Coucil: H.-Y. LEE Siberia, Novosibirsk: V. SHUBALOV Siberia, Tomsk: O. STUKACH Singapore: O. B. LEONG

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

South Africa: P. W. VAN DER WALT South Australia: H. HANSEN South Brazil: L. C. KRETLY Southeastern Michigan: L. M. ANNEBERG Southern Alberta: S. BOUMAIZA Spain: L. FE HARO Springfield: P. R. SIQUEIRA Sweden: A. RYDBERG Switzerland: J. HESSELBARTH Syracuse: E. ARVAS Taipei: C.-S. LU Thailand: M. KRAIRIKSH Toronto: G. V. ELEFTHERIADES Tucson: N. BURGESS/S. MORALES Turkey: O. A. CIVI Twin Cities: M. J. GAWRONSKI UK/RI: A. REZAZADEH Ukraine, Central Kiev: Y. POPLAVKO Ukraine, East: A. A. KIRILENKO Ukraine, Rep. of Georgia: R. ZARIDZE Ukraine, Vinnitsya: V. DUBOVOY Ukraine, West: I. ISAYEV Venezuela: M. PETRIZZELLI Victoria: A. MITCHELL Virginia Mountain: D. MILLER Washington DC/Northern Virginia: J. QIU Winnipeg: V. OKHMATOVSKI Yugoslavia: B. MILOVANOVIC

Associate Editors

ANDREAS CANGELLARIS KENJI ITOH STEVEN MARSH Univ. of Illinois, Urbana Champaign Mitsubishi Electronics Midas Consulting USA Japan U.K. email:[email protected] email: [email protected] email: [email protected] YOSHIO NIKAWA DAVID LINTON TADEUSZ WYSOCKI Queen’s Univ. Belfast Univ. of Wollongong Kokushikan Univ. Northern Ireland Australia Japan email: [email protected] email: [email protected] email: [email protected] RUEY-BEEI WU MANH ANH DO JOSÉ PEDRO Univ. of Aveiro National Taiwan Univ. Nanyang Technological Univ. Portugal Taiwan, R.O.C. Singapore email: jcp.mtted.av.it.pt email: [email protected] email: [email protected] ZOYA POPOVIC ALESSANDRO CIDRONALI VITTORIO RIZZOLI Univ. of Colorado, Boulder Univ. of Florence Univ. of Bologna USA Italy Italy email: [email protected] email: [email protected] email: [email protected] M. GOLIO, Editor-in-Chief, IEEE Microwave Magazine G. E. PONCHAK, Editor-in-Chief, IEEE Microwave and Wireless Component Letters

SANJAY RAMAN Virginia Polytech. Inst. and State Univ. USA email: [email protected] JENSHAN LIN Univ. of Florida USA email: [email protected] RICHARD SNYDER RS Microwave Company USA email: [email protected] ALEXANDER YAKOVLEV Univ. of Mississippi USA email: [email protected] T. LEE, Web Master

IEEE Officers MICHAEL R. LIGHTNER, President and CEO SAIFUR RAHMAN, Vice President, Publication Services and Products LEAH H. JAMIESON, President-Elect PEDRO RAY, Vice President, Regional Activities J. ROBERTO DE MARCA, Secretary DONALD N. HEIRMAN, President, IEEE Standards Association JOSEPH V. LILLIE, Treasurer CELIA L. DESMOND, Vice President, Technical Activities W. CLEON ANDERSON, Past President RALPH W. WYNDRUM, JR., President, IEEE-USA MOSHE KAM, Vice President, Educational Activities STUART A. LONG, Director, Division IV—Electromagnetics and Radiation

IEEE Executive Staff JEFFRY W. RAYNES, CAE, Executive Director & Chief Operating Officer DONALD CURTIS, Human Resources ANTHONY DURNIAK, Publications Activities JUDITH GORMAN, Standards Activities CECELIA JANKOWSKI, Regional Activities BARBARA COBURN STOLER, Educational Activities

MATTHEW LOEB, Corporate Strategy & Communications RICHARD D. SCHWARTZ, Business Administration CHRIS BRANTLEY, IEEE-USA MARY WARD-CALLAN, Technical Activities SALLY A. WASELIK, Information Technology

IEEE Periodicals Transactions/Journals Department Staff Director: FRAN ZAPPULLA Editorial Director: DAWN MELLEY Production Director: ROBERT SMREK Managing Editor: MONA MITTRA Senior Editor: CHRISTINA M. REZES IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (ISSN 0018-9480) is published monthly by the Institute of Electrical and Electronics Engineers, Inc. Responsibility for the contents rests upon the authors and not upon the IEEE, the Society/Council, or its members. IEEE Corporate Office: 3 Park Avenue, 17th Floor, New York, NY 10016-5997. IEEE Operations Center: 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331. NJ Telephone: +1 732 981 0060. Price/Publication Information: Individual copies: IEEE Members $20.00 (first copy only), nonmember $77.00 per copy. (Note: Postage and handling charge not included.) Member and nonmember subscription prices available upon request. Available in microfiche and microfilm. Copyright and Reprint Permissions: Abstracting is permitted with credit to the source. Libraries are permitted to photocopy for private use of patrons, provided the per-copy fee indicated in the code at the bottom of the first page is paid through the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923. For all other copying, reprint, or republication permission, write to Copyrights and Permissions Department, IEEE Publications Administration, 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331. Copyright © 2006 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved. Periodicals Postage Paid at New York, NY and at additional mailing offices. Postmaster: Send address changes to IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, IEEE, 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331. GST Registration No. 125634188.

Digital Object Identifier 10.1109/TMTT.2006.883076

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3453

Size Reduction of a MMIC Direct Up-Converter at 44 GHz in Multilayer CPW Technology Using Thin-Film Microstrip Stubs Loading Khelifa Hettak, Senior Member, IEEE, Gilbert A. Morin, and Malcolm G. Stubbs

Abstract—This paper proposes a novel compact multilayer 44.5-GHz coplanar waveguide (CPW) single-sideband (SSB) subharmonically pumped (SHP) mixer for direct up-conversion GaAs monolithic microwave integrated circuit. It uses previously developed thin-film microstrip (TFMS) and CPW structures for capacitive and inductive loading techniques to drastically reduce its size. The SSB SHP mixer uses 50-MHz in-phase and quadrature signals to directly modulate the second harmonic of a 22.25-GHz carrier to produce the required 44.55-GHz RF output. Two pairs of antiparallel diodes reduce feedthrough of the fundamental 22.25-GHz signal to the RF output while novel CPW/TFMS-based structures provide matching. This 1.2 1.5mm2 chip uses a lumped Wilkinson divider as a local-oscillator divider and a previously developed reduced size 90 coupler. The SSB SHP mixer acts as an up-converter with a measured conversion gain of 10 1 dB and the lower sideband suppression is greater than 23 dB across the RF bandwidth of 43.5–45.5 GHz. Additionally, it is shown that the RF port return loss is better than 20 dB, and 2 LO was suppressed by 21 dB over the same band. The circuit also does not require any dc bias. Compared to the conventional SSB SHP mixer, a 70% reduction in circuit area was achieved with better performances. Index Terms—Capacitive loading, coplanar waveguide (CPW), CPW series stubs, CPW shunt stubs, inductive loading, lower sideband (LSB), millimeter waves, monolithic microwave integrated circuit (MMIC), 90 coupler, size reduction, subharmonic mixer, thin-film microstrip (TFMS) shunt stubs, upper sideband (USB), Wilkinson divider.

I. INTRODUCTION HE DRIVE to reduce the complexity of digital wireless transmitter and receiver systems is generating interest in direct up- and down-conversion [1]. Such techniques can lead to savings in integrated circuit chip count and are gaining acceptance at low microwave frequencies. Direct up-conversion at millimeter waves can result, however, in severe filtering problems. Since the local-oscillator (LO) signal and up-converted sidebands are very close in frequency, it is very difficult to filter out any LO signal that appears at the single-sideband (SSB) mixer output. One approach that simplifies this filtering problem is to modulate a harmonic of a lower

T

Manuscript received February 11, 2006; revised May 8, 2006. K. Hettak and M. G. Stubbs are with the Communications Research Centre Canada, Ottawa, ON, Canada K2H 8S2 (e-mail: [email protected]). G. A. Morin is with the Defence Research and Development Canada–Ottawa, Ottawa, ON, Canada K2H 8S2. Color versions of Figs. 1–19 are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2006.879769

frequency LO signal. Any LO feedthrough is then well separated from the desired signal and can easily be removed [2]–[8]. A subharmonic mixer with an antiparallel diode pair simplifies this filtering problem because the fundamental LO signal is well separated from the desired signal and the problematic second harmonic signal is suppressed at the output by being confined within the diode pair. A subharmonically pumped mixer (SHP) with a well-matched pair of antiparallel diodes is an attractive candidate for direct up-conversion [9]–[13]. Furthermore, lower cost and more readily available oscillators can be used since the LO frequency required for an SHP mixer is half the LO for conventional mixers. SHP mixers do not require dc power and can be used for both direct up- and down-conversion [14], [15]. Therefore, the SSB mixers that use SHP mixers are desired for direct conversion [1]. SSB SHP mixer for both up- and down-conversion need an LO in-phase divider, quadrature hybrid at RF ports, and two SHP mixers [1]. However, the large size of many of the passive elements used in SSB SHP mixer prevents full cost savings from being realized. Indeed, the RF hybrid and Wilkinson splitter demand a relatively large amount of substrate area since each branch of the coupler and splitter is close to and (or , where is half ) in length, respectively. Similarly, the subharmonic mixer contains large passive structures: a open-circuit stub and short-circuit stub [18]–[22]. The miniaturization of such structures without sacrificing performance is, therefore, an important issue. We have previously introduced novel matching structures based on the integration of a thin-film microstrip (TFMS) and coplanar waveguide (CPW) on the same chip [16] and, in this paper, we expand on this concept by showing how such structures can be used to reduce the size of active monolithic microwave integrated circuits (MMICs). We have chosen to explore the size reduction of an SSB SHP mixer chip. In such circuits, the conversion loss and image rejection performance are extremely dependent on having passive dividers and combiners with accurate amplitude and phase characteristics, as well as on the stubs quality involving in SHP mixer design. In light of this, a novel compact multilayer 44.5-GHz CPW and TFMS SSB SHP mixer for a direct up-conversion GaAs MMIC is proposed in this paper. A picture of the MMIC is shown in Fig. 1(a) including the conventional structure in Fig. 1(b) and a block diagram in Fig. 2. The circuit was required to directly up-convert 50–200-MHz in-phase (I) and quadrature (Q) signals to an RF band of 43.5–45.5 GHz using a 21.75–22.75-GHz LO signal. It was

0018-9480/$20.00 © 2006 IEEE

3454

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 1. (a) Novel compact multilayer MMIC CPW SSB SHP mixer using TFMS at 44.5 GHz (1.2 and novel MMIC CPW SSB SHP mixer at 44.5 GHz.

manufactured with the 0.18- m OMMIC foundry process. As can be seen in Figs. 1 and 2, the SSB mixer requires the integration of two identical SHP mixers, a 90 hybrid coupler, and a Wilkinson power divider. This paper is divided into three sections. Section II discusses how the integrated TFMS and CPW structures for capacitive and

2 1.5 mm ). Not to scale. (b) Comparison between conventional

inductive loading can be used to reduce the size of the basic SHP mixer unit. The development of a 22-GHz lumped Wilkinson divider and reduced-size 90 branch-line coupler, which uses the same loading approach, will be outlined in Section III. Finally, the performance of the complete SSB SHP mixer will be presented in Section IV.

HETTAK et al.: SIZE REDUCTION OF MMIC DIRECT UP-CONVERTER AT 44 GHz IN MULTILAYER CPW TECHNOLOGY

Fig. 2. Block diagram of the SSB SHP mixer MMIC.

II. NOVEL MINIATURE SUBHARMONIC MIXER USING INDUCTIVELY AND CAPACITIVELY LOADED STUBS The SSB mixer requires two identical SHP mixers. These were designed in multilayer CPW and TFMS technology and used a pair of 0.18- m pseudomorphic high electron-mobility transistor (pHEMT)-based diodes (each with two 15- m cathode fingers) in an antiparallel configuration. The LO signal band is 21.75–22.75 GHz and the IF frequency range is 50–200 MHz. The second harmonic of the LO mixes with the IF signal to produce the desired 43.5–45.5-GHz RF signal. By virtue of the antiparallel diode configuration, all even-order , where is even) are mixing products ( suppressed. A photograph of the MMIC chip is shown in Fig. 3 and a simplified schematic of the mixer is shown in Fig. 4. The nonlinear model for the pHEMT diodes was provided by OMMIC. The circuit was designed using Libra for harmonic-balance simulation and Momentum [17] for electromagnetic field analysis. A. Circuit Description The IF input to the SHP mixer incorporates a low-pass filter that consists of a shunt capacitance followed by a length of CPW line. This combination presents a high impedance at the junction of the two diodes on the RF side (point X in Fig. 4) preventing the RF signal from appearing at the IF port. Fig. 5(a) shows the experimental and simulated results of the IF filter. The RF suppression at point X (Port 2) was measured to be 35 dB at 44 GHz. To isolate the RF port from the LO port, the conventional SHP open-circuit CPW shunt mixer includes two stubs, i.e., a short-circuit CPW shunt stub, as illustrated stub and a transmission lines are very in Fig. 1(b). However, the long and occupy a large substrate area. To make the stubs more compacts, each one was designed as a loaded CPW/TFMS combination, as outlined in [16]. short-circuit CPW shunt stub was reThe initially alized as a CPW short-circuit stub and was miniaturized by using an open-circuit TFMS shunt stub for capacitive loading. This is shown in Fig. 5(b). This reduced size stub is located on the LO port side of the pair of antiparallel diodes (point

3455

and a short at Y) and simultaneously provides an open at . The capacitively loaded shunt CPW stub was optimized to maximize the suppression at at the LO port side. open-circuit CPW shunt stub Similarly, the initially was implemented as a TFMS stub. Its size was reduced using a short-circuit CPW series stub for inductive loading, as shown in Fig. 5(c). This size reduced stub is located on the RF port side of the pair of antiparallel diodes (point X) and simultaneously and an open at provides a short at . The inductively loaded shunt microstrip thin-film stub . The length was optimized to minimize the insertion loss at stub to be , of the each stub was reduced from the corresponding to a reduction of approximately 50%. Fig. 5 illustrates the operation of these loaded stubs. The capacitively loaded shunt CPW stub at the LO port side allows the LO signal to pass with 1.1-dB attenuation with a return loss of 15.6 dB at 22.25 GHz. At the RF frequency of 44.45 GHz, the suppression is 20.4 dB. The LO signal, therefore, reaches the diodes with low attenuation, whereas the RF signal is suppressed at the LO port. The inductively loaded shunt microstrip thin-film stub at the RF port side effectively suppresses the LO signal with low attenuation at RF frequencies. The LO suppression is 17.5 dB at 22.25 GHz. At the RF frequency of 44.45 GHz, the RF signal passes with 0.8-dB attenuation with a return loss of 21.2 dB. The LO signal is, therefore, suppressed, but the RF signal is allowed to propagate to the output of the chip. open/short-cirIn comparison with the conventional cuit CPW stubs used in SHP mixer design (shown in Fig. 6), the advantages that may be derived from the use of the proposed new reduced stubs (Fig. 5) are: 1) additional freedom in the design (developed various stubs with better short and open characteristics and lower loss); 2) more size reduction than CPW or microstrip alone (at least 50% size reduction is achieved); and 3) better stub characteristics ( , loss, shorts, opens). The novel miniature SHP mixer also has an RF bandpass filter to isolate the RF and IF ports. This filter was realized by a series combination of lumped L–C elements. Fig. 7 illustrates the experimental and simulated results of the RF bandpass filter. B. Experimental Results of the Novel Miniature SHP Mixer The MMIC SHP mixer was measured on a semiautomatic wafer prober with ground–signal–ground probes and was found to provide optimum performance at an LO power of approximately 8 dBm, as shown in Fig. 8. The measured conversion gain was 12 0.5 dB over the 43.5–45.5-GHz frequency band, using a 50-MHz IF signal, as suppression versus RF freillustrated in Fig. 9. The quency is better than 28 dB, as shown in Fig. 9. The conversion gain versus IF frequency (50–200 MHz) is better than 12 dB, as illustrated in Fig. 10. The 1-dB compression point occurs with an IF input power level of 3 dBm, as shown in Fig. 11. Fig. 12 shows that the RF return loss is better than 15 dB over a 43.5–45.5-GHz frequency range.

3456

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 3. Novel compact multilayer MMIC CPW subharmonic mixer using TFMS at 44.5 GHz. Not to scale.

Fig. 4. Simplified schematic of subharmonic mixer.

III. WILKINSON POWER DIVIDER AND BRANCH-LINE 90 HYBRID In addition to the two SHP mixers, the complete SSB SHP mixer required an LO power divider and an RF quadrature hybrid. The LO is split in-phase by a Wilkinson power divider and is applied to each SHP mixer LO port, while I and Q baseband signals are applied to each SHP mixer IF port. The double-sideband suppressed carrier (DSBSC) RF signal from each mixer is combined in the 90 hybrid where one of the sidebands is cancelled. The carrier suppression is achieved within each antiparallel diode pair. A. Wilkinson Power Divider The quarter-wavelength arms of a 20-GHz distributed Wilkinson divider are quite large [26], [27]. To make it more

compact, a lumped-element approach was used for the SSB SHP mixer. In this case, the divider is composed of spiral inductors, lumped capacitors, and a lumped 100- resistor between the two output ports 1 and 2. The layout, equivalent circuit, and measured results of the divider are shown in Fig. 13. Isolation between the two output ports and input return loss of this circuit was measured to be better than 17 and 22 dB, respectively, over the 21.75–22.75-GHz LO range.

B. Branch-Line 90 Hybrid A quadrature hybrid combines the output from the two mixers from Section II and suppresses the unwanted sideband

HETTAK et al.: SIZE REDUCTION OF MMIC DIRECT UP-CONVERTER AT 44 GHz IN MULTILAYER CPW TECHNOLOGY

3457

Fig. 5. Measured and simulated (using Momentum) performance of: (a) IF filter, (b) capacitively loaded shunt TFMS shunt stub, and (c) inductively loaded shunt CPW shunt stub.

Fig. 6. Simulated (using Momentum) performance of conventional 

=4 open/short-circuit CPW stubs used in standard SHP mixer design.

[23]–[25]. The amount of sideband suppression is strongly dependent on the amplitude and phase imbalance of the 90 hybrid, as well as the symmetry between the two mixers. The quadrature hybrid was previously developed [16] and experimental results revealed that the measured phase unbalance varied from 0.4 to

2 over the design frequency range of 43.5–45.5 GHz and the measured amplitude difference varied between 1.17 and 1.29 dB for the two output ports. Experimental results of the input port return loss and isolation between the two output ports were both better than 20 dB over the RF range.

3458

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 9. Gain and suppression versus RF frequency of a single SHP mixer.

Fig. 7. Measured performance of RF bandpass filter and dc block. (a) Series combination of lumped L–C elements. (b) Performance.

Fig. 10. Gain and suppression versus IF power of a single SHP mixer.

Fig. 8. Gain and suppression versus LO power of a single SHP mixer. Fig. 11. Conversion gain versus IF input power of a single SHP mixer.

IV. PERFORMANCE OF THE COMPLETE SSB SHP MIXER Fig. 1(a) shows a photograph of the SSB SHP modulator chip fabricated using OMMIC’s ED02AH foundry process. The chip dimensions are 1.2 1.5 mm . The SSB SHP mixer was measured as an up-converter and was found to provide optimum performance at an LO power of 12 dBm. The difference in optimum LO power between the SHP mixer and SSB SHP mixer configuration is due to the LO Wilkinson power divider. Indeed, as each single SHP mixer is pumped with 8 dBm, the total required LO power for the SSB SHP mixer should be at least twice in order to compensate for the 3.4 dB lost through the LO Wilkinson power divider. The upper sideband (USB) was chosen as the RF desired signal.

Fig. 14 shows the dependence of the conversion gain characteristics on LO input power for the SSB SHP mixer. The frequency is fixed at 22.25 GHz for the LO and 50 MHz for I and Q at 16 dBm. Since the desired output was an SSB suppressed carrier signal, the suppression of both the lower sideband (LSB) and the second harmonic of the LO was important. Conversion gain nearly saturates for LO input power from 12 to 13 dBm. Conversion gain is 10 1 dB and the LSB suppression is greater than 21 dB across the RF bandwidth of 43.5–45.5 GHz, as illustrated in Fig. 15. For this measurement, the I and Q frequency was fixed at 50 MHz and the I and Q and LO input powers were set to 16 and 12 dBm, respectively.

HETTAK et al.: SIZE REDUCTION OF MMIC DIRECT UP-CONVERTER AT 44 GHz IN MULTILAYER CPW TECHNOLOGY

Fig. 12. RF-port return loss of a single SHP mixer.

3459

Fig. 14. Gain and suppression versus LO power of the SSB SHP mixer.

Fig. 15. Gain and suppression versus RF frequency of SSB SHP mixer.

Fig. 16. Gain and suppression versus I and Q frequency of SSB SHP mixer.

Fig. 13. Picture, schematic, and performance of the lumped Wilkinson.

In the RF bandwidth of 43.5–45.5 GHz, the difference in conversion gain from the results of the SHP mixer is due to the SSB SHP mixer configuration. The RF signal of SSB SHP mixer results form the summation of both RF signals coming from each individual SHP mixer, therefore, the conversion gain of the SSB

SHP mixer should be 3 dB less compared to a single SHP mixer. However, the conversion loss of the SSB SHP mixer includes the loss of the reduced size hybrid, which explains the 10 1 dB conversion gain obtained. I and Q frequency characteristics with LO frequency fixed at 22.25 GHz are shown in Fig. 16. The conversion gain of desired USB signals is nearly flat, but LSB suppression varies with I and Q frequency. The I and Q frequency response of the SSB SHP mixer MMIC is expected to be flat for variable I and Q, thus the LSB suppression variation is caused by small phase and amplitude imbalance of the I and Q signals.

3460

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

TABLE I COMPARISON BETWEEN THE STANDARD AND COMPACT SSB SHP MIXERS

Fig. 17. Gain and suppression versus I and Q input power of the SSB SHP mixer.

Fig. 18. 1-dB compression point of the SSB SHP mixer.

and secondly, the pertinence of the joint utilization of the TFMS and CPW to reduce its size topology compared to standard configuration. V. CONCLUSION

Fig. 19. RF-port return loss of the SSB SHP mixer.

Fig. 16 shows that, in comparison to the desired USB signal, the LSB was suppressed by at least 18 dB over the specified I and Q bandwidth of 50–200 MHz and that 2 was suppressed by 23 dB over the same bandwidth. Fig. 17 shows input/output characteristics at an LO input power of 12 dBm. The 1-dB compression point of the conversion gain is achieved with an I and Q input power of 5 dBm, as illustrated in Fig. 18. Fig. 19 shows that the RF return loss is better than 20 dB over the 43.5–45.5-GHz frequency range. Table I indicate, first, the validity of the novel SSB SHP mixer configuration and design method described in this paper,

A 1.2 1.5 mm novel SSB MMIC mixer with subharmonic LO has been successfully developed using TFMS and CPW structures for 44.5 GHz. This paper has demonstrated how the TFMS and CPW structures for capacitive and inductive loading techniques are effective in terms of reducing the size of SSB SHP and removing a number of limitations inherent to the conventional design approach. The mixer acts as an up-converter with low conversion loss and high LSB suppression over a wide frequency range. This possibility greatly expands the freedom in designing the original topology with good integration density. Compared to the conventional SSB SHP mixer, the advantages that may be derived from the use of the proposed framework are: 1) more degrees of freedom; 2) high compactness (70% size reduction); and 3) better performances. We believe that the SSB MMIC mixer with subharmonic LO presented here can be used to realize low-cost modules with high performance for advanced SATCOM terminal systems operating in the 20/44-GHz frequency bands.

HETTAK et al.: SIZE REDUCTION OF MMIC DIRECT UP-CONVERTER AT 44 GHz IN MULTILAYER CPW TECHNOLOGY

REFERENCES [1] A. A. Abidi, “Direct-conversion radio transceivers for digital communications,” IEEE J. Solid-State Circuits, vol. 30, no. 12, pp. 1399–1410, Dec. 1995. [2] C. J. Verver and M. G. Stubbs, “Development of a 20 GHz to baseband MMIC direct downconverter for advanced SATCOM terminal applications,” in Proc. ANTEM, Montreal, QC, Canada, 1996, pp. 639–642. -band satellite earth terminal development,” in Proc. 2nd [3] C. Pike, “ -Band Utilization Conf., Florence, Italy, 1996, pp. 24–26. [4] K. Itoh, M. Shimozawa, K. Kawakami, A. Lida, and O. Ishida, “Even harmonic quadrature modulator with low vector modulation error and low distortion for microwave digital radio,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1996, pp. 967–970. [5] S. A. Maas, Microwave Mixers, 2nd ed. Norwood, MA: Artech House, 1993. [6] K. Hettak, P. Béland, C. J. Verver, M. G. Stubbs, and G. A. Morin, “44.5 GHz MMIC direct I&Q modulator in coplanar waveguide technology for an EHF SATCOM terminal phased array,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 1315–1317. [7] H. I. Fujishiro, Y. Ogawa, T. Hamada, and T. Kimura, “SSB MMIC mixer with subharmonic LO and CPW circuits for 38 GHz band applications,” Electron. Lett., vol. 37, no. 7, pp. 435–436, Mar. 2001. [8] W. Philibert and R. Verbiest, “A subharmonically pumped I/Q vector modulator MMIC for -band satellite communication,” in IEEE RFIC Symp. Dig., Boston, MA, Jun. 2000, pp. 183–186. -band [9] S. Raman, F. Rucky, and G. Rebeiz, “A high-performance uniplanar subharmonic mixer,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 6, pp. 955–962, Jun. 1997. [10] M. Cohn, J. E. Degenford, and B. A. Newman, “Harmonic mixing with an antiparallel diode pair,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, no. 8, pp. 667–673, Aug. 1975. [11] M. V. Schneider and W. W. Snell, Jr., “Harmonically pumped stripline down-converter,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, no. 3, pp. 271–275, Mar. 1975. [12] K. Itoh, A. Lida, Y. Sasaki, and S. Urasaki, “A 40 GHz band monolithic even harmonic mixer with an antiparallel diode pair,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1991, pp. 879–881. [13] K. Hettak, C. J. Verver, G. A. Morin, and M. G. Stubbs, “A novel uniplanar 44 GHz MMIC subharmonic mixer using CPW series stubs,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, pp. 1157–1160. [14] A. Madjar, “A novel general approach for the optimum design of microwave and millimeter wave subharmonic mixers,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 11, pp. 1997–2000, Nov. 1996. [15] H. Okazaki and Y. Yamaguchi, “Wide-band SSB subharmonically pumped mixer MMIC,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2375–2379, Dec. 1997. [16] K. Hettak, G. A. Morin, and M. G. Stubbs, “The integration of thin film microstrip and coplanar technologies for reduced size MMICs,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 283–291, Jan. 2005. [17] Momentum. Agilent Technol., Palo Alto, CA, 2003. [18] K. Hettak, N. Dib, A. Omar, G. Y. Delisle, M. G. Stubbs, and S. Toutain, “A useful new class of miniature CPW shunt stubs and its impact on millimeter-wave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2340–2349, Dec. 1999. [19] K. Hettak, N. Dib, A. Sheta, and S. Toutain, “A class of novel uniplanar series resonators and their implementation in original applications,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1270–1276, Sep. 1998. [20] R. N. Simons, Coplanar Waveguide Circuits Components & Systems. New York: Wiley, 2001. [21] K. C. Gupta, R. Garg, and I. Bahl, Microstrip Lines and Slotlines. Dedham, MA: Artech House, 1979. [22] K. Chang, Microwave Ring Circuits and Antennas, ser. Microw. Opt. Eng. New York: Wiley, 1996. [23] T. Hirota, A. Minakawa, and M. Muraguchi, “Reduced-size branchline and rat-race hybrids for uniplanar MMIC’s,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 3, pp. 270–275, Mar. 1990.

Ka

Ka

Ka

W

3461

[24] R. B. Singhl and T. M. Weller, “Miniaturized 20 GHz CPW quadrature coupler using capacitive loading,” Microw. Opt. Technol. Lett., vol. 30, no. 1, pp. 3–5, Jul. 2001. [25] M. Gillick, I. D. Robertson, and J. S. Joshi, “Design and realization of reduced-size impedance transforming uniplanar MMIC branchline coupler,” Electron. Lett., vol. 28, pp. 1555–1557, Sep. 1992. [26] D. Kother, B. Hopf, T. Sporkmann, and I. Wolff, “MMIC Wilkinson couplers for frequencies up to 110 GHz,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1995, pp. 663–666. [27] K. Hettak, C. J. Verver, M. G. Stubbs, and G. A. Morin, “A novel compact uniplanar MMIC Wilkinson power divider with ACPS series stubs,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 59–62. Khelifa Hettak (M’05–SM’06) received the Dipl.-Ing. degree in telecommunications from the University of Algiers, Algiers, Algeria, in 1990, and the M.A.Sc. and Ph.D. degrees in signal processing and telecommunications from the University of Rennes 1, Rennes, France, in 1992 and 1996, respectively. In January 1997, he was with the Personal Communications Staff of the Institut National de la Recherche Scientifique (INRS)-Télécommunications. In October 1998, he joined the Electrical Engineering Department, Laval University, where he was an Associate Researcher involved in RF aspects of smart antennas. Since August 1999, he has been with Terrestrial Wireless Systems Branch, Communications Research Centre (CRC) Canada, Ottawa, ON, Canada, as a Research Scientist, where he is involved in the development of MMICs at 60 GHz, low-temperature co-fired ceramic (LTCC) packaging, and RF microelectromechanical systems (MEMS) switches.

Gilbert A. Morin received the B.Sc.A. degree in engineering physics from the École Polytechnique de Montréal, Montréal, QC, Canada, in 1977, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 1980 and 1987, respectively. Since 1987, he has been with Defence Research and Development Canada, Ottawa, ON, Canada, as a Defence Scientist with the Advanced Military Communications Systems Group. His research interests are GaAs MMICs, MEMS, LTCC packaging, reflector and lens antennas, phased arrays, and software-defined radio front-ends.

Malcolm G. Stubbs received the B.Eng., M.Eng., and Ph.D. degrees from the University of Sheffield, Sheffield, U.K., in 1970, 1972, and 1976, respectively. From 1975 to 1978, he was with the Communications Research Centre (CRC) Canada, Ottawa, ON, Canada. He then joined the Allen Clarke Research Centre, Caswell, U.K., where he was engaged in GaAs MMIC research. In 1981, he returned to the CRC Canada, where he is currently responsible for the development of planar microwave and millimeter-wave circuits. His interests include the modeling and application of MMICs, MEMS, and LTCC technologies to high-frequency components for communications systems. Dr. Stubbs was the recipient of a National Research Council Post-Doctoral Fellowship.

3462

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

A Wide Tuning-Range CMOS VCO With a Differential Tunable Active Inductor Liang-Hung Lu, Member, IEEE, Hsieh-Hung Hsieh, Student Member, IEEE, and Yu-Te Liao

Abstract—By utilizing a differential tunable active inductor for the LC-tank, a wide tuning-range CMOS voltage-controlled oscillator (VCO) is presented. In the proposed circuit topology, the coarse frequency tuning is achieved by the tunable active inductor, while the fine tuning is controlled by the varactor. Using a 0.18- m CMOS process, a prototype VCO is implemented for demonstration. The fabricated circuit provides an output frequency from 500 MHz to 3.0 GHz, resulting in a tuning range of 143% at radio frequencies. The measured phase noise is from 101 to 118 dBc/Hz at a 1-MHz offset within the entire frequency range. Due to the absence of the spiral inductors, the fully integrated VCO occupies an active area of 150 300 m2 . Index Terms—Differential active inductors, frequency tuning range, miniaturization, multistandard transceivers, phase noise, voltage-controlled oscillators (VCOs), wideband.

I. INTRODUCTION HE fast-growing market in wireless communications has led to the coexistence of a variety of services from several hundred megahertz to multigigahertz frequency bands. It is desirable to have a compact and power-efficient transceiver, which supports multiple wireless standards with maximum hardware sharing. One of the most challenging tasks for the realization of a multistandard transceiver is to implement a fully integrated voltage-controlled oscillator (VCO) with a wide frequency tuning range. In modern communication systems, VCOs are essential building blocks for frequency translation. Due to the superior performance in phase noise, LC-tank VCOs with spiral inductors and varactors are widely used for circuit implementation at radio frequencies. Theoretically, the VCO tuning range is determined by the maximum-to-minimum capacitance ratio of the varactor . For a typical capacitance ratio in a standard CMOS process, the tuning range of a LC-tank VCO is approximately limited within 30% [1]–[3], making it unattractive for wideband applications. Various techniques have been proposed to enhance the tuning range of the LC-tank VCO by switched capacitors [4], [5] and switched inductors [6], [7]. Though a wide frequency tuning range can be achieved, the

T

Manuscript received March 22, 2006; revised May 23, 2006. This work was supported in part by the National Science Council under Grant 94-2220-E002-026 and Grant 94-2220-E-002-009. L.-H. Lu and H.-H. Hsieh are with the Department of Electrical Engineering and Graduate Institute of Electronics Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: [email protected]). Y.-T. Liao was with the Department of Electrical Engineering and Graduate Institute of Electronics Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. He is now with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA. Digital Object Identifier 10.1109/TMTT.2006.880646

circuits suffer from a considerable increase in the chip area and the complexity of control mechanism. To overcome the limitations imposed on the tuning range, the concept of frequency tuning by active inductors has been introduced [8], [9]. Owing to the wide inductance range provided by the tunable active inductors, a frequency tuning range up to 120% has been reported for a single-ended VCO at multigigahertz frequencies [10]. In this paper, a novel circuit topology is proposed to further improve the performance of the wide tuning-range VCOs with active inductors. By utilizing a differential active inductor and a varactor for the LC-tank, the circuit exhibits a very wide frequency tuning range with a relatively low VCO gain while maintaining enhanced circuit performance in term of the output power and phase noise. The proposed VCO is designed and fabricated in a 0.18- m CMOS process, which is well suited for system integration in multistandard transceiver designs. This paper is organized as follows. Section II describes the proposed circuit topology. The theoretical analysis and circuit design of the wide tuning-range VCO are presented in Sections III and IV, respectively. The experimental results of the fabricated circuit are shown in Section V. Finally, a conclusion is given in Section VI. II. PROPOSED CIRCUIT TOPOLOGY A conceptual illustration of the proposed VCO is shown in Fig. 1(a), where the LC-tank is composed of a tunable active inductor and a varactor for frequency control, and the negative conductance is employed to compensate for the loss from the LC-tank. Since the equivalent inductance of an active inductor can be tuned over a wide range, it is employed as the mechanism for coarse frequency tuning or band selection. In addition, a varactor is included in the LC-tank for fine tuning, maintaining a relatively low tuning sensitivity to ensure the frequency stability. The complete schematic of the proposed VCO including all on-chip components is shown in Fig. 1(b). The tunable active inductor is implemented by transistors , where the gyrator-C architecture including a gain and a feedback element is adopted to emulate the current–voltage characteristics of an inductor. Instead of using the conventional one-port active inductors, a two-port circuit topology is adopted, allowing fully differential operation of the VCO. The equivalent inductance of the active inductor is controlled by . By optimizing the maximum-to-minimum inductance ratio of the tunable active inductor, the limitations on VCO tuning range can be alleviated. As a result, a wideband operation with continuous frequency tuning is realized. Accumulation-mode MOS devices

0018-9480/$20.00 © 2006 IEEE

LU et al.: WIDE TUNING-RANGE CMOS VCO WITH DIFFERENTIAL TUNABLE ACTIVE INDUCTOR

3463

Fig. 1. (a) Conceptual illustration of the VCO architecture. (b) Complete schematic of the wide tuning-range VCO. Fig. 2. (a) Small-signal equivalent circuit of the differential tunable active inductor. (b) Simplified circuit model of the active inductor.

are used as the varactor where the effective capacitance is controlled by . Since the varactor is only for the purpose of fine tuning, it can be optimized for the required tuning sensitivity without degrading the overall tuning range of the VCO. As for the loss compensation, nMOS cross-coupled transistors – are employed to provide the negative conductance. By stacking the cross-coupled pair with the differential active inductor, a current-reuse bias scheme is established to minimize the power consumption. The open-drain buffers and are employed to drive the 50- load of the testing instruments. III. CIRCUIT ANALYSIS A. Small-Signal Characteristics The operation of the wide tuning-range VCO is based on the design of the tunable active inductor. Hence, a small-signal analysis is performed to characterize the behavior of the differential active inductor. Fig. 2(a) shows the simplified small-signal equivalent circuit of the active inductor – . From a dc point-of-view, and form a cross-coupled pair, while and are in the common-drain configuration. At the quiescent bias point, it is obvious that transistors – are saturated. As for and , they can operate either in the saturation region or in the triode region, depending on the controlled voltage at

the gate . Therefore, and are modeled as and , respectively, representing the drain conductance at the associated bias point. By deriving the port voltage for a given input current , the input impedance at the differential port can be expressed as (1) , the input impedance of the differential For active inductor can be approximated by the small-signal model, as shown in Fig. 2(b), where (2) (3) (4) From (2), it is observed that the equivalent inductance depends on the circuit parameters including , , , , and

3464

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 3. Simplified small-signal model for the wide tuning-range VCO.

. An effective way for the inductance tuning is to manipulate the drain conductance by the gate voltage. Therefore, is used as the control mechanism for the tunable active inductor. In addition to the equivalent inductance, the factor of the active inductor is also evaluated from the small-signal analysis. Typically, the factor of an inductor is defined as the ratio of the imaginary part to the real part of the input impedance. Based on (1), the factor is given as

Fig. 4. Noise model for the wide tuning-range VCO.

Based on the circuit topology in Fig. 2, the active inductor and the cross-coupled pair share the same bias current. Therefore, the size and is thus determined by the required transconducof tance at the current level specified by the active inductor. C. Frequency Tuning Range

(5)

By setting the first derivative of to zero maximum factor and the associated frequency are

, the

(6) (7)

From the derivations, it is evident that the factor of the active inductor can be optimized at the frequencies of interest by properly choosing the circuit parameters of transistors – . B. Start-Up Conditions By adopting the small-signal model of the tunable active inductor, as shown in Fig. 2(b), a simplified equivalent circuit of the VCO is illustrated in Fig. 3. To ensure that the oscillation initiates, the negative conductance provided by the cross-couple pair – should be sufficiently large to compensate for the loss from the tank, which is primarily contributed by the equivalent conductance . The rule-of-thumb for the VCO design is to choose the negative conductance three times larger than the required value

(8)

In the VCO design, the coarse frequency tuning is achieved by the tunable active inductor, while the fine tuning is provided by the varactor. From (2), the equivalent inductance is strongly influenced by the drain conductance . As the controlled voltage increases from a low voltage level, transistors – are driven from the triode region toward the saturation region, leading to a decrease in the values of and . Consequently, the equivalent inductance of the active inductor increases, and the output frequency of the VCO decreases. Due to the nature of the inductance tuning characteristics, a very wide tuning range can be achieved for the VCO design with a simple control mechanism. The VCO fine tuning range is solely determined by the varactor. Typically, the tuning range of the varactor is degraded by the parasitic capacitances from the transistors, especially for – , due to the loading effect. By increasing the varactor size, a wide fine tuning range can be achieved at the expense of the maximum oscillation frequency. Therefore, a design tradeoff between the maximum operating frequency and the fine tuning range should be taken into account in determining the varactor size for the VCO implementation. D. Large-Signal Behavior Provided that the negative conductance from the crosscoupled pair is sufficiently large, the oscillation initiates at the resonant frequency of the LC-tank. As the oscillation amplitude grows, the effective bias current of the active inductor decreases due to the nonlinear operation of the transistors – , leading to a decrease in the transconductances and an increase in the drain conductance . From (2), it is observed that no significant variation is expected

LU et al.: WIDE TUNING-RANGE CMOS VCO WITH DIFFERENTIAL TUNABLE ACTIVE INDUCTOR

3465

Fig. 5. Small-signal equivalent circuit of the differential active inductor with noise current sources.

in the value of during the oscillation swing build-up. Therefore, the small-signal analysis provides a prediction to the oscillation frequency of the VCO with sufficient accuracy. Another important specification of the VCO, especially for wideband applications, is the output swing variation within the frequency tuning range. In a conventional LC-tank VCO where a constant bias current is employed for the cross-coupled pair, the output swing tends to decrease at the higher frequency bands due to the frequency response of the active devices. For the proposed VCO circuit, the coarse frequency tuning is controlled by the equivalent inductance of the active decreases, transistors – are driven inductor. As from the saturation region toward the triode region, resulting in a decreasing equivalent inductance and an increasing oscillation frequency. Meanwhile, the increase in the voltage headroom of transistors – leads to a larger bias current for the cross-coupled pair, providing an enhanced negative conductance to compensate for the output swing degradation at the higher frequency bands. As a result, a uniform output swing can be achieved over the wide frequency tuning range in the proposed VCO topology.

TABLE I CIRCUIT PARAMETERS OF THE VCO

E. Phase Noise In order to perform the noise analysis of the VCO circuit, a simplified circuit model is employed, as shown in Fig. 4, where the reactive components are noiseless, and the equivalent noise and represent the noise contributions from sources the LC-tank and cross-coupled pair, respectively. The analysis starts with the derivation of the equivalent noise current of the active inductor at the differential port. Fig. 5 shows the small-signal equivalent circuit of the active inductor with in– . Typically, dividual noise sources from the transistors the noise current of a MOSFET is modeled by the channelinduced noise and gate-induced noise as [11], [12]

Fig. 6. Microphotograph of the fabricated wide tuning-range VCO.

(9) Fig. 7. Coarse tuning characteristics of the VCO.

where and are the output conductance and transconductance, respectively, is a process-dependent constant,

and for short channel devices. Assuming and the noise sources are uncorrelated for the individual transis-

3466

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 8. Frequency tuning characteristics of the wide tuning-range VCO.

Fig. 10. (a) Measured output spectrum and (b) close-in phase noise of the VCO at 2.9 GHz.

where

(10) Fig. 9. (a) Measured output spectrum and (b) close-in phase noise of the VCO at 690 MHz.

tors, the input-referred noise current of the LC-tank can be expressed as

Other than the tunable active inductor, the cross-coupled pair – also contributes to the phase noise of the VCO, and the equivalent noise current is given by

(11)

LU et al.: WIDE TUNING-RANGE CMOS VCO WITH DIFFERENTIAL TUNABLE ACTIVE INDUCTOR

3467

TABLE II PERFORMANCE SUMMARY OF THE WIDE TUNING-RANGE VCO

By superimposing the results from (10) and (11), the overall noise current injected at the differential output nodes is simply

(12) According to [13] and [14], the phase noise of an oscillator can be derived by examining the excess phase at the output

(13)

is the unit impulse response for excess phase, where is the current impulse injected at the output nodes, is the maximum charge displacement across the capacitance on the nodes, and represents the impulse sensitivity function (ISF) of the oscillator. Based on (13), the total single-sideband phase noise spectral density at an offset frequency of is given by

(14)

where represents the coefficients in the Fourier series of the ISF. In order to minimize the equivalent inductance for a wide frequency tuning range, transistors – are typically biased at high overdrive voltage for maximum transconductance, which, in turn, leads to enhanced noise contributions from the active devices. Therefore, important design insights for the tradeoff between the frequency tuning range and phase noise of the VCO can be derived from the analysis. According to (10), simulation indicates that the coefficient exhibits a low-pass response versus frequency. Noise contributions from the transistor pair – have significant influences on the close-in phase noise at the VCO output. Hence, the transistor size and the bias current of and must be carefully designed to achieve an optimum phase noise for wideband applications.

IV. CIRCUIT DESIGN In order to demonstrate the wideband characteristics of the proposed circuit technique, a prototype VCO with a tuning range from several hundred megahertz to multigigahertz frequencies is implemented in a standard 0.18- m CMOS process. The design starts with the circuit parameters for the tunable active inductor, as indicated in (2)–(4). In consideration of the minimum inductance for the highest operating frequency, the controlled voltage is set to low and the transistors – are biased at a high overdrive voltage to achieve large transconductances with minimum gate capacitances. The transistor sizes for and are determined by (8) to ensure the oscillation at the highest operating frequency. As increases, the equivalent inductance increases and the VCO operation frequency decreases. Since the bias current of the cross-coupled pair drops during the frequency tuning, the lowest operation frequency is reached when the negative conductance is insufficient to compensate for the loss from the tank. The transistor sizes used for this design are tabulated in Table I. After the design of the tunable active inductor, a varactor with a maximum capacitance of 3 pF is chosen in this design to achieve the required resonant frequency and VCO gain. In the design of a wideband VCO with tunable active inductors, phase noise is one of the major concerns. The phase noise can be improved by increasing the channel length of the transistors. However, the excess parasitic capacitance degrades the frequency tuning range and the highest operating frequency. Therefore, MOS transistors with minimum channel length are utilized in this design to demonstrate the optimized tuning range for multistandard wireless applications. V. EXPERIMENTAL RESULTS Fig. 6 shows the microphotograph of the fabricated circuit. To ensure the fully differential operation, a symmetrical layout is used for the design. The total chip area is 0.40 0.63 mm including the pads, where the active area occupy only 0.15 0.30 mm . On-wafer probing was performed to characterize the circuit performance. The output spectrum and close-in phase noise of the VCO were measured by a spectrum analyzer. The circuit operates at a supply voltage of 1.8 V. As sweeps from 1.3 to 0.3 V, the fabricated VCO exhibits a coarse frequency tuning range from 500 MHz to 3.0 GHz with a of 2500 MHz/V. Within the frequency tuning

3468

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

range, the power consumption of the VCO core varies from 6 to 28 mW. Fig. 7 shows the output power and phase noise at 1-MHz offset versus oscillation frequency. The frequency fine tuning is achieved by the varactor. As sweeps from 0.6 to 1.8 V, an average of 108 MHz/V is obtained. The detailed frequency tuning characteristics are shown in Fig. 8. The output spectrum and close-in phase noise of the VCO at lower and higher frequency bands are shown in Figs. 9 and 10, respectively. When operating at an oscillation frequency of 690 MHz, the VCO delivers an output power of 17.8 dBm to the 50- test instrument with a phase noise of 118 dBc/Hz at 1-MHz offset frequency. When the VCO is operating at an oscillating frequency of 2.9 GHz, the output power and the phase noise at 1-MHz offset are 20 dBm and 102 dBc/Hz, respectively. The performance of the proposed circuit along with results from the state-of-the-art wideband VCO designs are summarized in Table II. VI. CONCLUSION With a differential tunable active inductor and a varactor for an LC-tank, a wide tuning-range VCO has been presented at radio frequencies. Using a 0.18- m CMOS process, a prototype VCO is implemented for demonstration. The fabricated circuit exhibits a very wide frequency tuning range from 500 MHz to 3.0 GHz while maintaining excellent circuit performance in terms of close-in phase noise and output power within the entire frequency range. It is suitable for the implementation of fully integrated RF transceiver for multistandard applications. ACKNOWLEDGMENT The authors would like to thank the National Chip Implementation Center (CIC), Hsinchu, Taiwan, R.O.C., for chip fabrication and the National Nano Device Laboratories (NDL), Hsinchu, Taiwan, R.O.C., for chip measurement. REFERENCES [1] B. Min and H. Jeong, “5-GHz CMOS LC VCOs with wide tuning ranges,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 5, pp. 336–338, May 2005. [2] R. S. Rana, X. D. Zhou, and Y. Lian, “An optimized 2.4 GHz CMOS LC-tank VCO with 0.55%/V frequency pushing and 516 MHz tuning range,” in IEEE Int. Circuits Syst. Symp., May 2005, pp. 4811–4814. [3] B. De Muer, N. Itoh, M. Borremans, and M. Steyaert, “A 1.8-GHz highly-tunable low-phase-noise CMOS VCO,” in IEEE Custom Integr. Circuits Conf., May 2000, pp. 585–588. [4] A. D. Berny, A. M. Niknejad, and R. G. Meyer, “A 1.8-GHz LC VCO with 1.3-GHz tuning range and digital amplitude calibration,” IEEE J. Solid-State Circuits, vol. 40, no. 4, pp. 909–917, Apr. 2005. [5] ——, “A wideband low-phase-noise CMOS VCO,” in IEEE Custom Integr. Circuits Conf., Sep. 2003, pp. 555–558. [6] F. Herzel, H. Erzgraber, and N. Ilkov, “A new approach to fully integrated CMOS LC-oscillators with a very large tuning range,” in IEEE Custom Integr. Circuits Conf., May 2000, pp. 573–576. [7] Z. Li and K. K. O, “A 1-V low phase noise multi-band CMOS voltage controlled oscillator with switched inductors and capacitors,” in IEEE Radio Freq. Integr. Circuits Symp. Dig., Jun. 2004, pp. 467–470. [8] J.-S. Ko and K. Lee, “Low power, tunable active inductor and its applications to monolithic VCO and BPF,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1997, pp. 929–932.

[9] T. Y. K. Lin and A. J. Payne, “Design of a low-voltage, low-power, wide-tuning integrated oscillator,” in IEEE Int. Circuits Syst. Symp., May 2000, pp. 629–632. [10] R. Mukhopadhyay, Y. Park, P. Sen, N. Srirattana, J. Lee, C.-H. Lee, S. Nuttinck, A. Joseph, J. D. Cressler, and J. Laskar, “Reconfigurable RFICs in Si-based technologies for a compact intelligent RF frontend,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 81–93, Jan. 2005. [11] D. Ham and A. Hajimiri, “Concepts and methods in optimization of integrated LC VCOs,” IEEE J. Solid-State Circuits, vol. 36, no. 6, pp. 896–909, Jun. 2001. [12] A. Hajimiri and T. H. Lee, “Design issues in CMOS differential LC oscillators,” IEEE J. Solid-State Circuits, vol. 34, no. 5, pp. 717–724, May 1999. [13] ——, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. [14] L. Jia, J.-G. Ma, K. S. Yeo, and M. A. Do, “9.3–10.4-GHz-band crosscoupled complementary oscillator with low phase-noise performance,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1273–1278, Apr. 2004. [15] Y.-H. Chuang, S.-L. Jang, J.-F. Lee, and S.-H. Lee, “A low voltage 900 MHz voltage controlled ring oscillator with wide tuning range,” in IEEE Asia–Pacific Circuits Syst. Conf., Dec. 2004, pp. 301–304. [16] Y. A. Eken and J. P. Uyemura, “A 5.9-GHz voltage-controlled ring oscillator in 0.18-m CMOS,” IEEE J. Solid-State Circuits, vol. 39, no. 1, pp. 230–233, Jan. 2004.

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

Hsieh-Hung Hsieh (S’05) was born in Taipei, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2004, and is currently working toward the Ph.D. degree in electronic engineering from National Taiwan University. His research interests include the development of low-voltage and low-power RF integrated circuits, multiband wireless systems, RF testing, and monolithic microwave integrated circuit (MMIC) designs.

Yu-Te Liao was born in Taichung, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 2003, the M.S. degree in electronics engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2005, and is currently working toward the Ph.D. degree at the University of Washington, Seattle. His research interests include RF integrated circuits, wideband frequency synthesizers, and wireless sensor network interface designs.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3469

Piecewise Pre-Equalized Linearization of the Wireless Transmitter With a Doherty Amplifier Wan-Jong Kim, Student Member, IEEE, Kyoung-Joon Cho, Shawn P. Stapleton, and Jong-Heon Kim, Member, IEEE

Abstract—This paper describes a predistorter (PD) based on piecewise pre-equalizers for use in multichannel wideband applications. The predistortion linearizer consists of piecewise pre-equalizers along with a look-up-table-based digital PD that together compensate for nonlinearities, as well as memory effects of power amplifiers (PAs). It takes advantage of multiple finite-impulse-response filters that significantly reduce the complexity when compared to memory polynomial methods. The proposed method was also compared with the conventional Hammerstein structure. A 300-W peak envelope power Doherty PA was first modeled by measured time-based samples in order to verify the adjacent channel power ratio (ACPR) performance in simulation using a multitone and single wideband code-division multiple-access (W-CDMA) carrier. Furthermore, the experimental results applying two W-CDMA carriers verify that the proposed method provided similar improvement to that of the memory polynomial approach. The experimental results verified the complexity reduction and superior ACPR performance over the conventional Hammerstein structure. Index Terms—Adjacent channel power ratio (ACPR), digital predistorter (PD), Doherty power amplifier (DPA), finiteimpulse-response (FIR) filter, Hammerstein, lookup table (LUT), memory effects, memory polynomial.

I. INTRODUCTION S A result of the increasing importance of spectral efficiency in mobile communications, the linearity and efficiency of RF power amplifiers (PAs) have been a critical design issue for nonconstant envelope digital modulation schemes, which have high peak-to-average power ratios (PARs). RF PAs have nonlinearities that generate amplitude modulation–amplitude modulation (AM–AM) and amplitude modulation–phase modulation (AM–PM) distortion at the output of the PA. These effects create spectral regrowth in the adjacent channels and in-band distortion, which degrades the error vector magnitude (EVM). The relation between linearity and efficiency is a tradeoff since power efficiency is very low when the amplifier operates in its linear region and increases as the amplifier is driven into its compression region. In order to enhance linearity and efficiency at the same time, one of the

A

Manuscript received August 9, 2005. This work was supported by the Ministry of Information and Communication, Korea under the Information Technology Research Center support program supervised by the Institute of Information Technology Assessment and by the Natural Sciences and Engineering Research Council of Canada. W.-J. Kim, K.-J. Cho, and S. P. Stapleton are with the School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 (e-mail: [email protected]; [email protected]). J.-H. Kim is with the Department of Radio Science and Engineering, Kwangwoon University, Seoul 139-701, Korea (e-mail: [email protected]). Color versions of Figs. 6, 14, and 15 are available online at http://ieeexplore. ieee.org. Digital Object Identifier 10.1109/TMTT.2006.880639

various linearization techniques should be applied to the RF PAs. Various linearization techniques have been proposed in the literature such as feedback, feedforward, and predistortion [1], [2]. The most promising linearization technique is baseband digital predistortion, which takes advantage of the recent advances in digital signal processors. Digital predistortion can achieve good linearity and good power efficiency with a reduced system complexity when compared to the widely used conventional feedforward linearization technique. Moreover, a software implementation provides the digital predistorter (PD) with reconfigurability suitable for the multistandards environments. In addition, a Doherty power amplifier (DPA) is able to achieve higher efficiencies than traditional PA designs, albeit at the expense of linearity. Therefore, combining digital predistortion with a DPA has the potential of maximizing system linearity and overall efficiency. However, most digital PDs presuppose that PAs have no memory or a weak memory [3]–[5]. This is impractical in wideband applications where memory effects describe the output signal as a function of the current, as well as past input signals. The sources of memory effects in PAs consist of self-heating of the active device (also referred to as long time constant or thermal memory effects) and frequency dependences of the active device related to the matching network or bias circuits (also referred to as short time constant or electrical memory effects) [6]. As signal bandwidth increases, memory effects of PAs become significant and will limit the performance of memoryless digital PDs. Various approaches have been suggested for overcoming memory effects in digital PDs. For short-term memory effects, a Volterra filter structure was applied to compensate memory effects using an indirect learning algorithm, but the number of optimization coefficients is very large as the order increases [7]. This complexity makes the Volterra-filter-based PD extremely difficult to be implemented in real hardware. The memory polynomial structure, which is a simplified version of the Volterra filter, has appeared in several publications in order to reduce the number of coefficients, but still requires a large computational load [8]–[11]. In addition, such a memory polynomial-based PD suffers from a numerical instability when higher order polynomial terms are included because a matrix inversion is required for estimating the polynomial coefficients. An alternative, yet equally complex structure based on orthogonal polynomials has been utilized to alleviate the numerical instability associated with the traditional polynomials [12]. To further reduce the complexity at the expense of the performance, the Hammerstein PD, which is a finite impulse response (FIR) filter or a linear time invariant (LTI) system followed by a memoryless polynomial PD, has been proposed by several authors [13]–[15]. The Hammerstein

0018-9480/$20.00 © 2006 IEEE

3470

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 1. Piecewise pre-equalized LUT PD system.

PD assumed that the PA models used follow a Wiener model structure, which is a memoryless nonlinearity followed by a FIR filter or an LTI system. This implementation means that the Hammerstein structure can only compensate for memory effects coming from the RF frequency response. Therefore, if the RF frequency response is quite flat, the Hammerstein PD cannot correct for any other types of memory effects such as bias-induced and thermal memory effects [16]. Most recently, a static lookup table (LUT) digital baseband PD cascaded with a subband filtering block has been published in order not to compensate electrical memory effects, but to combat gain and phase variation due to temperature changes of the PA after an initial setting for the fixed LUT PD [17]. To the best of our knowledge, a LUT-based approach for frequency-dependent memory effects able to compensate for not only RF frequency response memory effects, but also bias-induced or thermal memory effects has never been published. In this paper, we propose the piecewise pre-equalized LUT PD, which is a cascade of a LUT PD and piecewise pre-equalizers. This approach may be considered as the extended structure of the LUT-based Hammerstein PD, which has only one equalizer. The results show that the proposed method is superior to the Hammerstein PD in Sections IV and V. Our approach has the distinct advantage of simplicity and is easy to implement in real hardware. In Section II, the proposed piecewise pre-equalized LUT PD is described. The PA behavioral modeling based on measurements is presented in Section III, and the simulation results from the memoryless PD based on a LUT and the proposed PD with memory compensation polynomial PD are shown in Section IV. In Section V, experimental results using the proposed PD are compared to the alternative structures using two wideband code-division multiple-access (W-CDMA) carriers in the test bed. In Section VI, the complexity is evaluated for the proposed approach and the memory polynomial is discussed.

II. PIECEWISE PRE-EQUALIZERS-BASED LUT PREDISTORTION A. Proposed Piecewise-Equalizers-Based LUT PD Fig. 1 illustrates the structure of the piecewise pre-equalized filter coefficients in the LUT are LUT-based PD. by is the depth of the used to compensate for memory effects, taps (including , LUT and the FIR filter has is which is equal to 1). For a LUT-based Hammerstein PD, equal to 1. The piecewise pre-equalizers use a FIR filter rather than an infinite impulse response (IIR) filter because of stability issues. The output of the pre-equalizers can be described by

(1) is the th tap and th indexed coefficient where . corresponding to the magnitude of the input signal is also a function of , and is a function . For analysis purposes, the memoryless LUT of structure can be replaced by a polynomial model as follows: (2) where is the polynomial order and is a complex coefficient corresponding to the polynomial order. Moreover, it is noted that the tap coefficients and memoryless LUT coefficients depend on and , respectively.

KIM et al.: PIECEWISE PRE-EQUALIZED LINEARIZATION OF WIRELESS TRANSMITTER WITH DOHERTY AMPLIFIER

3471

Fig. 2. Corresponding block diagram of the proposed PD with the LUT replaced by a polynomial equation.

Therefore, the proposed model can be expressed using a polynomial equation by

(3) is the th tap coefficient with the th index where . Without loss of generality, the piecebeing a function of wise pre-equalizers can be defined similarly using a

Fig. 3. Graphical expressions. (a) Complex gain adjuster response. (b) Piecewise equalizers response. (c) Response of the cascade of complex gain adjuster and piecewise equalizers. (d) PA response. (e) Desired response from (c) and (d).

the memoryless nonlinearity PD LUT reduced as

, so that (5) can be

(6) (4)

th-order polynomial, where is the th tap and th-order coefficient. Fig. 2 illustrates the corresponding block diagram of the piecewise pre-equalizers PD when polynomial equations are utilized. From Figs. 1 and 2, it can be easily seen that (1) is , equivalent to (4). For illustrative purposes, we choose , and to represent a third odd-order polynomial model of a nonlinearity and expand (4) as follows:

(5) in (1) is in The first tap coefficient (5) and can be considered to be equal to 1 in order not to repeat

It can be clearly seen that (6) can be considered as the predistortion of a truncated Volterra series model for a PA; it is able to compensate for nonlinearity, as well as memory effects [18] (see the Appendix). However, (6) is based on a polynomial representation of the proposed model in order to show the piecewise pre-equalized LUT PD memory compensation. The polynomial representation requires too many complex multiplications similar to the Volterra series. The complexity is reduced when a piecewise pre-equalized LUT PD-based approach, as shown in Fig. 1, is utilized. For a graphical explanation of the piecewise pre-equalized LUT PD, we use Fig. 3. A typical memoryless PD response is shown in Fig. 3(a). Fig. 3(b) demonstrates the hysteresis created pieces. Since by the piecewise pre-equalizers divided into the hysterisis of the PA is not necessarily uniformly distributed over the whole input magnitude range, we propose the piecewise pre-equalizers. The cascade of Fig. 3(a) and (b) results in the piecewise pre-equalized LUT-based PD, as represented in Fig. 3(c). Fig. 3(d) shows the response of a typical PA response and Fig. 3(b) results in the piecewise pre-equalized LUT-based PD, as represented in Fig. 3(c). Fig. 3(d) shows the response of a typical PA response with memory. The desired linear response is achieved after Fig. 3(c) and (d) are cascaded.

3472

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

B. Algorithm of the Proposed and the Memory Polynomial PD Cavers has studied the optimal addressing of the LUT and has shown that addressing of the LUT by a uniform quantization has a close performance with optimal addressing [19]. Therefore, we decided to apply the linear magnitude addressing method for the LUT indexing as follows: round

(7)

where the function returns the nearest integer number and is the LUT size. The digital which is the index complex baseband input signal samples are multiplied prior to pre-equalization by complex coefficients drawn from LUT entries, as shown in (1)

Fig. 4. Indirect learning algorithm for the PD of the PA.

The input of the multiple equalizers in the feedback path is written in vector format as

(8) where is the complex coefficient corresponding to an input signal magnitude for compensating AM–AM and AM–PM distortions of the PA. , this signal is After digital-to-analog converting of up-converted to RF, amplified by the PA generating distortions, attenuated, down-converted to baseband, and then finally analog-to-digital converted and applied to the delay estimation algorithm. The feedback signal, i.e., the output of the PA with can be described by delay

(12) is the post LUT output, i.e., . where can be derived Therefore, the multiple FIR filter outputs in vector format using the following equations: (13) (14) where is a transpose operator. Adaptation of the tap coefficients of the pre-equalizers can be obtained as follows:

(9) where and is AM–AM and AM–PM distortions of the PA, respectively, and is the feedback loop delay. For estimating , a correlation technique was applied as follows:

(10) where is the delay variable and is the block size to correlate. After delay estimation, the memoryless LUT coefficients can be estimated by the following equation, which is the well-known least mean square (LMS) algorithm with indirect learning, as shown in Fig. 4 [7]: (11) where is the iteration number, is the stability factor and is . It should be pointed out that addressing already generated for (5) can be reused for indexing , which is a distorted signal able to cause another error due to incorrect indexing. During this procedure, the samples should bypass by the piecewise pre-equalizers. After convergence of this indirect learning LMS algorithm, the equalizers are activated. An indirect learning method with an LMS algorithm has also been utilized for adaptation of the piecewise filter coefficients.

(15) is the error signal between and , and where is the step size ( represents the complex conjugate). For the memory polynomial PD, indirect learning using a recursive least square (RLS) algorithm was applied as described in [7]. III. PA BEHAVIORAL MODELING In order to demonstrate the performance of the proposed PD in MATLAB simulations, the behavioral modeling based on timedomain measurement samples was first carried out. Among the PA models, the simplest truncated Volterra model is the diagonal Volterra model, called the memory polynomial model, where all off-diagonal terms are zero. Although this condition reduces the number of model parameters to be estimated dramatically, it also has significant consequences, which is decreased reliability of the model. This is because, in some PA cases, the off-diagonal terms are more important than the diagonal terms. For these cases, the condition should be relaxed by including near-diagonal terms. Therefore, the behavioral model chosen was based on the truncated Volterra model [18] as follows:

(16)

KIM et al.: PIECEWISE PRE-EQUALIZED LINEARIZATION OF WIRELESS TRANSMITTER WITH DOHERTY AMPLIFIER

3473

Fig. 5. Lineup for a 300-W PEP Doherty PA.

Fig. 6. Test bench setup for modeling of the PA.

where is the th-order Volterra kernel, represents the memory of the corresponding nonlinearity, and denotes the conjugate transpose. The model coefficients can be estimated via the least square method in order to minimize the cost function

Fig. 7. In-phase signal modeling result.

(17) A 300-W peak envelope power (PEP) DPA using two 170-W push–pull-type Motorola laterally diffused metal–oxide–semiconductor (LDMOS) at the final stage was designed and built with the whole lineup shown in Fig. 5 [20]. This DPA operates at a 2140-MHz band and has 61 dB of gain and 28% power-added efficiency (PAE) at an average 30-W output power. To construct the PA model based on measurements of the actual PA, the test bench, as shown in Fig. 6, was utilized [21], [22]. As a test signal, a single downlink W-CDMA carrier with 64 dedicated physical channels (DPCHs) of Test Model 1, which has 3.84 Mchip/s and 9.8 dB of a crest factor was used as the input signal in the measurements, in order to extract the model coefficients. The signal was uploaded to an Agilent ESG 4438C through a local area network (LAN) cable. The ESG generates the corresponding RF signal that is applied to the DPA and the output of the DPA is fed into a single channel vector signal analyzer (VSA) (VSA 89641A) after attenuation. The normalization and synchronization are performed to compare complex envelope both at the input and output of the DPA. 30 000 time-domain samples of data at the input and output of the DPA were used to construct the behavioral model of the truncated Volterra series. The truncated Volterra-series PA model of seventh order and memory length of four was

Fig. 8. Quadrature signal modeling result.

constructed and were extracted using in-house software implemented in MATLAB. The behavioral model extracted had 44.1 dB of normalized mean square error (NMSE). Figs. 7 and 8 show the time-domain results of in-phase (I) and quadrature (Q) components of the truncated Volterra-series behavioral model. The frequency-domain results of the behavioral model can be shown in Fig. 9. It is clearly seen that the behavioral model matches the measurement data in both the time and frequency domains.

3474

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 9. Frequency-domain modeling results.

Fig. 10. Linearization with memoryless LUT PD.

IV. SIMULATION RESULTS Based on the behavioral model constructed in Section III, we have simulated four types of PDs consisting of a memoryless LUT PD, a Hammerstein PD, the proposed piecewise pre-equalizers PD, and a memory polynomial PD. The adjacent channel power ratio (ACPR) performances are compared. The simulations for the PDs mentioned above were carried out in MATLAB based on the behavioral model for the PA. The LUT size was fixed to 128 entries through all simulations, which is a compromise size considering quantization effects and memory size. First, an eight-tone signal with 500-kHz spacing, which has 9.03 dB of PAR and 4-MHz bandwidth, which is comparable to a W-CDMA signal, was used for verifying the proposed method. Figs. 10 and 11 show the linearization results before and after linearization of the LUT PD and of the LUT Hammerstein PD. The Hammerstein PD deteriorates the performance above 10 MHz and improves it within a 10-MHz bandwidth. It should be noted that the Hammerstein PD with a LUT was not able to compensate for the memory effects. If

Fig. 11. Linearization with the LUT Hammerstein PD.

Fig. 12. Linearization with the proposed piecewise pre-equalizers PD.

the RF frequency response in the main signal path is quite flat, the Hammerstein PD is not able to correct any other memory effects, except for frequency response memory effects. There is no obvious improvement for reducing spectral regrowth using the conventional Hammerstein PD. From experimental results in Section V, it will be very clear that the ability of the Hammerstein PD for suppressing distortions coming from memory effects is quite limited. This is in agreement with the simulation results and conclusion in [13]. In Figs. 12 and 13, the performance of the proposed piecewise pre-equalizers PD (with two taps) and the memory polynomial PD (with fifth order and two memory terms) is shown. The proposed PD is comparable to the memory polynomial PD in terms of ACPR performance. A single W-CDMA carrier was applied to the LUT PD, LUT Hammerstein PD, proposed PD, and memory polynomial PD. Linearization results for all the PDs mentioned above are shown in Fig. 14. ACPR at a frequency offset ( 5 MHz) is evaluated and summarized in Table I. The conventional Hammerstein PD

KIM et al.: PIECEWISE PRE-EQUALIZED LINEARIZATION OF WIRELESS TRANSMITTER WITH DOHERTY AMPLIFIER

Fig. 13. Linearization with the memory polynomial PD.

3475

Fig. 15. Experimental results with the different PDs. (a) Without PD. (b) With LUT PD. (c) Hammerstein PD with a five-tap FIR filter. (d) Proposed PD with two taps. (e) Memory polynomial PD with fifth-order and two memory terms.

V. EXPERIMENTAL RESULTS

Fig. 14. Linearization with the different PDs. (a) Without PD. (b) With LUT PD. (c) Hammerstein PD with a five-tap FIR filter. (d) Proposed PD with two taps. (e) Memory polynomial PD with fifth-order and two memory terms.

TABLE I SUMMARY OF ACPR SIMULATION FOR THE DIFFERENT PDs

was unable to improve any distortions coming from memory effects over the memoryless PD. The proposed PD could suppress distortions due to nonlinearities, as well as memory effects of the PA.

After verifying the ACPR performance of the proposed PD in MATLAB simulations based on the behavioral PA model, an experimentation was performed using the actual DPA in our test bench. The experimental setup used to evaluate the performance of the various PDs is similar to the test bed in Fig. 5 without the switch. The transmitter prototype consists of an ESG, which has two digital-to-analog converters (DACs) and an RF up-converter, along with the PA. The receiver comprises an RF down-converter, a high-speed analog-to-digital converter, and a digital down-converter. This receiver prototype can be constructed by a VSA. For a host DSP, a PC with MATLAB and Agilent’s Advanced Design System (ADS) was used for delay compensation and the predistortion algorithm. As a test signal, two downlink W-CDMA carriers with 64 DPCH of Test Model 1, which has 3.84 Mchip/s and 9.8 dB of a crest factor, were used as the input signals in the measurements in order to verify the compensation performance of the different PDs. All coefficients of PDs are identified by an indirect learning algorithm, which is considered to be inverse modeling of the PA. During the verification process, a 256-entry LUT, a five-tap FIR filter for Hammerstein PD, the proposed piecewise pre-equalizers PD (with two taps), and a fifth-order two-delay memory polynomial were used. The choice of the number of taps was optimized from several measurements. Fig. 15 shows the linearization performance of all the different PDs. ACPR calculation at the output of the prototype transmitter is performed at a frequency offset (5 and 5 MHz) from the center frequency. The ACPR value for the transmitter with the Hammerstein PD having a five-tap FIR filter is approximately 1 dB better than a LUT PD on the upper ACPR (5-MHz offset) and the same at the lower ACPR ( 5-MHz offset), as summarized in Table II. The proposed PD and a fifth-order–two-memory polynomial PD show close compensation performance in terms of ACPR. Both are able to improve the ACPR approximately 4 and 6 dB more than Hammerstein PD and a memoryless LUT PD for the lower and upper ACPR, respectively.

3476

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

TABLE II SUMMARY OF ACPR MEASUREMENTS FOR THE DIFFERENT PDs

TABLE IV COMPLEXITY ESTIMATION OF MEMORY POLYNOMIAL PD

TABLE III COMPLEXITY ESTIMATION OF THE PROPOSED PIECEWISE PRE-EQUALIZED PD

B. Memory Polynomial PD For the memory polynomial method using an RLS indirect learning algorithm [7], the number of arithmetic operations are is equal to . For exgiven in Table IV where ample, and require 1342 real additions (subtractions), 1644 real multiplications per sample, and memory size of 24. In a comparison of the number of multiplications with the proposed PD, the memory polynomial PD requires 300 times more real multiplications per sample. Therefore, the complexity is significantly reduced by the proposed method. In addition, the number of real multiplication for the memory polynomial method grows as the square power of the polynomial order and memory length. VI. COMPLEXITY EVALUATION One of the crucial problems in digital predistortion is the complexity of the predistortion algorithm. Therefore, the complexity of the proposed method and the memory polynomial method is evaluated (neglecting LUT readings, writings, indexing, and calculation of the square root (SQRT) of the signal magnitude because LUT indexing depends not only on the methods, but also on the variable, e.g., magnitude, logarithm, power, etc., and the SQRT operation can be implemented in different ways). Therefore, the complexity is only estimated by counting the number of additions (subtractions) and multiplications per input sample. In order to consider a real hardware implementation, complex operations are converted into real operations and memory size is also considered. For example, one complex multiplication requires two real additions and four real multiplications. If is the number of LUT entries, memory size required is (I&Q LUTs). A. Piecewise Pre-Equalizers-Based LUT PD Table III gives the complexity of the proposed piecewise equalizers-based LUT PD for entries and coefficients per filter. If the LUT has 256 entries and the filters have two taps, the PD requires 40 real additions (subtractions), 54 real multiplications per sample, and a memory size of 1542. Our proposed PD requires the same number of additions and multiplications as the traditional Hammerstein PD, but requires more memory.

VII. CONCLUSION The piecewise pre-equalized LUT-based digital predistortion has been described, simulated, measured, and compared with the different PD structures. A MATLAB-based behavioral model of the PA has been constructed by time-domain measurements for a 300-W PEP DPA in our test bench. The results show a similar correction capability with the memory polynomial PD, and that the proposed PD is superior to the conventional Hammerstein approach, which has a limited capability, when eight tones with 500-kHz tone spacing and a single W-CDMA carrier with 3.84-MHz signal bandwidth are employed. Furthermore, the proposed method and the various PDs were experimentally verified with an actual DPA in the same test bed. Applying two W-CDMA carriers, around 4 dB more correction has been verified over the conventional Hammerstein PD. Moreover, the complexity of the proposed and memory polynomial method has been estimated and compared. The proposed piecewise pre-equalizers PD verified that it is equivalent to a memory polynomial PD, but requires much less complexity, as estimated in Section VI. The effectiveness of the piecewise pre-equalized LUT approach for compensating frequency-dependent memory effects was demonstrated in both simulations and measurements. In the future, the proposed method can be extended to correct long time constant memory effects generated from self-heating of the transistors.

KIM et al.: PIECEWISE PRE-EQUALIZED LINEARIZATION OF WIRELESS TRANSMITTER WITH DOHERTY AMPLIFIER

APPENDIX If the PA obeys the truncated Volterra model, it can be expressed as (16). To simplify the analysis, the simplest truncated Volterra model of the PA can be defined as

If the PD has a structure expressed in (4) and is reduced as much as possible without loss of generality, it follows that

We can replace with 1, with , with and with for the cancellation to be easily observed. can be inserted into , expanded, and finally simplified after removing the negligible high order terms as well as even order terms

Therefore, the desired output can be achieved via the , and cancel out , , and . PD, when ,

3477

[11] M. A. Nizamuddin, P. H. Balister, W. H. Tranter, and J. H. Reed, “Nonlinear tapped delayed line digital predistorter for power amplifiers with memory,” in Proc. Wireless Commun. Networking Conf., Mar. 2003, vol. 1, pp. 607–611. [12] R. Raich, H. Qian, and G. T. Zhou, “Orthogonal polynomials for power amplifier modeling and predistorter design,” IEEE Trans. Veh. Technol., vol. 53, no. 5, pp. 1468–1479, Sep. 2004. [13] L. Ding, R. Raich, and G. T. Zhou, “A Hammerstein linearization design based on the indirect learning architecture,” in Proc. IEEE Int. Acoust., Speech, Signal Process. Conf., May 2002, vol. III, pp. 2689–2692. [14] C. Eun and E. J. Powers, “A predistorter design for a memory-less nonlinearity preceded by a dynamic linear system,” in Proc. IEEE Global Telecommun. Conf., Nov. 1995, vol. 1, pp. 152–156. [15] H. W. Kang, Y. S. Cho, and D. H. Youn, “On compensating nonlinear distortions of an OFDM system using efficient adaptive predistorter,” IEEE Trans. Commun., vol. 47, no. 4, pp. 522–526, Apr. 1999. [16] H. Ku and J. S. Kenney, “Analysis of performance for memoryless predistortion linearizers considering power amplifier memory effects,” presented at the IEEE Top. PA for Wireless Commun. Workshop, Sep. 2003. [17] O. Hammi, S. Boumaiza, M. Jaidane-Saidane, and F. M. Ghannouchi, “Digital subband filtering predistorter architecture for wireless transmitters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1643–1652, May 2005. [18] A. Zhu and T. J. Brazil, “Behavioral modeling of RF power amplifiers based on pruned Volterra series,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 12, pp. 563–565, Dec. 2004. [19] J. K. Cavers, “Optimum table spacing in predistorting amplifier linearizers,” IEEE Trans. Veh. Technol., vol. 48, no. 5, pp. 1699–1705, Sep. 1999. [20] K. J. Cho, J. H. Kim, and S. P. Stapleton, “A highly efficient Doherty feedforward linear power amplifier for W-CDMA base-station applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 292–300, Jan. 2005. [21] K. Mekechuk, W. J. Kim, and S. P. Stapleton, “Linearizing power amplifiers using digital predistortion, EDA tools and test hardware,” High Freq. Electron., pp. 18–24, Apr. 2004. [22] S. Boumaiza and F. M. Ghannouchi, “Realistic power amplifiers characterization with application to baseband digital predistortion for 3G base stations,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 3016–3021, Dec. 2002.

REFERENCES [1] S. C. Cripps, Advanced Techniques in RF Power Amplifier Design. Norwood, MA: Artech House, 2002. [2] P. B. Kennington, High Linearity RF Amplifier Design. Boston, MA: Artech House, 2000. [3] J. K. Cavers, “Amplifier linearization using a digital predistorter with fast adaptation and low memory requirements,” IEEE Trans. Veh. Technol., vol. 39, no. 4, pp. 374–382, Nov. 1990. [4] R. Marsalek, P. Jardin, and G. Baudoin, “From post-distortion to predistortion for power amplifiers linearization,” IEEE Commun. Lett., vol. 7, no. 7, pp. 308–310, Jul. 2003. [5] M. Faulkner, T. Mattsson, and W. Yates, “Adaptive linearization using pre-distortion,” in Proc. IEEE 40th Veh. Technol. Conf., 1990, pp. 35–40. [6] J. Vuolevi and T. Rahkonen, Distortion in RF Power Amplifiers. Norwood, MA: Artech House, 2003. [7] C. Eun and E. J. Powers, “A new Volterra predistorter based on the indirect learning architecture,” IEEE Trans. Signal Process., vol. 45, no. 1, pp. 223–227, Jan. 1997. [8] 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, Nov. 2001. [9] L. Ding, G. T. Zhou, D. R. Morgan, Z. Ma, J. S. Kenney, 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. [10] G. Baudoin, P. Jardin, and R. Marsalek, “Power amplifier linearization using predistortion with memory,” in Proc. 13th Int. Czech–Slovak Sci. Radioelektron. Conf. , May 2003, pp. 193–196.

Wan-Jong Kim (S’03) received the B.S. and M.S. degrees in radio science and engineering from Kwangwoon University, Seoul, Korea, in 1999 and 2001, respectively, and is currently working toward the Ph.D. degree at Simon Fraser University, Burnaby, BC, Canada. His current research interests are wideband digital predistortion, peak-to-average power reduction techniques, and integrated RF/DSP design applications.

Kyoung-Joon Cho received the B.S. degree in information and communication engineering from Anyang University, Anyang, Korea, in 1998, and the M.S. and Ph.D. degrees in radio science and engineering from Kwangwoon University, Seoul, Korea, in 2000 and 2004, respectively. In 2004, he joined Simon Fraser University, Burnaby, BC, Canada, as Post-Doctoral Fellow with the School of Engineering Science, where he is currently involved with RF integrated circuit (RFIC) and monolithic-microwave integrated-circuit (MMIC) PA developments. His research interests are highly efficient PA design and linearization techniques.

3478

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Shawn P. Stapleton received the B.S., M.S., and Ph.D. degrees in engineering from Carleton University, Ottawa, ON, Canada, in 1982, 1984, and 1988, respectively. Since 1988, he has been a Professor with the School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada. His research has focused on integrated RF/DSP applications for wireless communications, GaAs MMIC circuits, and PA linearization. Prior to joining Simon Fraser University, he was involved in a wide variety of projects including optical communications, RF/microwave communications systems, and adaptive array antenna. While with Simon Fraser University, he has developed numerous adaptive linearization techniques ranging from feedforward active-biasing work-function predistortion to digital baseband PDs. He has developed phase-locked loop (PLL), linearization, amplifier, and communications design software for the wireless industry. He has been a Staff Scientist with Scientific Atlanta (an external partner to Agilent Technologies), founder of Prophesi Technologies, and a consultant on PA enhancement techniques. He has authored or coauthored numerous technical papers on linearization and has given many presentations at various companies.

Jong-Heon Kim (M’95) received the B.S. degree in electronic communication engineering from Kwangwoon University, Seoul, Korea, in 1984, the M.S. degree in electronic engineering from Ruhr University, Bochum, Germany, in 1990, and the Ph.D. degree in electronic engineering from Dortmund University, Dortmund, Germany, in 1994. Since 1995, he has been a Professor with the Department of Radio Science and Engineering, Kwangwoon University. He is also currently a Research Associate with Simon Fraser University, Burnaby, BC, Canada, where he is involved with DSP techniques of PAs for the wireless industry. His current interests include digital linearization of PAs and transmitters, smart PAs, and integrated RF/DSP design.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3479

Propagation Characteristics for Periodic Waveguide Based on Generalized Conservation of Complex Power Technique Hui Kan Liu, Student Member, IEEE, and Tian Lin Dong

Abstract—The propagation characteristics of surface waves with periodic waveguides is examined based on the characterization of a unit cell through the generalized conservation of complex power technique (GCCPT). The bounded approach, in which the periodic structure is enclosed by perfectly conducting walls, is applied to facilitate discussion. Comparison among different matching techniques, including the mode-matching technique (MMT), conservation of complex power technique (CCPT), improper MMT with power decrement or increment (PD/PI-MMT), is taken to derive some properties for the generalized scattering matrix (GSM) in conjunction with these techniques. It is shown that the improper PD/PI-MMT, in contrast to the equivalence between the MMT and CCPT in the context of a closed waveguide, is in contradiction to the principle of conservation of complex power, and cannot be employed to express GSM for discontinuity. Further analysis, however, suggests that PI-MMT and PD-MMT are in duality symmetry, and can, in combination, restore the continuity of complex power in a periodic environment, which gives rise to the GCCPT. The propagation constants are determined from the eigenvalues of a transmission matrix for a unit cell with the aid of a solution selection rule. A number of numerical results are shown to demonstrate the efficiency and accuracy of our approach. Index Terms—Conservation of complex power technique (CCPT), discontinuity properties, duality symmetry, eigenvalues, generalized conservation of complex power technique (GCCPT), generalized scattering matrix (GSM), mode-matching technique (MMT), periodic waveguide.

I. INTRODUCTION ERIODIC structures are of considerable applications in the microwave, millimeter-wave, and optic-wave regions [1], [2]. Although the general quantitative behavior characterizing periodic waveguide has been well understood and rigorously formulated [3], theoretical and numerical analysis of the Floquet propagation constants is still a topic of current research [4]–[7]. This is especially the case for many applications such as distributed feedback lasers, optical filters, Bragg-feedback modulators, and photonic-bandgap materials, which involve opera-

P

Manuscript received December 1, 2005; revised March 23, 2006. H. K. Liu was with the Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China. He is now with the Radiation Laboratory, Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor, Ann Arbor, MI 48109 USA (e-mail: [email protected]). T. L. Dong is with the Radiation Laboratory, Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor, Ann Arbor, MI 48109 USA (e-mail: [email protected]) . Digital Object Identifier 10.1109/TMTT.2006.880647

tion near the stopbands region corresponding to Bragg reflection. Apart from the rigorous modal approach and its associated transmission-line theory [3], [8], another group of methods, which regard the periodic corrugations as consisting of step discontinuities connected via a length of a waveguide, have also been well developed [6], [7], [9]–[11]. In this group, the modematching technique (MMT) is employed to model the discontinuity, and the transmission matrix for a unit cell is then derived. Starting from the building block, one may obtain the scattering characteristics of a periodic structure of finite length by a cascading algorithm, and propagation constants of infinite periodic corrugation from eigenvalues of a matrix. In [4], the bounded approach and conservation of complex power technique (CCPT) [12]–[14] are invoked to obtain the scattering coefficients at waveguide discontinuity, and numerical results about waveguiding properties of surface (bound) waves with a periodic dielectric structure are provided. In this paper, the generalized conservation of complex power technique (GCCPT) is proposed to analyze discontinuity based on the comparison of properties in connection with different matching techniques or, specifically, conventional MMT, MMT through CCPT, and “ill-matched” MMT with power increment/decrement (PI/PD-MMT). As already been proven [14] and as will be verified in this paper, MMT and CCPT are formally equivalent, thereby producing no difference in deriving a generalized scattering matrix (GSM). In contrast, another MMT with an improper choice of mode-matching equations incorrectly enforces the discontinuity of transverse electric/magnetic fields, leading to a violation of the principle of conservation of complex power, and fails to obey the usual constraints on a GSM such as reciprocity, orthogonality, and unitariness [11], [15]. It is furthermore shown that PD-MMT and PI-MMT are in duality symmetry, and the continuity of complex power is restored in the context of a periodic unit cell if PI-MMT and PD-MMT are combined to yield the GCCPT. In addition, the duality symmetry can be exploited to reduce computational effort, enabling us to specify the GSMs for two discontinuities in a unit cell by calculating only one GSM. Numerical examples are provided for the case of a corrugated rectangular waveguide to confirm our conclusions, as well as the efficiency of the approach. II. ANALYSIS OF TRANSVERSE DISCONTINUITY WITH GCCPT A. Choice of Mode-Matching Equations The bounded approach is used here to enclose the open waveguide with either perfectly conducting walls so that the integra-

0018-9480/$20.00 © 2006 IEEE

3480

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Suppressing a propagation factor and eigenmodes for waveguide and have -field matching matrix equations

and retaining , respectively, we

(3)

Fig. 1. Transverse discontinuity between two inhomogeneously filled waveguide.

where are the incident and reflected scattering coefficients of waveguide A and are corresponding coefficients of waveguide B. The continuity can be fulfilled by two sets of testing functions, i.e., or , and the following two equations arise in matrix form: (4-1)

tion over the continuous nonsurface-wave spectrum can be replaced by a summation over discrete nonsurface modes. A detailed discussion on a bounded multilayer planar dielectric configuration can be found in [16] and [17]. Consider the dielectric waveguide discontinuity bounded by perfectly conducting walls, as shown in Fig. 1, where the two regions and are inhomogeneously filled with a lossless medium, and we assume that the dielectric constants of the two regions , have one-dimensional variation along the -axis. Besides, to avoid the complexity due to possible complex modes [18], the decoupling of TE and TM modes is also assumed to hold [19]. Under these assumptions, we have the following. 1) No complex modes can exist. 2) Two alternative orthogonal relations in the sense of “selfreaction” and “complex power,” i.e.,

(1)

can be shown equivalent [18], where superscript signifies complex conjugate. Thus, in the following discussion, we will use the latter orthonormality relation, which incorporates the conjugation of the -field to accord with the formulation of complex power. It is worth noticing that the former orthonormality relation can be utilized in favor of the possible existence of complex modes, provided the structure is lossless. 3) Without loss of generality, transverse electric modes are investigated. Let and denote the normalized transverse modal function of waveguide and corresponding to the th and th eigenmodes, respectively, we get

(4-2) where

(5) describes the “inter-waveguide orthonormality” of modes with respect to the different waveguide, and denotes the Hermitian transpose operator. By a similar token, matrix equations enforcing the continuity of tangential magnetic fields are as follows: (6-1) (6-2) where is a diagonal matrix whose th diagonal element is the characteristic admittance of the th eigenmode in the waveguide given by [16]

(7) where is the relative dielectric constant. The complex power passing from waveguide across the junction may be written as

(8) and the complex power

flowing into waveguide

is

(9) (2)

where is the Kronecker delta symbol and section of the bounded waveguide.

is the cross

It can be easily shown after some matrix manipulation that (4-1) and (6-2) as a whole form linearly independent equations that: 1) sufficiently and uniquely determine the modes scattering at the discontinuity and 2) conserve the continuity of complex power on account of (8) and (9), i.e., , not nec-

LIU AND DONG: PROPAGATION CHARACTERISTICS FOR PERIODIC WAVEGUIDE BASED ON GCCPT

3481

essarily including an infinite number of modes [15]. In a similar manner, we can verify that (4-2) and (6-1) together represent linearly independent equations satisfying the two above-mentioned properties. Thus far, we have obtained two alternative sets of conventional MMT equations to determine the scattering matrix of transverse discontinuity, namely, (10-1) (10-2) and (11-1) (11-2) The following then applies. 1) Both (10) and (11) automatically satisfy the principle of complex power conservation, indicating that MMT and CCPT are essentially equivalent formally and numerically. This equivalence actually has been confirmed in previous studies [14], [15]. 2) Equations (10) and (11) are not equivalent when matrix is truncated to a finite size, as will be shown in the Appendix. 3) The resulting GSM satisfies the following relations [15], [20]: Symmetry (or Reciprocity) Orthogonality Unitariness.

(12-1) (12-2) (12-3)

Note that (12-3) holds only for the network incorporating no evanescent modes [20]. Relations (12) are quite useful for checking not only the correctness of GSM, but also, as will be shown later, the correctness of choosing testing functions when enforcing fields continuity. At this stage, a natural and interesting issue is what if we choose (4-1) and (6-1) [or (4-2) and (6-2)] as a set alternative to (10) [or (11)] to derive the scattering matrix, which will be taken via a closer examination in Section II-B.

Fig. 2. Duality symmetry of PD-MMT and PI-MMT for characterizing discontinuity in the context of a periodic unit cell.

2) The properties of the GSM, as stated by (12), as an immediate consequence, do not hold any more [11], specifically (14-1) (14-2) (14-3) 3) The complex power difference can be given by (8), (9), and (13) as follows:

(15)

Now we consider a couple of symmetric discontinuities as shown in Fig. 2, which are typical structures in a unit cell of periodic waveguide. To get linear independent equations, we require that the same number of eigenmodes are taken for waveguide and , i.e., . Combing (4-1) and (6-1), we have the GSM in connection with discontinuity plane I

which implies that part of complex power, due to improper mode matching, “vanishes” across the abrupt discontinuity. Since the mode-matching equations (13) leading to “power decrement (PD)” are in contradiction with principle of conservation of complex power, it may be appropriately called the PD-MMT, and cannot be applied to the analysis of discontinuity. In the context of symmetric discontinuities, however, this matching technique can be applied together with its dual form (MMT with power increment or PI-MMT) to preserve the continuity of complex power across a unit cell instead of a transverse discontinuity. Actually, it is important to notice that (13) is also applicable to determine wave scattering at discontinuity plane II with exactly the same formulation, i.e.,

(13-1)

(16-1)

(13-2)

(16-2)

B. Introduction of GCCPT

In accordance with the above similar argument, we observe the following. 1) The principle of conservation of complex power (in turn, the principle of conservation of energy) is violated in virtue of (8) and (9), i.e., .

or, in a more convenient form, (17-1) (17-2)

3482

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

which gives rise to the power increment

(18) Equation (18), in contrast to (15), suggests the existence of power increment cross the discontinuity. Furthermore, (18) and (15) are in duality symmetry, indicating the restoration of continuity of complex power for a unit cell, which may be viewed as a GCCPT. A quite useful consequence of this duality is that GSM for II can be directly obtained from that of discontinuity I. Simple algebraic operation yields

1) The dispersion curves of a perturbed structure will follow closely those of an unperturbed one in the light of the mode coupling argument [21]. 2) The bi-directionality property of periodic waveguides [5], which suggests that if is the solution of (23), is also a solution. Specifically, the discrete eigenvalues corresponding to propagation factors are pairs of reciprocal numbers referred to waves traveling along opposite directions with the same propagation factor. Such a property would be very important to the determination of a possible propagation factor. In fact, if are a pair of genuine eigenvalues of , corresponding propagation factors can be expressed as

(19)

(24)

and (20)

which defines the forbidden band (or stopband) of the periodic section as

where

(25)

(21) Therefore only one GSM need be calculated for a unit cell, which is advantageous to conventional MMT, which requires calculation of two GSMs. It should be appreciated that the duality symmetry of PI-MMT and PD-MMT and the derivation GCCPT, although aimed to facilitate the analysis of the periodic structure, are generally valid for any two symmetric abrupt discontinuities. III. MODE PROPAGATION THROUGH PERIODIC CORRUGATIONS WITH FINITE OR INFINITE LENGTH A cascading procedure from GSMs of step discontinuities and of the waveguide transmission line is straightforward and well formulated [7], [11] and is omitted here; we only present a GSM of a length of waveguide [20] for propagating modes for evanescent modes

(22-1) (22-2)

Upon obtaining GSMs for the discontinuity planes, we may get a transmission matrix for a unit cell by network conversion [4]. The propagation constant are then derived from classical matrix eigenvalues of in the form [6], [9]

(23) where is length of unit cell and is the propagation constant. The solution selection rule (SSR) [4] used to find eigenvalues corresponding to surface modes can be established based on the following well-known facts.

Since modes are retained in (13), (23) and (24) yield propagation constants, and the surface wave propagation factor is determined in the “modes filtering” process by the SSR. Here, we take the unperturbed approximation by replacing the periodic layer by a homogeneous one having the average dielectric constant. It is well known from the theory of mode coupling that the dispersion curves of a perturbed structure will follow closely those of an unperturbed one, except in the vicinity of the forbidden gap where space harmonics are strongly coupling. Accordingly, the SSR requires that having the closest value to the unperturbed estimation should be chosen to represent the propagation constant corresponding to the slow wave. The propagation characteristics of periodic waveguide of finite length, as previously mentioned, can be readily obtained by connecting unit cells with a cascading procedure and taking guided power into account [22], [23].

IV. COMPUTATIONAL CONSIDERATIONS AND NUMERICAL EXAMPLES OF PERIODIC WAVEGUIDE A basic criterion for the choice of was established in [4]: all slow modes are physically and numerically significant and should thus be included in the calculation of the propagation constants. In the below discussion, we shall focus our attention on the surface waves supported by the planar grating structures, and the surface-wave condition is assumed to be valid [3]; denotes the bounding height involved in the bounded approach. A. Dispersion Diagram Comparison We apply the foregoing argument and formulation to describe the dispersion characteristics of the periodic dielectric waveguide with various geometrical configurations, whose constructions are shown through Figs. 3–5. For the sake of brevity, we

LIU AND DONG: PROPAGATION CHARACTERISTICS FOR PERIODIC WAVEGUIDE BASED ON GCCPT

3483

Fig. 3. Dispersion curves around the first Bragg interaction region. Here, k d is shown as a function of . The result is compared numerically with an alternative one presented by Tamir and Zhang [3]. The parameters M = 2 and T = 6:0.

Fig. 5. Dispersion curves around the first Bragg interaction region. k d is shown as a function of . The geometry of the structure is sketched in Fig. 3. Here, t = 1:0, t = 0:5, d = d = 0:5=2, and " = 1:0 , other parameters being given in Fig. 3. M = 20 and T = 6:0.

Fig. 4. Dispersion curves around the first Bragg interaction region. k d is shown as a function of . The geometry of the structure is sketched in Fig. 3. Here, t = 0:5, t = 0:3, and d = d = 0:1; other parameters being given in Fig. 3. The truncation order M = 10, and the bound location T = 6:0.

Fig. 6. Convergence of the magnitude of attenuation constant  as a function of M for TE mode in the case shown in Fig. 3. Here,  = 1:55.

B. Convergence Test focus our attention on the TE case, but the extension to the TM case is trivial. Figs. 3 and 4 display the situation near the first-order Bragg regime for a weakly modulated structure. Only a single TE surface mode is supported in either slab section. The frequency range over which no propagation occurs can be viewed as a true surface-wave stopband corresponding to the Bragg reflection in which and while varies between 0 to a large maximum value and back to 0 [3]. As Prof. Tamir kindly supplied us with his calculated data, we present in Figs. 3 and 4 the comparison between his results and ours as a check on our theory. It is evident that the agreement is excellent and, for these cases, in fact, the difference between the two curves cannot be distinguished on the given scale. In Fig. 5, we establish the propagation characteristics and especially the stopband regions of strongly modulated periodic structure. The result is shown in Fig. 5, and the agreement is quite close. Throughout Figs. 3–5, the error of this approach is collectively small inside the stopband, but more pronounced in the edge of stopband where undertakes abrupt variation; in any case, the difference is smaller than the tolerances involved in designing and fabricating practical devices.

To test the convergence of this technique, the magnitude of attenuation constant is plotted as a function of the size of the matrix size (also the truncation order) for the TE case with different heights of perfectly conducting walls in Figs. 6–8. The following points associated with the convergence behavior are noted. 1) Generally the results show rapid convergence with matrix size of ten or greater, giving accuracy to approximately 1–3 significant figures. 2) It can be observed that larger matrix size is needed for stronger periodic modulation with larger or larger difference between the dielectric constants of the materials forming the grating layer. In particular, in the case of a weakly modulated periodic structure, as indicated by Fig. 6, is sufficient to obtain accurate to three significant figures; however, in the case of a strongly modulated periodic structure, as shown in Fig. 8, can marginally achieve an accuracy of two significant figure. Higher order of truncation is required for TM modes [3]. 3) Faster convergence occurs for the smaller value of , while unduly small values of may produce errors, as illustrated

3484

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 7. Convergence of the magnitude of attenuation constant  as a function of M for TE mode in the case shown in Fig. 4. Here,  = 1:3869.

Fig. 9. Convergence of the magnitude of attenuation constant  versus M with different matching techniques for the same point as Fig. 8.

implemented by taking advantage of duality symmetry of the PI-MMT and PD-MMT, which separately violate the continuity of complex power across the transverse discontinuity, but together maintain the principle of conservation of complex power in a periodic unit cell. The duality symmetry permits calculation of the GSM only once for two abrupt discontinuities. Excellent agreement with previously results, as well as a rapid convergence rate has demonstrated the effectiveness of the current approach. It is expected that this approach and its essential viewpoints regarding the duality symmetry are potentially applicable to other situations involving symmetric discontinuities. Fig. 8. Convergence of the magnitude of attenuation constant  as a function of M for the case shown in Fig. 5. Here,  = 3:2593.

by Fig. 6, where the curve corresponding to converges most rapidly, although to a larger value than the asymptotic one. However, this error is tolerable. In practice, could give computationally efficient and numerically accurate results. Finally, we compare various convergence curves related to different matching techniques (CCPT, GCCPT, and PD-MMT), shown in Fig. 9, and conclude that PD-MMT (or PI-MMT) produce large errors for lower number of truncation order because it essentially violates the principle of conservation of energy, but in the limiting case that tends to infinity, it also converges to a correct value (see the Appendix for a rigorous proof); GCCPT converges in a more smooth and rapid fashion than CCPT, but the difference is negligible when becomes large; more importantly, only one GSM need be calculated for GCCPT due to the duality symmetry, while CCPT requires calculating two distinct GSMs. V. CONCLUSION The main contribution of this study has been the development of a theoretically compact and computationally efficient field matching technique called the generalized CCPT in the context of a periodic structure. Based upon a different combination of various matching equations, this technique has been

APPENDIX Referring to the notations of Section II, we may show that the following equations cannot exactly hold unless : (26) which give rise to

(27) or in matrix notation

(28) The solution of (10) is

(29-1)

(29-2)

LIU AND DONG: PROPAGATION CHARACTERISTICS FOR PERIODIC WAVEGUIDE BASED ON GCCPT

while (11) can be solved as follows:

(30-1)

(30-2) For (29) and (30) to be satisfied simultaneously, must hold, which implies that (10) and (11) are not equivalent when the matrix is truncated to a reduced size. In a similar manner, we may obtain the solution of (13) as follows:

(31-1)

(31-2) It can be easily proven that (31) is consistent with (29) and (30) only in the limiting case of for which the GSM can satisfy the formal check (12). ACKNOWLEDGMENT The authors wish to thank Prof. T. Tamir, Polytechnic University of New York, Brooklyn, for useful comments and for his cooperation in comparing numerical data with those obtained by his technique. REFERENCES [1] C. Elachi, “Waves in active and passive periodic structure: A review,” Proc. IEEE, vol. 64, no. 12, pp. 1666–1698, Dec. 1976. [2] J. Lightw. Technol. (Special Issue), vol. 15, no. 8, Aug. 1997. [3] T. Tamir and S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightw. Technol., vol. 14, no. 5, pp. 914–926, May 1996. [4] H. K. Liu and T. L. Dong, “Guidance of surface waves with periodic dielectric waveguide based on transmission matrix,” in Proc. Asia–Pacific Microw. Conf., Suzhou, China, Dec. 2005, pp. 2329–2332. [5] D. Pissoort and F. Olyslager, “Study of eigenmodes in periodic waveguides using the Lorentz reciprocity theorem,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 542–543, Feb. 2004. [6] S. Amari, R. Vahldieck, J. Bornemann, and P. Leuchtmann, “Spectrum of corrugated and periodically loaded waveguides from classical matrix eigenvalues,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 3, pp. 453–460, Mar. 2000. [7] H. A. Jamid and M. N. Akram, “Analysis of deep waveguide gratings: An efficient cascading and doubling algorithm in the method of lines framework,” J. Lightw. Technol., vol. 20, no. 7, pp. 1204–1209, Jul. 2002. [8] S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, no. 1, pp. 123–133, Jan. 1975. [9] M. Tsuji, S. Matsumoto, H. Shigesawa, and K. Takiyama, “Guided wave experiments with dielectric waveguides having finite periodic waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 4, pp. 337–344, Apr. 1983. [10] T. E. Rozzi, F. Chiaraluce, and M. Lanari, “A rigorous analysis of DFB lasers with large and aperiodic corrugations,” IEEE J. Quantum Electron., vol. 27, no. 2, pp. 212–223, Feb. 1991.

3485

[11] R. Schmidt and P. Russer, “Modeling of cascaded coplanar waveguide discontinuities by the mode-matching approach,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 12, pp. 2910–2917, Dec. 1995. [12] R. Safavi-Naini and R. H. MacPhic, “On solving waveguide junction scattering problem by the conservation of complex power technique,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 4, pp. 337–343, Apr. 1981. [13] R. R. Mansour and R. H. MacPhic, “Scattering at an n-furcated parallel-plate waveguide junction,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 9, pp. 830–835, Sep. 1985. [14] J. D. Wade and R. H. Macphie, “Conservation of complex power technique for waveguide junctions with finite wall conductivity,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 4, pp. 373–378, Apr. 1990. [15] G. V. Eleftheriades, A. S. Omar, L. P. B. Katehi, and G. M. Rebeiz, “Some important properties of waveguide junction generalized scattering matrices in the context of the mode matching technique,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 10, pp. 1896–1903, Oct. 1994. [16] S. T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides: Part 1—Mathematical formulations,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 9, pp. 843–855, Sep. 1981. [17] G. H. Brooke and M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 5, pp. 760–770, May 1982. [18] A. S. Omar and K. Schünemann, “Complex and backward-wave modes in inhomogeneously and anisotropically filled waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 3, pp. 268–275, Mar. 1987. [19] T. F. Jabłon´ ski, “Complex modes in open lossless dielectric waveguides,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 11, pp. 1272–1282, Apr. 1994. [20] A. Morini and T. Rozzi, “On the definition of the generalized scattering matrix of a lossless multiport,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 160–165, Jan. 2001. [21] S. T. Peng, “Rigorous analysis of guided waves in doubly periodic structures,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 7, pp. 1448–1456, Aug. 1990. [22] S. Zhang and T. Tamir, “Analysis and design of broadband grating couplers,” IEEE J. Quantum Electron., vol. 29, no. 11, pp. 2813–2824, Nov. 1993. [23] H. K. Liu and M. Xia, “Guided-wave characteristics with periodic dielectric waveguides of finite length,” Int. J. Infrared Millim. Waves, vol. 26, pp. 1565–1581, Nov. 2005.

Hui Kan Liu (S’05) received the B.Eng. and M.Eng. degrees in electrical engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2004 and 2006, respectively, and is currently working toward the Ph.D. degree at The University of Michigan at Ann Arbor. He is currently with the Radiation Laboratory, The University of Michigan at Ann Arbor. His research interests include guided-wave theory, periodic structures, and electromagnetic compatibility.

Tian Lin Dong received the M.Sc. degree and Ph.D. degrees from the Polytechnic University of New York, Brooklyn, in 1982 and 1985, respectively. He is currently a faculty member with the Department of Electronics and Information Engineering, Huazhong University of Science and Technology (HUST), Wuhan, China. His research interests include electromagnetic theory and its microwave and RF applications.

3486

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Monostatic Reflectivity Measurement of Radar Absorbing Materials at 310 GHz Anne Lönnqvist, Aleksi Tamminen, Juha Mallat, and Antti V. Räisänen, Fellow, IEEE

Abstract—This paper presents monostatic reflectivity measurements of radar absorbing materials at 310 GHz in a phase-hologram-based compact range. The radar cross-section method was used and the backscattered reflection was measured with horizontal and vertical polarizations in plane-wave conditions. Transmission was also studied. The reflectivity was measured over an incidence angle of 0 –45 . The reflectivity of Thomas Keating Terahertz RAM at normal incidence was found to be 56 dB—the smallest of the studied materials. The reflectivity of carpet material measured was also below 40 dB and it was found to be suitable for use as an absorber. The results are in line with those available from previous studies of reflectivity and complement them with new materials, frequency, and angle information. Index Terms—Compact range, monostatic reflectivity, radar absorbing materials, radar cross-section (RCS) method.

I. INTRODUCTION ADAR absorbing materials are needed to suppress unwanted reflections, e.g., in compact test ranges and to minimize radar cross section (RCS) of a target. Absorbers can also be used as beam dumps in quasi-optical systems and as calibration loads in radiometers [1], [2]. Application-specific absorbing materials have been designed for these different purposes. Attenuation can be caused by electric or magnetic losses and by the structure of the absorber. In this paper, attention is concentrated on absorbers, which can be used when building a compact antenna or RCS measurement range for submillimeter wavelengths. Characteristics of absorbing materials need to be known when building a compact range. With proper placement of absorbers, the amount of absorbers needed can be minimized and low reflectivity level of the background can be assured. Possible standing waves can also be suppressed. For this purpose, we have characterized four commercially available radar absorbing materials and three carpets. The use of carpets as absorbers at submillimeter wavelengths is appealing due to their low price compared to the commercially available absorbing materials. In most of the absorbers used in anechoic chambers, carbon-impregnated polyurethane foam is used to provide

R

Manuscript received January 23, 2006; revised June 1, 2006. This work was supported in part by the Academy of Finland and Tekes under their Centre-ofExcellence Program. The work of A. Lönnqvist was supported by the Graduate School in Electronics, Telecommunication, and Automation, by the Jenny and Antti Wihuri Foundation, by the Foundation of the Finnish Society of Electronic Engineers, by the Nokia Foundation, by the Foundation of Technology (Finland), and by the Emil Aaltonen Foundation. The authors are with MilliLab, Radio Laboratory, The Smart and Novel Radios Research Unit, Helsinki University of Technology, Espoo FI-02015 TKK, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.881023

loss. The absorbers are usually shaped so that the geometrical transformation from the free space to lossy medium provides a dielectric gradient and reduces reflections. Pyramidal and wedge-shaped painted absorbers are commonly used. A layer of low reflection paint is used to provide protection and to reduce the amount of carbon dust in the measurement range. These kinds of absorbers scatter more energy to the directions satisfying the grating equation, as shown with measurements in [3] for pyramidal and wedge-shaped absorbers below 18 GHz and in [4] for a wedge-shaped Far-Infrared Radiation Absorbing Material (FIRAM) at 584 GHz. However, usually electromagnetic simulations cannot exactly predict the scattering behavior of absorbers and, therefore, it needs to be verified with measurements before designing their placement in a compact range. Earlier, bistatic, and specular measurements of the absorbers investigated here have been carried out at MilliLab, Helsinki University of Technology, Espoo, Finland [5], [6]. However, these measurements were done in near field conditions. Now the campaign is completed with monostatic compact range measurements, where the absorber sample is placed into a planewave region and measured over an angle range of 45 from the normal of the absorber. The transmission of the absorbers is also measured. In [1] and [2], the reflectivity has been measured only in the direction of the normal of the absorber, but for compact antenna test range (CATR) design, more information on the characteristics of the absorber are needed. In the RCS method [7] used here, the absorber sample is fastened on a heavy metal backing plate, the plate is installed on a rotating fixture, and the RCS pattern of the ensemble is recorded. The perfect reflection from the backside of the plate can be used as a reference and the characteristics of the absorber can be evaluated by comparing the reflection from the absorber to the reference. Since the result is obtained by comparing these two reflections, the absolute amplitude of the RCS pattern does not need to be calibrated. The measurement range itself will be presented in more detail in Section II. In Section III, details of the absorber materials and the measurement are described. The results, discussion, and conclusion are presented in Sections IV–VI, respectively. II. COMPACT RCS MEASUREMENT RANGE The measurement range used for absorber characterization has originally been designed for measuring RCS of scaled models [8]. In a compact range, we are able to evaluate the absorber samples in more realistic conditions than with near-field measurements. The plane-wave region needed for RCS evaluation is created with a phase hologram [9], which transforms

0018-9480/$20.00 © 2006 IEEE

LÖNNQVIST et al.: MONOSTATIC REFLECTIVITY MEASUREMENT OF RADAR ABSORBING MATERIALS AT 310 GHz

3487

2

Fig. 1. Hologram-based test range. (RX = receiver; TX = transmitter).

Fig. 3. 5 5 cm samples of FIRAM, TERASORB, TK THz RAM, Eccosorb, carpet #1, carpet #2, and carpet #3. The size of the samples measured was 10 10 cm .

2

III. ABSORBERS AND MEASUREMENT SYSTEM

Fig. 2. Closer view of the radar setup.

the spherical wave radiated by the feed into a plane wave. As a diffractive element, the phase hologram is narrowband so when wanting to cover a wide range, several holograms need to be designed and manufactured. The region where the scaled model, or here, a sample of an absorber, is placed is what is called the quiet zone (QZ). The setup is shown in Fig. 1. The distance from the receiver/transmitter to the hologram and also from the hologram to the target is 1 m. The phase hologram is realized as a groove structure on a thin Teflon plate. In this case, the phase-hologram operation is optimized for operation at 310 GHz, which is suitable for our compact ranges. The changing depth of the grooves causes the phase modulation of the transmitted field. The hologram structure of size 28 cm 24 cm was fabricated on a 5-mm-thick Teflon plate. The amplitude and phase variations of the QZ field are approximately 2 dB and 10 peak-to-peak, respectively. The diameter of the QZ is 12 cm. Outside the QZ, the amplitude of the field drops quickly, approximately 10 dB when moving from a distance of 6 cm ( diameter) to 8 cm from the axis of the QZ. The instrumentation is based on a millimeter-wave vector network analyzer (AB Millimètre MVNA-8–350) with submillimeter-wave extensions. Corrugated horns are used as transmitting and receiving antennas. A dielectric slab with 3-dB power division is used as a directional coupler. The load absorber, which is made of Thomas Keating Terahertz (TK THz) RAM, is placed on a translation stage to enable its proper placement. TK THz RAM is also used around the transmitter and receiver, as can be seen in Fig. 2. Additional absorbers were placed around the setup before measurements to reduce reflections.

Seven different materials presented in Fig. 3 were selected for investigation: FIRAM-500 and TERASORB-500 by the Submillimeter-Wave Technology Laboratory, University of Massachusetts at Lowell; Space-qualified Tessalating Terahertz RAM by Thomas Keating Ltd., Billingshurst, West Sussex, U.K., unpainted Eccosorb VFX-NRL-2 by Emerson and Cuming Microwave, Company, Westerlo, Belgium, and three carpets, referred to here as carpets #1–#3. These were also studied [6] with bistatic and specular near-field measurements. The sample sizes were 10 10 cm . FIRAM-500 is a wedge-type iron loaded silicone absorber designed for submillimeter wavelengths; it is available as sheets of size 61 61 cm . TERASORB-500 has the same wedge-type structure, but it is made of carbon loaded ethylene vinyl acetate. The size of the interlocking tiles is 10 10 cm . Both materials have a groove opening angle of 22.5 , groove spacing of 1.55 mm, groove depth of 3.8 mm, and thickness of 7.6 mm. TK THz RAM is made of carbon loaded polypropylene plastic. It consists of small pyramids of 1.5-mm height and 1-mm spacing. The size of the interlocking tiles is 2.5 2.5 cm and the thickness is 7.5 mm. It has been designed for frequencies of 100–1000 GHz. The unpainted Eccosorb VFX-NRL-2 for millimeter wavelengths is a pyramidal carbon loaded polyurethane foam absorber, which consists of pyramids with a height of 38 mm and a spacing of 19 mm. The thickness of the material is 58 mm. Unpainted material was selected since it had been previously noted that the paint itself could increase the reflection when the absorber was used at 310 GHz. Carpet #1 consists of woven knots with a separation of approximately 2 mm. The pile fiber is 100% polyamide. The knots are bound to an intermediate layer, which is glued to a felt-like base layer. The double calendared vinyl coating is stabilized with glass fiber reinforcement. The thickness of the material is 5.5 mm. Carpet #2 has knots woven made of polypropylene fiber, which are glued onto a synthetic rubber backing. The thickness of the carpet is 9 mm. The carpet #3 is made of polypropylene fibers, which are held together with glue and a rubber backing. The thickness of the material is 8 mm.

3488

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 6. Reflection pattern of TERASORB-500 at horizontal polarization at 310 GHz. Grey line wedges vertically, black line wedges horizontally.

=

=

Fig. 4. Sample of FIRAM-500 placed on a sample holder. Front and back views. On the metal plate, the laser beam also used to assure accurate alignment of the sample can be seen (indicated with an arrow).

Fig. 7. Reflection pattern of TK THz RAM at horizontal polarization at 310 GHz.

Fig. 5. Reflection pattern of FIRAM-500 at horizontal polarization at 310 GHz. Grey line wedges vertically, black line wedges horizontally.

=

=

The sample of the absorber was placed on a sample holder (see Fig. 4). The support is a column made of extruded polystyrene foam (Styrofoam). The dielectric constant of Styrofoam was measured and found to be 1.048 at 310 GHz. Thus, the reflections from the support column are so low that with current measurement setup, it is not possible to separate them from the background reflection. We have been able to measure RCS down to 42 decibels relative to a squre meter (dBsm) for the vertical polarization and 36 dBsm for the horizontal polarization [8]. The support column itself was placed on a rotation/translation stage, which enables rotation of the target and also movement of the target in the -direction. The reflection from the absorber is separated from the background reflection by moving the target in the -direction and, as a result, a periodic response is obtained. The field component caused by the moving absorber can be evaluated from the variation of the amplitude and the phase [8]. An aluminum plate was placed behind the absorber sample. A laser beam was used to assure proper alignment of the plate and absorber. The beam was pointed to the aluminum plate from a distance of 1.2 m and the position of the plate was tuned until the transmitted and reflected beams converged. It was calculated that the angular alignment precision was better than 0.12 . The transmission of the absorbers at 310 GHz was measured during QZ testing. The measured amplitude of the QZ field with

Fig. 8. Reflection pattern of Eccosorb VFX-NRL-2 and carpet #1 (grey line) at horizontal polarization at 310 GHz.

Fig. 9. Reflection patterns of two different samples of carpet #1.

an absorber (without the metal back plate) placed in front of the field probe was compared to the amplitude without the absorber. The measured amplitude was averaged over 120 samples taken in a period of 1 min. IV. MEASUREMENT RESULTS The reflectivity of the absorber samples was measured over an incidence angle of 0 –45 . The measurements were done both at horizontal and vertical polarization at 310 GHz. The wedge-type absorbers were measured in two positions, i.e.,

LÖNNQVIST et al.: MONOSTATIC REFLECTIVITY MEASUREMENT OF RADAR ABSORBING MATERIALS AT 310 GHz

3489

TABLE I MAXIMA AND MINIMA OF REFLECTIVITES OF THE INVESTIGATED MATERIALS AT 310 GHz

wedges horizontally and wedges vertically. Measured patterns for FIRAM, TERASORB, TK THz RAM, Eccosorb, and carpet #1 are shown in Figs. 5–9. The maxima and minima of the reflectivity and transmission of all the materials investigated are gathered in Table I. We were able to measure the reflectivity down to 70 dB compared to the reflection from the reference. When the wedges of FIRAM are vertical, i.e., against the polarization, the absorber forms a reflection grid according to Bragg’s equation (1)

(1) where is the distance between wedges (i.e., parallel slits), is the angle of the maxima, is an integer, and is the wavelength. Peaks predicted by (1) can be seen around angles 18 ( 40 dB) and 39 ( 45 dB) (see Fig. 5). Between these maxima, the reflectivity level is below 60 dB. The level of reflectivity in the direction of the normal of the absorber is 31 dB. When the wedges are horizontal, the reflectivity is below 60 dB, except in the direction of the normal of the absorber it is a maximum of 30 dB. The transmission is quite high, i.e., 24 dB. TERASORB has a fingerlike pattern at both positions (see Fig. 6). The performance is clearly better near the normal of the absorber when the wedges are vertical, i.e., against the polarization, i.e., 41 dB versus 29 dB when the wedges are horizontal. The peaks caused by the grid structure can be seen, but their level is a lot lower than for FIRAM, approximately 50 dB. The transmission is also lower, i.e., 33 dB. The fingerlike structure of the reflection pattern caused by the pyramidal structure of TK RAM can be seen in Fig. 7. The level of reflectivity in the direction of the normal of the absorber is 51 dB and below 50 dB in the other directions. The transmission is low, below 50 dB. The same kind of fingerlike pattern was not seen when measuring Eccosorb (see Fig. 8). At submillimeter wavelengths, the pyramids of Eccosorb are very large compared to wavelength, therefore the diffraction peaks are so close to each other that they cannot be seen in Fig. 8. As the absorber is made of foam material, its structure is also not as uniform as, for example, the structure of TK RAM. The reflectivity of Eccosorb is low, below 50 dB for all angles, and the transmission is also below

50 dB. Overall performance of the Eccosorb absorber is adequate for use at submillimeter wavelengths. The monostatic reflectivity level of carpet #1 is about the same to all directions (see Figs. 8 and 9), i.e., it scatters energy to all directions. The reflectivity is surprisingly low, below 40 dB. The transmission of carpet #1 is higher than that for commercial absorbers, i.e., 17 dB. The levels of reflectivity of different samples of the same material are alike, as can be seen in Fig. 9. The samples are from different manufacturing batches and also the color of the samples is different. The reflection pattern of carpet #2 is very similar. It does have one advantage, the transmission is lower, approximately 25 dB. Carpet #3 has the worst performance with the reflection in the direction of the normal of the carpet being 20 dB, which can partly be due to its high transmission, measured to be 5 dB, so the metal plate can partly be seen through the carpet. To the other directions, the performance is about the same as for the other carpets, i.e., scatters to all directions. V. DISCUSSION In these monostatic measurements, TK THz RAM performed best among the materials investigated. Clearly it works better to certain reflection angles and with proper placement of the absorbers, reflectivity level better than 60 dB can be expected. In [2], the reflection in the direction of the normal has been measured to be 48 dB at 406 GHz, and in [1], 42 dB at 576 GHz and 35 dB at 672 GHz, thus the result obtained here; 51– 56 dB at 310 GHz is very well in line with the previous measurements. In [1] and [2], there was no information on reflectivity to other angles than in the direction of the normal of the absorber. Eccosorb VFX-NRL-2 also performed very well. It seems to scatter energy to all directions just like the carpet materials investigated. In [6], the material was investigated without the pyramids, i.e., the flat side of the absorber was measured, so the results cannot be compared. Also, as stated in [3], if the pyramids are large compared to wavelength, even though absorber tips tend to scatter coherently, the absorber walls built of several absorber panels scatter incoherently and the reflectivity level of a wall is considerably lower than that of a single panel, or as here, part of one panel. From the carpet materials, the best choice would be carpet #2 due to its high attenuation in transmission measurements and

3490

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

low overall reflectivity level. This carpet had the best performance among the carpets also in bistatic measurements. The material is inexpensive compared to the commercial absorber materials (by the order of tens of times). Since the need of absorbing material in a large-sized compact range can be over 500 m , this type of difference reduces the costs significantly. The reflectivity is still higher than for the best materials so the placement of the carpets has to be designed carefully when using them together with better absorber materials. The performance of TERASORB and FIRAM is strongly polarization dependent. For both, the performance is good when excluding the peaks caused by the wedge-type structure. With this kind of material, even more attention has to be paid to proper placement since even a small mistake in placement can cause a 20-dB difference. These absorbers did not perform as good as expected. However, this can be due to their design, which was optimized to frequencies higher than 500 GHz. For FIRAM, the reflection in the direction of the normal was approximately 30 dB at 310 GHz and, in [4], it was measured to be approximately 38 dB at 584 GHz. TERASORB and FIRAM are at their best when used with one polarization only. In compact ranges, this very seldom is the situation. The reflection coming back to the transmitter was not measured in [6] so it is not possible to do a direct comparison of the results, but the order of superiority is the same as in [6] with the exception of Eccosorb, which was measured without the pyramids in the previous study. In the far-field situation, the maximum measured reflectivity is clearly lower than what was measured with near-field measurements. In the monostatic measurements reported here, it was possible to eliminate the effect of background reflections better than those reported in [6] and, furthermore, direct coupling was not a problem in the measurements reported here. For compact ranges, absorber measurements done in plane-wave conditions can be estimated to resemble a real-life situation better than results of measurements done in the near field of the absorber. The test samples were relatively small and they filled the QZ almost entirely. Even though the metal plate was totally covered with the absorber, diffraction from the edges could have caused some uncertainty to the measurement result though any clear indication of this kind of phenomena was not found. Getting larger samples of the materials to the tests or a smaller QZ diameter would eliminate the possibility of this kind of effect showing in the measurement results. In the future, also making bistatic measurements in far-field conditions would be of interest. With a phase hologram setup containing two holograms on a moving axis, this would be possible. This kind of measurement would give a better understanding of the scattering behavior of the absorber. The crosspolarization performance of absorbers should also be tested. Low reflectivity may be due to energy transforming from one polarization to another. In this study, we have tested absorbers at only one frequency, namely, at 310 GHz. This is due to the narrowband behavior of our hologram-based RCS measurement setup. However, we believe that the results obtained are useful, and also demonstrate how a relatively simple measurement setup can be used to test absorbers in far-field conditions at submillimeter wavelengths.

VI. CONCLUSION Monostatic reflectivity of a set of absorbing materials was investigated at 310 GHz at an angle range from the normal of the absorber to 45 . As expected, TK THz RAM manufactured by Thomas Keating Ltd. was found to have the best performance. Eccosorb VFX-NRL-2 also performed well, better than 50-dB attenuation to all angles. This is better than expected for an absorber that is designed for the millimeter-wave region. The Eccosorb material investigated here was unpainted so carbon dust can cause trouble in some applications. Since its pyramids are large compared to wavelength and absorber walls scatter incoherently, it can be expected to perform even better when a large absorber wall is built. A floor carpet had the next best performance, over 40-dB attenuation to all angles. This can be found adequate at least for the noncritical places in the compact range. A combination of all of these three, i.e., TK THz RAM, Eccosorb, and carpets, is a good compromise. Use of the carpets considerably brings down the costs of building a measurement range.

ACKNOWLEDGMENT The authors thank the members of the Millimeter Wave Group, MilliLab/TKK Radio Laboratory, Helsinki University of Technology, Espoo, Finland, for their support and useful conversations during this study. The authors also thank J. Häkli for the transmission measurements. The authors further thank E. Noponen for designing the hologram used in these experiments.

REFERENCES [1] A. Murk, N. Kämpfer, and N. J. Keen, “Baseline measurements with a 650 GHz radiometer,” in Proc. 2nd Millim. Wave Technol. Applicat.: Antennas, Circuits, Syst, Workshop, Espoo, Finland, May 27-29, 1998, pp. 121–126. [2] A. Murk, N. Kämpfer, and N. J. Keen, “Baseline issues in an airborne 650 GHz radiometer,” in COST-712 Microw. Tech. Meteorol. Workshop, Bern, Switzerland, Dec. 1999, pp. 42–51. [3] B. T. DeWitt and W. D. Burnside, “Electromagnetic scattering by pyramidal and wedge absorber,” IEEE Trans. Antennas Propag., vol. 36, no. 7, pp. 971–984, Jul. 1988. [4] R. H. Giles, A. J. Gatesman, J. Fitzgerald, S. Fisk, and J. Waldman, “Tailoring artificial dielectric materials at terahertz frequencies,” in Proc. 4th Int. Space Terahertz Technol. Symp., Los Angeles, CA, Apr. 1993, pp. 124–133. [5] J. Säily, J. Mallat, and A. V. Räisänen, “Reflectivity measurements of several commercial absorbers in the 200–600 GHz range,” Electron. Lett., vol. 37, no. 3, pp. 143–145, 2001. [6] J. Säily and A. V. Räisänen, “Characterization of submillimeter wave absorbers from 200–600 GHz,” Int. J. Infrared Millim. Waves, vol. 25, no. 1, pp. 71–88, Jan. 2004. [7] E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section, 2nd ed. Norwood, MA: Artech House, 1993. [8] A. Lönnqvist, J. Mallat, and A. V. Räisänen, “Phase hologram based compact RCS test range at 310 GHz for scale models,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 2391–2397, Jan. 2006. [9] J. Meltaus, J. Salo, E. Noponen, M. M. Salomaa, V. Viikari, A. Lönnqvist, T. Koskinen, J. Säily, J. Häkli, J. Ala-Laurinaho, J. Mallat, and A. V. Räisänen, “Millimeter-wave beam shaping using holograms,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1274–1280, Apr. 2003.

LÖNNQVIST et al.: MONOSTATIC REFLECTIVITY MEASUREMENT OF RADAR ABSORBING MATERIALS AT 310 GHz

Anne Lönnqvist was born in Somero, Finland, in 1977. She received the Master of Science (Tech.) (with honors) and Licentiate of Science (Tech.) degrees in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 2001 and 2004, respectively, and is currently working toward the Doctor of Science (Tech.) degree at TKK. Since 2000, she has been a Research Assistant and Research Engineer with the Radio Laboratory, TKK. Her current research interests include millimeter-wave measurement techniques with a focus on hologram applications.

Aleksi Tamminen was born in Ruotsinpyhtää, Finland, in 1982. He received the Bachelor’s (Tech.) degree in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 2005 , and is currently working toward the Master of Science (Tech.) degree at TKK. He is currently a Research Assistant involved with millimeter-wave measurement projects with the Radio Laboratory, TKK.

Juha Mallat was born in Lahti, Finland, in 1962. He received the Master of Science (Tech.) (with honors), Licentiate of Science (Tech.), and Doctor of Science (Tech.) degrees in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 1986, 1988, and 1995, respectively. Since 1985, he has been with the Radio Laboratory (and its Millimeter Wave Group), TKK, as a Research Assistant, Senior Teaching Assistant, and Research Associate until 1994. From 1995 to 1996, he was a Project Manager and Coordinator involved with an education project between TKK and the Turku Institute of Technology. Since 1997, he has been a Senior Scientist with the Millimetre Wave Laboratory of Finland (MilliLab), European Space Agency (ESA) External Laboratory, Helsinki, TKK, with the exception of a period of one year from 2001 to 2002 when he served as a Professor (protem) of radio engineering with TKK. His research interests and experience cover various topics in radio-engineering applications and measurements, especially in millimeter-wave frequencies. He has also been involved in building and testing millimeter-wave receivers for space applications.

3491

Antti V. Räisänen (S’76–M’81–SM’85–F’94) received the Master of Science (Tech.), Licentiate of Science (Tech.), and Doctor of Science (Tech.) degrees in electrical engineering from the Helsinki University of Technology (HUT), Espoo, Finland, in 1973, 1976, and 1981, respectively. In 1989, he was appointed Professor Chair of Radio Engineering, HUT, after holding the same position as an Acting Professor in 1985 and 1987–1989. He has been a Visiting Scientist and Professor with the Five College Radio Astronomy Observatory (FCRAO), University of Massachusetts at Amherst (1978–1981), Chalmers University of Technology, Göteborg, Sweden (1983), Department of Physics, University of California at Berkeley (1984–1985), Jet Propulsion Laboratory, California Institute of Technology, Pasadena (1992–1993), and Paris Observatory and University of Paris 6 (2001–2002). He currently supervises research in millimeter-wave components, antennas, receivers, microwave measurements, etc. at the Radio Laboratory, HUT, and Millimetre Wave Laboratory of Finland (MilliLab—European Space Agency (ESA) External Laboratory). The Smart and Novel Radios Research Unit (SMARAD), HUT (which he leads), obtained in 2001 the national status of Center of Excellence in Research from The Academy of Finland after competition and international review. He has authored and coauthored over 400 scientific or technical papers and six books, most recently, Radio Engineering for Wireless Communication and Sensor Applications (Artech House, 2003). He also coauthored the chapter “Radio-Telescope Receivers” in Radio Astronomy (Cygnus-Quasar Books, 1986, 2nd ed.). Dr. Räisänen was secretary general of the 12th European Microwave Conference in 1982. He was chairman of the IEEE Microwave Theory and Techniques (MTT)/Antennas and Propagation (AP) Chapter in Finland from 1987 to 1992. He was conference chairman for the 22nd European Microwave Conference in 1992, and for the “ESA Workshop on Millimeter Wave Technology and Applications” in 1998. From 1995 to 1997, he served on the Research Council for Natural Sciences and Engineering, Academy of Finland. From 1997 to 2000, he was vice-rector for research and international relations of HUT. From 2002 to 2005, he was an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.

3492

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Distortion Mechanisms in Varactor Diode-Tuned Microwave Filters Bruce E. Carey-Smith and Paul A. Warr

Abstract—This paper examines the broadband distortion behavior in flexible filters employing varactor-diode tuning elements. Series- and parallel-resonant varactor-loaded transmission-lines, both commonly used in bandpass and bandstop microwave filters, are analyzed. Nonlinear Volterra-series analysis is employed to determine the second- and third-order distortion ratios dependent on the frequencies of the incident signals. It is shown that in a bandpass filter (employing parallel tuned resonators), maximum distortion occurs in the passband, while in a bandstop filter (employing series tuned resonators), minimum distortion occurs at the minimum-loss passband. The analysis is verified by practical measurement of filters employing the two modes of resonators. Index Terms—Distortion, microwave filters, tunable filters.

I. INTRODUCTION HE USE of semiconductor devices as tuning elements in flexible filters is attractive, as they are economical and easily integrated into miniature circuits. However, because their inherent transfer characteristics are nonlinear, they distort the incident signals. Tunable filters must be judged according to their linearity since their primary purpose is to either remove the signals that could cause distortion or to remove the distortion itself. In order to arrive at useful conclusions about the characteristics of the distortion within semiconductor-tuned filters, it is necessary to have an understanding of the nonlinear behavior of the circuit elements and subsystems. Firstly, an examination of the nonlinear properties of two of the most common semiconductor devices used as the tuning components in microwave filters is presented. This leads to a detailed analysis of two varactor-tuned transmission-line resonator topologies, which are the common subsystems within tunable bandpass and bandstop filters (BSFs). The linear and nonlinear properties of these resonators are examined in order to draw conclusions about the nonlinear performance of their associated filters. In filters that contain semiconductor elements, the nonlinearity is not memoryless and, thus, the method of describing it must incorporate time-dependent circuit information, making Taylor-series analysis inadequate. The Volterra series is employed for the nonlinear analysis in this paper, as it is suitable for modeling memory effects. Its elements (or “kernels”) are progressively higher order convolution integrals encompassing greater numbers of excitation signals [1].

T

Manuscript received October 31, 2005; revised March 3, 2006. The authors are with the Centre for Communications Research, University of Bristol, Bristol BS8 1UB, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.881024

Narrowband distortion measurements of tunable resonators [2] and filters [3] have been reported previously. In contrast, this paper focuses on broadband performance, using mathematical analysis to explain the observed trends in nonlinearity. II. SEMICONDUCTOR ELEMENTS IN TUNABLE MICROWAVE FILTERS In order to establish the distortion produced in semiconductor-tuned filters, the mechanisms by which it is generated in the components should be examined. The most common semiconductor devices used as tuning elements in microwave filters are varactor and p-i-n diodes. It is established here that the contribution of varactor diodes to the nonlinear characteristics is highly dominant over p-i-n diodes. A. p-i-n Diodes p-i-n diodes are constructed by sandwiching a relatively thick layer of intrinsic semiconductor material between a pn junction. The device behaves like a current controlled resistance to highfrequency signals. Provided there is sufficient dc-bias current, p-i-n diodes will display almost linear behavior for small RF signals. As long as the displacement of junction charge due to the high-frequency excitation is much smaller than the stored charge due to the dc bias, the high-frequency resistance of the junction will not be modulated by the high-frequency signal [4]. It is shown in [5] that the stored charge and the charge due to the incident RF signal in the intrinsic layer is given as follows by (1) and (2), respectively: (1) (2) is the forward bias current, is the minority charge where carrier lifetime, and the RF signal has frequency and amplitude . Since it is desirable that be much greater than , an expression for the relative magnitude of required forward bias current can be found as (3) Typical small-signal RF p-i-n diodes have charge carrier lifetimes in the order of 300 ns–1.5 s. The logarithmic ratio representing the level of RF modulation of the stored charge is plotted in Fig. 1 for an RF carrier frequency of 800 MHz. The equivalent series resistance is dominated by a proportional relationship to so that any variation in the overall stored charge in the

0018-9480/$20.00 © 2006 IEEE

CAREY-SMITH AND WARR: DISTORTION MECHANISMS IN VARACTOR DIODE-TUNED MICROWAVE FILTERS

3493

Fig. 1. Relative level of high-frequency charge displacement q against normalized bias current I =i .

p-i-n diode junction will result in modulation of the resistance. This is a nonlinear effect and will cause distortion. However, from Fig. 1, it can be seen that, when the RF signal is small, it is relatively easy to provide enough forward bias current to reduce below 40 dB. the variation in If designed correctly, p-i-n diodes will not introduce levels of distortion that are significant in all but the most sensitive applications. For this reason, they will not be considered further in this paper. However, should be maintained at a sufficient level; this may be an issue in certain low-power applications.

Fig. 2. Normalized coefficients for a Taylor-series expansion of depletion capacitance around dc-bias point V .

The Taylor series expansion of the depletion capacitance is given by

(5) where is given by (5) and the coefficients by the th-order derivative of

–

are given

B. Varactor Diodes A varactor diode is a voltage-controlled device. The reversebiased junction capacitance can be adjusted by varying the reverse-biased voltage. Two main components make up the capacitance across the semiconductor junction: diffusion and depletion capacitance. Diffusion capacitance is dominant under forward bias and is due to the injected minority charge in the junction. Under reverse bias, this injected charge tends to zero and the depletion capacitance becomes dominant. The depletion capacitance is due to the stored charge in the depletion region and is influenced by the magnitude of the reverse-bias voltage. The equation that describes the depletion capacitance is [6] (4)

where is the zero-bias junction capacitance, is the reverse-bias voltage, and is the junction potential, which is related to the thermal voltage and doping levels of the junction. is the junction grading or doping coefficient and varies between 0.33 (linear junction) and 5.0 (hyper-abrupt junction) depending on the doping profile. The varactor diode depletion capacitance can be directly modulated by an RF signal. Since the depletion capacitance is a nonlinear function of reverse-bias voltage, this modulation will result in distortion products being generated. The depletion capacitance nonlinearity can be modeled by a memoryless Taylor-series expansion around the dc-bias point. Weak nonlinearities, as seen in small-signal varactor diodes, typically have small variation around a quiescent point and can be described with good accuracy by only a few terms in the Taylor series.

(6) The terms and are the dc and ac components of the reverse-bias voltage . The coefficients can be normalized to the dc or linear component so that

(7) The magnitude of the first- and second-order normalized coefficients is plotted in Fig. 2 for a typical small-signal varactor diode with V and pF. The depletion capacitance of the varactor diode decreases with increasing reverse bias. Fig. 2 indicates that for larger values of reverse voltage (and, hence, lower varactor capacitance), the relative level of distortion reduces. The remaining analysis and discussion in this paper is concerned with varactor diodes and their impact on linearity within the context of tunable filters. Despite recent advances in topologies to lower the distortion generated in varactor diodes [7], this element will continue to dominate the observed distortion. III. DISTORTION ANALYSIS OF VARACTOR TUNED FILTERS The common building blocks used in semiconductor tunable microwave filters are the varactor diode and the transmission line. Used together they can be connected to form a relatively high- resonant circuit whose resonant frequency can be tuned by altering the varactor bias voltage. Both series and parallel resonant structures can be formed in this way. Two of the most commonly used structures are shown in Fig. 3.

3494

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 3. Varactor-tuned transmission-line resonators. (a) Parallel resonant. (b) Series resonant.

Fig. 5. Magnitude of the voltage across the varactor diode in a series-resonant varactor-loaded transmission line. The specified value of corresponds to the transmission-line quarter-wavelength.

X

of capacitive loading. These attenuation poles correspond to the half-wavelength frequency of the transmission lines. The maximum voltage across the varactor remains constant in the parallel configuration, regardless of capacitance value. This means that the distortion in the filter passband will follow the relationship shown in Fig. 2, decreasing for larger capacitive reactance. B. AC Analysis of Series Resonant Varactor-Tuned Structure Fig. 4. Magnitude of the voltage across the varactor diode in a parallel-resonant varactor-loaded transmission line. The specified value of corresponds to the transmission-line quarter-wavelength.

X

In order to study distortion in filter structures, it is of benefit to analyze these simple resonant structures when driven from a resistive source. Of particular interest is the voltage developed across the varactor diode since this contributes directly to the distortion. A. AC Analysis of Parallel Resonant Varactor- Tuned Structure By undertaking simple ac analysis of the parallel resonant configuration in Fig. 3(a), it is possible to plot the voltage across the varactor diode. Fig. 4 shows the magnitude of the ac voltage across the varactor versus the phase length of the transmission line for several values of capacitive reactance . Source voltage and resistance and transmission-line impedance are set to unity. The point of maximum voltage corresponds to the parallel resonant condition when the impedance of the resonator approaches infinity. Since the distortion induced in the varactor due to modulation of the depletion capacitance is proportional to the ac voltage across it, for a fixed value of dc bias, the point of maximum distortion will always correspond to the parallel resonant frequency of the resonator. Resonators whose primary resonant mode is high impedance are used widely in bandpass filter design. This is the most common form of varactor-tuned filter [8]–[10]. In this case, using filter transfer function terminology, the maximum distortion will occur at the transmission pole frequencies, which are distributed across, and define, the passband of the filter. Conversely, minimum distortion will occur at the attenuation poles of the filter, which are independent of the value

A simple ac analysis can also be performed on the series resonator configuration shown in Fig. 3(b). The voltage information for this topology is shown in Fig. 5. The series resonant mode occurs when there is maximum voltage across the varactor. At this point, the resonator impedance approaches zero and the distortion introduced by the varactor diode will be highest. This type of resonator is used predominantly in BSFs [11], [12] leading to the important conclusion that maximum distortion will occur when the excitation signals are at the center of the stopband. The grounded transmission-line series topology has the characteristic that its point of maximum impedance will always occur at the frequency where the transmission line is a quarter-wavelength. At this frequency, the voltage across the varactor is at a minimum, independent of the actual value of capacitive reactance. This relationship can be seen mathematically by deriving the expression for the voltage across the varactor when the resonator is loaded with a resistance . In this situation, the voltage across the varactor is given by (8) The magnitude of this function is at a maximum at resonance, when , and drops to zero when , as the transmission-line impedance tends to infinity. Referring to Fig. 5, as the capacitive reactance is increased, and the series resonant point moves toward the quarter-wavelength frequency of the transmission line, the maximum voltage across the varactor also increases. This would lead to an increase in distortion generated by the varactor. However, recalling the

CAREY-SMITH AND WARR: DISTORTION MECHANISMS IN VARACTOR DIODE-TUNED MICROWAVE FILTERS

discussion on normalized coefficients for the Taylor series expansion of the depletion capacitance in Section II-B, this increase will be partially offset by the decrease in distortion as the bias voltage is increased to bring about the change in varactor capacitance. C. Volterra-Series Analysis of Varactor-Loaded Transmission-Line Resonators Here we use the Volterra-series approach to describe the varactor-loaded transmission-line resonators of Fig. 3. The analysis is developed from the form given in [12]. 1) Parallel Resonant Topology: The Volterra kernels, which describe the parallel resonant circuit configuration, can be derived by summing the currents in the circuit such that

(9) is given by (4) with is the voltage across where the varactor, is the voltage source conductance, and the normalized Taylor series coefficients etc. are found from (7). Due to the charge being less than unity and the decreasing size of the capacitance coefficients, the terms in (9) quickly diminish in magnitude as their order increases. In order to simplify the following analysis, the capacitance function is limited to third order, under the assumption that the higher order components will have only modest effect on the results. The nonlinear transfer functions are found sequentially by exciting the circuit with an increasing numbers of noncommensurate frequency tones and solving (9) for the voltage [7]. Only the contributions that relate to the particular transfer function being solved need be considered. The first-, second-, and third-order transfer functions for the parallel configuration resonant circuit are given by

(10)

and

(11)

(12) where

(13) and (14)

3495

Using these functions, it is possible to calculate any first-, second-, or third-order signal component by selecting the appropriate values for and . The parallel resonant configuration finds greatest application in bandpass filters. In this context, the primary concern, when dealing with nonlinearity, will be situations where distortion products fall within the passband of the filter. In this case, the distortion products have the potential to reduce the integrity of the wanted signal. Where distortion products fall in the stopband, they will generally be attenuated sufficiently to avoid any detrimental impact on performance. a) Third-Order Intermodulation Distortion (IMD): Both even- and odd-order distortion can potentially generate products in-band, depending on the spectral location of the exciting signals. The component of greatest interest is the third-order IMD, which produces distortion products near to the excitation signals. The relative magnitude of the third-order intermodulation components can be found from the ratio of (10) and (11) by setting and equal to , and equal to , or vice versa. The third-order intermodulation rejection ratio (3IMRR) for the third-order component at is given by

(15) is the magnitude of the tones used to excite the circuit. where The two tones are assumed to have equal amplitude. The thirdorder component at can be found by interchanging and in (15). Equation (15) is used to determine the relative level of the third-order intermodulation component for a typical small-signal varactor diode connected in parallel with a 90transmission line and driven from a 50- source. The results are shown in Fig. 6. The excitation signals were swept such that either the upper or lower third-order intermodulation product occurred at the parallel resonant frequency of the resonator. The circuit was tuned such that the parallel resonant frequency occurred at 600 MHz and the transmission-line half-wavelength occurred 2.06 GHz. The required varactor-bias voltage was 0.55 V. An equivalent parallel resonant circuit was built using an Infineon BB857 varactor diode and a grounded section of 90microstrip fabricated on low-loss microwave laminate ( mm, ). Using a buffered two-tone source at 1.4 dBm per tone, the third-order IMD was measured at the circuit input using a directional coupler and high dynamic-range spectrum analyzer. The results are plotted in Fig. 6. The difference in the measured and modeled results is due to two factors. Firstly, a resonance associated with the varactor diode bias network is causing reduction in low-frequency distortion. Secondly, the parasitic inductance of the varactor diode results in the frequency of the second parallel resonance being reduced below that of the transmission-line half-wavelength. This effect lowers the measured distortion above the primary parallel resonance. The significant point, however, is that the maximum distortion occurs as the two excitation signals converge on the parallel resonant frequency. Conversely, distortion is at a minimum

3496

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 6. Third-order IMD component relative to the excitation signal in a varactor-tuned parallel transmission-line resonator.

when the excitation signals coincide with the transmission-line half-wavelength frequency. The first significant null in the magnitude of the third-order distortion products at the parallel resonant frequency occurs when the upper of the two excitation signals is coincident with the transmission-line half-wavelength frequency. The increase in magnitude above this point is due to the upper excitation signal reaching the second parallel resonance of the circuit. b) Second-Order IMD: The two second-order intermodulation components generated from a two-tone excitation signal will occur at and . The second-order intermodulation rejection ratio (2IMRR) for the second-order component at , neglecting higher order components, is given by (16) is the magnitude of the tones used to excite the cirwhere cuit. The second-order component at can be found by reversing the sign of in (16). Using the same setup as described in Section III-C.1a, the relative level of the second-order IMD products was calculated and measured. Again, the excitation signals were swept such that either the upper or lower second-order IMD product ( or ) occurred at the parallel resonant frequency of the resonator. The results are given in Fig. 7. Note that, in this case, the rejection ratio is plotted against the frequency of the lowest excitation tone. Maximum second-order IMD occurs when either of the excitation tones approaches either the first or second parallel resonant frequency of the circuit. A null occurs when the higher excitation tone reaches the half-wavelength of the transmission line. The data is discontinuous between 0.3–0.6 GHz because the distortion products cannot fall at the parallel resonant frequency when the lowest frequency tone is between these values. Although the level of the second-order IMD component is higher than that of the third (see Fig. 6) when a resonator of this topology is used in a filter, the second-order products are

Fig. 7. Second-order IMD relative to excitation signal. The excitation signal consists of two equal amplitude tones spaced so that the intermodulation component always falls at the parallel resonant frequency.

not so important. To generate in-band second-order distortion, at least one of the excitation tones must be far from the parallel resonant frequency, which, in the case of a bandpass filter, will result in considerable attenuation. 2) Series Resonant Topology: The analysis of the series resonant circuit of Fig. 3(b) is similar to the parallel resonant topology of Fig. 3(a). However, it is simplified by describing the voltage across the varactor as a function of charge. In this way, the voltages in the circuit can be summed such that

(17) where and are first-, second-, and third-order nonlinear coefficients of the capacitor’s elastance and is given by the time integral of the current from to . The elastance coefficients can be found by rearranging the equation for depletion charge (4) [14]

(18) as a function of charge , and then to give reverse voltage differentiating with respect to charge. The first- and secondorder nonlinear coefficients are thus given by (19) and (20) The first-, second-, and third-order transfer functions can be found in a similar manner described in Section III-C.1. They are given by

(21)

CAREY-SMITH AND WARR: DISTORTION MECHANISMS IN VARACTOR DIODE-TUNED MICROWAVE FILTERS

3497

(22) and

(23) where

Fig. 8. 2IMRR of the series resonator topology. The excitation signal comprises two equal-amplitude tones. The first, i.e., ! , is held at the series resonant frequency of the resonator while the other, i.e., ! , is swept above and below the first.

(24) and

(25) a) Second-Order IMD: In contrast to bandpass filters, the second-order IMD generated by the varactor diode is important when considering the application of BSFs. If the IMD product of two high-level interfering signals falls in the band of interest, it is possible that it may inhibit the detection of the wanted signal. The worst case scenario occurs when at least one of the interferers is located at the center of the filter stopband, which is likely since the purpose of the BSF is to attenuate interfering signals. The 2IMRR for the transmission-line grounded topology can be found from the ratio of the first- and second-order currents, which, with the appropriate excitation, can be written in terms of the magnitude of the excitation signals and the first- and second-order transfer functions from (20) and (21)

(26) The second-order IMD results for the series configuration are plotted in Fig. 8. The graph shows the two-tone second-order IMD performance when one of the tones occurs at the series resonant frequency and the other is swept across frequency. The same circuit components and test setup as described in Section III-C.1a were used with the exception that the transmission-line quarter-wavelength frequency was set at 1.36 GHz. The varactor capacitance was tuned such that the series resonance occurred at 1.0 GHz ( V). Maximum second-order distortion occurs at 2 GHz when both excitation tones occur close to the series resonant frequency. At this frequency, there will be maximum current flowing through the varactor diode and maximum voltage across it. The second-order IMD passes through a null at the quarter-wavelength frequency. At this point, the transmis-

sion line presents high impedance, inhibiting the flow of distortion current. The null at 0.36 GHz is also related to the quarter-wavelength and occurs when the second tone reaches this quarter-wavelength frequency. b) Third-Order IMD: The highest level of third-order IMD occurs when at least one of the excitation signals is located at the series resonant frequency. The relative level of the third-order intermodulation components at and can be found by substituting (21) and (23) into (15). The modeled and measured results for both components are shown in Fig. 9. The response of the component is virtually symmetrical around the series resonant frequency. It reaches a minimum at 0.64 and at 1.36 GHz; in the first case, when the excitation signal passes through the quarter-wavelength frequency of the transmission line, and in the second case, when the distortion product itself passes through that frequency. The component reaches a minimum at 1.36 GHz when it passes through the quarter-wavelength frequency of the transmission line. A second minimum occurs at 1.72 GHz (not shown) when the excitation tone reaches that frequency. Below the series resonant frequency, the component exhibits a gentle decay in the absence of any resonances. The results presented in Fig. 9 have significant implications for BSFs constructed using the varactor-coupled transmission-line resonators from Fig. 3(b). In this context, maximum distortion will be generated in the resonator when the excitation tones occur in the stopband of the filter, which will be their likely location. However, in many applications, the concern over distortion is its masking effect of a wanted signal. The results above show that if the minimum loss passband (transmission-line quarter-wavelength frequency) can be designed to coincide with the wanted signal band, then the second- and third-order distortion products occurring in this frequency band will be minimized. IV. DISTORTION MEASUREMENTS OF VARACTOR-TUNED FILTERS To verify the wideband distortion analysis above and to extend previous reported narrowband measurements [3], a

3498

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 10. Measured third-order IMD plotted against mean interference frequency for two filter tuning positions. The filter insertion loss for each tuning point is also shown.

2 0

2 0

Fig. 9. Third-order IMD components: (a) ! ! and (b) ! ! relative to the fundamental component of the current in a series-resonant varactor-tuned transmission line. The excitation signal comprises two equal-amplitude tones. The first, i.e., ! , is held at the series resonant frequency of the resonator, while the other, i.e., ! , is swept above and below the first.

combline bandpass filter using parallel-coupled transmission-line resonators and a BSF comprising two series-coupled transmission-line resonators coupled via a quarter-wave through-line were constructed and measured. A five-element varactor-tuned combline filter was constructed in microstrip to give a 5% fractional bandwidth between 1.8–2.1 GHz. The filter employed resonators with commensurate length and width, short circuited at one end and terminated in a series combination of a lumped 1.2-pF capacitor and varactor (Infineon BB839) capable of tuning between 1–18 pF. In order to gather third-order IMD data on the constructed filter, two equal-amplitude interfering signals were introduced at the filter input and the amplitude of the third-order distortion product was measured at the output. The relationship between the frequencies of the two interfering signals was chosen so that the distortion product would always fall within the passband of the filter. The results are shown in Fig. 10 for two filter tuning positions, i.e., 1.85 and 2.0 GHz. The measured results agree with the analysis of Section III-C.1a. The increase in distortion level at the lower center-frequency tuning position follows from the results of Section III-A, where it was shown that, as the reactance of the varactor increases in proportion to the transmission-line impedance, the voltage across the varactor will increase, leading to more distortion. In most cases, a BSF will be used to reduce the level of a large unwanted signal. This means that there will be at least one

0

+

Fig. 11. Measured second-order IMD (f f and f f ) of a varactor-tuned BSF relative to the excitation signal power at the input. The excitation signal comprises two equal-amplitude tones. The first is held at the filter stopband, while the other is swept above and below the first.

large signal located in the stopband. Second-order IMD measurements were made on a varactor tuned BSF, and the results are shown in Fig. 11 for a two-tone excitation signal. Where needed in order to maintain the dynamic range of the measurement, a tunable notch filter was used to attenuate the excitation tones after the filter under test. Resistive attenuators were placed on either side of this notch filter to minimize its loading effect. The constructed filter was an inverter coupled two-element BSF formed from varactor-coupled quarterwave-length resonators of the form shown in Fig. 3(b). The construction details of this filter may be found in [12]. The results show the same general trend as those calculated theoretically using Volterra series (see Fig. 8). The minimum second-order intermodulation level occurs at the quarter-wavelength of the resonator transmission line (in this case, 1.41 GHz) and the maximum occurs around the stopband of the filter. The local minimum at the stopband center frequency occurs because

CAREY-SMITH AND WARR: DISTORTION MECHANISMS IN VARACTOR DIODE-TUNED MICROWAVE FILTERS

Fig. 12. Measured third-order IMD of a varactor-tuned BSF. The first tone, i.e., f , is held constant at the center of the stopband, while the second excitation tone, i.e., f , is swept above and below the first. The filter insertion loss is also shown.

the distortion is measured at the output of the filter; the distortion component at the output will be reduced in proportion to the filter stopband attenuation. The third-order IMD performance of the BSF was also measured and the results are presented in Fig. 12. The measured results gave good agreement with the calculated results in Section III-C.2b. Below the filter stopband, the product decreases more rapidly than its counterpart, as predicted by the theoretical and measured results for the single resonator (Fig. 9). The quarter-wavelength frequency of the resonators in the measured filter were set to approximately 1.9 GHz; hence, the minima in the third-order distortion results are out of range in the case of the measured results.

V. CONCLUSION The presented analysis has shown the distortion performance of two common varactor-tuned transmission-line configurations and the implications for tunable filter design. Simple ac analysis on the parallel varactor transmission-line configuration, common in bandpass filters, reveals that the maximum signal voltage across the varactor diode occurs at the parallel resonant frequency. Thus, in a bandpass filter based on this configuration, maximum distortion will occur in the passband of the filter. This is demonstrated further through a Volterra-series analysis of the varactor-tuned parallel transmission-line resonator. Third-order distortion products are identified as being the most problematic for bandpass filters based on this topology and the third-order products are shown to be highest when the excitation signals are located in the passband. Similar analysis of a series varactor-coupled transmission-line topology demonstrated that, when used as the building

3499

block in a BSF, much more favorable distortion characteristics are observed. In this case, maximum signal voltage across the varactor diode, and correspondingly, maximum distortion, will occur at the series resonant frequency. In a BSF based on this resonator configuration, this will correspond to the stopband of the filter. AC analysis also showed that the varactor voltage passes through a minimum at the transmission-line quarter-wavelength frequency. For the series configuration, this is significant since, in a BSF, the minimum-loss passband will occur at this frequency. Volterra-series analysis of the varactor-tuned series transmission-line resonator showed that both the second- and third-order IMD products were at a maximum at the series resonant frequency. Notably, however, their magnitude passed through a minimum at the quarter-wavelength frequency (filter passband). The conclusion here is that, as long as the wanted signal is centred at the minimum-loss passband, varactor-tuned filters using this configuration may be useful in situations where only very low levels of distortion can be tolerated. The nonlinear characteristics of bandpass and BSFs were measured in order to verify the analysis; good concurrence was shown between theoretical and measured results.

REFERENCES [1] D. Mirri, G. Iuculano, F. Filicori, G. Pasini, G. Vannini, and G. Gualtieri, “A modified Volterra series approach for nonlinear dynamic systems modelling,” IEEE Trans. Circuits Syst. I—Fundam. Theory Appl., vol. 49, no. 8, pp. 1118–1128, Aug. 2002. [2] K. B. Ivlev, “Nonlinear distortions in passive varactor tunable resonators,” in Proc. IEEE–Russia High Power Microw. Electron.: Meas., Identification, Appl. Conf., 1999, pp. I34–I39. [3] I. C. Hunter and S. R. Chandler, “Intermodulation distortion in active microwave filters,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 145, no. 1, pp. 7–12, Feb. 1998. [4] R. H. Caverly and G. Hiller, “Distortion in p-i-n diode control circuits,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 5, pp. 492–501, May 1987. [5] The PIN Diode Circuit Designer’s Handbook. Watertown, MA: Microsemi-Watertown, 1998. [6] I. Bahl and P. Bhartia, Microwave Solid State Circuit Design. New York: Wiley, 1988. [7] K. Buisman, L. de Vreede, L. Larson, M. Spirito, A. Akhnouk, T. Scholtes, and L. Nanver, “Distortion-free varactor diode topologies for RF adaptivity,” IEEE MTT-S Int. Microw. Symp. Dig. Jun. 12–17, 2005, 4 pp. [8] H. Dayal, “Variable bandwidth wide tunable frequency voltage tuned filter,” Int. J. RF Microw. Comput.-Aided Eng., vol. 14, no. 1, pp. 64–72, Jan. 2004. [9] A. R. Brown and G. M. Rebeiz, “A varactor-tuned RF filter,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 7, pp. 1157–1160, Jul. 2000. [10] I. C. Hunter and J. D. Rhodes, “Electronically tunable microwave bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 9, pp. 1354–1360, Sep. 1982. [11] S. Toyoda, “Quarter-wavelength coupled variable bandstop and bandpass filters using varactor modes,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 9, pp. 1387–1389, Sep. 1982. [12] B. Carey-Smith and P. A. Warr, “Broadband configurable bandstop filter with composite tuning mechanism,” Electron. Lett., vol. 40, pp. 1587–1589, Dec. 2004. [13] S. Maas, Nonlinear Microwave Circuits. Piscataway, NJ: IEEE Press, 1997, pp. 178–186. [14] J. C. Pedro and N. Borges Carvalho, Intermodulation Distortion in Microwave and Wireless Circuits. Norwood, MA: Artech House, 2003.

3500

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Bruce E. Carey-Smith received the B.E. degree in electrical and electronic engineering from the University of Canterbury, Christchurch, New Zealand, in 1995, and is currently working toward the engineering Ph.D. degree at the University of Bristol, Bristol, U.K. From 1995 to 2002, he was with Tait Electronics Ltd., Christchurch, New Zealand, where he was involved in the design of RF circuits and systems for mobile radio applications. He subsequently joined the University of Bristol, as a Research Associate with the Centre for Communications Research. His current research interests are in the areas of tunable microwave circuits and amplifier linearization for software reconfigurable radios.

Paul A. Warr received the B.Eng. degree in electronics and communications from The University of Bath, Bath, U.K., in 1994, and the M.Sc. degree in communications systems and signal processing and Ph.D. degree from The University of Bristol, U.K., in 1996 and 2001, respectively. His doctoral research concerned octave-band linear receiver amplifiers. He is currently a Lecturer of radio frequency engineering with The University of Bristol. He was with the Marconi Company, where he was involved with secure high-redundancy cross-platform communications. His research concerns the front-end aspects of software (reconfigurable) radio and diversity-exploiting communication systems, responsive linear amplifiers, flexible filters, and linear frequency translation. His research has been funded by the U.K. Engineering and Physical Science Research Council (EPSRC) alongside the European Commission (EC) and industrial collaborators. Dr. Warr is a member of the Executive Committee of the Institution of Electrical Engineers (IEE) Professional Network on Communication Networks and Services.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3501

Design of Compensated Coupled-Stripline 3-dB Directional Couplers, Phase Shifters, and Magic-T’s—Part II: Broadband Coupled-Line Circuits Slawomir Gruszczynski, Krzysztof Wincza, and Krzysztof Sachse

Abstract—The problem of discontinuities in broadband multisection coupled-stripline 3-dB directional couplers, phase shifters, high-pass tapered-line 3-dB directional couplers, and magic-T’s, regarding the connections of coupled and terminating signal lines, is comprehensively investigated in this paper for the first time. The equivalent circuit of these discontinuities proposed in Part I has been used for accurate modeling of the broadband multisection and ultra-broadband high-pass coupled-stripline circuits. It has been shown that parasitic reactances, which result from the connections of signal and coupled lines, severely deteriorate the return losses and the isolation of such circuits and also—in case of tapered-line directional couplers—the coupling responses. Moreover, it has been proven theoretically and experimentally that these discontinuity effects can be substantially reduced by introducing compensating shunt capacitances in a number of cross sections of coupled and signal lines. Results of measurements carried out for various designed and manufactured coupled-line circuits have been very promising and have proven the efficiency of the proposed broadband compensation technique. The theoretical and measured data are given for the following coupled-stripline circuits: a decadebandwidth asymmetric three-section 3-dB directional coupler, a decade-bandwidth three-section phase-shifter compensator, and a high-pass asymmetric tapered-line 3-dB coupler. Index Terms—Asymmetric multisection coupled-line directional couplers, asymmetric tapered-line directional couplers, broadband magic-T’s, discontinuities, multisection coupled-line phase shifters and compensators, technique of capacitive compensation.

I. INTRODUCTION OUPLED-LINE 3-dB directional couplers, phase shifters, and magic-T’s are often used in microwave integrated circuits and large microwave networks and systems. For many broadband networks and systems, a one-octave bandwidth of the most simple quarter-wavelength single-section coupled-line circuits becomes insufficient. There are two groups of coupled-line circuits operating in wider frequency bands. In one, multioctave bandwidth can be achieved by means of cascading several quarter-wavelength sections of uniformly coupled lines [1]–[3]. Unfortunately, the physical junctions of various quarter-wavelength sections in the usually employed

C

Manuscript received January 20, 2006; revised May 25, 2006. This work was supported by the Wroclaw Division, Telecommunications Research Institute. The authors are with the Department of Electronics, Institute of Telecommunications, Teleinformatics and Acoustics, Wroclaw University of Technology, 50-370 Wroclaw, Poland (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.880649

TEM strip- and quasi-TEM microstrip conductor geometries contribute reactive discontinuities, which significantly degrade the circuits’ performance [4]–[6]. In another group, multioctave bandwidth can be achieved if several cascaded quarter-wavelength sections of uniformly coupled lines are replaced by a section of nonuniformly coupled lines, in which the coupling coefficient changes continuously along the whole section length [7]. Theoretically, in view of their high-pass response behavior, these tapered-line directional couplers and magic-T’s can operate in frequency bands significantly wider than these of the multisection coupled-line circuits. Although the methods of compensation of discontinuities in transmission lines within a wide frequency range have been reported [8], no methods of the multioctave coupled-strip- and microstripline circuit’s design that take under consideration parasitic reactances associated with the regions of connected coupled and signal lines are known. Such discontinuities may be the reason of deterioration of circuits’ frequency-dependent characteristics. Until now, the problem of discontinuities in the broadband coupled-stripline and microstrip line circuits has not been reported in the literature, neither design methods are known that would take these phenomena under consideration. We have found only one paper [9] describing a design technique for raising upper frequency limit of a wideband magic-T consisting of tandem-connected 8.34-dB tapered coupled-stripline directional couplers, where frequency-dependent characteristics have been considerably improved by introducing an air-gap section near the magic-T junction, compensating for parasitic fringing capacitances. In this paper, the problem of discontinuities in broadband multisection coupled-stripline 3-dB directional couplers, phase shifters, high-pass tapered-line 3-dB directional couplers, and magic-T’s, regarding the connections of coupled and terminating signal lines, is comprehensively investigated for the first time. The equivalent circuit of these discontinuities, proposed in Part I of this paper [10], the parameters of which can be computed in a process of fitting curves of the circuit and electromagnetic (EM) analyses, has been used for accurate modeling of the broadband multisection and ultra-broadband high-pass coupled-stripline circuits. It has been shown that parasitic reactances, which result from the connections of signal and coupled lines, severely deteriorate the return losses and the isolation of such circuits and also, in the case of tapered-line directional couplers and magic-T’s, the couplings at coupled

0018-9480/$20.00 © 2006 IEEE

3502

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

TABLE I STRIP WIDTHS AND STRIP OFFSETS OF THE COUPLED-LINE SECTIONS USED IN THE 3-dB DIRECTIONAL COUPLER

Fig. 1. Circuit models of the: (a) uncompensated and (b) compensated asymmetric three-section directional couplers.

and direct ports. Moreover, it has been proven theoretically and experimentally that these discontinuity effects can be substantially reduced by introducing compensating shunt capacitances in a number of cross sections of coupled and signal lines. Very promising results of measurements carried out for various designed and manufactured coupled-line circuits have proven the efficiency of the proposed broadband capacitive compensation technique. The theoretical and measured data are given for the following coupled-stripline circuits: 1) decade-bandwidth asymmetric three-section 3-dB directional coupler; 2) decade-bandwidth three-section phase-shifter compensator, which in conjunction with the developed asymmetric three-section 3-dB coupler can constitute a decade-bandwidth magic-T [11]–[13]; 3) high-pass asymmetric tapered-line 3-dB coupler, which in conjunction with two transmission-line sections can constitute a high-pass tapered-line magic-T [14]. Until now, the tapered-line 3-dB couplers and magic-T’s were only designed in a tandem connection of two tapered-line 8.34-dB couplers [9], [14]. The above-mentioned circuits are individually described in Sections II–IV. Conclusions are drawn in Section V.

II. ASYMMETRIC THREE-SECTION 3-dB DIRECTIONAL COUPLER Asymmetric multisection coupled-line directional couplers are often used circuits in microwave engineering because of their broadband characteristics. Parasitic reactances associated with the connection of signal and coupled lines, however, have a stronger deteriorative influence on the asymmetric coupler characteristics for a given coupling C than on these of a single-section coupled-line directional coupler with the same C. This is related to the fact that edge sections of coupled lines in asymmetric couplers from the side of tight coupling have significantly stronger coupling than in case of single-section couplers. Here, we show the possibility of compensation of parasitic reactances in a broadband asymmetric three-section 3-dB directional coupler. Fig. 1(a) presents a circuit model of the asymmetric three-section 3-dB directional coupler constituted by three sections of ideal coupled lines and equivalent circuits of the transition regions. For a given bandwidth, the even-mode characteristic impedances of coupled lines are as fol-

TABLE II COMPENSATING CAPACITANCES OBTAINED IN AN OPTIMIZATION PROCESS OF THE ASYMMETRIC THREE-SECTION 3-dB DIRECTIONAL COUPLER

lows: , , and the odd-mode characteristic impedances can be found from (1) is the characteristic impedance of terminating signal where lines. The coupler has been designed in an offset coupled stripline technique, such as in Part I of this paper. Coupled lines were etched on both sides of a thin laminate with the thickness m and the dielectric constant . This laminate was placed between two laminates with the thickness mm and the same dielectric constant . Strip widths and strip offsets of the coupled-line sections are given in Table I. The values of parasitic reactances have been calculated in a similar way as in Part I—by comparison of -parameters of the proposed equivalent circuit with -parameters of the transition region obtained from EM analysis. In the asymmetric coupler, the values of these reactances are different at each end of the coupler due to the differences between coupling coefficients and geometries of coupled-line sections on both ends. We have found the following values of elements representing the parasitic reactances: pF, nH, nH for the tightest coupled-line section and pF, nH, and nH for the section in which the coupling is the weakest. Fig. 1(b) presents a schematic diagram of the coupler with additional compensating capacitances. Values of the compensating capacitances obtained by optimization of the return losses and the isolation of the coupler are given in Table II. Calculated frequency-dependent characteristics of the couplers are shown in Fig. 2. Fig. 2(a) presents the calculated results of the uncompensated coupler. As we can see, parasitic reactances associated with the transition regions severely deteriorate the return losses and the isolation of the coupler. Fig. 2(b) shows the calculated frequency-dependent characteristics of the compensated coupler. One can see that the presented compensation technique allows for great improvement of the return loss and isolation characteristics.

GRUSZCZYNSKI et al.: DESIGN OF COMPENSATED COUPLED-STRIPLINE 3-dB DIRECTIONAL COUPLERS, PHASE SHIFTERS, AND MAGIC-T’s—PART II

3503

Fig. 2. Frequency characteristics of the: (a) uncompensated and (b) compensated couplers. Results of circuit analysis. Fig. 3. Measurement results of the: (a) uncompensated and (b) compensated 3-dB three-section directional couplers.

In the presented design, the improvement exceeds even 20 dB in a range of higher frequencies. The measurement results of both uncompensated and compensated couplers are shown in Fig. 3(a) and (b), respectively. A significant improvement has been achieved. The return loss, as well as isolation characteristics of the compensated coupler, do not exceed 25 dB, whereas those of the uncompensated one are barely below 18 dB. Fig. 4 shows a photograph of the compensated coupler in which additional capacitive elements connected to both signal and coupled lines are visible. The number and location of the compensating capacitances differ from what was shown by the circuit analysis and have been taken from the EM optimization process. Results of circuit analysis, however, can serve as a good starting point to EM analysis. III. THREE-SECTION PHASE-SHIFTER COMPENSATOR AND ITS CONJUNCTION WITH ASYMMETRIC THREE-SECTION 3-dB COUPLER A similar technique of compensation can be applied in case of multisection coupled-line fixed phase shifters [2] and phase compensators, which, in conjunction with asymmetric multisection 3-dB couplers, can constitute broadband magic-T networks. As an example, let us consider a three-section phase compensator with a decade bandwidth for which the even-mode impedances of the coupled-line sections have been found in [13] to be the following: , and . This phase shifter was designed in the

Fig. 4. Coupler’s strip patterns etched on both sides of a thin laminate.

TABLE III STRIP WIDTHS AND STRIP OFFSETS OF COUPLED-LINE SECTIONS USED IN THE PHASE SHIFTER

offset coupled-stripline structure described in Section II. Dimensions of the coupled-line sections are listed in Table III. A schematic diagram of the phase shifter with parasitic reactances and compensating capacitances is shown in Fig. 5. The following values of the elements representing parasitic reactances have been found: pF, nH, nH, and the values of compensating capacitances are listed in Table IV. The calculated return-loss characteristics of the uncompensated and compensated phase shifters are shown in Fig. 6. The improved return-loss characteristic is considerably below the

3504

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 5. Circuit model of the compensated three-section phase shifter.

TABLE IV COMPENSATING CAPACITANCES OBTAINED IN AN OPTIMIZATION PROCESS OF THE THREE-SECTION PHASE SHIFTER

Fig. 7. Measurement results of the: (a) uncompensated and (b) compensated three-section phase shifters. Return and insertion losses.

Fig. 6. Results of circuit analysis of the three-section phase shifter modeled by the ideal coupled-line sections and equivalent circuits of the transition regions. Return and insertion losses.

original one. Measured return-loss characteristics of the uncompensated and compensated phase shifters are shown in Fig. 7(a) and (b), respectively. These results prove the possibility of compensation of coupled-line phase shifters using the proposed technique. Fig. 8 shows a photograph of the compensated three-section phase shifter. As in the case of an asymmetric three-section 3-dB coupler, the number and location of compensating capacitances have been optimized electromagnetically. It is worth mentioning that the influence of the proposed capacitive compensation technique on the phase shifters’ response is negligible. Fig. 9 shows the simulated and measured sum and difference characteristics of the magic-T network consisting of the asymmetric 3-dB coupler described in Section II and two identical phase compensators described above. IV. ASYMMETRIC TAPERED-LINE 3-dB DIRECTIONAL COUPLER AS A MAGIC-T In asymmetric tapered coupled-line couplers, the coupling coefficient, starting from a high value, changes continuously along the coupled lines down to zero. From all of the couplers with a given coupling C, the asymmetric tapered-line couplers

Fig. 8. Phase shifter’s strip patterns etched on both sides of a thin laminate.

6

1

Fig. 9. Sum and difference characteristics of a magic-T being a connection of the asymmetric three-section coupled-line 3-dB coupler and two three-section coupled-line phase compensators.

require the strongest coupling. Consequently, the transition regions in which the signal lines and coupled lines are connected introduce parasitic reactances having greater impact on the frequency-dependent circuit’s characteristics than in case of other

GRUSZCZYNSKI et al.: DESIGN OF COMPENSATED COUPLED-STRIPLINE 3-dB DIRECTIONAL COUPLERS, PHASE SHIFTERS, AND MAGIC-T’s—PART II

3505

TABLE V COUPLING COEFFICIENTS, EVEN- AND ODD-MODE IMPEDANCES, COMPENSATING CAPACITANCES, COUPLED-LINE SECTION STRIP WIDTHS AND OFFSETS OF THE ASYMMETRIC TAPERED-LINE 3.3-dB DIRECTIONAL COUPLER

types of couplers with the same coupling C. To illustrate this problem, a design of asymmetric tapered-line 3.3-dB directional coupler1 is briefly presented. Such a coupler has the properties of a magic-T when 50- transmission lines of appropriate length are added. This directional coupler was designed in the coupled-line structure described in Section II. For the analysis we used the method given in [7] and analyzed this coupler as a cascade of 35 sections of coupled transmission lines. For the purpose of the coupler’s design, the following dependence of the normalized even-mode impedance of the tapered-coupled lines constituting the coupler versus normalized distance has been chosen:

(2) where is the distance measured in the direction of weak coupling and is the coupler’s overall length. 1This is our experience, that the coupling to direct port in a directional coupler with losses, especially in a tapered-line long length loss coupler, is weaker than in case of the same coupler without losses. This behavior can decrease the assumed unbalance between the two output ports and it is the reason that we assumed 3.3-dB coupling instead of 3.01-dB.

Fig. 10. Circuit model of the tapered-line directional coupler with parasitic reactances and compensating capacitances.

Formula (2) is a new one and has been obtained in an optimization process aimed at minimizing the designed 3.3-dB coupler’s cutoff frequency for both amplitude and phase characteristics in a way that would ensure the widest bandwidth of the magic-T. The coupling coefficients, even- and odd-mode characteristic impedances, strip widths, and offsets of the coupled-line sections are listed in Table V. A schematic diagram of the tapered-line coupler with the parasitic reactances and compensating capacitances is shown in Fig. 10, whereas Fig. 11 presents the calculated amplitude and phase characteristics of the coupler with additional 50- transmission lines. For the analyzed coupler, the following values of elements representing parasitic reactances have been found: pF, nH, nH.

3506

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 13. Measurement results of the compensated tapered-line directional coupler.

Fig. 11. Frequency characteristics of the ideal 3.3-dB tapered-line directional coupler. Coupling coefficient distribution is given by (2). (a) Amplitude characteristics. (b) Phase characteristic.

6

1

Fig. 14. Sum and difference characteristics of the magic-T network being a connection of the 3.3-dB tapered-line directional coupler and transmissionline sections.

Fig. 15. Layout of the tapered-line coupler with compensating capacitances.

the magic-T are shown in Fig. 14. Layout of the compensated coupler is presented in Fig. 15. As in all previous cases, number and location of compensating capacitances have been optimized electromagnetically; at all frequencies, the return losses and the isolation of the coupler were better than 28 dB. Fig. 12. Calculated results of the tapered-line directional coupler together with parasitic reactances obtained from circuit analysis.

V. CONCLUSIONS

Calculated characteristics of the ideal tapered-line coupler with parasitic reactances are shown in Fig. 12 and, as in all previous cases, the return loss and isolation deteriorate gradually with the increase of frequency. To improve the performance of this coupler, compensating capacitances have been added (values are listed in Table V) and the calculation results of the compensated coupler are also shown in Fig. 12. The measurement results of the compensated tapered-line coupler are shown in Fig. 13 and the sum and difference characteristics of

In this paper, we have shown a significant influence of parasitic reactances associated with the connection of signal and coupled lines on the frequency-dependent characteristics of broadband coupled-line directional couplers and phase shifters. These circuits suffer from poor return losses and poor isolations, becoming unattractive for application in high-performance microwave networks and systems. As a solution to this problem, we have proposed a compensation technique that allows us to significantly improve the performance of the designed broadband directional couplers, phase shifters, and magic-T’s. The method is an extension of that presented in

GRUSZCZYNSKI et al.: DESIGN OF COMPENSATED COUPLED-STRIPLINE 3-dB DIRECTIONAL COUPLERS, PHASE SHIFTERS, AND MAGIC-T’s—PART II

Part I and is based on introducing compensating capacitive elements in a number of coupled lines’ and signal lines’ cross sections. Theoretical analyses have shown the possibility of return loss and isolation improvement in all presented examples; in particular, in case of the asymmetric tapered-line 3-dB directional coupler, the return losses and isolation can be improved even by 25 dB in a wide range of frequencies. The measurement results of compensated circuits are in good agreement with the calculated ones and confirm the usefulness of the proposed capacitive compensation technique. In case of the asymmetric three-section 3-dB directional coupler, the improvement of return loss and isolation has been achieved in the whole frequency band of 0.45–4.5 GHz . Until now, the tapered-line 3-dB couplers and magic-T’s have only been designed as a tandem connection of two asymmetric tapered-line 8.34-dB couplers. We trust that our attempt to design high-performance compensated coupled-stripline circuits will also turn out to be useful in designing high-performance compensated coupled-microstripline circuits. ACKNOWLEDGMENT The authors would like to thank S. Gulinska and D. Skrzypek, both with the Wroclaw Division, Telecommunications Research Institute, Wroclaw, Poland, for manufacturing and assembling all experimental models. REFERENCES [1] E. G. Cristal and L. Young, “Theory and tables of optimum symmetrical TEM-mode coupled-transmission-line directional couplers,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 9, pp. 544–558, Sep. 1965. [2] J. P. Shelton and J. A. Mosko, “Synthesis and design of wideband equal ripple TEM directional couplers and fixed phase shifters,” IEEE Trans. Microw. Theory Tech., vol. MTT-14, no. 10, pp. 462–473, Oct. 1966. [3] K. Sachse, A. Sawicki, and G. Jaworski, “Novel, multilayer coupledline structures and their circuit applications,” in Proc. MIKON 13th Int. Microw., Radar, Wireless Commun. Conf., Wroclaw, Poland, May 2000, pp. 131–155, pt. 3 (invited paper). [4] M. Nakajima, E. Yamashita, and M. Asa, “New broadband 5-section microstrip-line directional coupler,” in IEEE MTT-S Int. Microw. Symp. Dig., 1990, pp. 383–386. [5] C. Person, J. P. Coupez, S. Toutain, and M. Morvan, “Wideband 3-dB/90 coupler in multilayer thick-film technology,” Electron. Lett., vol. 31, no. 10, pp. 812–813, May 1995. [6] C. Person, L. Carre, E. Rius, J. P. Coupez, and S. Toutain, “Original techniques for designing wideband 3-D integrated couplers,” in IEEE MTT-S Int. Microw. Symp. Dig., 1998, pp. 119–122. [7] C. P. Tresselt, “Design and computed theoretical performance of three classes of equal-ripple nonuniform line couplers,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 4, pp. 218–230, Apr. 1969. [8] B. M. Kats, V. P. Meschanov, and A. L. Khvalin, “Synthesis of superwide-band matching adapters in round coaxial lines,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 575–579, Mar. 2001. [9] M. Nakajima and H. Tanabe, “A design technique for raising upper frequency limit of wideband 180 hybrids,” in IEEE MTT-S Int. Microw. Symp. Dig., 1996, pp. 879–882. [10] S. Gruszczynski, K. Wincza, and K. Sachse, “Design of compensated coupled-stripline 3-dB directional couplers, phase shifters, and magic T’s—Part I: Single-section coupled-line circuits,” IEEE Trans. Microw. Theory Tech., submitted for publication. [11] D. J. Kraker, “Asymmetric coupled-transmission-line magic-T,” IEEE Trans. Microw. Theory Tech., vol. MTT-12, no. 11, pp. 595–599, Nov. 1964. [12] S. Gruszczynski and K. Sachse, “A coupled-transmission-line multisection asymmetric 3 dB directional coupler in conjunction with phase compensation networks as a broadband 3 dB=0 =180 coupler and a sum-difference circuit,” in Proc. Mediterranean Microw. Symp., Caceres, Spain, Jun. 2002, pp. CP:1–CP:4. [13] S. Gruszczynski, “Wide band asymmetric coupled-transmission-line magic-T and sum-difference circuit,” in Proc. MIKON 15th Int. Microw., Radar, Wireless Commun. Conf., Warsaw, Poland, May 2004, pp. 162–165.

3507

[14] R. H. DuHamel and M. E. Armstrong, “The tapered-line magic-T,” in Parallel Coupled Lines and Directional Couplers, L. Young, Ed. Dedham, MA: Artech House, 1972, pp. 207–233.

Slawomir Gruszczynski was born in Wroclaw, Poland, on December 14, 1976. He received the M.Sc. degree in electronics and telecommunications and Ph.D. degree in electronics and electrical engineering from Wroclaw University of Technology, Wroclaw, Poland, in 2001 and 2006, respectively. From 2001 to 2006, he has been with the Wroclaw Division, Telecommunications Research Institute, where he was involved in numerous projects for military applications. In 2005, he joined the Institute of Telecommunications and Acoustics, Wroclaw University of Technology, becoming an Assistant Professor in 2006. He is currently a Principal Researcher involved in the CRAFT-016927 Project within the European Union Sixth Framework Programme. He has coauthored 17 scientific papers, including journal, European Microwave Conference, and IEEE Antennas and Propagation Symposium papers. His research interests include passive ultra-broadband microwave circuits such as directional couplers, power dividers, magic-T networks and also multibeam antennas, Butler matrices, and integrated planar and conformal antenna arrays.

Krzysztof Wincza was born in Walbrzych, Poland, on May 27, 1979. He received the M.Sc. degree in electronics and telecommunications from the Wroclaw University of Technology, Poland, in 2003, and is currently working towards the Ph.D. degree at Wroclaw University of Technology. He is currently a Principal Researcher involved in the CRAFT-016927 Project within the European Union Sixth Framework Programme. He is an expert on the European Union COST 284 Project: Innovative Antennas for Emerging Terrestrial and Space-based Applications. He has coauthored 17 scientific papers, including journal, European Microwave Conference, and IEEE Antennas and Propagation Symposium papers. His scientific interests include multibeam antennas, Butler matrices, lightweight and highly integrated antenna arrays, conformal antennas, spaceborne antennas, reconfigurable arrays, and satellite communication. Mr. Wincza was the recipient of The Youth Award presented at the 10th National Symposium of Radio Sciences (URSI) and the Third-Place Award for the best M.Sc. thesis in the field of microwaves, antenna, and radar technology (awarded by the IEEE Polish Chapter) in 2001 and 2004, respectively.

Krzysztof Sachse was born in Cracow, Poland, on January 10, 1942. He received the M.Sc. degree in telecommunications from the Gdansk University of Technology, Gdansk, Poland, in 1965, and the Ph.D. degree in electronics and electrical engineering and Doctor of Sciences degree from the Wroclaw University of Technology, Wroclaw, Poland, in 1974 and 1991, respectively. From 1965 to 1967, he was a Research Fellow with the Electronics and Electrical Engineering Department, Gdansk University of Technology. In 1967, he joined the Institute of Telecommunications and Acoustics, Wroclaw University of Technology, becoming an Assistant Professor in 1974. Since 1996, he has been a Full Professor. For eight months in 1979, he was a Visiting Scholar with the Microwave and Semiconductor Laboratory, Lille Technical University, Lille, France. He currently lectures on microwave engineering. Since 1988, he has been a Chair of the Microwave Theory and Technique Division, Institute of Telecommunications and Acoustics. His field of research is the solution of EM boundary problems for microstrip-like transmission lines and waveguides, passive microwave integrated-circuit design, and design of microwave feed antenna systems and components. His current research is related to a design of planar antenna filters for modern communication systems, as well as of beam-forming networks for phased-array antennas. Dr. Sachse is a member of the Polish Society of Electrical and Electronic Engineering.

3508

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Compact Planar Microstrip Branch-Line Couplers Using the Quasi-Lumped Elements Approach With Nonsymmetrical and Symmetrical T-Shaped Structure Shry-Sann Liao, Member, IEEE, and Jen-Ti Peng

Abstract—A class of the novel compact-size branch-line couplers using the quasi-lumped elements approach with symmetrical or nonsymmetrical T-shaped structures is proposed in this paper. The design equations have been derived, and two circuits using the quasi-lumped elements approach were realized for physical measurements. This novel design occupies only 29% of the area of the conventional approach at 2.4 GHz. In addition, a third circuit was designed by using the same formula implementing a symmetrical T-shaped structure and occupied both the internal and external area of the coupler. This coupler achieved 500-MHz bandwidth while the phase difference between 21 and 31 is 1 . Thus, the bandwidth is not only 25% wider within 90 than that of the conventional coupler, but occupies only 70% of the circuit area compared to the conventional design. All three proposed couplers can be implemented by using the standard printed-circuit-board etching processes without any implementation of lumped elements, bonding wires, and via-holes, making it very useful for wireless communication systems. Index Terms—Branch-line coupler, compact size, nonsymmetrical T-shaped structure, quasi-lumped elements, symmetrical T-shaped structure.

I. INTRODUCTION TRADITIONAL branch-line coupler is composed of four quarter-wavelength transmission-line sections at a designated frequency. It has been used extensively in the design of balanced mixers, image-rejection mixers, antenna array feed networks, balanced amplifiers, power combiners, and power dividers. However, at the lower frequencies of the microwave band, the size of a conventional branch-line coupler is too large for practical use. It is also too large for monolithic-microwave integrated-circuit (MMIC) applications, as a larger dimension results in higher chip cost. Due to recent market demands, multifunctional, small-sized, and inexpensive wireless communication devices, circuit miniaturization and low-cost fabrication have become major design considerations that need to be taken into account. Several design techniques have been reported to reduce the coupler size. The complete lumped-element approach [1]–[7] provides a significant size reduction and is more suitable for

A

Manuscript received January 19, 2006; revised April 18, 2006. This work was supported by the National Science Council of Taiwan under Project NSC 93-2213-E-035-025. The authors are with the Department of Communication Engineering, Feng Chia University, Taichung, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.880650

MMIC applications. This technique needs precise lumped-element models based on careful measurements. Another approach considers the combinations of shunt lumped capacitors and short high-impedance transmission lines [8]–[11]. In those cases, metal–insulator–metal capacitors are needed for the MMICs, which increase cost and complexity of fabrication. Other methods using only a distributed element have been reported. For example, meandering of transmission lines through folding and bending has been used to realize the couplers [12], [13], and has proven very effective for low-frequency applications. However, at higher frequencies, these methods will increase the effect of parasitic coupling between lines, making the design impractical. Photonic-bandgap (PBG) structures are another way to miniaturize the circuits [14]–[16]. However, the existence of many defected cells on the ground plane may limit the use of this technique. Compensated spiral compact microstrip resonant cells (SCMRCs) show a similar performance to that of PBG. Although a good size reduction and a harmonic suppression can be achieved by using SCMRC to design a branch-line coupler, this technique leads to less isolation [26]. Recently, [17] suggested a branch-line coupler design using only microstrip lines. A compact size was achieved by adding artificial transmission lines, which consisted of microstrip lines periodically loaded with open shunt stubs. Unfortunately, the size of this coupler is only 63% compared to a conventional design. This is due to the use of the unoccupied area in the couplers interior and reveals an overlapping problem, as shown in Fig. 1(a), which constrains the size-reduction process. Considering the symmetrical structure, the traditional design method of the branch-line coupler is simulated based on the normal-mode analysis with both even- and odd-mode analysis techniques [18]. Moreover, a transmission line shorter than the quarter-wavelength results in a lower inductance and a lower capacitance. The loss of series inductance can be compensated by increasing the characteristic impedance of the line or adding the distributed inductive loading. The loss of shunt capacitance can be compensated by adding lumped capacitors or capacitive loading [7], [8], [19]–[21]. The conventional distributed capacitive loading is the open-end stub line [17], [22], [23]. In the proposed compact-sized branch-line coupler design, the length of transmission lines is reduced by using open-end stub lines. The overlapping between the stubs must be resolved in order to further reduce the size of couplers. In general, three methods can be implemented in order to avoid the overlapping,

0018-9480/$20.00 © 2006 IEEE

LIAO AND PENG: COMPACT PLANAR MICROSTRIP BRANCH-LINE COUPLERS USING QUASI-LUMPED ELEMENTS APPROACH

Fig. 1. Various topologies of using unoccupied interior area of the couplers to reduce circuit size. (a) Overlapping to limit size reduction. (b) Moving the symmetrical T-shaped structure to the exterior of the coupler. (c) Using quasilumped elements approach to reduce size further with symmetrical T-shaped structure. (d) Shifting the stub from the middle to near the end of the lines in the nonsymmetrical T-shaped structure and quasi-lumped elements approach. Not all these structures are analyzed in this paper, but the approach is similar for all of them. (Color version available online at http://ieeexplore.ieee.org.)

3509

Fig. 3. Designed curves. (a)  versus  if M is given. (b)  versus K if  and M are known.

Fig. 2. Equivalent T-shaped structure of quarter-wavelength transmission line.

which are: 1) moving the symmetrical T-shaped structure to the exterior of the coupler, as shown in Fig. 1(b), resulting in a wider bandwidth but less compactness; 2) keeping the locations of the stubs at the middle, but miniaturizing the size again with the quasi-lumped elements approach, as shown in Fig. 1(c); and 3) shifting the location of the stubs from the middle of the lines to near the end of the reduced lines, as shown in Fig. 1(d). This quasi-lumped elements approach will further decrease the size. To illustrate these ideas, three new branch-line couplers were designed and fabricated using the planar microstrip-line technology. The design descriptions of the compact-size branchline coupler are first discussed in Section II, followed by the implementation of the couplers in Section III. These couplers without any lumped elements, bonding wires, and via-holes can be easily fabricated on an FR4 substrate. The circuit areas of

Fig. 4. (a) Stub- line. (b) Equivalent circuit of (a). (c) Structure of the further reduced line. (d) Equivalent circuit of (c).

these couplers are smaller with good performances. These performances are comparable to those of a conventional design at 2.4 GHz. In addition, good agreement between the measured and simulated results were observed.

3510

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 5. Design curves of Z versus  for given  , Z , and  .

II. DESIGN DESCRIPTIONS The equivalent T-shaped structure of the reduced line is shown in Fig. 2, where , , , , , and represent the characteristic impedances and the electrical lengths of the reduced line, respectively. is the characteristic impedance of the quarter-wavelength transmission line. The relations between the quarter-wavelength transmission line and reduced line can be estimated using matrices. The design equations are [24] (1) (2) with , , and . Equations (1) and (2) determine , , , , , and . For simplicity, is assumed. The electrical length is plotted against the electrical length for different values of , as shown in Fig. 3(a). It appears that the total electrical length of

the reduced line decreases as increases. Moreover, the locations of all the open-end stub lines are constrained within the interior of the branch-line coupler in our scheme, thus a value of between and is possible. However, while (1) stays unchanged with , and are interchanged if is given, hinting at the existence of a second solution, i.e., type II [24]. and can be calculated when and are known. Fig. 3(b) shows the designed curves of and if and are given in the case of . The stub can be further reduced with the quasi-lumped-elements approach. The structures of the original and the further reduced stub lines, as well as their corresponding equivalent circuits, are shown in Fig. 4(a)–(d). By applying a matrix formulation, the -parameters of the proposed circuit can be derived as

(3)

LIAO AND PENG: COMPACT PLANAR MICROSTRIP BRANCH-LINE COUPLERS USING QUASI-LUMPED ELEMENTS APPROACH

3511

The values of and in the T-equivalent circuit, shown in Fig. 4(b), are obtained as follows [25]: (4) (5) and the shortened electrical lengths of the inductor sections and the capacitor sections are (6) (7) respectively. , and represent the characteristic impedances and the electrical lengths of the further reduced line, respectively, and are shown in Fig. 4(c). Their corresponding equivalent circuits are shown in Fig. 4(d). To simplify the calculation and analysis process without losing generality, the higher order terms of were omitted, under the condition of . This results in

(8) Assuming the value used for each inductor and capacitor are equal, then (9) (10) From (9) and (10), and (4)–(7), (8) can be rewritten as (11) is plotted against From (11), the characteristic impedance the electrical length for the different values of when and are given and shown in Fig. 5(a)–(d), where the subscript and denote the main line and branch line, respectively. However, the total electrical length of the further reduced line must be less than , which sets an additional constraint to the value of . To minimize the effects of conductor loss, radiation loss, and prevention of spurious modes, the width of the microstrip line is limited, thus limiting the range of characteristic impedances that can be implemented. In this study where FR4 was used, the width of the microstrip line was limited to 0.2 mm and the corresponding characteristic impedance was approximately 113 . III. DESIGN AND IMPLEMENTATION OF COMPACT-SIZE COUPLERS For verification, a class of compact-size microstrip branchline couplers operating at 2.4 GHz was designed, simulated, and fabricated. The parameters were calculated following the design description discussed in Section II. A full-wave Sonnet em simulator was used for all simulations, and the couplers were constructed using an FR4 substrate with , , thickness mm, and metal thickness mm.

Fig. 6. Symmetrical T-shaped structure: W = 4:4 mm, L = 5:2 mm, = 0:6 mm, L = 3:3 mm, W = 0:3 mm, W = 4:4 mm, L = 3:6, W = 1:4 mm, and L = 3:0 mm. L = 3:9 mm, W

A. Symmetrical T-Shaped Structure of a Branch-Line Coupler With a Partially Occupied Internal Region The characteristic impedances of the conventional branch-line coupler are 35 and 50 . The parameters with of the reduced main lines were first selected from Fig. 3(a). Substituting all these data into (2) and assuming , then was obtained. Similarly, the parameters of the reduced branch line were obtained , , , and . Due to the layout of the enclosed circuits, selection of the line impedances, ports matching, and fabricated tolerances were slightly adjusted as necessary in the last step. The final circuit is shown in Fig. 6. The simulated and measured results of the -parameters and the phase response are shown in Fig. 7(a) and (b), respectively. This coupler reveals a bandwidth of approximately 500 MHz, while thephase difference between and are within 90 1 . Thus, its bandwidth is 25% wider than that of a conventional coupler, and it occupies 70% of the circuit area of the conventional design. B. Symmetrical T-Shaped Structure With Quasi-Lumped Elements Approach The quasi-lumped elements were implemented to further reduce size and to avoid overlapping. Substituting , , and the value of in (11), the parameters for the reduced line were then obtained from Fig. 5(a) and (b), respectively. The parameters for the reduced lines and were obtained from Fig. 5(a) and (b), respectively. i.e., in the case of , let , then , ; in the case of , let , then and . The final circuit after the slight adjustment is shown in Fig. 8. The simulated and measured results of the -parameters and the phase response are shown in Fig. 9(a) and (b), respectively. The circuit occupies only 29% of the circuit area of the conventional design. C. Nonsymmetrical T-Shaped Structure With Quasi-Lumped Elements Approach The size of the coupler that is shown in Fig. 6 can be further reduced by using the nonsymmetrical T-shaped structure with the quasi-lumped elements approach. To avoid any overlap, the

3512

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 7. Simulated and measured results of Fig. 6. (a) S -parameters. (b) Phase response. Fig. 9. Simulated and measured results of Fig. 8. (a) S -parameters. (b) Phase response.

Fig. 8. Symmetrical T-shaped structure and quasi-lumped elements approach: W = 0:6 mm, L = 9:0 mm, W = 0:3 mm, L = 11:0 mm, W = 1:4 mm, L = 3:0 mm, W = 0:6 mm, L = 4:6 mm, W = 0:2 mm, L = 0:5 mm, W = 0:9 mm, L = 7:6 mm, W = 0:2 mm, and L = 0:5 mm.

locations of the open-end stub line are placed as close to the end of the reduced line as possible, as shown in Fig. 1(d). With the same process previously described, and assuming and , the parameters for the reduced lines and are obtained from Fig. 5(c)–(d), i.e., in the case of ,

Fig. 10. Nonsymmetrical T-shaped structure and quasi-lumped elements approach: W = 0:6 mm, L = 9:2 mm, W = 0:3 mm, L = 10:7 mm, W = 1:4 mm, L = 3:0 mm, W = 4:6 mm, L = 0:6 mm, W = 0:2 mm, L = 0:4 mm, W = 5:9 mm, L = 0:3 mm, W = 0:2 mm, L = 0:4 mm, G = 0:5 mm, G = 0:3 mm, G = 0:2 mm, and G = 0:2 mm.

let

, then , let

and , then

and

. In the case of .

LIAO AND PENG: COMPACT PLANAR MICROSTRIP BRANCH-LINE COUPLERS USING QUASI-LUMPED ELEMENTS APPROACH

3513

IV. CONCLUSION The design and implementation of a class of planar compact branch-line coupler has been described and verified. The branch-line couplers operating at 2.4 GHz consisted of only distributed components without bonding wires and via-holes. The design formulas have been discussed under ideal lossless situations. In the quasi-lumped elements approach, the size of the prototype couplers has been reduced to 29% in the circuit area compared to the conventional design. The circuit area of the symmetrical T-shaped structure was approximately 70% that of a conventional design, but bandwidth was increased by 25%. These couplers can be easily fabricated using standard printed circuit board (PCB) etching processes, making this novel design suitable for microwave integrated circuit (MIC) and MMIC applications. ACKNOWLEDGMENT The authors would like to thank the reviewers for their helpful comments and suggestions. REFERENCES [1] M. Caulton, B. Hershenov, S. P. Knight, and R. E. Debrecht, “Status of lumped elements in microwave integrated circuits—Present and future,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, no. 7, pp. 588–599, Jul. 1971. [2] R. W. Vogel, “Analysis and design of lumped- and lumped- distributedelement directional couplers for MIC and MMIC applications,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 2, pp. 253–262, Feb. 1992. [3] J. Hongerheiden, M. Ciminera, and G. Jue, “Improved planar spiral transformer theory applied to a miniature lumped element quadrature hybrid,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 4, pp. 543–545, Apr. 1997. [4] Pieters, Vaesen, Brebels, S. Mahmoud, De Raedt, Beyne, and Mertens, “Accurate modeling of high- spiral inductors in thin film multilayer technology for wireless telecommunication applications,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 4, pp. 589–599, Apr. 2001. [5] Y. C. Chiang and C. Y. Chen, “Design of a wideband lumped-element 3-dB quadrature coupler,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 476–479, Mar. 2001. [6] I. Sakagami, K. Sagaguti, M. Fujii, M. Tahara, and Y. Hao, “On a lumped element three-branch 3-dB coupler with Butterworth and Chebyshev characteristic,” in IEEE Int. Midwest Circuits Syst. Symp., 2004, pp. III-21–III-24. [7] T. Hirota, A. Minakaw, and M. Muraguchi, “Reduced-size branchline and rat-race hybrids for uniplanar MMIC’s,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 3, pp. 270–275, Mar. 1990. [8] R. B. Singh and T. M. Weller, “Miniaturized 20 GHz CPW quadrature coupler using capacitive loading,” Microw. Opt. Technol. Lett., vol. 30, no. 1, pp. 3–5, Jul. 2001. [9] M. C. Scardelletti, G. E. Ponchak, and T. M. Weller, “Miniaturized Wilkinson power dividers utilizing capacitive loading,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 1, pp. 6–8, Jan. 2002. [10] W. S. Tung, H. H. Wu, and Y. C. Chiang, “Design of microwave wideband quadrature hybrid using planar transformer coupling method,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp. 1852–1856, Jul. 2003. [11] G. E. Ponchak, “Experimental analysis of reduced-sized coplanar waveguide transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 971–974. [12] S. S. Liao, J. T. Peng, and C. Y. Gun, “A super compact planar microstripline rat-race coupler,” Microw. Opt. Technol. Lett., vol. 47, no. 3, pp. 302–304, Nov. 2005. [13] E. G. Cristal, “Meander-line and hybrid meander-line transformers,” IEEE Trans. Microw. Theory Tech., vol. MTT-21, no. 2, pp. 69–76, Feb. 1973. [14] A. Saib, R. Platteborze, and I. Huynen, “Experimental demonstration of the origin photonic bandgap creation and associated defected modes in microwave planar circuits,” Microw. Opt. Technol. Lett., vol. 41, no. 1, pp. 5–9, Apr. 2004.

Q

S

Fig. 11. Simulated and measured results of Fig. 10. (a) -parameters. (b) Phase response.

TABLE I PERFORMANCE COMPARISON OF COUPLERS

The final circuit after the slight adjustment is shown in Fig. 10. The simulated and measured results of the -parameters and the phase response are shown in Fig. 11(a) and (b), respectively. The circuit occupies approximately 29% of the circuit area of the conventional design. The simulated and measured results of the direct and coupling power, the high losses by the metal structure, and the radiation loss did not reveal a significant influence of the performance in the frequencies range of interest. The comparison of the performance of these couplers with a conventional coupler is shown in Table I. Cases where or will be reported in a future study.

3514

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

[15] F. R. Yang, K. P. Ma, and T. Itoh, “A uniplanar compact photonicbandgap structure and its applications for microwave circuit,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1509–1514, Aug. 1999. [16] Q. Xue, K. M. Shum, and C. H. Chan, “Novel 1-D microstrip PBG cells,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 403–405, Oct. 2000. [17] K. W. Eccleston and S. M. Ong, “Compact planar microstripline branch-line and rat-race couplers,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 10, pp. 2119–2125, Oct. 2003. [18] J. Reed and G. J. Wheeler, “A method of analysis of symmetrical fourport networks,” IRE Trans. Microw. Theory Tech., vol. MTT-4, no. 10, pp. 246–252, Oct. 1956. [19] K. Hettak, C. J. Verver, M. G. Stubbs, and G. A. Morin, “A novel compact uniplanar MMIC Wilkinson power divider with ACPS series stubs,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, vol. 1, pp. 59–62. [20] 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. [21] J. S. Hong and M. J. Lancaster, “Capacitively loaded microstrip loop resonator,” Electron. Lett., vol. 30, no. 8, pp. 1494–1495, Sep. 1994. [22] M. Gillick, I. D. Robertson, and J. S. Joshi, “Design and realization of reduced-size impedance transforming uniplanar MMIC branch-line coupler,” Electron. Lett., vol. 28, pp. 1555–1557, Sep. 1992. [23] K. Hettak and G. Y. Delistle, “A new miniature uniplanar lowpass filter using series resonators,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1998, pp. 1193–1196. [24] S. S. Liao, P. T. Sun, N. C. Chin, and J. T. Peng, “A novel compact-size branch-line coupler,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 9, pp. 588–590, Sep. 2005. [25] D. M. Polar, Microwave Engineering, 3rd ed. New York: Wiley, 2005, ch. 8. [26] J. Gu and X. Sun, “Miniaturization and harmonic suppression of branch-line and rat-race hybrid coupler using compensating spiral compact microstrip resonant cell,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1211–1214.

Shry-Sann Liao (M’02) was born in Taiwan, R.O.C., in 1954. He received the B.S. and M.S. degrees in physics from Tamkang University, Tamshui, R.O.C., in 1976 and 1978, respectively, and the Ph.D. degree in physics from the University of Texas at Dallas, in 1988. From 1979 to 1983, he was a Nuclear Engineer with the Taiwan Power Company, Taiwan, R.O.C., where he was responsible for providing education and training to engineers who had worked in nuclear power plants. Since 1988, he had been with the Department of Electronic Engineering, Feng Chia University (FCU), Taichung, Taiwan, R.O.C., as an Associate Professor. He has taught at various levels and has developed specialized courses in RF circuit design at both the undergraduate and graduate levels. In 2002, he joined the Department of Communication Engineering, FCU. His research is currently focused on microwave circuits design.

Jen-Ti Peng was born in Taipei, Taiwan, R.O.C., in October 1980. He received the B.S. degree in electronic engineering from Feng Chia University (FCU), Taichung, Taiwan, R.O.C., in 2004, and is currently working toward the M.S. degree in communication engineering at FCU. His research interests concerns RF/microwave active and passive circuits design.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3515

A New Look at the Practical Design of Compact Diplexers Antonio Morini, Member, IEEE, Tullio Rozzi, Fellow, IEEE, Marco Farina, Member, IEEE, and Giuseppe Venanzoni

Abstract—This paper presents a method for the design of compact diplexers, where the common junction is considered part of the filters, thus reducing the diplexer length by roughly one cavity. Starting from two filter prototypes satisfying the desired specifications, the technique provides the design equations in closed form. Effectiveness and accuracy of the proposed method are demonstrated by means of a practical example. Index Terms—Diplexer, filter, prototype.

I. INTRODUCTION HE SYNTHESIS of diplexers classically starts from two filters and a three-port junction that are separately designed and combined in such a way as to maximize performance. Unfortunately, simply connecting filters and junction produces an unacceptable deterioration of the response of the two filters in their passbands, which can be restored by modifying the first elements of each filter, as indicated in [1]–[3]. Alternatively, some years ago, we introduced a simple analytical formula, providing the line lengths at which two filters have to be connected to an arbitrary three-port junction, in order to guarantee an optimal design [4], [5]. This approach featured a clear segmentation of the problem and the provision of simple formulas for the combination of filters and junction. Although the performances of the original filters were not completely preserved in the final diplexer, the response of the channels in the resulting device were very satisfactory, at least in the noncontiguous case and for junctions endowed with some properties. Moreover, in practical cases, the same screws used to tune the cavities could be used to easily compensate for the resulting small deviation. As a matter of fact, the method ought to be seen as providing a near-optimal starting point from where to apply some efficient optimization. There is, however, a further aspect that impacts on maximum achievable performance. In fact, in the above approach, two line sections are interposed between the first coupling elements of the two filters and the junction. This makes the diplexer longer than the sum of the filters lengths, possibly challenging the constraints on maximum acceptable size and increasing losses.

T

The new method allows to eliminate these additional sections, while preserving the original electrical behaviors of the two filters and the simplicity of design. The latter is guaranteed by means of practical closed formulas, derived in Section II, as no modification of the original prototypes of the two filters is required. The basic idea is that the junction can be seen as a part of each channel filter. In other words, the junction can be designed so as to mimic the effect of the first coupling elements, i.e., the first -inverters, of each filter. This task is not very difficult to accomplish, as shown in Section II. Section III describes a practical example: in the first part, the proposed approach is validated by means of a full-wave analysis performed by Ansoft’s High Frequency Structure Simulator (HFSS) [7], while in the second part, we report pictures and measurements of the actual final device. II. ANALYSIS It is expedient to split each filter into two cascading blocks, called, respectively, “head,” constituted by the first -inverter, and “tail,” involving the remaining part [6]. In this study, the heads are included into the three-port junction, whose scattering matrix is given by

(1)

The diplexer is obtained when ports 1 and 2 of this new special junction are connected to the tails of two filters, say, T1 and T2, respectively. At , the midband frequency of filter F1, the junction must of the original filter F1. The behave as the first inverter indicates filter , and denotes the th superscript in inverter of such a filter. Considering that port 2 of the junction is connected to tail T2, whose reflection is , the scattering matrix of the resulting two-port is given by (2)

Manuscript received February 13, 2006; revised May 19, 2006. This work was supported by the MITEL Srl, Milan, Italy. The authors are with the Dipartimento di Elettromagnetismo e Bioingegneria, Università Politecnica delle Marche, 60100 Ancona, Italy (e-mail: [email protected]). Color versions of Figs. 2, 3, 7, and 8 are available online at http://ieeexplore. ieee.org. Digital Object Identifier 10.1109/TMTT.2006.879770

Note that , as filter F2 is in its out-band at , and, for the same reason, . In fact, the equality would mean that port 2 is matched, i.e., F2 is in-band. In addition, since the junction is assumed to be lossless, we can focus on .

0018-9480/$20.00 © 2006 IEEE

3516

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Therefore, condition (4) simply becomes

(7)

Fig. 1. In its simplest form, the three-port junction to be used is formed by the connection of a Y-junction and the first two K - inverters of each filter (represented in the figure by their ABCD -parameters).

By the same token, at the midband frequency is required that

of filter F2, it

According to our strategy, the junction replaces the first -inverter of filter F1; consequently, its scattering matrix should coincide with the one of , whose expression is

(3)

(8)

Equations (7) and (8) provide the design formulas for the junction. Note that the right-hand sides of (7) and (8) are real, while the reflection coefficients of a three-port are generally complex. Therefore, phase correction at ports 1 and 2 is required. This is accomplished by inserting a section of line featuring a negative electrical length

Therefore, we require that

(9)

(4)

The left-hand side of the above expression has to be carefully analyzed. It is noted that the equality is immediately satisfied if and the reflection coefficient at port 1 of the junction, namely, , equates that of the first -inverter. This occurs, for instance, when the junction is formed by an ideal Y-junction whose arms 1 and 2 are connected to the first inverters of the corresponding filters, as shown in Fig. 1. In fact the matrix of the two-port 1 and 2, obtained by terminating port 3 on its characteristic impedance, is given by

(5)

Therefore,

(6) as required. Actually, equality (6) also holds when the “core” junction is more involved than just a simple Y. It turns out that the only needed requirement is that the two inverters be connected to it, as shown in Fig. 1.

in front of ports 1 and 2. The corresponding physical lengths are given by

(10) In practice, the above negative lengths are absorbed by the first cavities of each filter, thus reducing even more the overall diplexer size. In conclusion, the junction is designed by considering just the amplitudes of the reflections at ports 1 and 2, at the midband frequencies of the two filters and , provided that transmission between the two ports is negligible, as discussed above. In addition, note that when the passbands of the filters are contiguous and bandwidths coincide, their first -inverters are . In this case, it is quite easy almost equal, namely, to design the junction. For instance, in waveguide technology, this can be done either by placing a metal insert, as shown in Fig. 2, or by using an iris, as will be shown in Section III. The insert is adjusted in order to obtain the reflection magnitudes required by (7) and (8) at the two filter midband frequencies. This task is easily handled with the help of a full-wave solver, such as Ansoft’s HFSS [7]. The phases are finally adjusted by shortening the waveguide sections, which separate the first two -inverters of each filter, according to (10). It deserves to be emphasized that the common belief [8] that the junction has to be as neutral as possible is basically wrong. Of course, the same principle also applies to different technologies, such coaxial and microstrip.

MORINI et al.: NEW LOOK AT PRACTICAL DESIGN OF COMPACT DIPLEXERS

3517

Fig. 2. H -plane metal insert placed into a classical T-junction creates an arbitrary reflection at the main arm, thus satisfying the design conditions expressed by (7) and (8).

Fig. 4. Prototype response of the two designed six-cavity Chebyshev filters.

TABLE I DIMENSIONS (IN MILLIMETERS) OF THE TWO WR75 H-WINDOW SIX-POLE FILTERS DESIGNED. A AND C ARE THE WIDTH OF THE iTH APERTURE AND THE LENGTH OF THE iTH CAVITY, RESPECTIVELY. ALL THE WINDOW THICKNESSES ARE 0.5 mm

Fig. 3. In the X -band diplexer built, the compact junction was obtained using an H -plane window of the same kind employed for the two six-cavity filters.

III. RESULTS In order to demonstrate the effectiveness of the proposed method, here we report the details of the design and the results for a WR75-band diplexer, whose sketch is shown in Fig. 3. The diplexer was meant for practical use in a communication system for which a set of specifications was available. These specifications included the possibility to shift the channels by means of tuning screws. The responses of the Chebyshev ideal filter prototypes, featuring 23-dB minimum return loss and six cavities, are shown in Fig. 4. The bandwidth is 32 MHz for both filters, while the midband frequencies are 12.961 and 13.227 GHz, respectively, corresponding to the higher operating frequencies. Actually, it is well known that the insertion of a screw into a cavity can only shift down its resonant frequency. The filters were implemented by using inductive asymmetric windows (thickness: 0.5 mm) as coupling elements. Their dimensions are calculated by using in-house software, which provides the reflections of the windows evaluated at the midband frequencies and . In particular, the scattering matrices for a discrete set of apertures are stored in a database, and are interpolated during the optimization process. Table I reports the resulting dimensions The junction is obtained by using an -plane window of the same thickness as those employed for the filters, as shown in

Fig. 5. Sketch of the junction designed. The arrows point to the apertures between the common arm and first cavities of each filter. Note that the waveguide sections connecting the filters to the junction are no longer required, yielding a compact device.

Fig. 5. This choice is quite suitable for the current purpose, as it allows to obtain a (practically) arbitrary reflection. At this stage, it would be possible to accurately consider the effects of the corner roundness, produced in the fabrication by

3518

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

TABLE II AMPLITUDES AND PHASES OF THE SCATTERING PARAMETERS OF THE JUNCTION OF FIG. 5, CALCULATED AS FUNCTIONS OF THE APERTURE A’S, AT THE MIDBAND FREQUENCY OF THE FIRST FILTER (f 1 = 12:961 GHz), CONSIDERING 0.1-mm APERTURE STEPS. THE WAVEGUIDES ARE WR75, WHILE THE WINDOW THICKNESS IS 0.5 mm. THE VALUES HAVE BEEN COMPUTED BY ANSOFT’S HFSS [7]

Fig. 6. Ansoft’s HFSS simulation of the designed diplexer. This is exactly what results from the design process illustrated, without any adjustment or optimization, which, of course, would significantly improve performance.

the milling machine. However, we have purposely neglected them. The reason is that, as stated above, the two filters were required to be tunable, by means of tuning screws penetrating cavities and couplings windows. Therefore, we did not spend any effort in optimizing the design, being aware that the implemented structure would deviate somewhat from the designed one. One could argue that this choice weakens the validation of our approach. Actually, the validation is performed by means of Ansoft’s full-wave solver HFSS, thus immunizing the proof from the effects of nonideal materials, imperfect realization, and whatever affects real-world devices. Instead, the realization shows the usefulness of the approach, being a step-by-step example of a real-life diplexer. The design of the compact junction was obtained by interpolating the data set of junctions computed by Ansoft’s HFSS in order to satisfy (7) and (8). In particular, the values of the -inverters, corresponding to the first apertures of the two filters are

Therefore, the compact junction, designed according to (7) and (8), must feature GHz and GHz In addition, negligible transmission is required between the two ports, i.e., . Note also that, in the current case, the two reflections are nearly the same. This simplifies the design. The first step is to select a junction that could meet the requirements, possibly by varying a parameter at a time.

Fig. 7. WR75 diplexer built, as it appears when the lateral walls are removed.

The junction shown in Fig. 5 has the features we are looking for since, for practical values of the apertures, the transmission between ports 1 and 2 is negligible and each aperture only affects the reflection of the corresponding port (and, of course, of the common one). A lookup table (Table II) showing the magnitude of for different apertures, calculated by Ansoft’s HFSS at the midband frequency of the first filter (where the two are assumed to be identical), clearly shows apertures this point. The calculation of the aperture corresponding to the first -inverter of the second filter F2 is conceptually identical. It only requires the recalculation of the above table at the midband frequency of filter 2, i.e., . In practice, it matters little whether the two apertures and are equal or slightly different. As can be easily checked by direct inspection, the transmission between ports 1 and 3 is

MORINI et al.: NEW LOOK AT PRACTICAL DESIGN OF COMPACT DIPLEXERS

3519

The complete diplexer was simulated by Ansoft’s HFSS, obtaining the results shown in Fig. 6. They are noteworthy, being obtained without any optimization of the overall diplexer. The diplexer in question was finally built, as shown in Figs. 7 and 8, including the tuning screws and with rounded corners. Fig. 9 shows the measured scattering parameters. As can be observed, the bandwidths of the two filters were deliberately widened up to 40 MHz in order to reduce losses. This was easily accomplished by increasing the penetration of the screws controlling the couplings. IV. CONCLUSION

Fig. 8. Diplexer realized as it appears at the common port.

Diplexer design has been improved by looking at the device as a whole, in contrast to what has been usually done, namely, by considering separately junction and filters. In this way, the junction itself plays the role of the first -inverter of each filter if suitably designed. As a result, devices are more compact. We have introduced closed-form design formulas for designing the junction and, consequently, the entire diplexer. The validity of the proposed approach has been assessed by means of full-wave simulations, performed by Ansoft’s HFSS. A practical design example, along with its realization and experimental characterization, has also been discussed. ACKNOWLEDGMENT The authors are indebted to M. Villa, MITEL Srl, Milan, Italy, not only for helpful discussions and suggestions, but also for the realization and measurement of the prototype shown. REFERENCES

Fig. 9. Measurements of the diplexer designed. The bands of the two filters were slightly widened (40 MHz) by increasing the couplings in order to reduce losses.

essentially independent of the aperture placed across ports 2 and 3. The resulting window apertures corresponding to the above reflections are as follows. • Filter 1: mm. mm. • Filter 2: Correspondingly, the lengths of the first cavities are as follows. • Filter 1: mm. mm. • Filter 2: As can be observed, the latter differ just slightly from the starting values, namely, those of the filters reported in Table I.

[1] G. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures. Dedham, MA: Artech House, 1980, pp. 595–595. [2] J. D. Rhodes and R. Levy, “A generalized multiplexer theory,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 2, pp. 99–111, Feb. 1979. [3] ——, “Design of general manifold multiplexers,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 2, pp. 111–122, Feb. 1979. [4] A. Morini and T. Rozzi, “Constraints to the optimum performances and bandwidth limitations of diplexers employing symmetric threeport junctions,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 2, pp. 242–248, Feb. 1996. [5] A. Morini, T. Rozzi, and M. Morelli, “New formulas for the initial design in the optimization of T-junction manifold multiplexers,” in IEEE MTT-S Microw. Symp. Dig., San Diego, CA, Jun. 1997, pp. 1025–1028. [6] A. Morini and G. Cereda, “Re-configurable reciprocal multiplexers (r-mux) for terrestrial radio links,” in Proc. 31th Eur. Microw. Conf., Milan, Italy, Sep. 2002, pp. 105–107. [7] HFSS. ver. 10, Ansoft, Pittsburgh, PA, 2006. [8] Uher, J. Bornemann, and U. Rosenberg, Waveguide Components for Antenna Feed Systems. Norwood, MA: Artech House, 1993. Antonio Morini (M’96) received the Laurea degree (summa cum laude) in electronics and Ph.D. degree in electromagnetism from the University of Ancona, Ancona, Italy, in 1987 and 1992, respectively. He is currently an Associate Professor of applied electromagnetism with the Università Politecnica delle Marche, Ancona, Italy. His research activity is mainly devoted to the modeling and design of passive microwave components such as filters and antennas.

3520

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Tullio Rozzi (M’66–SM’74–F’90) received the Dottore degree in physics from the University of Pisa, Pisa, Italy, in 1965, the Ph.D. degree in electronic engineering from The University of Leeds, Leeds, U.K., in 1968, and the D.Sc. degree from the University of Bath, Bath, U.K., in 1987. From 1968 to 1978, he was a Research Scientist with Philips Research Laboratories, Eindhoven, The Netherlands. In 1975, he spent one year with the Antenna Laboratory, University of Illinois at Urbana–Champaign. In 1978, he became the Chair of Electrical Engineering with the University of Liverpool. In 1981, he became the Chair of Electronics and Head of the Electronics Group, University of Bath, where he was also the Head of the School of Electrical Engineering on an alternate three-year basis. Since 1988, he has been a Professor with the Dipartimento di Elettromagnetismo e Bioingegneria, Università Politecnica delle Marche, Ancona, Italy, where he is also Head of the department. Dr. Rozzi was the recipient of the 1975 Microwave Prize presented by the IEEE Microwave Theory and Technique Society (IEEE MTT-S).

Marco Farina (M’98) received the M. Eng. (summa cum laude) degree in electronics and Ph.D. degree from the University of Ancona, Ancona, Italy, in 1990 and 1995, respectively. From 1991 to 1992, he was a Technical Officer in the Italian Army. Since 1992, he has been with the Department of Electromagnetics and Bioengineering, University of Ancona, where he is an Assistant Professor. In 2002, he became an Associate Professor. He is also a Consulting Engineer in electronics. He coauthored Advanced Electromagnetic Analysis of Passive and Active Planar Structures (IEE Press, 1999). He developed the full-wave software package for three-dimensional (3-D) structures EM3DS.

Giuseppe Venanzoni received the Laurea and Ph.D. degrees in electronic engineering from the Università di Ancona, Ancona, Italy, in 2001 and 2004, respectively. He is currently a Post-Doctoral student with the Università Politecnica delle Marche, Ancona, Italy, where he is involved with electromagnetism and microwaves. His research interests include the design of waveguide passive devices, microwave filters, and antennas.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3521

A 3-dB Quadrature Coupler Suitable for PCB Circuit Design Jui-Chieh Chiu, Chih-Ming Lin, and Yeong-Her Wang, Member, IEEE Abstract—A quadrature 3-dB coupler, which combines the advantages of a coplanar waveguide and microstrip line structure suitable for single-layer substrate printed circuit board (PCB) circuit design is proposed. As compared to the conventional Lange coupler, the proposed coupler with the advantages of increasing the coupled linewidths and coupling spacing without using extra bonding wires can solve the drawbacks of Lange coupler. In addition, the proposed structure can easily be realized in a single-layer substrate by PCB manufacturing processes to eliminate the effects and uncertain factors from a multilayer substrate. Good agreements between the simulation and measurement in the frequency range from 0.45 to 5 GHz can be seen. With the operation bandwidth ranging from 1.8 to 2.8 GHz, the measured results of the return loss are better than 18.2 dB and insertion losses of coupled and direct ports are approximately 3 0.1 dB; the relative phase difference is approximately 89.8 . The dimension of the circuit is 2.7 cm 1.6 cm 0.08 cm. Index Terms—Coplanar waveguide (CPW), Lange coupler, microstrip line, 3-dB coupler, quadrature.

I. INTRODUCTION UADRATURE 3-dB coupler is an important component widely applied in microwave circuit design such as balanced amplifiers, phase shifters, mixers, etc. It can separate an incoming signal into two parts with equivalent amplitude and quartered relative phase difference. Generally speaking, the 3-dB couplers are realized by using tightly coupled microstrip lines such as a Lange coupler [1]–[3] or using multilayer coupled structures [4]–[7]. Among most of the couplers, the Lange coupler is the most wildly used because the circuit is compatible with monolithic microwave integrated circuit (MMIC) technology. The strip linewidths and the coupling spacing of the Lange coupler can easily be implemented by thin-film technology [8]. However, when it comes to printed circuit board (PCB) design, the manufacture is difficult to realize. This is due to the limitation of the tightly coupled structure and wirebonding processes that are not easily to be realized by using PCB processes. In order to realize the tightly coupled structure, the conductor widths and the spacing between conductors become prohibitively small. For example, the minimum linewidth and space, in general, of the PCB process is approxi-

Q

Manuscript received February 24, 2006; revised May 16, 2006. This work was supported in part by the National Science Council under Contract NSC94-2215-E-006-001, by the Ministry of Education Program for Promoting Academic Excellence of Universities under Grant A-91E-FA08-1-4, and by the Foundation of Chen under the Jieh-Chen Scholarship. The authors are with the Institute of Microelectronics, Department of Electrical Engineering, Advanced Optoelectronic Technology Center, National Cheng-Kung University, Tainan, 70101 Taiwan, R.O.C. (e-mail: [email protected]). Color versions of Figs. 4–6 are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2006.879772

mately 0.1 mm and a conventional 3-dB Lange coupler cannot easily be fabricated on a 0.8-mm-thick substrate because it required approximately 0.05-mm spacing between the coupled lines for signal coupling. After the circuit has been processed, the circuit still needs a wire bonder for bonding wires, which requires extra cost and also suffers extra yield loss [9]. Though there have been some design methods, such as using a multilayer structure to fabricate a floating potential conductor over a dielectric overlay signal line or using broadside-coupled coplanar waveguides (CPWs) [10], [11]. These circuits also require some special requirements such as bonding wires or multilayer substrates. These methods are also limited and inconvenient to PCB circuit design [8]. A novel 3-dB coupler that is designed on a single-layer substrate without using bonding wires, which is compatible for general printed circuit processes, is proposed. The novel structure can solve the manufacturing problems of conventional Lange couplers in PCB processes by increasing conductor widths and coupling spacing and without using bonding wires to crossover signal lines, and the proposed circuit only requires a single-layer substrate. These solutions are benefits for the circuits to be realized by general PCB processes. The design concept of the proposed couplers is to utilize a CPW structure to combine with microstrip line structures. This is due to the CPW circuit having more geometrical parameters and not being limited by the substrate thickness. The conventional five-finger microstrip Lange coupler structure will be substituted by a three-finger CPW coupled line structure. Therefore, the widths and spacing of coupled lines could be modified by adjusting the position of the CPW ground plane to match the minimum requirements of the printed circuit processes. Since the bottom metal of the CPW coupled lines is eliminated, the signal lines that need to connect to the other side can be realized by using via-holes to connect the bottom metal without using bonding wire processes. Furthermore, the circuit can also be housed with a milled pocket to prevent the back side from touching the ground plane [10], [17]. Comparisons of the proposed structure with the other reported couplers are summarized in Table I. The advantages of the proposed coupler are: 1) no bonding wires are needed; 2) it can be fabricated on a single-layer substrate; 3) the spacing of coupled lines can be increased; 4) three-finger coupled lines are used; and 5) the width of coupled lines can be increased. With the above-mentioned improvements, it is convenient to implement the circuit by using general printed circuit processes. II. CIRCUIT LAYOUT AND ANALYSIS The design concept of the proposed circuit is to combine the three-finger CPW coupled lines with microstrip line circuits to

0018-9480/$20.00 © 2006 IEEE

3522

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

TABLE I COMPARISON OF THE QUADRATURE 3-dB COUPLERS

Fig. 2. Schematic drawing of the three-finger CPW coupled line.

*A CPW coupled line structure to increase the widths of coupled lines. **The coupler using overlapped structure of multilayer to increase the coupled linewidths.

Fig. 1. (a) Layout of the proposed 3-dB quadrature coupler. (b) Cross-sectional view of the proposed 3-dB quadrature coupler.

realize tightly coupled structures. The layout of the proposed circuit is illustrated in Fig. 1. On the top layer, the circuit layout consists of two L-type metals connecting to isolated or coupled ports, one S-type metal connecting from input port to direct port, and two ground plane metals outside the CPW three-finger coupled lines connecting to the bottom metal by via-holes. In order to form a CPW structure with the back side ground connected to the top side, the back side metal is moved behind the top side

ground pattern, leaving only a small overlap to connect the back side ground to the top side using via-holes. In the proposed circuit, eight via-holes were used to connect to the ground plane to ensure that the CPW structure has a quasi-static TEM mode. The CPW coupled region is formed by the three metals and the ground plane, as shown in Fig. 1. The length of the coupled region is designed to match the quarter-wavelength of the center frequency to get the maximum signal coupling. On the bottom layer, because the ground plane below the CPW thee-finger coupled lines is eliminated, the two L-type metals can connect to two tape strip lines at the bottom layer by using four via-holes. The proposed structure can provide an equal power split between coupled and direct ports, while having a quadrature phase output. The schematic diagram of the CPW coupled region is expressed in Fig. 2. The schematic coupler consists of lines 1–3 and two ground planes on the top metal. From the proposed 3-dB coupler, lines 2 and 3 are connected to each other on both ends to form the four-port coupler structure with line 1. and are terminating admittances, respectively. To discuss the coupled region, a capacitance matrix analyzing method was used. In 1966, Cristal proposed the design concept of unequal odd- and even-mode admittance for each individual line of a two-line coupler [15]. The method used the capacitances per unit length or the odd- and even-mode admittances to specify the electrical characteristics of the coupled line. In the proposed method, the coupled lines’ parameters, including the coupling ratios and the terminating admittances, are developed. Based on the electrical characteristics of asymmetric coupled lines, Perlow and Presser proposed the design procedure for the general three-line coupler [16]. A model using a capacitance matrix to discuss the characteristics of three-line coupler is then developed. From the methods of Cristal and Perlow and Presser, the proposed coupler circuit can also be expressed as the capacitance matrix to analyze the circuit, as shown in Fig. 3(a). The parameters of and are mutual capacitance, while , , , and are self-capacitance per unit length. The proposed three-finger coupled line structure can use the capacitance per

CHIU et al.: 3-dB QUADRATURE COUPLER SUITABLE FOR PCB CIRCUIT DESIGN

3523

Fig. 4. Proposed 3-dB coupler. (a) Top side. (b) Bottom side. The dimension is 2.7 cm 1.6 cm.

2

Fig. 3. (a) Capacitance matrix model of the coupled lines for the proposed coupler. (b) Equivalent-circuit model.

unit length or the odd- and even-mode admittance to specify the electrical characteristics of the coupled line. In the proposed circuit, the widths of line 2 and 3 and the spacing to the ground were made equal, the coupler becomes symmetric, and the capacitances can be reduced as , , and as the capacitance matrix equivalent-circuit model shown in Fig. 3(b), where

Fig. 5. Measured and simulated results of the insertion loss and isolation as a function of frequencies.

The proposed circuit is symmetric in lines 2 and 3, which leads to a similar model as developed by Perlow and Presser. The oddand even-mode admittance can be achieved [16]. The proposed circuit is to design a 3-dB coupler in a 50system ms , the terminating admittance can be set to be ms. The line impedances of lines 1–3 can be calculated as and . These results can then be used to design the 3-dB coupler for different substrate thicknesses. III. IMPLEMENTATION OF THE 3-DB QUADRATURE COUPLER A three-asymmetric coupled line model (CPW3LINA) and a cylindrical via model (VIA) of AWR software were used to calculate the dimension of the proposed circuits. The impedance of coupled lines was determined using time-domain refelectometry (TDR) analysis. A low-cost FR-4 PCB with a dielectric constant of 4.4 and a thickness of 0.8 mm was used to implement the 3-dB coupler at the center operation frequency of 2.4 GHz. To confirm whether the circuit is suitable for general PCB circuit design rules, 0.1-mm processes are used with the recommended minimum spacing of the PCB as the spacing of the coupling linewidth to design the circuit. The spacing can also easily be realized by a milling machine. Based on the above equations, the proposed circuit can adjust the distance between the signal lines and the ground plane to keep the coupled line gap as 0.1 mm to calculate the initial dimensions of three signal lines ,

, and . To reduce the dissipation losses and improve the return losses, the discontinuous interfaces with access to the 3-dB power split were modeled by the electromagnetic (EM) simulator Zeland IE3D and considered in the process of circuit design. Finally, the entire layout was verified by the EM simulator to analyze the desired performance. The final dimensions of the circuit are obtained as mm, mm, and mm and the length of the coupled lines is 20.23 mm. The photograph of the fabricated circuit is shown in Fig. 4. The overall dimension of the circuit is 2.7 cm 1.6 cm. All the measurements of the couplers were taken by an Agilent PNA E8364A network analyzer using standard subminiature A (SMA) connectors. A thru-reflect-line (TRL) calibration method was used to deembed the coupler’s -parameters from the measured data. The performance for the measured and simulated results of the insertion loss and isolation are shown in Fig. 5. Good agreements between the simulation and measurement can be seen. From the measured results, the insertion loss for the direct port and the coupled port is approximately 3 0.1 dB; the coupling values at the two output ports are quite similar. This evidence indicates the power divider can successfully separate an incoming signal into two equivalent amplitude outputs. The isolation between the input port and the isolated port is also shown. The measured isolation between ports 4 and 1 is also better than 20 dB. It implies that an incoming signal

3524

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

of 20 dB at low frequency. At operation frequency, maximum coupling of the incoming signals is detected at the coupled port, corresponding to the obtained 3-dB power division shown in ports 2 and 3. While the operation frequency is gradually higher than the quadrature wavelength, the coupled voltage begins to deteriorate. Subsequently, the quantities of return loss and isolation also start to degrade. At the same time, the relative phase difference between ports 2 and 3 is increased [18]. IV. CONCLUSIONS

Fig. 6. Measured and simulated results of the return loss as a function of frequency.

A quadrature 3-dB coupler has been proposed and demonstrated. With the advantages of CPW coupled line structures, the proposed coupler can reduce the drawbacks of a conventional thin-film microstrip line Lange coupler manufacturing by PCB processes. The spacing and linewidths of the proposed circuit can be increased, which can easily be fabricated by printed circuit processes or a milling machine. The proposed circuit structure also eliminated the extra bonding wires. Therefore, the structure can not only save manufacturing cost, but also enhance the yield of manufacture. A model for the proposed structure can be used for the assessments of the coupler with very good agreement between the simulation and measurement. The proposed compact and practical circuit is easily realized on a single-layer substrate without using multilayer substrates or bonding wires. REFERENCES

Fig. 7. Measured and simulated results of relative phase differences.

from port 1 could have good isolation to the other signal from port 4. The measured and simulated results of the return loss as a function of frequencies are shown in Fig. 6. For the proposed circuit, layout is symmetric in and , the simulated results of return loss are the same. Due to the variation from fabrication and assembly, the measured results show a slight difference between or . Though the signal of port 2 needs to be coupled and transformed through via-holes to the bottom layer to connect to port 4, the return loss is still better than 18.2 dB and is better than 10 dB until 4 GHz. Since port 1 is connected to port 3 directly, the return loss is better than 20 dB at the center frequency. The good measured results look very promising. The measured and simulated results of the relative phase difference between the direct port and coupled port are shown in Fig. 7. The phase difference is 89.8 at the center frequency of 2.4 GHz and approximately 90 0.6 from 45 MHz to 4 GHz, which meets the design requirements of the proposed circuit. Based on the measured results, the performance is remarkably good. A good quadrature phase balance can be achieved. At very low frequency, all the incoming signals will go to the direct port (port 3), and no signals will be coupled at the coupled port (port 2). As such, the isolation will be better than that

[1] J. Lange, “Interdigitated stripline quadrature hybrid,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 12, pp. 1150–1151, Dec. 1969. [2] D. Kajfez, Z. Paunovic, and S. Pavlin, “Simplified design of Lange coupler,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 10, pp. 806–808, Oct. 1978. [3] R. Waugh and D. LaCombe, “Unfolding the Lange coupler,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 11, pp. 777–779, Nov. 1972. [4] L. Krishnamurthy, V. V. Tuyen, R. Sloan, K. Williams, and A. A. Rezazadeh, “Broadband CPW multilayer directional couplers on GaAs for MMIC applications,” in High-Freq. Postgraduate Student Colloq., Sep. 2004, pp. 183–188. [5] M. Nedil, T. A. Denidni, and L. Talbi, “CPW multilayer slot-coupled directional coupler,” Electron. Lett., vol. 41, pp. 706–707, Jun. 2005. [6] S. Al-Taei, P. Lane, and G. Passiopoulos, “Design of high directivity directional couplers in multilayer ceramic technologies,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, pp. 51–54. [7] Q. H. Wang, T. Gokdemir, D. Budimir, U. Karacaoglu, A. A. Rezazadeh, and I. D. Robertson, “Fabrication and microwave characterization of multilayer circuits for MMIC applications,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 143, pp. 225–232, Jun. 1996. [8] D. Willems and I. Bahl, “An MMIC-compatible tightly coupled line structure using embedded microstrip,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 12, pp. 2303–2310, Dec. 1993. [9] D. P. Andrews and C. S. Aitchison, “Wide-band lumped element quadrature 3-dB couplers in microstrip,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2424–2431, Dec. 2000. [10] C. V. B. Maciel, D. F. M. Argollo, and H. Abdalla, “Broadside suspended stripline 3 dB couplers,” in SBMO Int. Microw. Conf., São Paulo, Brazil, Aug. 1993, vol. 1, pp. 117–122. [11] 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. [12] M. Nakajima, E. Yamashita, and M. Asa, “New broadband 5-section microstrip line directional coupler,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1990, pp. 383–386. [13] V. Tulaja, B. Schiek, and J. Kohler, “An interdigitated 3-dB coupler with three strips,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 9, pp. . 643–645, Sep. 1978. [14] A. Sawicki and K. Sachse, “A novel directional coupler for PCB and LTCC applications,” in IEEE MTT-S Int. Microw. Symp. Dig., 2002, pp. 2225–2228.

CHIU et al.: 3-dB QUADRATURE COUPLER SUITABLE FOR PCB CIRCUIT DESIGN

[15] E. G. Cristal, “Coupled-transmission-line directional couplers with coupled lines of unequaI characteristic impedances,” IEEE Trans. Microw. Theory Tech., vol. MTT-4, no. 7, pp. 337–346, Jul. 1966. [16] S. M. Perlow and A. Presser, “The interdigitated three-strip coupler,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 10, pp. 1418–1422, Oct. 1984. [17] D. P. Andrews and C. S. Aitchison, “Wide-band lumped-element quadrature 3-dB couplers in microstrip,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2424–2431, Dec. 2000. [18] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. Jui-Chieh Chiu was born in Taoyuan Taiwan, R.O.C., on October 30, 1976. He received the B.S. degree from the National Taiwan Ocean University, Keelung, Taiwan, R.O.C., in 2001, and the M.S. degree from National Cheng-Kung University, Tainan, Taiwan, R.O.C., in 2003, both in electrical engineering, and is currently working toward the Ph.D. degree in electrical engineering at the Institute of Microelectronics, National Cheng-Kung University. His current research involves microwave millimeter-wave circuits, hybrid microwave integrated circuits (HMICs), and MMIC design.

Chih-Ming Lin was born in Chiayi, Taiwan, R.O.C., on May 6, 1981. He received the B.S. degree from Chinese Culture University, Taipei, Taiwan, R.O.C., in 2004, and is currently working toward the Master’s degree at the Institute of Microelectronics at National Cheng-Kung University, Tainan, Taiwan, R.O.C. His research concerns microwave power amplifier circuit design including MMICs, power amplifiers, and modules.

3525

Yeong-Her Wang (M’89) was born on December 2, 1956. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from National Cheng-Kung University, Tainan, Taiwan, R.O.C., in 1978, 1980 and 1985, respectively. His doctoral research concerned molecular-beam epitaxy (MBE) and its application to bulk barrier devices. From 1982 to 1985, he was an Instructor, and from 1985 to 1992, he was an Associate Professor of electrical engineering with National Cheng-Kung University. In 1992, he became a Professor. From 1989 to 1991, he was a Post-Doctoral Researcher with AT&T Bell Laboratories, Murray Hill, NJ, where he was involved with the MBE growth for the study of vertical-cavity surface-emitting lasers. He was appointed the Associate Chairman of the Institute of Electrical Engineering from 1993 to 1996 and Director of the Electrical Factory from 1995 to 1996. From 1996 to 1999, he also served as the Chairman of the Department of Electrical Engineering, National Cheng-Kung University. He is currently a Distinguished Professor with the Institute of Microelectronics and Department of Electrical Engineering, National Cheng-Kung University. His research and teaching activities are focused on semiconductor devices and physics, and the development and modeling of III–V compound semiconductor devices. His current interests include developing new techniques for oxide materials for GaAs-, InGaAs-, InGaP-, and GaN-based metal–oxide–semiconductor field-effect transistors (FETs) or heterostructure field-effect transistors (HFETs), and GaN/AlGaN quantum-cascade interband transition lasers by MBE. He is responsible for MMIC design and fabrication. He has authored or coauthored over 200 international journal papers and 160 conference papers. He currently holds 93 Taiwan, R.O.C. patents and 13 U.S. patents.

3526

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Novel Dual-Mode Bandpass Filters Using Hexagonal Loop Resonators Rui-Jie Mao, Student Member, IEEE, and Xiao-Hong Tang

Abstract—Novel dual-mode hexagonal loop resonators with the patch and notch perturbation are proposed in this paper. The field patterns of the degenerate modes and their mode splitting characteristics are investigated. The applications of the dual-mode hexagonal loop resonator as bandpass filters are presented. It is shown that different responses can be realized using only a single dualmode resonator by changing the position of the perturbation. A coupling and routing scheme is presented to model the operations of these filters. Two- and four-pole filters of this type are demonstrated for the first time. All the theoretical analysis and design procedures have been verified by experimental results. Index Terms—Dual-mode filters, dual-mode resonators, hexagonal loop, microstrip filters.

I. INTRODUCTION ICROSTRIP filters at the wireless communication frequency band are a highly active area of research. To meet the demands for compact size and high-performance filters, there has been a growing interest in the planar dual-mode filters and their components, i.e., the dual-mode resonators. A dual-mode resonator can be seen as an asynchronously tuned resonant circuit, the two degenerate modes of which can be excited by a perpendicular feed structure or by introducing different forms of perturbations. The first planar dual-mode filter was presented by Wolff [1]. After that, numerous researchers have proposed various configurations for the dual-mode filter [2]–[11]. As is well known, there are several commonly used dual-mode microstrip resonators including the circular ring [1], square loop [2], meander loop [3], and circular disk [4], and square patch [5]. Recently, the dual-mode triangular patch and loop resonators have been reported [6], [7]. However, thus far, dual-mode resonators with side numbers more than four have received little attention. Previous research on the hexagonal loop has been confined to its single-mode operation [8]. Besides the resonator type, the topology of a dual-mode filter deserves more consideration. By proper arrangement of the feed lines or the perturbation, a bandpass filter based on the dualmode resonator can achieve a certain response [9], [10]. In [11], Görür claimed that either a Chebyshev or quasi-elliptic response can be obtained using one resonator by changing the size of the perturbation with respect to that of the reference elements. However, further comments in [12] pointed out that the Chebyshev response in [11] was actually a response with two real axis transmission zeros. In addition, two possible coupling and routing schemes were discussed in [12].

M

Manuscript received March 20, 2006; revised May 25, 2006. The authors are with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.881025

Fig. 1. (a) Dual-mode hexagonal loop resonator using the patch perturbation. (b) Dual-mode hexagonal loop resonator using the notch perturbation. Geometrical dimensions are given in this figure for theoretical modeling.

In this paper, the research results on the dual-mode hexagonal loop resonators and their applications as bandpass filters are reported for the first time. The advantage of this new type of filter over conventional ones is that it facilitates the realization of different responses within one dual-mode resonator. The number and location of the transmission zero can be controlled simply by varying the position of the perturbation with respect to those of the input and output. II. DUAL-MODE HEXAGONAL LOOP RESONATOR USING PATCH PERTURBATION AND ITS APPLICATION AS FILTERS A. Dual-Mode Hexagonal Loop Resonator Using Patch Perturbation Fig. 1(a) shows the geometry of a dual-mode hexagonal loop resonator using the patch perturbation. A regular hexagonal loop consisting of six identical arms forms the basic element of the resonator. A rhombus-shaped patch perturbation with point angle 120 and diagonal length is attached to a corner of the loop for splitting the two degenerate modes. According to Wheeler’s cavity model [13], where the top and bottom of the cavity are the perfect electric walls and the remaining sides are the perfect magnetic walls, the electromagnetic (EM) fields inside the loop cavity can be expanded in terms of modes (where is perpendicular to the ground plane). A full-wave EM eigenmode simulator [14] was used to characterize the electric field patterns for a dual-mode hexagonal loop resonator using the patch perturbation. Fig. 2(a)–(c) depicts the simulated electric field vector between the metal strip and ground plane at the resonance frequency before and after mode splitting, respectively. As is seen from Fig. 2(a), when (without perturbation), the field pattern exhibits a rotational antisymmetric property with respect to the -axis. When , the perturbation is introduced and the two degenerate modes are split. The electric field pattern at the lower resonance frequency is illustrated in Fig. 2(b), where the

0018-9480/$20.00 © 2006 IEEE

MAO AND TANG: NOVEL DUAL-MODE BANDPASS FILTERS USING HEXAGONAL LOOP RESONATORS

3527

Fig. 3. Simulated resonance frequencies of the degenerate modes against the perturbation size where e = 9:5; h = 0:635 mm, a = 5 mm, and w = 1 mm.

to 4099 MHz, while the resonance frequency of the odd mode increases from 4447 to 4844 MHz. It should be mentioned that, without the perturbation, namely, mm, the hexagonal loop resonator acts as a single-mode resonator and no mode splitting is observed. The coupling coefficient between the two modes should be computed according to the theory of the coupled asynchronously tuned resonators as

Fig. 2. Simulated electric field patterns for a dual-mode hexagonal loop resonator using the patch perturbation with e = 9:5; h = 0:635 mm, a = 5 mm, p = 2:15 mm, and w = 1 mm. Darker sections indicate the maxima of the field. Lighter sections indicate the minima of the field. Arrows denote the direction of the electric field vector. (a) Electric field pattern at the resonance frequency before mode splitting. (b) Electric field pattern at the lower resonance frequency. (c) Electric field pattern at the higher resonance frequency.

maxima of the field are moved to the upper and lower arms of the resonator. Moreover, it is observed from the direction of the electric field vector that the field is symmetric with respect to the symmetry axis – , the mode thus behaves as an even mode. Fig. 2(c) shows the electric field pattern at the higher resonance frequency, where the maxima of the field are located along the left and right arms. The field distribution in Fig. 2(c) is similar to that before mode splitting, except the difference in magnitude. Apparently, the mode is an odd mode about the symmetry axis – . Since the field pattern of the odd mode can not be obtained by rotating the field of the even mode, the polarization of the degenerate modes is quite different from that of a square or circular loop resonator, where the two modes are perpendicular to each other. To observe the mode splitting, the dual-mode hexagonal loop resonator has been simulated using a full-wave EM eigenmode simulator with different perturbation size . The simulated resonance frequencies of the degenerate modes are plotted in Fig. 3 as a function of the perturbation size . As is shown in Fig. 3, when increases from 1.15 to 3.65 mm, the resonance frequency of the even mode decreases almost linearly from 4447

(1) and are the two mode splitting frequencies, and where and are the two self-resonant frequencies of the resonator, which can by obtained by placing a magnetic or electric wall along the symmetry axis during simulation, respectively [15]. Simulated results denote that and , which lead to the conclusion that , i.e., there is no coupling between the two degenerate modes. B. Bandpass Filter Using Dual-Mode Hexagonal Loop Resonator With Patch Perturbation Due to the geometry of a regular hexagonal loop, the position of the perturbation with respect to the input and output is very flexible. Fig. 4 shows three available configurations of the two-pole dual-mode hexagonal loop resonator filter using the patch perturbation, along with their simulated frequency responses. The dimensions of these structures are all the same, except a different position of the perturbation. The geometrical parameters of these filters are listed in Table I. The circumference of each hexagonal loop is approximately one wavelength. The size of the perturbation corresponds to the mode-splitting frequencies of 4385 MHz for the even mode and 4562 MHz for the odd mode, respectively. The resonator is fed by a pair of perpendicular 50- feed lines and each feed line is connected to a V-shaped coupling arm. The gap between the resonator and

3528

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 5. Responses of the coupling matrix in (2) (solid lines) and of the coupling matrix in (4) and (5) (dash lines).

Fig. 4. Configurations and simulated frequency responses of the dual-mode filters with different perturbation position. (a) Structure I. (b) Structure II. (c) Structure III. (d) Their simulated frequency responses.

TABLE I DIMENSIONS OF THE TWO-POLE DUAL-MODE HEXAGONAL LOOP RESONATOR FILTERS USING THE PATCH PERTURBATION (IN MILLIMETERS)

coupling arms was selected in consideration of strong coupling and etching tolerance. As is seen from Fig. 4(a), the patch perturbation is located at the right upper corner of the hexagonal loop. By comparing the configuration with the electric field patterns in Fig. 2, it is observed that port 1 is coupled to both modes and so is port 2. The filter exhibits an asymmetric quasi-elliptic frequency response, denoted by the solid lines in Fig. 4(d). The simulated response has a bandwidth of 160 MHz at the center frequency of 4480 MHz. It should be mentioned that the designed center frequency of the filter is slightly higher than the fundamental resonance frequency of the resonator before mode splitting (4447 MHz). This is because the added patch perturbation decreases the effective circumference of the loop. The two transmission zeros are located at 3660 and 4662 MHz, respectively. As analyzed above, there is no coupling between the two degenerate modes. To realize the response with transmission zeros, it seems that the coupling between the input and output cannot be ignored. Simulated results of the symmetric structure without the perturbation denote that there is a very weak coupling between the input and output. The transmission coefficient is in the order of 60 dB at the positions of the transmission zeros

in Fig. 4(d). The weak coupling between the input and output might be attributed to the presences of higher modes and surface waves. Based on the premise, the coupling and routing scheme is modeled as shown in the inset of Fig. 5. The dark circles represent the two degenerate modes and the empty ones represent the input and output, respectively. The input is coupled to both modes by the admittance inverters, represented by the solid lines, and so is the output. There is no coupling between the two modes. In addition, the input and output are coupled through the weak coupling, denoted by the dash line. Synthesis of such coupling scheme follows the approach in [16]. For the response with two transmission zeros at normalized frequencies of the complex -plane ( , where is the normalized angular frequency) and and an in-band return loss of 12.5 dB, the obtained coupling matrix is

(2)

, Note that the indices of the matrix elements and refer to the input, even mode, odd mode, and output, respectively. The minus signs of the matrix entries in (2) are due to the different coupling nature. The nonzero diagonal elements of the matrix in (2) are fixed to the normalized values of the two mode-splitting frequencies. Moreover, the relationships indicate that the even and odd mode couple differently to the input and output. The transmission coefficient and reflection coefficient between the input and output are given by (3a) (3b) (3c) where

is a 4 4 matrix whose only nonzero entries are is the normalized angular frequency variable, and is similar to the 4 4 identity matrix, except that

MAO AND TANG: NOVEL DUAL-MODE BANDPASS FILTERS USING HEXAGONAL LOOP RESONATORS

3529

[16]. Shown in Fig. 6 as the solid lines is the response of the coupling matrix in (2), which agrees well with the simulated result in Fig. 4(d). The position of the perturbation in Fig. 4(b) is at the right lower corner of the loop. Although ports 1 and 2 are still coupled to both modes, it is found according to the maxima of the field in Fig. 2 that the coupling strength between port 1 and the even mode is stronger than that between port 1 and the odd mode, and so is the condition between port 2 and the two modes. The simulated response of the configuration is illustrated in Fig. 4(d), denoted by dash lines, which has one transmission zero at the lower passband. The filter has a fractional bandwidth of 162 MHz at the center frequency of 4500 MHz. If the coupling matrix in (2) is modified according to the configuration in Fig. 4(b), the resulting coupling matrix is (4) Comparing (4) with (2), it is seen that the magnitude of the coupling coefficients between the even mode and ports in (2), i.e., 0.5246, are interchanged with those between the odd modes and ports (0.7196). It is because the ports of the filter in Fig. 4(b) are mainly coupled to the even mode, which is different from the coupling arrangement of the configuration in Fig. 4(a), where the ports are mainly coupled to the odd mode. The relationships and still come into existence, which indicate the even- and odd-mode properties. The nonzero diagonal elements in (4) are updated due to the slight differences between the center frequency and bandwidth of these filters. Furthermore, the signs of the other coupling coefficients are modified according to the even and odd modes of the field distributions. A direct analysis of the coupling matrix in (4) yields the dash lines in Fig. 5, which coincide with the response in Fig. 4(d) well. It is possible to determine the transmission zeros of the resulting coupling matrix, which are found to be and . As to the filter in Fig. 4(c), it has the same frequency response with the configuration in Fig. 4(b). The reason is summarized as follows. When the perturbation is moved to the opposite corner, the coupling nature between port 1 and both modes is changed due to the reversed direction of the electric field vector , which yields the opposite signs of the coupling coefficients. At the same time, the coupling between port 2 and the modes is also changed. It is these two factors that keep the response unaffected. To verify the theoretical analysis, the coupling matrix in (4) has been rearranged as the following: (5) Analysis of this coupling matrix yields the same response with that obtained by (4), which is shown in Fig. 5 as dash lines. C. Experimental Results To confirm the bandpass characteristics of the proposed dualmode hexagonal loop resonator using the patch perturbation,

Fig. 6. (a) Fabricated two-pole dual-mode hexagonal loop resonator filter using the patch perturbation. (b) Measured and simulated narrowband frequency responses. (c) Measured and simulated wideband frequency responses.

a two-pole filter having the configuration of Fig. 4(b) was designed and fabricated on a CER10 Teflon substrate, which has a thickness of 0.635 mm and a relative dielectric constant of 9.5. Fig. 6(a) shows a photograph of the filter. The size of the filter is very compact, which only amounts to 0.71 0.83 , where is the guided wavelength on this substrate at the center frequency. The measured frequency response, which was obtained using an Agilent E8363B network analyzer, along with the simulated one are shown in Fig. 6(b) and (c). Good agreement between them is observed, except for the frequency shift, which is

3530

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 8. Simulated resonance frequencies of the degenerate modes against the perturbation sizes where e = 9:5; h = 0:635 mm, a = 5 mm, and w = 1 mm.

Fig. 7. Simulated electric field patterns for a dual-mode hexagonal loop resonator using the notch perturbation with e = 9:5; h = 0:635 mm, a = 5 mm, l = 1:5 mm, n = 0:75 mm, and w = 1 mm. (a) Electric field pattern at the lower resonance frequency. (b) Electric field pattern at the higher resonance frequency.

mainly caused by the fabrication tolerance and simulation precision. The measured bandwidth is approximately 4.1% at the center frequency of 4640 MHz. The minimum passband insertion loss is 2.24 dB. This is mainly due to the conductor and dielectric losses of the substrate. The in-band return loss is greater than 12.5 dB. The first spurious response is at 8740 MHz, at about twice the center frequency. The transmission zero at lower stopband is located at 4150 MHz with 37.2-dB attenuation. It is interesting to notice that there is an additional 48-dB attenuation at 7700 MHz. This transmission zero is mainly caused by the harmonic effects of the distributed transmission line and should be useful for the rejection of the interference in the stopband. III. DUAL-MODE HEXAGONAL LOOP RESONATOR USING NOTCH PERTURBATION AND ITS APPLICATION AS FILTERS A. Dual-Mode Hexagonal Loop Resonator Using Notch Perturbation Besides the patch, other types of perturbation can be used for mode splitting of the hexagonal loop resonator. Fig. 1(b) shows the geometry of a dual-mode hexagonal loop resonator using the notch perturbation. A V-shaped notch is cut from a corner of the regular hexagonal loop for splitting the two degenerate modes. It is seen from the simulated electric field patterns in Fig. 7(a) that the maxima of the electric field at the lower resonance frequency are located along the left- and right-hand-side arms of the resonator, and the field is antisymmetric with respect to the symmetry axis – . From Fig. 7(b), it is observed that the maxima of the electric field at the higher resonance frequency

are at the upper and lower arms and the field is symmetric about the symmetry axis – . Obviously, the field in Fig. 7(a) is an odd mode, while that in Fig. 7(b) is an even mode. The condition is thus different from that described in Section II, where the odd mode has a higher resonance frequency than the even mode. The simulated mode-splitting characteristics of the resonator using the notch perturbation in Fig. 8 imply that the resonance frequencies of the two degenerate modes are separated off from each other with the increase of the perturbation sizes. The self-resonant frequencies and can be obtained by placing in turn an electric or magnetic wall on the symmetry axis – during simulation, respectively. By applying the same analysis method in Section II, it is found that there is actually no coupling between the two modes. B. Bandpass Filter Using Dual-Mode Hexagonal Loop Resonator With Notch Perturbation Before the application of the dual-mode hexagonal loop resonator with the notch perturbation as filters is studied, the following question arises: By similarity, would the filter have various configurations that can realize different responses, as is the case in Section II? Unfortunately, the answer to the question is no. As mentioned above, the resonance frequency of the even mode is higher than that of the odd mode within the resonator using the notch perturbation, which is different from the condition of the resonator using the patch perturbation. The resulting coupling structures are thus quite different, and so are the responses. Two feasible configurations constructed by the dual-mode hexagonal loop resonator using the notch perturbation are presented in Fig. 9(a) and (b). They have the same geometrical parameters, but an opposite position of the perturbation. The dimensions of these filters are listed in Table II. The sizes of the perturbation correspond to the mode-splitting frequencies of 4360 MHz for the odd mode and 4500 MHz for the even mode, respectively. By inspecting the field patterns in Fig. 7, it is observed that each port of the configuration in Fig. 9(a) is

MAO AND TANG: NOVEL DUAL-MODE BANDPASS FILTERS USING HEXAGONAL LOOP RESONATORS

3531

Fig. 9. Configurations and frequency responses of the dual-mode filters with the notch perturbation. (a) Structure I. (b) Structure II. (c) Their simulated and theoretical frequency responses.

TABLE II DIMENSIONS OF THE TWO-POLE DUAL-MODE HEXAGONAL LOOP RESONATOR FILTERS USING THE NOTCH PERTURBATION (IN MILLIMETERS)

coupled to both modes, while the coupling strength between the ports and even mode is stronger than that between the ports and odd mode. When the configuration in Fig. 9(b) is compared with the field patterns in Fig. 7, the same conclusion can be drawn. Simulated results of the two filters exhibit the same frequency response, shown in Fig. 9(c) as the solid lines. The simulated response has a fractional bandwidth of 4% at the center frequency of 4.4 GHz. It is worth mentioning that since the notch perturbation increases the effective circumference of the loop resonator, the designed center frequency of the filter is a little lower than the fundamental resonance frequency of the resonator before mode splitting (4447 MHz). The two transmission zeros are located at 3650 and 4752 MHz, respectively. The reason for the identical responses of the two different configurations can be attributed to the field distributions of the degenerate modes. When the position of the notch is moved to the opposite corner of the loop, the coupling nature between port 1 and both modes is changed, and so is that between port 2 and the modes. These factors cancel out and the response remains invariable.

Fig. 10. (a) Fabricated two-pole dual-mode hexagonal loop resonator filter using the notch perturbation. (b) Measured and simulated narrowband frequency responses. (c) Measured and simulated wideband frequency responses.

The coupling and routing scheme in the inset of Fig. 5 is still used to model the configurations in Fig. 9. For the simulated response in Fig. 9(c), where the two transmission zeros at normalized frequencies of the complex -plane are and and the in-band return loss is 20 dB, the synthesized coupling matrix is given by

(6)

3532

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 11. (a) Fabricated four-pole dual-mode hexagonal loop resonator filter. (b) Its coupling and routing scheme.

Note that the indices of the matrix elements , and refer to the input, odd mode, even mode, and output, respectively. The nonzero diagonal elements of the matrix are equal to the normalized values of the two mode-splitting frequencies. The relationships are due to the odd and even field distribution. The response of this coupling matrix is shown in Fig. 9 as the dash lines, which agrees very well with the simulated one. If the coupling matrix in (6) is rearranged according to the configuration in Fig. 9(b), the resulting matrix is

(7)

By comparing the matrices in (6) and (7), it is seen that the signs of the matrix entries are all changed, except those of and the diagonal elements. A direct analysis of this coupling matrix yields the same response with the one obtained from the matrix in (6), which verifies the theoretical analysis. It is interesting to mention that the theoretical response in Fig. 9(c) is similar to the solid lines in Fig. 5. Since the coupling strength between the modes and ports is decided by the dimensions of the coupling arms and their position about the symmetry axis, the values of the nondiagonal elements are all the same, except the differences of signs. The nonzero diagonal elements denote the different mode-splitting properties. Given the narrow bandwidth of the filter, they only affect the center frequency and bandwidth. Thus, it is not strange that these different configurations have similar responses.

Fig. 12. (a) Measured and simulated narrowband frequency responses of the four-pole dual-mode hexagonal loop resonator filter. (b) Measured and simulated wideband frequency responses.

C. Experimental Results A dual-mode bandpass filter having the configuration in Fig. 9(a) was fabricated to justify the study. A photograph of the filter is shown in Fig. 10(a). It was realized on a CER-10 substrate. The measured and simulated frequency responses are shown in Fig. 10(b) and (c). Again, the two results agree well. The filter has an asymmetric quasi-elliptic frequency response. The measured bandwidth is approximately 4.5% at the center frequency of 4515 MHz. The minimum insertion loss is 2.59 dB and the in-band return loss is greater than 12.5 dB. The two transmission zeros are located at 3642 MHz with 49-dB attenuation and 4760 MHz with 29.8-dB attenuation, respectively. IV. FOUR-POLE DUAL-MODE HEXAGONAL LOOP RESONATOR FILTER After the two-pole dual-mode hexagonal loop resonator filters are demonstrated, a four-pole filter has been developed. Fig. 11(a) shows a photograph of the fabricated four-pole dualmode hexagonal loop resonator filter on a CER10 substrate. The filter is constructed by two cascaded hexagonal loop resonators described in Section II; a reactive element is inserted between

MAO AND TANG: NOVEL DUAL-MODE BANDPASS FILTERS USING HEXAGONAL LOOP RESONATORS

the resonators as the nonresonating node (NRN) [18]. Note that the input and output are both mainly coupled to the even mode of the resonators. By such configuration, the selectivity of the filter can be improved [17]. The distance between the resonators is kept large enough, i.e., mm, to prevent unwanted coupling between the resonators. A possible coupling and routing scheme is shown in Fig. 11(b) with all the coupling coefficients presented in Section II. The dark circles represent the degenerate modes within the resonators and the empty ones represent the input and output. The reactive element is denoted by the gray circle. The nodes in the coupling scheme, either resonating or nonresonating, are connected by the admittance inverters, represented by the solid and dash lines. The measured and simulated frequency responses of the four-pole filter are plotted in Fig. 12. The measured result has a fractional bandwidth of 4.3% at the center frequency of 4640 MHz. The passband insertion loss is approximately 3.6 dB and the in-band return loss is greater than 12 dB. The filter has four transmission zeros at the stopband, their locations are at 4453, 4950, 4047, and 5351 MHz, respectively. The first and second deeps are close to the passband, which yield a very sharp cutoff rate. The rejection level of the two remaining transmission zeros is greater than 60 dB, which improve the stopband rejection significantly. It is worth mentioning that there are two additional resonant peaks at 3348 and 6350 MHz, respectively. These resonant peaks might be due to the structure of the reactive element and can be further suppressed by optimal design of the length . V. CONCLUSION A novel dual-mode hexagonal loop resonator and its applications as filters have been thoroughly studied in this paper. The field patterns of the degenerate modes and their mode-splitting characteristics have been investigated. The dual-mode hexagonal loop resonator filters have been presented for the first time. It is found that this type of filter operates differently from the dual-mode square or circular loop resonator filter. By varying the position of the perturbation with respect to those of the input and output, the number and location of the transmission zero can be controlled within one dual-mode resonator. They not only offer flexible realization of the desired filter characteristics, but also feature compact size. A coupling and routing scheme has been proposed to model the performances of these filters. It is expected that the dual-mode hexagonal loop resonator filter will find applications in low-loss, low-cost, and compact circuit design.

3533

[3] ——, “Microstrip bandpass filter using degenerate modes of a novel meander loop resonator,” IEEE Microw. Guided Wave Lett., vol. 5, no. 11, pp. 371–372, Nov. 1995. [4] J. A. Curtis and S. J. Fiedziuszko, “Miniature dual mode microstrip filters,” in IEEE MTT-S Int. Microw. Symp. Dig., 1991, pp. 443–446. [5] R. R. Mansour, “Design of superconductive multiplexers using singlemode and dual-mode filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 7, pp. 1322–1338, Jul. 1996. [6] J.-S. Hong and S.-Z. 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. [7] R.-B. Wu and S. Amari, “New triangular microstrip loop resonators for bandpass dual-mode filter application,” in IEEE MTT-S Int. Microw. Symp. Dig., 2005, pp. 941–944. [8] K. F. Chang, K. W. Tam, W. W. Choi, and R. P. Martins, “Novel quasi-elliptic microstrip filter configuration using hexagonal open-loop resonators,” in IEEE Int. Circuits Syst. Symp., May 2002, pp. 863–866. [9] L. Zhu, P. M. Wecowski, and K. Wu, “New planar dual-mode filter using cross-slotted patch resonator for simultaneous size and loss reduction,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 5, pp. 650–654, May 1999. [10] L.-W. Hsieh and K. Chang, “Dual-mode quasi-elliptic-function bandpass filters using ring resonators with enhanced-coupling tuning stubs,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 5, pp. 1340–1345, May 2002. [11] A. Görür, “Description of coupling between degenerate modes of a dual-mode microstrip loop resonator using a novel perturbation arrangement and its dual-mode bandpass filter application,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 671–677, Feb. 2004. [12] S. Amari, “Comments on ‘Description of coupling between degenerate modes of a dual-mode microstrip loop resonator using a novel perturbation arrangement and its dual-mode bandpass filter application’,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2190–2192, Sep. 2004. [13] H. A. Wheeler, “Transmission line properties of a strip on a dielectric sheet on a plane,” IEEE Trans. Microw. Theory Tech., vol. MTT-25, no. 8, pp. 631–647, Aug. 1977. [14] HFSS. ver. 8.0, Ansoft Software Inc., Pittsburgh, PA, 2001. [15] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001, pp. 251–257. [16] S. Amari, U. Rosenberg, and J. Bornermann, “Adaptive synthesis and design of resonator filters with source/load-multiresonator coupling,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 8, pp. 1969–1977, Aug. 2002. [17] A. Cassonese, M. Barra, G. Panariello, and R. Vaglio, “Multi-stage dual-mode cross-slotted superconducting filters for telecommunication application,” in IEEE MTT-S Int. Microw. Symp. Dig., 2001, pp. 491–494. [18] S. Amari and U. Rosenberg, “New building blocks for modular design of elliptic and self-equalized filters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 721–736, Feb. 2004. Rui-Jie Mao (S’06) was born in Sichuan Province, China, in 1978. He received the B.S. degree in physical electronics and M.S. degree in electronic engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2001 and 2005, respectively, and is currently working toward the Ph.D. degree in electrical engineering at UESTC. His research interests include microwave planar filters and millimeter-wave devices design.

ACKNOWLEDGMENT The authors would like to thank Y. Zhang, University of Electronic Science and Technology of China (UESTC), Chengdu, China, for his technical assistance, and S.-H. Han, UESTC, for his helpful discussions. REFERENCES [1] I. Wolff, “Microstrip bandpass filter using degenerate modes of a microstrip ring resonator,” Electron. Lett., vol. 8, pp. 302–303, Jun. 1972. [2] J.-S. Hong and M. J. Lancaster, “Bandpass characteristics of new dualmode microstrip square loop resonators,” Electron. Lett., vol. 31, pp. 891–892, May 1995.

Xiao-Hong Tang was born in Chongqing, China, in 1962. He received the M.S. and Ph.D. degrees in electromagnetism and microwave technology from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1983 and 1990, respectively. In 1990, he joined the School of Electronic Engineering, UESTC, as an Associate Professor, and became a Professor in 1998. He has authored or coauthored over 80 technical papers. His current research interests are microwave and millimeter-wave circuits and systems, microwave integrated circuits, and computational electromagnetism.

3534

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Simple Analysis and Design of a New Leaky-Wave Directional Coupler in Hybrid Dielectric-Waveguide Printed-Circuit Technology José Luis Gómez-Tornero, Member, IEEE, Sofía Martínez-López, and Alejandro Álvarez-Melcón, Member, IEEE

Abstract—This paper presents a new theory to analyze and design a novel leaky-wave coupler based on a hybrid dielectric-waveguide printed-circuit technology. The analysis theory is based on a combined method of moments (MoM) geometrical optics (GO) technique, which makes use of the leaky mode complex propagation constant. This MoM–GO approach allows to easily obtain the response of the coupler, and to illustratively understand the working mechanism of the proposed structure. A coupler is designed to work at 5.5 GHz. Leaky mode results are compared with three-dimensional finite-element method simulations, showing the validity and accuracy of the new theory developed. This novel directional coupler presents interesting advantages when compared to previous wave couplers. Index Terms—Directional coupler, geometrical optics (GO), leaky waves, method of moments (MoM).

I. INTRODUCTION N ORIGINAL directional coupler, based on hybrid dielectric-waveguide printed-circuit technology, was presented in [1]. The design procedure was illustrated using three-dimensional finite-element method (FEM) simulations, and the frequency response was obtained, showing the ability to control the coupling level. In this paper, an original theory based on the leaky mode complex propagation constant is described. With this theory, the response of the coupler can be obtained in a much faster way since the formulation is totally analytical and based on the leaky mode expressions. In this way, a straightforward design can be performed following the same procedure described in [1], but avoiding three-dimensional FEM simulations (which are very time consuming for large structures, as with the coupler studied in this paper). Moreover, the proposed theory helps to better understand the coupling working mechanism inherent to this coupler, illustrating the importance of higher order effects. The structure of the proposed leaky-wave coupler is shown in Fig. 1. Two radiating dielectric waveguides, laterally shielded

A

Manuscript received March 17, 2006; revised May 25, 2006. This work was supported in part under Spanish National Project ESP2001-4546-PE and Spanish National Project TEC2004-04313-C02-02-TCM and under the Regional Séneca Project 2002 PB/4/FS/02. J. L. Gómez-Tornero and A. Álvarez-Melcón are with the Department of Information Technologies and Communications, Escuela Técnica Superior de Ingenieros de Telecomunicación, Polytechnic University of Cartagena, 30202 Cartagena, Spain (e-mail: [email protected]; [email protected]). S. Martínez-López is with the Communications and Electronic Department, École Nationale Supérieure de Telécommunications, 75014 Paris, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879773

Fig. 1. Scheme of the proposed leaky-wave coupler.

by parallel plates, are faced toward each other in order to transfer energy from one to the other. Planar slot circuits are printed at the broad sides of the dielectric guides in order to excite a leaky-wave mode and to control its propagation and radiation properties. From a technological point-of-view, the proposed coupler presents a more flexible and simpler design than conventional grating couplers since the dielectric guide structure must not be mechanized to obtain a maximum efficiency design [1]. Only the printed circuit must be tapered [1]–[3]. In addition, the new coupler is based on radiation from the main space harmonic. In this way, the structure dimensions are minimized as compared to grating couplers, which are based on higher order radiating harmonics [4], [5]. II. THEORY Here, the theory to analyze and design the proposed coupler is described for the first time. The working mechanism of the proposed coupler will be explained, describing the theory to analyze and design each part of the circuit in order to obtain a successful response. The theory presented in this paper is novel, and it is based on a combination of the method of moments (MoM) and geometrical optics (GO) approaches. A. MoM Analysis and Design of Dielectric Waveguides and Slot Printed Circuits Rectangular dielectric waveguides of width , height , and relative permittivity are used to propagate the main mode. These dimensions control the mode cutoff and, therefore, determine the fast-wave frequency region where this mode can become leaky. The width of the slot, printed at the dielectric–air interface, also controls the leaky-wave region. A

0018-9480/$20.00 © 2006 IEEE

GÓMEZ-TORNERO et al.: SIMPLE ANALYSIS AND DESIGN OF NEW LEAKY-WAVE DIRECTIONAL COUPLER

3535

Fig. 4. Necessity of tapering the printed circuit.

Fig. 2. Normalized phase constant of TE

mode obtained with MoM.

to obtain a cosine shaped beam [see Fig. 4(b)]. The MoM analysis approach allows to obtain the variation of the leaky mode complex longitudinal propagation constant along the waveguides length ( -axis, see Fig. 1) for both the TX and RX couplers (1) (2) The normalized illuminations at the TX and RX antenna apertures and can be computed from the following well-known equations [6]:

Fig. 3. Asymmetry radiation and coupling mechanism.

(3)

MoM technique, especially conceived for the analysis of hybrid waveguide-planar transmission lines [2], is used to obtain the leaky mode phase and leakage constant. Fig. 2 shows the normalized phase constant of the mode for three different slot widths (see the inset of Fig. 2). As it can be seen in Fig. 2, the fast wave region of the mode (dark area in Fig. 2) extends from 4.8 GHz (cutoff region) to 6.5 GHz (bound wave region) for a dielectric guide of dimensions mm, mm, and , and a centered slot of width mm. In addition, the slot must be uncentered with respect to the parallel plates (see Fig. 3) in order to launch the leaky wave. This asymmetry radiation mechanism is explained in detail in [3], where interesting leaky-wave antennas (LWAs) were designed using this technology. In this way, no coupling will occur between the two dielectric guides if the separation between them is large enough, and the printed slot-circuit is centered, as sketched in Fig. 3(a) and (b). However, if the slot is uncentered, there will be radiation and coupling no matter the value of , as illustrated by Fig. 3(c). To compute the coupling between the input waveguide [transmitter (TX)] and the output waveguide [receiver (RX)], an “overlapping integral” along the whole structure length can be computed, as proposed in [4] and [5]. Note that the total length is divided into according to Fig. 1, where is the radiating slot length and is the spatial offset between the TX and the RX shown in Fig. 1. This integral computes the overlapping between the beams created by each waveguide at the reference plane located in the middle height. As sketched in Fig. 4, the emitted beams must be shaped to obtain maximum overlapping and, therefore, maximum coupling [5]. For this purpose, the printed slot circuit must be tapered along the coupler length, as explained in [3], in order

(4) (5)

(6) Fig. 5(a) shows the complex illumination functions created by cosine-tapered LWAs for both the TX and RX couplers at the level of their apertures (the TX LWA is located at position , while the RX LWA is located at , see Fig. 4). The illumination is plotted along the coupler normalized length ( -axis, see Fig. 1). As can be seen in Fig. 5, cosine-shaped beams can be synthesized by tapering the printed-circuit dimensions of the LWAs. The design procedure is described in detail in [3], and it is based on the MoM analysis of the dispersion curves of the leaky mode. The TX and RX LWAs are usually designed to point at a given elevation direction, which can be computed from the leaky mode phase constant . The TX and RX pointing angles and can be obtained from as [6] (7) (8) where is the free-space wavenumber in vacuum. As is illustrated in Fig. 4, the TX and RX angles should be unvarying along the antennas length in order to avoid unwanted beam dispersion. For these purposes, it can be seen from (7) and (8) that the leaky

3536

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

and the position and 4. As can be seen in Table I, the width of the slot [see Fig. 3(b)] must be tapered along the waveguide length to obtain the constant-pointing cosine-shaped beams. The frequency of design is 5.5 GHz, and the length of the LWAs is mm. B. GO Analysis The beam created at the LWAs apertures is able to travel along the parallel-plates stub of height , as shown in Fig. 4. The transverse plane located at will be used as the reference plane to compute the coupling coefficients. For this purpose, a simple GO model [7] is applied. Due to the fact that a uniform propagation direction is obtained for all the length of the synthesized is almost constant in Fig. 5(b)], a simple displacebeams [ ment can be used to obtain the illuminations created by the TX and RX antennas at the reference plane (9) (10) Fig. 5. Complex illumination functions for the beams created by the TX and RX LWAs at their apertures.

TABLE I DIMENSIONS (IN MILLIMETERS) OF THE LWAs TAPERED PRINTED SLOT CIRCUIT

modes’ phase constants should keep unchanged along the antenna aperture. Following the design procedure described in [3], a nearly constant pointing direction can be obtained for all the emitted beam length, as shown in Fig. 5(b). Table I represents the dimensions of the printed-slot circuit obtained in the design of the TX and RX LWAs. Both antennas are identical, but the RX LWA must be located facing the TX LWA, at a height and an offset , as illustrated in Figs. 1

(11) and are the mean values for the beam pointing where direction of the TX and RX LWAs, respectively [see Fig. 5(b)]. Once the beams complex illuminations have been computed at the reference plane , a normalized overlapping integral can be used to compute the coupling between these two beams along all the coupler length [5], shown in (12) at the bottom of this page. In order to have high coupling coefficient , the beams must match in the same position with the same amplitude function and with conjugated phases. This will depend on the amplitude and phases of the emitted beams ( and ), but it will also depend on the height of the coupler , the length of the radiating aperture , and the offset between the LWAs . Fig. 6 illustrates different scenarios. The first-order coupling computed by occurs when the TX and RX beams overlap, as sketched in Fig. 6(c). A wrong shift of the beams will decrease the overlapping, as shown in Fig. 6(b) and (d). The energy of the TX, which is not coupled to the RX, will be reflected back from the RX waveguide to the TX waveguide. In Fig. 6(a), a higher order coupling scenario is sketched. The reflected beam will be retransmitted up to the RX LWA, as is shown in Fig. 6(a). In this way, a second-order coupling coefficient can be computed from the new incident beam, shown in (13) at the bottom of the following page, where the illumination

(12)

GÓMEZ-TORNERO et al.: SIMPLE ANALYSIS AND DESIGN OF NEW LEAKY-WAVE DIRECTIONAL COUPLER

Fig. 6. Four different coupling scenarios. (a) Higher order coupling. (b)–(d) Influence of beam shifting on first-order coupling.

created by the TX beam after the multiple reflections is computed using the displacement

3537

and (18)]. In our case, we assure good radiation efficiency for all the designs with . In a practical LWA coupler, the TX and RX antennas are arranged to have direct coupling (high ) at the design frequency, as shown in Fig. 6(c). Due to the frequency beam scanning response of LWA [6], the coupling decreases for frequencies below [see Fig. 6(b)] and above [see Fig. 6(d)] the center frequency. For quite low frequencies, higher order couplings can be very important, as shown in Fig. 6(a). In any case, the energy that has not been coupled after higher order couplings reaches the structure far end at , where the perfect absorber must be located to avoid unwanted standing-wave phenomena. In Section III, a practical design of an LWA coupler in hybrid waveguide printed-circuit technology is performed using the previously described theory. Results obtained with a three-dimensional finite-element electromagnetic solver (FEM) will be presented to validate the theory. III. RESULTS

(14) This process can be recursively repeated so that the total amount of energy coupled from the TX to the RX can be computed, after multiple reflections, as (15) In (15), represents the total number of reflections considered, and is the maximum coupling coefficient, computed from the TX and RX antennas radiation efficiencies and . They are obtained from their respective leakage functions and ), computed in a lossless scenario and in the absence of the other antenna. The procedure for the calculation of and is detailed in [6] as follows: (16) (17)

(18) To obtain the highest coupling, the radiation efficiency of the two LWAs, and [see (16)–(18)] must be maximized. Once the leakage rate of the LWA is known (in our case, it is typically ), the radiating slot length must be chosen to fulfill the high-efficiency requirements of a particular application [the higher is, the higher is obtained, see (17)

Once the LWAs have been designed to synthesize long cosine-shaped constant pointing beams (see Fig. 5), the total coupler can be analyzed and designed using the combined MoM–GO theory. As demonstrated in [1], the height of the parallel plates must be high enough to avoid direct coupling due to the evanescent bound wave (with mm, the direct coupling is below 50 dB). In this way, the coupling mechanism can only be due to the emitted leaky-wave beams. For a given central frequency, which, in our case, is 5.5 GHz, the TX and RX slot circuits of length must be displaced at an offset [see Figs. 1 and 4(a)] in order to make the emitted beam to overlap with the trace of the receiver [5]. The theoretical value of , which provides this overlapping, is given by

(19) where [if the design is correct, as shown in Fig. 5(b)]. The radiating angle obtained from the leaky mode dispersion curve for the design frequency (5.5 GHz) is [see Fig. 5(b)] and is chosen to be mm. This gives rise to a theoretical offset (19) mm. Fig. 7 shows the variation of the coupling as a function of for the designed -long cosine-shaped slot-circuits, pointing at at 5.5 GHz. The results obtained with the leaky mode theory described in this paper are compared with a commercial FEM solver. It is checked that the maximum coupling takes place between mm and mm. For other values of , the traces are shifted, obtaining a smaller overlapping integral, which is translated into a lower coupling, as can be seen

(13)

3538

Fig. 7. Design of longitudinal offset between couplers L = 90 mm, L = 10 = 545:5 mm).

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

L

(f = 5:5 GHz,

Fig. 9. LWAs pointing direction and radiation efficiency; dependence with frequency.

Fig. 8. Beam illumination along coupler length (z = 0) for different frequencies (L = 70 mm, L = 90 mm, L = 10 = 545:5 mm). Fig. 10. Frequency response of coupling and direct output.

in Fig. 7. Very good agreement is observed between the two different analysis approaches (leaky mode and FEM). When frequency is varied around the design point (5.5 GHz), a beam-shifting phenomenon occurs. Due to the frequency-scanning response of LWAs [6], the beams are displaced and the overlapping is reduced. This phenomenon is illustrated in Fig. 8, where the traces created by the TX and RX LWAs at the reference level are plotted for different frequencies. The offset between the two antennas has been chosen to be mm. The beams obtained with the combined MoM–GO theory [see (9)–(11)] are compared to the plots obtained with FEM analysis. Very good agreement is observed for the beam shape and frequency shift. Fig. 9 shows the dependence with frequency of the pointing direction and of the radiation efficiency for the designed tapered LWAs. The pointing direction is derived from the mean value applied to (7) and (8), while the radiation efficiency is computed with (17) and (18). The leaky mode results are compared with FEM analysis, showing excellent agreement. As can be seen in Fig. 9, the antennas are designed to point at for GHz with nearly . For higher frequencies, the radiation tends to the endfire direction [ , see Fig. 6(c)], while the radiation efficiency decreases [6]. For lower frequencies, the beam

, keeping points closer to the broadside radiation the nearly 100% radiation efficiency. Below 4.8 GHz, the mode is at cutoff, and above 6.8 GHz, the leaky mode becomes a bound wave [6]. These three regimes can also be identified in the frequency response of the coupler shown in Fig. 10, where both the coupling and direct output responses are shown. Coupling and direct output are very low below 4.8 GHz due to the cutoff behavior of the mode. The leaky-wave region corresponds to the range between 4.8–6.8 GHz, where it can be seen in Fig. 10 that the maximum coupling occurs. Above 6.5 GHz, the mode becomes a bound wave, making most of the injected energy to travel to the OUT port (see Fig. 1 for the definition of the ports), therefore decreasing the coupling. In the leaky-wave band, it is observed that the maximum coupling is obtained at the design frequency (5.5 GHz). Around this frequency, the coupling decreases due to the frequency beam-shifting phenomenon illustrated in Fig. 8. However, another maximum appears at 5.1 GHz. This maximum is created by a second-order coupling, as was described and illustrated in Fig. 6(a). For frequencies above 5.5 GHz, the emitted beams diverge [see Fig. 6(b)] and coupling decreases as frequency increases, as it is checked

GÓMEZ-TORNERO et al.: SIMPLE ANALYSIS AND DESIGN OF NEW LEAKY-WAVE DIRECTIONAL COUPLER

3539

Fig. 12. Convergence of the MoM–GO coupling model with the number of significant contributions N . Fig. 11. Computation of overlapping factors (C quency using the MoM–GO model.

;C ;C

) as a function of fre-

in Fig. 10. Very good agreement is observed between the FEM analysis and the results obtained with the MoM–GO model. The total coupling was obtained from (15) using three higher order integrals. Moreover, the couplings to the OUT port were obtained assuming perfect matching and perfect isolation to the ISOLATED port (it was checked with three-dimensional FEM analysis that the return losses and isolation are below 15 dB in the entire coupling band) COUPLING(dB) OUT(dB)

(20) (21)

Fig. 11 shows the absolute values of the first three overlapping factors ( , , and ), computed with the MoM–GO model using (9)–(14). The first-order overlapping dominates at the design frequency ( at 5.5 GHz) for which there is direct sight between the TX and RX LWAs. For lower frequencies, the TX and RX beams overlap after higher order reflections due to the nearly normal angle of emission [see Fig. 6(a)]. This phenomenon is checked by the high values of and for the frequency of 5.1 GHz. As a result, another maximum of total coupling is obtained at 5.1 GHz, as can be seen in Fig. 10. It is necessary to take into account these higher order overlapping factors to accurately model the response of the designed LWA directional coupler. Moreover, Fig. 11 helps to understand the election of the index in (15), which stands for the total number of reflections considered. As can be seen in Fig. 11, there are two significant contributions to the total coupling: the one from the direct beam at 5.5 GHz and the higher order reflections ( and ) at 5.1 GHz, just after the cutoff frequency. Although the value for is quite high at 5.1 GHz , one can readily see that, according to (15), does not contribute much to the total response. This is due to the fact that most of the energy is already coupled to the second-order coefficient , leaving very little energy left ( in (15) to be coupled to the third-order reflection. This phenomenon explains the results shown in Fig. 12, where the convergence of the total coupling (15) frequency response is shown as a function of the index . It can be seen

Fig. 13. Adjustment of the coupling level by varying the radiation efficiency of the TX LWA.

coefficients, the total response has almost how with converged. All the results shown in this paper are for to assure convergence. All the previous results correspond to a directional coupler designed to obtain a coupling level of dB at the design frequency of 5.5 GHz. The maximum coupling level is controlled by the radiation efficiency of the TX and RX LWAs, and [see (17) and (18)]. Once the first-order overlapping integral (12) is maximized at the design frequency , the coupling level (15) can be computed as (22) If the RX LWA is designed with maximum radiation efficiency , the coupling level can be determined by the TX LWA efficiency . In this way, the input guide printed circuit will be responsible for controlling the coupling level of the entire directional coupler. Fig. 13 shows the results obtained for three couplers with maximum overlapping designed at 5.5 GHz. One coupler is designed for a 95% coupling level ( , dB), the other for 52% of coupling ( , , dB), and the last one for a coupling level of 22% ( , , dB). On the right-hand side of Fig. 13, the layout of the three TX LWAs printed circuits is sketched, which is designed using the asymmetry radiation

3540

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 14. Illumination of TX (continuous line) and RX (dotted line) LWAs along the coupler longitudinal direction (y ) at the reference plane (z = 0) for four different frequencies in the coupling band and for three different beam shapes: (a) cosine, (b) triangular, and (c) square cosine.

principle described in [3]. To decrease the radiation efficiency of the TX LWA , the asymmetry of the tapered slot-circuit must be decreased, as can be seen in the layouts of Fig. 13. The results obtained with the proposed MoM–GO theory are compared with FEM analysis, observing very good agreement. For all cases, it is observed the aforementioned higher order coupling at low frequencies (another maximum of coupling around 5 GHz). Finally, a study on the influence of the slot profiles on the coupling characteristics was performed. The authors have obtained similar coupling frequency responses by using three different symmetrical beam shapes, namely, cosine-beam, triangular-beam, and square-cosine beam. Each shape is synthesized using a different tapered slot profile, as explained in [3]. The maximum coupling level (obtained at the frequency of design) does not depend on the beam shape, but on the radiation efficiency of the TX and RX LWAs [see (22)], provided the correct alignment between the two radiating waveguides is assured [see (19)]. Fig. 14 shows the beams created by the TX and RX LWAs at the reference plane ( , where the overlapping integrals are computed) for the aforementioned beam shapes and for four different frequencies of the coupling band (5.1, 5.5, 6, and 6.5 GHz). At the design frequency (5.5 GHz), the desired beam shape is synthesized, and perfect overlapping is achieved by the correct alignment of the couplers [due to , see (19)]. For other frequencies, the overlap decreases due to the beamfrequency-shifting effect, and also due to the distortion of the emitted beams. The graphs shown in Fig. 14 are obtained with the MoM–GO approach, and have been compared to FEM results, showing very good agreement. The frequency response does not change much for these three different slot profiles, as illustrated by Fig. 15, where 95% coupling efficiency couplers were designed using the three aforementioned beam shapes. A slight difference in the coupling frequency response is observed. This is due to the fact that the evolution of the overlapping factor with frequency varies from one beam shape to another, as can be depicted from Fig. 14. Fig. 15 shows FEM simulations, together with results from MoM–GO model for the cosine case (the results for the

Fig. 15. Variation of the coupling frequency response for three different beam shapes.

triangle and square cosine shapes are similar, and they are not shown for the shake of clarity). It must be emphasized that the proposed theory provides fast and accurate results for complex three-dimensional structures due to its simplicity and analytical nature. This simplicity has been achieved by splitting the analysis of the whole three-dimensional structure into several steps. First, the evolution along the -axis of the leaky-wave complex propagation constants is obtained for the TX and RX radiating waveguides by means of a two-dimensional leaky mode analysis ( – -plane, see Fig. 1). The complex illumination functions along the longitudinal axis are then straightforwardly derived for both the emitted and received beams. Overlapping integrals only in the -direction (one-dimensional analysis) can be applied without losing accuracy. The beam coupling does not depend on the -axis due to the fact that the field distribution along the -direction is uniform (the radiated beam is basically a horizontally polarized oblique plane wave, as explained in [3] and [10]). A simple GO model in one dimension ( -axis) can also be applied without losing accuracy due to the uniform pointing direction of the synthesized beams. Moreover, the mutual influence between the TX and RX waveguides is negligible, provided that the parallel

GÓMEZ-TORNERO et al.: SIMPLE ANALYSIS AND DESIGN OF NEW LEAKY-WAVE DIRECTIONAL COUPLER

plates’ height is higher than . Under this condition, it is a good approximation to compute the normalized illumination functions considering that TX and RX are isolated. The analysis and design of these types of couplers is much more time consuming if generic three-dimensional numerical electromagnetic solvers are used as FEM analysis. The development of a specific and simpler theory is very convenient due to the large electrical dimensions of this type of traveling-wave couplers. This is also the case for other couplers based on nonradiative dielectric (NRD) technology [8], [9]. For instance, the coupler designed in this paper has a total length , a width , and a total height (see Fig. 1). The simple combination of the MoM and GO gives accurate results due to the particular configuration of the studied leaky-wave coupler, which is laterally shielded by metallic walls. Other authors have used other approaches to analyze other type of couplers such as the perturbation theory [4], Marcuse’s coupling formula [8], or the coupled mode theory [9]. In [5], a similar overlapping integral approach was used to obtain a scattering matrix representation of a dielectric grating coupler. However, in [5], only the conditions to obtain maximum coupling at the design frequency were studied. With the proposed theory, the response of a leaky-wave coupler can be accurately obtained, for the first time, for any frequency in the leaky-wave region, using a simple GO analysis based on the dispersion of the synthesized beams.

IV. CONCLUSION A new theory to analyze and design a new leaky-wave directional coupler in hybrid dielectric-waveguide printed-circuit technology has been presented. A MoM approach has been used to obtain the rectangular dielectric guide dimensions and the tapered geometry of the printed slot circuit in order to leaky control the complex propagation constant of the mode. In this way, a cosine shaped beam can be synthesized to obtain maximum coupling efficiency between the input and output couplers. Using GO, the displacement of the transmitted beams as they propagate through the coupler can be computed. In this way, total coupling is obtained from the computation of the overlapping integrals in the reference plane. It has been demonstrated that the offset between the transmitter and receiver leaky-wave radiating slots must be properly chosen to maximize the overlapping at the design frequency. Results obtained using a three-dimensional electromagnetic solver based on the FEM have been compared to predictions based on this leaky-wave MoM–GO approach, showing very good agreement. It has been confirmed that it is important to take into account higher order coupling effects due to reflections in the structure. Finally, it has been shown how the printed circuit can also control the radiation efficiency of the transmitter slot, allowing for the adjustment of the coupling level. The influence of the slot profile has also been studied by comparing the coupling for three different radiated beam shapes (cosine, triangular, and

3541

square cosine). The theory developed in this paper allows to easily design these types of couplers using the information of the complex propagation constant of the radiating leaky mode. This modal approach has been validated by comparisons with full-wave three-dimensional FEM simulations. REFERENCES [1] J. L. Gómez, S. Martínez, D. Cañete, J. Pascual, and A. A. Melcón, “A new leaky-wave directional coupler in hybrid dielectric-waveguide printed-circuit technology,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 11–16, 2006, pp. 1718–1721. [2] J. L. Gómez and A. A. Melcón, “Nonorthogonality relations between complex-hybrid modes: An application for the leaky-wave analysis of laterally-shielded top-open planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 760–767, Mar. 2004. [3] J. L. Gómez, A de la Torre, D. Cañete, M. Gugliemi, and A. A. Melcón, “Design of tapered leaky-wave antennas in hybrid waveguide-planar technology for millimeter waveband applications,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2563–2577, Aug. 2005. [4] T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys., vol. 14, pp. 235–254, 1977. [5] R. Ulrich, “Efficiency of optical-grating couplers,” J. Opt. Soc. Amer., vol. 63, no. 11, pp. 1419–1431, Nov. 1973. [6] A. A. Oliner, , R. Johnson, Ed., Leaky-Wave Antennas in Antenna Engineering Handbook, 3rd ed. New York: McGraw-Hill, 1993, ch. 10. [7] D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction, ser. Microw. Library. Norwood, MA: Artech House, 1990. [8] T. Yoneyama, N. Tozawa, and S. Nishida, “Coupling characteristics of nonradiative dielectric waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 8, pp. 648–654, Aug. 1983. [9] D.-C. Niu, T. Yoneyama, and T. Itoh, “Analysis and measurement of NRD-guide leaky wave coupler in ka band,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 12, pp. 2126–2132, Dec. 1993. [10] J. L. Gómez and A. A. Melcón, “Radiation analysis in the space domain of laterally-shielded planar transmission lines. Part II: Applications,” Radio Sci., vol. 39, no. RS3006, pp. 1–10, Jun. 2004.

José Luis Gómez-Tornero (M’06) was born in Murcia, Spain, in 1977. He received the Telecommunications Engineer degree from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 2001, and the Ph.D. degree (laurea cum laude) in telecommunication engineering from the Technical University of Cartagena (UPCT), Cartagena, Spain, in 2005. In 1999, he joined the Radiocommunications Department, UPV, as a Research Student, where he was involved in the development of analytical and numerical tools for the study and automated design of microwave filters in waveguide technology for space applications. In 2000, he joined the Radio Frequency Division, Industry Alcatel Espacio, Madrid, Spain, where he was involved with the development of microwave active circuits for telemetry, tracking, and control (TTC) transponders implicated in many different spatial missions for the European Space Agency (ESA), National Aeronautics Space Administration (NASA) and other space agencies. In 2001, he joined the Technical University of Cartagena, as an Assistant Professor, where he currently develops his teaching activities. Since October 2005, he has been Vice Dean for students and lecture affairs with the Telecommunication Engineering Faculty, Technical University of Cartagena. His scientific research is focused on the analysis and design of leaky-wave antennas for millimeter-waveband applications and the development of numerical methods for the analysis of novel passive radiating structures in planar and waveguide technologies. His scientific interests also include the study of active devices for microwave and millimeter wavebands such as oscillators and active antennas. Dr. Gómez-Tornero was the recipient of the 2004 Second National Award presented by the EPSON-Ibérica Foundation for the best doctoral project in the field of technology of information and communications (TIC). He was also the recipient of the 2006 Vodafone Foundation Colegio Oficial de Ingenieros de Telecomunicación (COIT/AEIT) Award presented to the best Spanish doctoral thesis in the area of advanced mobile communications technologies.

3542

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Sofía Martínez-López was born in Murcia, Spain, in 1983. She received the Telecommunications Engineer degree from the Polytechnic University of Cartagena (UPCT), Murcia, Spain, in 2005, and is currently working toward the Ph.D. degree at the Ecole Nationale Supérieure de Télécommunications (ENST), Paris, France. Her research interests are in the fields of antenna array signal processing and radio channel parameter estimation.

Alejandro Álvarez-Melcón (M’99) was born in Madrid, Spain, in 1965. He received the Telecommunications Engineer degree from the Polytechnic University of Madrid (UPM), Madrid, Spain, in 1991, and the Ph.D. degree in electrical engineering from the Swiss Federal Institute of Technology, Lausanne, Switzerland, in 1998. In 1988, he joined the Signal, Systems and Radiocommunications Department, UPM, as a Research Student, where he was involved in the design, testing, and measurement of broadband spiral antennas for electromagnetic measurements support (EMS) equipment. From 1991 to 1993, he was with the Radio Frequency Systems Division, European Space Agency (ESA/ESTEC), Noordwijk, The Netherlands, where he was involved in the development of analytical and numerical tools for the study of waveguide discontinuities, planar transmission lines, and microwave filters. From 1993 to 1995, he was with the Space Division, Industry Alcatel Espacio, Madrid, Spain, where he was involved with the ESA and collaborated in several ESA/ESTEC contracts. From 1995 to 1999, he was with the Swiss Federal Institute of Technology, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, where he was involved in the field of microstrip antennas and printed circuits for space applications. In 2000, he joined the Technical University of Cartagena, Cartagena, Spain, where he currently develops his teaching and research activities. Dr. Alvarez-Melcón was the recipient of the Journée Internationales de Nice sur les Antennes (JINA) Best Paper Award for the best contribution to the JINA’98 International Symposium on Antennas, and the Colegio Oficial de Ingenieros de Telecomunicación (COIT/AEIT) Award for the best doctoral thesis in basic information and communication technologies.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3543

Design of Composite Right/Left-Handed Coplanar-Waveguide Bandpass and Dual-Passband Filters Shau-Gang Mao, Member, IEEE, Min-Sou Wu, Student Member, IEEE, and Yu-Zhi Chueh

Abstract—The design and implementation of novel coplanarwaveguide (CPW) bandpass and dual-passband filters that consist of the composite right/left-handed short-circuited stubs are proposed. In contrast to the conventional short-circuited stub, whose input impedance repeats every half-wavelength, the composite right/left-handed CPW stub combines the phase-lead composite right/left-handed CPW and the phase-lag uniform CPW to achieve the arbitrary resonant frequencies with compact size. The equivalent bandpass and dual-passband LC resonators are established by investigating the frequency responses of the composite right/left-handed CPW stubs. The composite right/left-handed CPW bandpass filter is 62.5% more compact than the conventional quarter-wavelength shunt-stub CPW bandpass filter. The composite right/left-handed CPW dual-passband filter, which possesses the sharp rejection between the two asymmetric passbands, is also developed. The synthesis procedures of the composite right/left-handed CPW bandpass and dual-passband filters are successfully validated by measurement and full-wave simulation. Index Terms—Bandpass filter, composite right/left-handed structure, dual-passband filter, short-circuited stub, synthesis technique.

I. INTRODUCTION EFT-HANDED metamaterials (LHMs), an emerging class of artificially designed and structured materials with simultaneously negative permittivity and permeability, have received keen interest from physicists and engineers [1]. Although the theoretical analysis of the negative refractive index material was presented in 1968 [2], such material was not practically realized until 2000 [3]. The composite right/left-handed microstrip line and coplanar waveguide (CPW), which comprise an interdigital capacitor in series with a shunt short-circuited stub inductor to yield a broad left-handed passband and small loss, were recently proposed to facilitate a wide variety of LHM-based microwave circuit applications. Many resonators, hybrids, couplers, phase shifters, and antennas in the composite right/left-handed configuration have shown unusual radiating and guiding properties [4]–[10]. However, the development on LHM-based filters is still lacking [11], [12]. Filters with compact size and multiple passbands are essential for the portable devices operating on various RF standards.

L

Manuscript received April 5, 2006; revised June 2, 2006. This work was supported in part by the National Science Council of Taiwan, R.O.C., under Grant NSC 95-2221-E-027-018, Grant 95-2221-E-027-031, and Grant 95-2752-E-002-002-PAE. The authors are with the Graduate Institute of Computer and Communication Engineering, National Taipei University of Technology, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.880652

Recently, research concerning the size reduction and dual-passband characteristic of filters has drawn considerable interest in the microwave community. Unlike the conventional dual-passband filter combining both the bandpass and bandstop filters [13], several advanced configurations of the dual-passband filters have been reported [14]–[18]. Nevertheless, these constructions using coupling resonators inevitably increase the complexity of the circuit and are not suitable for flexible filter specifications. Moreover, the dual-passband filters based on the association of different shunt stubs to represent the dual-behavior resonators have been proposed [19]–[24]. However, these topologies produce the undesired bandpass regions below the specified passbands. In this paper, the short-circuited stubs comprising the composite right/left-handed CPW and the uniform CPW are characterized and applied to fulfill the parallel resonators for realizing the bandpass and dual-passband filters. The arbitrary passband–stopband characteristics of the periodic structures based on the finite repetition of the composite right/left-handed short-circuited stubs have been proposed by the authors [25], and the significant miniaturization of these structures were achieved as compared to the conventional quarter-wavelength shunt-stub filter. This study further investigates the design methodologies of the CPW bandpass and dual-passband filters using the composite right/left-handed short-circuited stubs. First, the composite right/left-handed CPW short-circuited stubs are analyzed based on the equivalent-circuit model. In contrast to the input admittance of the conventional short-circuited transmission line, which is repeated periodically as frequency increases, the input admittances of the proposed composite right/left-handed CPW short-circuited stubs possessing the bandpass and dual-passband characteristics are obtained by properly adjusting the dimensions of the composite right/left-handed CPW and the uniform CPW. Second, the analytical formula for determining the CPW line length of the composite right/left-handed short-circuited stub at a definite resonant frequency is obtained. Third, two configurations of the composite right/left-handed resonators are presented for realizing the bandpass and dual-passband composite right/left-handed CPW filters. The type-I composite right/left-handed CPW stub has the frequency response of the shunt LC resonator within the limited frequency range and, thus, is suitable for the bandpass filter application. The type-II composite right/left-handed CPW stub is equivalent to the parallel-connected series LC and shunt LC resonators, which represents two passbands with one stopband in between the two. Hence, the dual-passband composite right/left-handed

0018-9480/$20.00 © 2006 IEEE

3544

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

and are the characteristic admittance and the phase where constant of the uniform lossless CPW, respectively, and

(2) , is the input admittance on plane , and , as shown in Fig. 1(b). In the low-loss condition, the real part of is small and the composite right/left-handed short-circuited stub becomes an open circuit when the imaginary part of , i.e., , is zero, and thus, can be calculated by

(3) where

Fig. 1. (a) Physical configuration of the composite right/left-handed CPW short-circuited stub. (W = 1:4 mm, W = 1:55 mm, W = 0:7 mm, S = 0:4 mm, ` = 2:2 mm, ` = 1:8 mm, and ` = 0:7 mm. ` and ` are variables. The other strip and slot widths are all 0.2 mm.) (b) Equivalent-circuit model of the composite right/left-handed CPW short-circuited stub (from [25]).

CPW filter is constructed by cascading the admittance inverters and the type-II composite right/left-handed short-circuited stubs with specific CPW line lengths. Finally, the measurement, full-wave simulation, and equivalent-circuit model of the bandpass and dual-passband composite right/left-handed CPW filters are presented to support the usefulness of the proposed design methodologies. II. MODELING OF COMPOSITE RIGHT/LEFT-HANDED CPW STUB The basic configuration of the composite right/left-handed CPW short-circuited stub is presented her, and then its input admittance with arbitrary frequency response is characterized. Fig. 1(a) displays the layout of the short-circuited stub implemented with a composite right/left-handed CPW section. In theoretical modeling, this structure is decomposed into two parts, i.e., the composite right/left-handed CPW and the uniform CPW. The corresponding equivalent-circuit model is depicted in Fig. 1(b). Note that the equivalent-circuit elements of the unit-cell composite right/left-handed CPW are extracted based on the effective medium approach [26]. To characterize this composite right/left-handed CPW stub, the input admittance is determined by

(1)

Hence, when the line length of the uniform CPW is chosen, can then be obtained using (3) at an arbitrary frequency corresponding to . The analytical formula for determining is validated by comparing the measured and calculated results of the composite right/left-handed CPW short-circuited stub, as shown in [25, Fig. 2]. All the circuits presented in this study are fabricated on the RT/Duroid 6010.2 substrate with 1-oz copper cladding, 0.635-mm substrate thickness, and a dielectric constant of 10.2. The full-wave simulator IE3D is used to examine all the structures. In contrast to the conventional short-circuited stubs with periodically repeated input susceptances, the zero and infinite input susceptances of the proposed composite right/left-handed stubs can be shifted in frequency arbitrarily by changing the CPW line lengths, i.e., and . Fig. 2 depicts the calculated input susceptances for the conventional quarter-wavelength CPW short-circuited stub and the two types of the composite right/left-handed CPW short-circuited stubs with mm and (type I) and mm and mm (type II). Results show that the frequencies of zero and infinite input susceptances are approximately the same for both the conventional CPW and the type-I composite right/left-handed CPW stubs. This indicates that the type-I composite right/left-handed stub with a length of 3.6 mm possesses the same frequency response as that of the conventional quarter-wavelength short-circuited stub with a length of 8.2 mm, but the type-I stub is 56.1% more compact in size. Moreover, making of the composite right/lefthanded stub larger (type-II case) shifts the zeros and poles of the input susceptances toward lower frequencies, as well as reduces the spacing between the zero and pole of input susceptance. Therefore, the bandpass and dual-passband frequency responses of the type-I and type-II composite right/left-handed

MAO et al.: DESIGN OF COMPOSITE RIGHT/LEFT-HANDED CPW BANDPASS AND DUAL-PASSBAND FILTERS

3545

Fig. 2. Calculated input susceptances of the type-I and type-II composite right/ left-handed CPW stubs, as well as the conventional CPW short-circuited stub with length of 8.2 mm (data from [25]).

CPW short-circuited stubs are observed, respectively. The corresponding equivalent circuits of both the type-I and type-II composite right/left-handed stubs can then be developed. A. Type-I Composite Right/Left-Handed CPW Stub According to the frequency characteristic of the input susceptance of the type-I stub shown in Fig. 2, the composite right/lefthanded CPW short-circuited stub with can be regarded as a shunt LC resonator when the operating frequency is near 4.25 GHz. By properly replacing the short CPW line with the series inductance and shunt capacitance and applying the series-to-shunt transformation of the resonant network [27], the type-I stub with input admittance can be equivalent to the series connection of two shunt LC resonators, as shown in Fig. 3(a). The element values are given as follows:

Fig. 3. (a) Equivalent circuit of the type-I composite right/left-handed CPW stub. (b) Calculated susceptances of Y , blocks A and B. (L = 0:39 nH, L = 0:32 nH, C = 0:58 pF, L = 1:99 nH, and C = 0:63 pF).

(4) (5) (6) (7) (8) (9) where

By considering the individual susceptances of blocks A and B in Fig. 3(a), it is obviously concluded that the susceptance of block A is significantly larger than that of block B at 4.5 GHz, as shown in Fig. 3(b). Thus, the input susceptance of the type-I can be approximated composite right/left-handed CPW stub as that of a shunt and circuit, when its operating frequency is near 4.5 GHz. B. Type-II Composite Right/Left-Handed CPW Stub With reference to Fig. 2 again, the input susceptance of the type-II composite right/left-handed stub represents two zeros at 3.95 and 6 GHz with a pole at 4.98 GHz in between the two. This frequency characteristic reveals that the type-II composite right/left-handed stub is equivalent to the dual-passband resonators, i.e., the parallel-connected shunt and series resonators shown in Fig. 4. The input admittance is

(10)

3546

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 5. Equivalent bandpass filter circuit with admittance inverters and shunt resonators.

Fig. 4. Equivalent circuit of the type-II composite right/left-handed CPW stub.

The zeros and pole of found as

are

,

, and

with and the cascaded quarter-wavelength admittance inverters, is obtained as follows [28]: (14)

, and can be

(11) (12) (13) dB

(15)

where

Hence, the zeros and poles of the input susceptance of the type-II composite right/left-handed stub can be adjusted to approach , , and of by properly adjusting the physical configuration of the composite right/left-handed stub. Thus, the equivalence between the type-II stub and the dual-passband network (Fig. 4) can be achieved within the specific frequency ranges. III. FILTER DESIGN AND IMPLEMENTATION In a previous study [25], the CPW filters loaded periodically with the composite right/left-handed short-circuited stubs have been proposed. The presented CPW bandpass filter is quite compact and shows the adjustable passband and stopband responses. Here, the synthesis of the composite right/left-handed CPW filters with a wide variety of prescribed frequency characteristics is further investigated. In Section II, since the operations of the type-I and type-II composite right/left-handed CPW stubs are found very similar to those of the bandpass and dual-passband resonators, respectively, the design procedures of the bandpass and dual-passband filters based on the composite right/lefthanded CPW resonators can then be developed. Additionally, the implementation and measurement of these filters are presented in here. A. Composite Right/Left-Handed CPW Bandpass Filter Once the center frequency , the 3-dB bandwidth , and the passband response are specified, the generalized bandpass filter circuit of Fig. 5, which uses the identical shunt type-I resonators

where ’s are the coefficients of the equivalent low-pass filter prototype with cutoff frequency 1 rad/s, are the terminating admittances of the circuit, and is given in (9). A bandpass filter is designed with an of 4.5 GHz, a 3-dB bandwidth of 30%, and a system impedance of based on the third-order 0.1-dB Chebyshev response. The corresponding circuit parameters are obtained using (3), (9), and (14) as follows: mm pF mm mm Considering the bandwidth limit of the composite right/left-handed CPW equivalent circuit and the parasitic effect of the CPW cross junction, is optimized to 0.8 mm and the physical configuration of the bandpass CPW filter is determined in Fig. 6(a) with the corresponding equivalent circuit depicted in Fig. 6(b). The measured, full-wave simulated, and equivalent-circuit modeled results are presented in Fig. 7 and a good agreement among them is observed. The measured is at 4.47 GHz, with a minimum insertion loss of 0.74 dB, a return loss of 19.4 dB, and a 3-dB bandwidth of 31.1%. This filter with the shunt-stub length is 62.5% more compact than the conventional third-order shunt-stub filter. Additionally, the coupling between the composite right/left-handed CPW short-circuited stubs is investigated. In Fig. 8, the transmission coefficient of the neighboring composite right/left-handed CPW stubs with mm [see Fig. 6(a)] is lower than

MAO et al.: DESIGN OF COMPOSITE RIGHT/LEFT-HANDED CPW BANDPASS AND DUAL-PASSBAND FILTERS

3547

Fig. 8. Simulated S of the neighboring composite right/left-handed CPW short-circuited stubs with d = 1:9 mm shown in Fig. 6(a).

stub essentially behaves like that of the parallel-connected series LC and shunt LC resonators of Fig. 4 over a limited frequency range. Hence, the composite right/left-handed CPW dual-passin band filters can be realized when the shunt resonators Fig. 5 are implemented by the identical type-II stubs. For the dual-passband filter design methodology [29], the required four parameters that define the dual-passband transformation are the lower and upper limits of the two passbands, i.e., and , respectively. The admittance inverters (Fig. 5) and the capacitors and inductors (Fig. 4) can then be found as Fig. 6. (a) Geometry of the composite right/left-handed CPW bandpass filter with ` = 0:8 mm, W = W = 0:8 mm, and W = W = 0:28 mm. (The dimensions of the unit-cell composite right/left-handed CPW are the same as in Fig. 1.) (b) Equivalent-circuit model of the composite right/left-handed CPW bandpass filter. The LHM CPW is the circuit of Fig. 1(b) located at plane AA .

(16) (17) (18) (19) (20) where

Fig. 7. Measured, full-wave simulated, and equivalent-circuit modeled S -parameters of the composite right/left-handed CPW bandpass filter.

40 dB around 5 GHz, which indicates that the insignificant coupling effect between the composite right/left-handed stubs can be neglected. B. Composite Right/Left-Handed CPW Dual-Passband Filter In Section III-A, it has been shown that the frequency characteristic of the type-II composite right/left-handed short-circuited

The above described synthesis technique is employed to implement the composite right/left-handed CPW dual-passband

3548

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

two passbands, i.e., 3.75 and 5.25 GHz, respectively. The return losses are better than 10 dB for both passbands. Unlike the dual-passband filters using dual-behavior resonators [19]–[24], the proposed filter indicates that the spurious passband on the lower side of the bandpass regions is eliminated. Good agreement among the full-wave simulated equivalent-circuit modeled and measured results is observed and, thus, validates the proposed design procedure of the composite right/left-handed CPW dual-passband filter. The measured and full-wave simulated exhibit the transmission peaks around 7.5 GHz, which may due to the discontinuities in this structure resonating at approximately this frequency. The return loss is somewhat different from that of the typical Chebyshev response and the passband and stopband regions are shifted, which is mainly caused by the fabrication tolerance in composite right/left-handed stubs with bonding wires. IV. CONCLUSION

Fig. 9. (a) Geometry of the composite right/left-handed CPW dual-passband filter with ` = 1:2 mm, ` = 8:6 mm, W = W = 0:94 mm, and W = W = 0:84 mm. The inverter impedances are Z = Z = 63:2

and Z = Z = 66:2 , and the lengths are ` = ` = ` = ` = 8 mm. (The dimensions of the unit-cell composite right/left-handed CPW are as the same as in Fig. 1.) (b) Measured, full-wave simulated, and equivalent-circuit modeled S -parameters of the composite right/left-handed CPW dual-passband filter.

filter. The first and second passbands of the filter are specified as GHz, GHz, GHz, and GHz with the third-order 0.5-dB Chebyshev passband characteristic. By using (16)–(20), the quarter-wavelength admittance inverters are obtained, and and of the composite right/left-handed stub are determined by approximating of (1) and of (10). The equivalent-circuit model of the dual-passband composite right/left-handed CPW filter can then be established and its geometry is determined, as depicted in Fig. 9(a). The measured and calculated -parameters are shown in Fig. 9(b). The measured in-band insertion losses are only 1 and 0.9 dB at the center frequencies of the

Novel composite right/left-handed CPW bandpass and dual-passband filters have been investigated theoretically and experimentally. The design procedures and implementation of these circuits have been presented. The composite right/left-handed CPW short-circuited stubs with various CPW line lengths are equivalent to the bandpass and dual-passband LC resonators within the specific frequency range. The bandpass and dual-passband filter circuits, which consist of a cascade of the quarter-wavelength admittance inverters alternating with the shunt composite right/left-handed CPW stubs, can then be derived. The flexibility in choosing the dimensions of the composite right/left-handed stubs, such as the impedance and length of uniform CPW and the construction of the unit-cell composite right/left-handed CPW, results in the design and realization of the bandpass and dual-passband filters with a wide variety of specifications. The presented design methodologies can be applied to the multiple passband and stopband composite right/left-handed CPW filters. The composite right/left-handed CPW resonators may be useful in many dual-band applications of modern wireless communication such as the dual-band antennas and oscillators. REFERENCES [1] T. Itoh and A. A. Oliner, Eds., “Guest editorial,” IEEE Trans. Microw. Theory Tech. (Special Issue), vol. 53, no. 4, pp. 1413–1417, Apr. 2005. [2] V. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp., vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [3] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, no. 18, pp. 4184–4187, May 2000. [4] C. Caloz and T. Itoh, “Metamaterials for high-frequency electronics,” Proc. IEEE, vol. 93, no. 10, pp. 1744–1752, Oct. 2005. [5] M. A. Antoniades and G. V. Eleftheriades, “A broadband series power divider using zero-degree metamaterial phase-shifting lines,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 808–810, Nov. 2005. [6] I.-H. Lin, M. D. Vincentis, C. Caloz, and T. Itoh, “Arbitrary dual-band components using composite right/left-handed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1142–1149, Apr. 2004. [7] S.-G. Mao, S.-L. Chen, and C.-W Huang, “Effective electromagnetic parameters of novel distributed left-handed microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1515–1521, Apr. 2005.

MAO et al.: DESIGN OF COMPOSITE RIGHT/LEFT-HANDED CPW BANDPASS AND DUAL-PASSBAND FILTERS

[8] S.-G. Mao and Y.-Z. Chueh, “Broadband composite right/left-handed coplanar waveguide power splitters with arbitrary phase responses and balun and antenna applications,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 243–250, Jan. 2006. [9] K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 135–146, Jan. 2006. [10] S.-G. Mao and S.-L. Chen, “Characterization and modeling of lefthanded microstrip lines with application to loop antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1084–1091, Apr. 2006. [11] S.-G. Mao, M.-S. Wu, Y.-Z. Chueh, and C. H. Chen, “Modeling of symmetric composite right/left-handed coplanar waveguides with applications to compact bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3460–3466, Nov. 2005. [12] J. Bonache, I. Gil, J. García-García, and F. Martín, “Novel microstrip bandpass filters based on complementary split-ring resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 265–271, Jan. 2006. [13] L.-C. Tsai and C.-W. Hsue, “Dual-band bandpass filters using equallength coupled-serial-shunted lines and z -transform technique,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1111–1117, Apr. 2004. [14] J.-T. Kuo, T.-H. Yeh, and C.-C. Yeh, “Design of microstrip bandpass filters with a dual-passband response,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1331–1337, Apr. 2005. [15] 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. [16] J. Lee, M. S. Uhm, and I.-B. Yom, “A dual-passband filter of canonical structure for satellite applications,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 271–273, Jun. 2004. [17] H. Uchida, H. Kamino, K. Totani, N. Yoneda, M. Miyazaki, Y. Konishi, S. Makino, J. Hirokawa, and M. Ando, “Dual-band-rejection filter for distortion reduction in RF transmitters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2550–2556, Nov. 2004. [18] R. J. Cameron, M. Yu, and Y. Wang, “Direct-coupled microwave filters with single and dual stopbands,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3288–3297, Nov. 2005. [19] C. Quendo, E. Rius, and C. Person, “Narrow bandpass filters using dual-behavior resonators based on stepped-impedance stubs and different-length stubs,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 1034–1044, Mar. 2004. [20] A. Manchec, C. Quendo, E. Rius, C. Person, and J.-F. Favennec, “Synthesis of dual behavior resonator (DBR) filters with integrated low-pass structures for spurious responses suppression,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 1, pp. 4–6, Jan. 2006. [21] C. Quendo, E. Rius, and C. Person, “An original topology of dual-band filter with transmission zeros,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 1093–1096. [22] V. Palazzari, S. Pinel, J. Laskar, L. Roselli, and M. M. Tentzeris, “Design of an asymmetrical dual-band WLAN filter in liquid crystal polymer (LCP) system-on-package technology,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 3, pp. 165–167, Mar. 2005. [23] C.-M. Tsai, H.-M. Lee, and C.-C. Tsai, “Planar filter design with fully controllable second passband,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3429–3439, Nov. 2005. [24] X. Guan, Z. Ma, P. Cai, Y. Kobayashi, T. Anada, and G. Hagiwara, “Synthesis of dual-band bandpass filters using successive frequency transformations and circuit conversions,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 3, pp. 110–112, Mar. 2006. [25] S.-G. Mao and M.-S. Wu, “Compact composite right/left-handed coplanar waveguide filters with arbitrary passband and stopband responses,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, to appear in. [26] ——, “Equivalent circuit modeling of symmetric composite right/lefthanded coplanar waveguides,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1953–1956.

3549

[27] J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [28] G. Mattaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures. Norwood, MA: Artech House, 1980. [29] G. Macchiarella and S. Tamiazzo, “Design techniques for dual-passband filters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3265–3271, Nov. 2005.

Shau-Gang Mao (S’97–M’98) received the Ph.D. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1998. From October 1998 to July 2000, he fulfilled military service with the Department of Communication, Electronics, and Information, Coast Guard Administration, Executive Yuan, Taiwan, R.O.C., where he conducted projects on coastal surveillance and communication systems. From August 2000 to January 2002, he was with the Department of Electrical Engineering, Da-Yeh University. Since February 2002, he has been a faculty member with the Graduate Institute of Computer and Communication Engineering, National Taipei University of Technology, Taipei, Taiwan, R.O.C., where he is currently an Associate Professor. His research areas of interest include microwave and millimeter-wave circuits and antennas. Dr. Mao was the secretary of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Taipei Chapter in 2001. He is a member of the Editorial Boards of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He was the recipient of the Best Paper Award presented at the 2001 Asia–Pacific Microwave Conference (APMC) and the International Union of Radio Science (URSI) Young Scientist Award presented at the 2004 International Symposium on Electromagnetic Theory.

Min-Sou Wu (S’06) was born in Taoyuan, Taiwan, R.O.C., on December 25, 1979. He received the B.E. degree in electronic engineering and M.S. degree from the Institute of Computer and Communication Engineering, National Taipei University of Technology, Taipei, Taiwan, R.O.C., in 2003 and 2005, respectively, and is currently working toward the Ph.D. degree at the Graduate Institute of Computer and Communication Engineering, National Taipei University of Technology. His research interests include the design and analysis of microwave circuits and composite right/left-handed CPW structures.

Yu-Zhi Chueh was born in I-Lan, Taiwan, R.O.C., on August 30, 1981. He received the B.E. degree in electronic engineering from the National Chin-Yi Institute of Technology, Taichung, Taiwan, R.O.C., in 2004, the M.S. degree from the Institute of Computer and Communication Engineering, National Taipei University of Technology, Taipei, Taiwan, R.O.C., in 2006, and is currently working toward the Ph.D. degree at the Graduate Institute of Computer and Communication Engineering, National Taipei University of Technology. His research interests include design and analysis of microwave circuits and their applications.

3550

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Design of Dual- and Triple-Passband Filters Using Alternately Cascaded Multiband Resonators Chi-Feng Chen, Ting-Yi Huang, and Ruey-Beei Wu, Senior Member, IEEE

Abstract—A novel method for designing multiband bandpass filters has been proposed in this paper. Coupling structures with both Chebyshev and quasi-elliptic frequency responses are presented to achieve dual- and triple-band characteristics without a significant increase in circuit size. The design concept is to add some extra coupled resonator sections in a single-circuit filter to increase the degrees of freedom in extracting coupling coefficients of a multiband filter and, therefore, the filter is capable of realizing the specifications of coupling coefficients at all passbands. To verify the presented concept, four experimental examples of filters with a dual-band Chebyshev, triple-band Chebyshev, dual-band quasi-elliptic, and triple-band quasi-elliptic response have been designed and fabricated with microstrip technology. The measured results are in good agreement with the full-wave simulation results. Index Terms—Coupling coefficient, external quality factor, microstrip filter, stepped-impedance resonator, transmission zero.

I. INTRODUCTION N MODERN wireless and mobile communication systems, RF/microwave filters are always important and essential components. Planar filters are particularly popular structures because they can be fabricated using printed circuit technology and are suitable for commercial applications due to their compact size and low-cost integration [1]. Therefore, many applications to planar filters such as parallel- and cross-coupled bandpass filters have been extensively used in microwave communication systems due to their high-practicality, high-performance, and simple synthesis procedures [2]–[6]. Recently, in exploring advanced dual-band wireless systems, filters with dual-band operation for RF devices have become quite popular. Therefore, dual-band filters had been widely studied in several papers [7]–[17]. In [7], a dual-band filter was implemented by the combination of two individual filters with two specific single passbands. Extra impedance-matching networks must be used to design the input and output structure of the filter. In [8], three open stubs in parallel are introduced to create three transmission zeros for separating the two passbands. In [9], an optimization scheme based on hybrid-coded genetic-algorithm techniques is presented to design dual-band filters. However, physical meanings are less addressed in the design process. In [10], a dual-band bandpass filter is achieved by a cascade connection of a wideband bandpass filter and a bandstop filter, resulting in a large circuit size. Furthermore, dual-band resonators with open stubs in series and parallel [11],

I

Manuscript received April 12, 2006. This work was supported in part by the National Science Council under Grant NSC 93-2752-E-002-003-PAE. The authors are with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.880653

dual-band stepped-impedance resonator filters [12]–[16], and a dual-band filter using coupled resonator pairs [17] are also presented to achieve dual-passband responses. With the rapid evolution of multiband and multiservice communication systems, filters with a multiband response will be largely required. Unfortunately, to our knowledge, triple-band filters have not been presented in past literature. The aforementioned design techniques for dual-band filters based on a single filter circuit, however, are still challenging to the designer because it is difficult to exactly extract the desired coupling coefficients and external quality factors to simultaneously fit the specifications at both bands due to the limited degrees of freedom in the design parameters. Therefore, the concept of a dual-band filter based on a single circuit is quite difficult to extend to the design of triple-band filters. In order to conquer this problem without a significant increase in circuit size, this paper presents a new coupling structure by adding some extra coupled resonator sections to the single filter circuit. This novel structure can not only be used to design dual-band filters, but is also capable of designing triple-band filters. Both Chebyshev and quasi-elliptic response structures are proposed. Four experimental examples of filters with a dual-band Chebyshev, triple-band Chebyshev, dual-band quasi-elliptic, and triple-band quasi-elliptic response have been designed and implemented with microstrip technology to verify the proposed concept. This paper is organized as follows. Section II describes the proposed design concept. Several new coupling structures for dual- and triple-band filters have been presented. Section III characterizes the theory of a stepped-impedance resonator. The design graphs for determining the structural parameters of a stepped-impedance resonator are provided. Sections IV–VII provide the design procedures for dual- and triple-band microstrip bandpass filters with Chebyshev and quasi-elliptic function responses. The experimental data are presented and compared with the simulated results. Finally, Section VIII draws some brief conclusions. II. COUPLING STRUCTURES OF DUAL- AND TRIPLE-PASSBAND FILTERS Fig. 1(a) shows the coupling structure of a dual-band parallelcoupled filter with an th-order response, where each node represents a resonator and the solid lines represent the direct coupling routes. The odd-number resonators are able to operate simultaneously at the center frequencies of the first and second passbands ( and ). The even-number resonators with superscripts I and II are able to operate only at and , respectively. It should be noted that each pair of the even-number resonators operate at different frequencies, mutual coupling interference between them

0018-9480/$20.00 © 2006 IEEE

CHEN et al.: DESIGN OF DUAL- AND TRIPLE-PASSBAND FILTERS USING ALTERNATELY CASCADED MULTIBAND RESONATORS

3551

Fig. 2. Coupling structures of: (a) dual- and (b) triple-band cross-coupled filters with fourth-order response.

Fig. 1. Coupling structures of: (a) dual- and (b) triple-band parallel-coupled filters.

will not occur. This implies that each of the passbands can be implemented individually, i.e., each of the coupling coefficients between adjacent resonators can be extracted separately to meet the specifications. Degrees of freedom in extracting the coupling coefficients at the two specific passbands are then increased while compared to the conventional single-circuit dual-band filters. The design process is, therefore, more flexible and possible to be extended to achieve a multiband response. Fig. 1(b) shows the extension of this concept, i.e., the coupling structure of a triple-band parallel-coupled filter with an th-order response. In a similar manner as Fig. 1(a), the oddnumber resonators are able to operate simultaneously at the center frequencies of the first, second, and third passbands ( , , and ), while the even-number resonators with superscripts I–III are designed to operate only at , , and , respectively. As described above, each of the passbands in this structure can also be implemented individually. In other words, each of the coupling coefficients between adjacent resonators can be extracted separately at the three specific passbands. In order to improve the selectivity of filters, the coupling structures of dual- and triple-band filters with a quasi-elliptic frequency response have also been proposed. Fig. 2(a) shows the coupling structure of a dual-band cross-coupled filter with a fourth-order response, where each node represents a resonator. The solid and dashed line represent the direct coupling and the cross-coupling routes, respectively. The multipath effect achieved by the combination of a direct coupling path and a cross-coupling path introduces a single pair of transmission zeros near the upper and lower sides of the passband edges at finite frequency, thus a much sharper filter skirt and higher selectivity can be achieved. In this structure, resonators 1 and 4 are able to operate simultaneously at and , while the other resonators with superscripts I and II are designed to operate only at and , respectively. As previously mentioned, each of the coupling coefficients between adjacent resonators can also be extracted separately at the two specific passbands. The same concept is also extended to the design of a triple-band quasi-elliptic filter, whose coupling structures are shown in Fig. 2(b). It should be noted that the concept can be extended theoretically to design a filter with more passbands, however, problems caused by insufficient degrees of freedom of design parameters

Fig. 3. Structure of the stepped-impedance resonator. (a) 1. = (b)

K

Z =Z >

K = Z =Z1 < 1.

will arise in an actual design. For example, in the design of an input/output coupling structure, this will cause the restriction in designing the bandwidth of each passband. Moreover, in order to construct the odd-number resonators in Fig. 1 and resonators 1 and 4 in Fig. 2 for the simultaneous resonance at two or three specific frequencies, a simple kind of resonator that can control all the resonant frequencies, i.e., a stepped-impedance resonator, is used. This also limits the range of passband center frequency ratios if too many resonant frequencies are needed to be simultaneously controlled. Detailed descriptions for a steppedimpedance resonator will be presented in Section III. III. CHARACTERISTICS OF STEPPED-IMPEDANCE RESONATORS The stepped-impedance resonator was originally presented not only to control spurious responses, but also to reduce resonator size [18], [19]. Fig. 3(a) and (b) shows the typical structures of the half-wavelength stepped-impedance resonator for the cases of and , respectively, where is the impedance ratio defined as (1) If the length ratio fined as

of the stepped-impedance resonator is de-

(2) substituting (1) and (2) into the resonance conditions of the stepped-impedance resonator yields (3) and

(4)

3552

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 4. Design graph for stepped-impedance resonators.

where (3) and (4) correspond to the odd- and even-mode resonances, respectively. It should be noted that the fundamental frequency and the other higher order mode frequencies can be determined by properly choosing a suitable combination of the impedance and the length ratios of the stepped-impedance resonator. Fig. 4 shows the design graph for stepped-impedance resonators. Ratios of the first and second higher order resonant frequencies to the fundamental frequency ( and ) for half-wavelength stepped-impedance resonators are plotted with different values of and . It can be observed that, with typical values of and , realizable is between 1.2–3.8, and realizable is between 1.5–5.8. The case of , i.e., point A in Fig. 4, denotes a traditional half-wavelength unit-impedance resonator whose spurious frequencies are the multiples of its fundamental frequency. For example, if a resonator with and are required, one can locate point B with and to satisfy the given specifications. It is also evident from Fig. 4 that and for the cases of . The smaller the impedance ratio is, the larger and will be. On the contrary, if or is required, then the cases of must be chosen. As described in Section II, resonators that are able to operate simultaneously at two or three specific frequencies are demanded in the design processes. Since the objective of our design is to look for this kind of resonators, Fig. 4 is quite useful for designing dual- or triple-mode resonators. As a result, when two or three resonant frequencies of a resonator are specified, a suitable combination of and for the stepped-impedance resonator can be easily chosen from Fig. 4 and the dimensions of the resonator can, therefore, be determined. IV. DESIGN OF A DUAL-PASSBAND CHEBYSHEV FILTER To demonstrate the usefulness of the proposed design concept for a multipassband filter, a dual-passband filter with a third-order Chebyshev frequency response and 0.1-dB ripple

Fig. 5. (a) Coupling structure, (b) schematic layout, and (c) equivalent circuit of the dual-band Chebyshev filter.

level was designed and fabricated with the following specifications. The center frequencies of two bands are GHz and GHz. The lower and higher fractional bandwidths are and , respectively. The filter was designed to be fabricated using copper metallization on a Rogers RO4003 substrate with a relative dielectric constant of 3.38, a thickness of 0.508 mm, and a loss tangent of 0.002. Fig. 5(a) and (b) shows the coupling structure and circuit configuration of the filter. The equivalent-circuit model of the filter is shown in Fig. 5(c). The admittance inverter represents the coupling between resonators, where denotes the characteristic admittance. The first and last resonators (resonators 1 and 3) have to operate simultaneously at 2.8 and 4.2 GHz and, hence, their physical parameters can then be easily determined from Fig. 4. One of the suitable solutions is to choose the combinations of and . Besides, resonators 2 and 2 have to operate at 2.8 and 4.2 GHz, respectively. A half-wavelength unit-impedance resonator is employed to realize these two resonators and each of them are folded for a compact design. Since filter order and ripple level are specified and identical at the two passbands, the value of can be obtained by (5) where subscript 1 and 2 denotes the first and second passbands, respectively. The external quality factor can be characterized by (6)

CHEN et al.: DESIGN OF DUAL- AND TRIPLE-PASSBAND FILTERS USING ALTERNATELY CASCADED MULTIBAND RESONATORS

Fig. 6. Ratio

1 =1

versus the feeding point of input/output resonator.

3553

Fig. 7. Simulated and measured results of the dual-band Chebyshev filter.

where and represent the resonant frequency and the 3-dB bandwidth of the input or output resonator. The feeding points at the input and output resonators directly relate the filter bandwidth and external quality factor. Fig. 6 illustrates the simulated results of versus the feeding point of the input/ output resonator. As can be seen, once is specified, the feeding point can be easily determined. As indicated by this figure, when is adjusted from 2 to 9 mm, a wide range of , i.e., from 10 to 0.2, can be obtained. It should be noted that, once this figure is reconstructed for specific substrates, the other procedures in this paper can be repeated directly. As described in Section II, each of the passbands of the filter can be implemented individually. This is equivalent to the design of two single band filters independently, thus the traditional design procedure of a single band filter described in [5] can be followed directly. The design parameters of bandpass filter, i.e., the coupling coefficients and external quality factor, can be obtained from circuit elements of a low-pass prototype filter. The lumped circuit element values of the low-pass prototype filter are found to be , , , , and . In order to obtain the physical dimensions of the filter, the coupling coefficients and input/output external quality factors can then be found as and for the first passband and and for the second passband. A full-wave simulator has been used to extract the above parameters. When two synchronously tuned coupled resonators have a close proximity, the coupling coefficient can be evaluated from the two dominant resonant frequencies. If and are defined to be the lower and higher of the two resonant frequencies, respectively, the coupling coefficient can be obtained by

the feeding position of the first/last resonator from Fig. 6 to meet the specific or . The final step is to adjust the coupling gaps and , the overlapped lengths and to meet the specific coupling coefficients at the first and second passbands, respectively. Realizable design geometric parameters have been obtained as mm, mm, mm, mm, and mm. The prototype circuit size of the filter is approximately 49.6 mm 18.2 mm, i.e., approximately by , where is the guided wavelength on the substrate at the center frequency of the first passband. Measurement was carried out using an Agilent E5071B network analyzer. The measured and simulated results of the filter are illustrated in Fig. 7. As can be seen, a transmission zero at the lower side of each passband edge is observed. Due to the cross coupling between nonadjacent resonators, the electrical behavior is similar to the traditional trisection filters, as described in [5], thus the selectivity on the lower side is better than that on the higher side at each passband. It is also obvious that two additional transmission zeros occur at approximately 3.5 and 4.8 GHz, which are the frequencies when the lengths between the tapped points and open-ends of the input/output resonators approximate a quarter guided wavelength. The measured lower and higher fractional bandwidths are and , respectively. The measured passband return losses at both bands are below 19 dB, while the passband insertion losses are approximately 2.9 and 2.3 dB at the first and second passbands, respectively. The insertion losses would be attributed mainly to the conductor and dielectric losses. The measured results are in good agreement with the simulated predictions. V. DESIGN OF A DUAL-PASSBAND QUASI-ELLIPTIC FILTER

(7) represents the coupling coefficient between reswhere onators and . The overall design procedure of this example can be summarized as follows. First, determine the dimensions of each resonator to meet the respective frequencies and then determine

In order to improve the selectivity of dual-band filter, a quasielliptic filter with a dual-passband response has been designed. The circuit configuration of the filter is shown in Fig. 8(a), and its coupling structure is shown in Fig. 2(a). Fig. 8(b) shows the equivalent circuit model of the filter. It is obvious from Fig. 8(a) that compact folded stepped-impedance resonators and U-shaped unit-impedance resonators are used as the building

3554

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 9. Ratio

Fig. 8. (a) Schematic layout and (b) equivalent circuit of the dual-band quasielliptic filter.

block. This special arrangement of resonators helps not only to create the cross coupling between nonadjacent resonators, but also to reduce the circuit size. It can be anticipated that a single pair of transmission zeros near each passband at finite frequency will occur due to the multipath effect. In addition, the skew-symmetrical (0 ) feeding structure [20] is introduced to realize the input and output ports of the filter. A filter with a skew-symmetrical feeding structure could create extra transmission zeros in the stopband. Thus, the selectivity and out-of-band rejection of the filter can be significantly improved. The proposed filter is fabricated on the same Rogers RO4003 substrate as mentioned in the above section. The quasi-elliptic dual-band filter was designed with the following specifications. The center frequencies and fractional bandwidths are GHz, GHz, and , respectively. The lumped circuit element values of the low-pass prototype filter are found to be , , , , and . The required coupling coefficients and external quality factors are found to be , , , and for the first and second passbands. As previously mentioned, both passbands can also be designed individually, therefore, a single band filter design can be applied based on the knowledge of the coupling coefficients of the three basic coupling structures, i.e., electric, magnetic, and mixed couplings, which has been well documented in [5]. In the same manner as Section IV, since the filter-order and lumped circuit element values of the low-pass prototype filter are specified and identical at the two passbands, can be obtained by (5). Fig. 9 illustrates versus the feeding

1 =1

versus the feeding point of input/output resonator.

point of the input/output resonator. Once is specified, the feeding point can be easily determined. To implement this filter example, the first step is to determine the dimensions of each resonator to meet the respective frequencies. Resonators 1 and 4 based on a folded steppedimpedance resonator are designed to operate simultaneously at 2 and 3.1 GHz, resonators 2 and 3 based on a folded steppedimpedance resonator are design to operate at 2 GHz, and resonators 2 and 3 based on a U-shaped unit-impedance resonator are designed to operate at 3.1 GHz. The input and output feeding points are then selected for proper or . The last step is to individually determine the coupling gaps between resonators to match the specific coupling coefficients listed above. Additionally, it is worth mentioning that when the gap between the nonadjacent resonators (resonators 1 and 4) is given, the corresponding coupling coefficients at two passbands are unlikely the same. Fortunately, the required coupling coefficients between nonadjacent resonators are very small, which are about one order less than the others between adjacent resonators. Thus, a slight error ( 20%) in this coupling coefficient hardly affects passband performance. Fig. 10(a) presents a photograph of the filter. The size of the filter is 39.3 mm 20 mm, i.e., only approximately by , where is the guided wavelength on the substrate at the center frequency of the first passband. The measured and simulated results of the filter are illustrated in Fig. 10(b). As expected, there is a single pair of transmission zeros near each passband due to cross-coupling effects, i.e., transmission zeros at approximately 1.9, 2.1, 2.9, and 3.2 GHz can be observed. It can also be clearly observed that there are three extra transmission zeros at approximately 1.6, 2.45, and 3.6 GHz over the measured frequency range due to the skew-symmetrical feeding structure, resulting in a good stopband response. The measured lower and higher fractional bandwidths are and , respectively. The measured passband return losses are both below 15 dB at two passbands. The passband insertion losses are approximately 2.1 and 2.2 dB at the first and second passbands, respectively, which would be attributed

CHEN et al.: DESIGN OF DUAL- AND TRIPLE-PASSBAND FILTERS USING ALTERNATELY CASCADED MULTIBAND RESONATORS

3555

Fig. 11. (a) Schematic layout and (b) equivalent circuit of the triple-band quasielliptic filter.

Fig. 10. (a) Fabricated filter. (b) Simulated and measured results of the dualband quasi-elliptic filter.

mainly to the conductor and dielectric losses. The measured results are in good agreement with the simulated predictions. Compared with the dual-band quasi-elliptic filter in [15], there are three major advantages arising from the proposed configuration shown in Fig. 8. The first advantage is the flexible design process in our structure because it has increased the degrees of freedom in extracting the design parameters. The second one is the smaller circuit size in our structure due to the special arrangement of resonators. Moreover, extra impedance transformers at the input/output ports are required in [15], which are not needed in our design. Fig. 12. Fractional bandwidth design graph for the filter.

VI. DESIGN OF A TRIPLE-PASSBAND QUASI-ELLIPTIC FILTER In this design example, a quasi-elliptic filter with a triple-passband response is realized with compact folded stepped-impedance resonators and U-shaped unit-impedance resonators as the building block. The circuit configuration of the filter is shown in Fig. 11(a), and its coupling structure is shown in Fig. 2(b). The equivalent-circuit model of the filter is shown in Fig. 11(b). Similarly, this special arrangement of resonators helps not only to create the cross-coupled effects, but also to reduce the circuit size. Also, it can be anticipated that a single pair of transmission zeros near each passband will occur due to the multipath effect and extra transmission zeros in the stopband will occur due to the skew-symmetrical feeding structure.

Since filter order and lumped circuit element values of the low-pass prototype filter are specified and identical at the three passbands, the fractional bandwidth ratios can be written as

and

(8)

Fig. 12 illustrates the fractional bandwidth design graph for the filter. Obviously, once and are specified from the design curve, the feeding point can be easily determined. because there is only a single design parameter in designing the input/output tapped-line coupling structure, the range of realizable bandwidths is limited, as indicated.

3556

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

The proposed filter is fabricated on the same Rogers RO4003 substrate. The quasi-elliptic triple-band filter was designed with the following specifications. The center frequencies of three passbands are GHz, GHz, and GHz, and the fractional bandwidths are , , and . The same lumped circuit element values of the low-pass prototype filter, as mentioned in Section V, are used to design the filter. The coupling coefficients and the external quality factors are found to be , , , and for the first passband, , , , and for the second passband, and , , , and for the third passband. The design procedures of this example are as follows. The first step is to determine the dimensions of each resonator to meet the respective frequencies. Resonators 1 and 4 based on a folded stepped-impedance resonator are designed to operate simultaneously at 2.5, 3.6, and 5.1 GHz, resonators 2 and 3 based on a folded stepped-impedance resonator are designed to operate at 2.5 GHz, resonators 2 and 3 based on a U-shaped unit-impedance resonator are designed to operate at 3.6 GHz, and resonators 2 and 3 based on a U-shaped unit-impedance resonator are designed to operate at 5.1 GHz. The second step is to determine the feeding position of the first/last resonator from Fig. 12 to meet the specific and . The final step is to individually determine the coupling gaps between resonators to match the specific coupling coefficients listed above. Fig. 13(a) presents a photograph of the filter. The size of the filter is 32.5 mm 29.1 mm, i.e., only approximately by , where is the guided wavelength on the substrate at the center frequency of the first passband. The measured and simulated results of the filter are illustrated in Fig. 13(b). As expected, there are a single pair of transmission zeros near each passband due to cross-coupled effects, which can be observed at approximately 2.4, 2.6, 3.5, 3.7, 4.7, and 5.5 GHz. It can also be clearly observed that there are three extra transmission zeros at approximately 2.1, 2.9, and 4.2 GHz over the measured frequency range due to the skew-symmetrical feeding structure, resulting in a good stopband response. The measured fractional bandwidths are , , and . The measured passband return losses are all below 17 dB, while the passband insertion losses are approximately 2.9, 2.7, and 2.3 dB at the first, second, and third passbands, respectively. Again, the insertion losses would be attributed mainly to the conductor and dielectric losses. The measured results are in good agreement with the simulated predictions. As a result, the triple-band quasi-elliptic filter with the advantages of compact size and high selectivity is quite useful for multiband and multiservice applications in future mobile communication systems. VII. DESIGN OF A TRIPLE-PASSBAND CHEBYSHEV FILTER In order to overcome the problem of limited range of bandwidths in Section VI, the coupled-line coupling structure is employed to design the input/output coupling structure because it has more degrees of freedoms in the design process. Therefore, in the last design example, a triple-passband filter based

Fig. 13. (a) Fabricated filter. (b) Simulated and measured results of the tripleband quasi-elliptic filter.

on coupled-line coupling structure with third-order Chebyshev frequency response and a 0.1-dB ripple level was designed and fabricated with the given specifications. The center frequencies of three bands are GHz, GHz, and GHz. The fractional bandwidths are , , and . Fig. 14(a) and (b) shows the coupling structure and circuit configuration of the filter. The equivalent-circuit model of the filter is shown in Fig. 14(c). As can be seen, the input and output are coupled through coupled lines to resonators 1 and 3, and there are three design parameters ( , and ) in designing the coupled-line feeding structure. Thus, this solution is able to provide more choices in designing the bandwidths for three passbands. The filter is designed to be fabricated on a Rogers RO4003 substrate with a thickness of 1.524 mm. Fig. 15 illustrates the fractional bandwidth ratio design graph for the filter with mm, which is the simulated results evaluated from (8). More design data can be obtained by choosing different combinations of and . As a result, there are more choices in designing the bandwidths for three passbands when compared with that in Fig. 12. It should be noted that, when the substrate or filter structure is changed, the design graph should be restructured. However, once reconstructed, other procedures in this section can be followed directly. The same lumped circuit element values of the low-pass prototype filter, as mentioned in Section IV, are used to design the filter. Thus, the coupling coefficients and single-loaded external quality factors of the triple-band filter can then be found to be

CHEN et al.: DESIGN OF DUAL- AND TRIPLE-PASSBAND FILTERS USING ALTERNATELY CASCADED MULTIBAND RESONATORS

3557

Fig. 16. (a) Fabricated filter. (b) Simulated and measured results of the tripleband Chebyshev filter.

Fig. 14. (a) Coupling structure, (b) schematic layout, and (c) equivalent circuit of the triple-band Chebyshev filter.

Fig. 15. Fractional bandwidth design graph for the filter.

and and and band.

and

for the first passband, for the second passband, for the third pass-

The design procedure of this example is to at first determine the dimensions of each resonator to meet the respective frequencies. The first and last resonators (resonators 1 and 3) are designed to operate simultaneously at 2.3, 3.7, and 5.3 GHz. Their physical parameters can be determined by finding a combination of and (in this example, and ) from Fig. 4. Resonators and based on a folded unit-impedance resonator are designed to operate at 3.7 and 5.3 GHz, respectively. Resonator is designed to operate at 2.3 GHz, which is based on a folded stepped-impedance resonator due to compactness consideration. The second step is to determine the coupled-line feeding structure at the input/output of the first/last resonator from Fig. 15 to satisfy the specific and . Thirdly, determine the coupling gap to match the specific coupling coefficient of the third passband and, finally, adjust the parameters of and , and to match the specific coupling coefficients of the first and second passbands, respectively. Fig. 16(a) is the photograph of the filter. Geometric parameters for the filter are mm, mm, mm, mm, mm, mm, mm, and mm. The resulting circuit size is approximately 54.1 mm 22.9 mm, i.e., approximately by , where is the guided wavelength on the substrate at the center frequency of the first passband. The measured and simulated results of the filter are illustrated in Fig. 16(b). A transmission zero at the lower side of each passband is also observed, which is due the cross coupling between nonadjacent resonators. The measured fractional bandwidths are , , and . The measured passband return losses are at least below 18 dB, and the passband insertion

3558

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

losses are approximately 2.5, 1.9, and 2.9 dB at the first, second, and third passbands, respectively. The insertion losses would be attributed mainly to the conductor and dielectric losses. The measured results are again in good agreement with the simulated predictions. VIII. CONCLUSION In this paper, novel coupling structures have been presented to design dual- and triple-band microstrip bandpass filters without a significant increase in circuit size. Both theory and experiments have been provided. Several layouts of filters with dualband Chebyshev, dual-band quasi-elliptic, triple-band Chebyshev, and triple-band quasi-elliptic response have been designed and implemented with microstrip technology to demonstrate the proposed concepts. Different input/output coupling structures have been designed and compared in the two triple-band filter examples, showing that a filter with the coupled-line feeding structure is able to provide more choices in designing bandwidths. For the four examples, the measured results are all in good agreement with simulated predictions. No extra matching networks were needed for all four filters. As a result, these compact-size filter circuits are particularly suitable for multiband and multiservice applications in future mobile communication systems. REFERENCES [1] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998, ch. 8. [2] S. B. Cohn, “Parallel-coupled transmission-line-resonator filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-6, no. 4, pp. 223–231, Apr. 1958. [3] E. G. Cristal and S. Frankel, “Hairpin-line and hybrid hairpin-line/halfwave parallel-coupled-line filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 11, pp. 719–728, Nov. 1972. [4] J. S. Hong and M. J. Lancaster, “Design of highly selective microstrip bandpass filters with a single pair of attenuation poles at finite frequencies,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 7, pp. 1098–1107, Jul. 2000. [5] ——, Microstrip Filter for RF/Microwave Application. New York: Wiley, 2001. [6] S. Y. Lee and C. M. Tsai, “New cross-coupled filter design using improved hairpin resonators,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2482–2490, Dec. 2000. [7] H. Miyake, S. Kitazawa, T. Ishizaki, T. Yamada, and Y. Nagatom, “A miniaturized monolithic dual band filter using ceramic lamination technique for dual mode portable telephones,” in IEEE MTT-S Int. Microw. Symp. Dig., 1997, vol. 2, pp. 789–792. [8] C. Quendo, E. Rius, and C. Person, “An original topology of dual-band filter with transmission zeros,” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, vol. 2, pp. 1093–1096. [9] M. I. Lai and S. K. Jeng, “Compact microstrip dual-band bandpass filters design using genetic-algorithm technique,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 160–168, Jan. 2006. [10] L. C. Tsai and C. W. Hsue, “Dual-band bandpass filters using equallength coupled-serial-shunted lines and Z -transform technique,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1111–1117, Apr. 2004. [11] C. M. Tasi, H. M. Lee, and C. C. Tsai, “Planar filter design with fully controllable second passband,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3429–3439, Nov. 2005. [12] H. M. Lee, C. R. Chen, C. C. Tsai, and C. M. Tsai, “Dual-band coupling and feed structure for microstrip filter design,” in IEEE MTT-S Int. Microw. Symp. Dig., 2004, vol. 3, pp. 1971–1974. [13] S. Sun and L. Zhu, “Compact dual-band microstrip bandpass filter without external feed,” IEEE Microw. Wireless Compon. Lett., vol. 15, pp. 644–646, Oct. 2005.

[14] M. L. Chuang, “Concurrent dual band filter using single set of microstrip open-loop resonators,” Electron. Lett., vol. 41, pp. 1013–1014, Sep. 2005. [15] J. T. Kuo and H. S. Cheng, “Design of quasi-elliptic function filters with a dual-passband response,” IEEE Microw. Wireless Compon. Lett., vol. 14, pp. 472–474, Oct. 2004. [16] J. T. Kuo, T. H. Yeh, and C. C. Yah, “Design of microstrip bandpass filters with a dual-passband response,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1331–1337, Apr. 2005. [17] C. C. Chen, “Dual-band bandpass filter using coupled resonator pairs,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 259–261, Apr. 2005. [18] M. Makimoto and S. Yamashita, “Bandpass filters using parallel coupled stripline stepped impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. 28, no. 12, pp. 1413–1417, Dec. 1980. [19] M. Sagawa, M. Makimoto, and S. Yamashita, “Geometrical structures and fundamental characteristics of microwave stepped-impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1078–1085, Jul. 1997. [20] C. M. Tsai, S. Y. Lee, and C. C. Tsai, “Performance of a planar filter using a 0 feed structure,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2362–2367, Oct. 2002. Chi-Feng Chen was born in PingTung, Taiwan, R.O.C., on September 3, 1979. He received the B.S. degree in physics from Chung Yuan Christian University, Taoyuan, Taiwan, R.O.C., in 2001, the M.S. degree in electrophysics from the National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 2003, and is currently working toward the Ph.D. degree in communication engineering at National Taiwan University, Taipei, Taiwan, R.O.C. His research interests include the design of microwave filters and associated RF modules for microwave and millimeter-wave applications.

Ting-Yi Huang was born in Hualien, Taiwan, R.O.C., on November 12, 1977. He received the B.S. degree in electrical engineering and M.S. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2000 and 2002, respectively, and is currently working toward the Ph.D. degree in communication engineering at National Taiwan University. His research interests include computational electromagnetics, the design of microwave filters, transitions, and associated RF modules for microwave and millimeter-wave applications.

Ruey-Beei Wu (M’91–SM’97) received the B.S.E.E. and Ph.D. degrees from National Taiwan University, Taipei, Taiwan, R.O.C., in 1979 and 1985, respectively. In 1982, he joined the faculty of the Department of Electrical Engineering, National Taiwan University, where he is currently a Professor. He is also with the Graduate Institute of Communications Engineering, which was established in 1997. From March 1986 to February 1987, he was a Visiting Scholar with IBM, East Fishkill, NY. From August 1994 to July 1995, he was with the Electrical Engineering Department, University of California at Los Angeles (UCLA). From 1998 to 2000, he was also appointed Director of the National Center for High-Performance Computing, and has served as Director of Planning and Evaluation Division since November 2002, both under the National Science Council. His areas of interest include computational electromagnetics, transmission line and waveguide discontinuities, microwave and millimeter-wave planar circuits, and interconnection modeling for computer packaging.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3559

Tunable Bandstop Defected Ground Structure Resonator Using Reconfigurable Dumbbell-Shaped Coplanar Waveguide Amr M. E. Safwat, Member, IEEE, Florence Podevin, Philippe Ferrari, and Anne Vilcot, Member, IEEE

Abstract—A modification of the conventional dumbbell-shaped coplanar waveguide defected ground structure (DGS) is proposed. This modification permits the continuous tuning of the rejected frequencies by using reconfiguration technique and it allows the control of the DGS equivalent-circuit model. The modified DGS possesses two-dimensional symmetry, hence, it has been studied under different symmetry conditions and the corresponding equivalent-circuit model in each case has been developed. Based upon this study, a tunable bandstop DGS resonator is proposed. 19% tuning range centered at 3.7 and 7.4 GHz, respectively, is achieved. The equivalent-circuit model of the resonator is also developed. All proposed structures have been fabricated. Measurements as well as three-dimensional simulations are found to be in a very good agreement with theoretical predictions. Index Terms—Coplanar waveguide (CPW), defected ground structure (DGS), tunable bandstop resonator.

I. INTRODUCTION ECENTLY, defected ground structures (DGSs) have been used numerously and they have shown increasing potential for implementation in several applications [1]–[5]. Different DGSs have been proposed in either microstrip or coplanar waveguide (CPW) to achieve lower rejected frequency for smaller defect dimensions [6]–[12]. Among these structures, the two-dimensional (2-D) periodic dumbbell- and L-shaped DGSs were proposed by one of the authors [7], [11]. Both structures were etched on the ground of the CPW and they preserved the 2-D symmetry. As in a typical DGS, the propagation of a certain band of frequencies was prevented and the equivalent-circuit model was the parallel resonance circuit. In this paper, a modification of the conventional dumbbellshaped DGS is proposed. The modification results in a new equivalent-circuit model that depends on the symmetry of the DGS. The model varies from the conventional single resonator to three resonators connected in series. The relation between the model and the DGS symmetry has been formulated. Moreover, this modification allows a reconfiguration technique to tune the

R

Manuscript received November 19, 2005; revised March 6, 2006. A. M. E. Safwat is with the Faculty of Engineering, Electronics and Communications Engineering Department, Ain Shams University, 11517 Cairo, Egypt (e-mail: [email protected]). F. Podevin, P. Ferrari, and A. Vilcot are with the Institut de Microélectronique, Electromagnétisme et Photonique, Ecole Nationale Supérieure d’Electronique et de Radioélectricité de Grenoble, Grenoble Cedex 38016, France (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.880654

Fig. 1. Schematic diagram of: (a) standard dumbbell-shaped DGS, (b) modified dumbbell-shaped DGS, and (c) tunable bandstop filter.

rejected frequency. The tuning range extends up to 19% of the central frequency. This paper is organized as follows. Section II presents the basic theory of the proposed modification, the reconfiguration technique, the simulation, and the measurement results that confirm the idea. In Section III, the symmetry of the modified defected ground structure (M-DGS) is studied. Different configurations are analyzed using both simulation and measurement. For each configuration, the equivalent-circuit model is also developed. The tunable bandstop DGS resonator and its equivalent-circuit model are presented in Section IV, which also includes simulation and measurement results of the fabricated filter. II. MODIFIED DUMBBELL-SHAPED DGS The dumbbell-shaped DGS, shown in Fig. 1(a), was first proposed in [1]. It is symmetrical with respect to the center line and it has three design parameters, i.e.: 1) the square area ; 2) the gap opening ( ); and 3) the separation between the square and the edge of the ground ( ). The equivalent-circuit model is a parallel inductor ( ), capacitor ( ), and resistor ( ) whose values depend on the dimensions of the defect. Generally, the capacitance is dominated by and , while the inductance is dominated by and , which is due to the electromagnetic (EM) field distribution in the defect [11]. The resistance corresponds to the different types of losses that may exist in the defect, which include radiation and conductor losses. In the M-DGS, instead of removing the metal from inside the defect, only a strip of width is removed along the circumference keeping a smaller metal square (patch) inside the defect,

0018-9480/$20.00 © 2006 IEEE

3560

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

as shown in Fig. 1(b). In the structures presented in this paper, was chosen to be equal to , the gap opening. The presence of the square patch allows the insertion of two terminal components, as shown in Fig. 1(c), such that one terminal is connected to the patch while the second is connected to the ground. Moreover, bias can be supplied to the patch to control the operation of the inserted devices. In such a case, the current path in the ground will be perturbed creating a resonance circuit in which the parameters are variable and, hence, a tunable bandstop DGS resonator is achieved. The current generated on the borders of the patch will have an opposite direction to the one existing on the CPW ground. However, for a large , the current on the CPW ground will be slightly affected. Hence, the performance of the M-DGS is expected to be very close to the conventional DGS. The current distributions of both, the conventional and the M-DGS, are shown in Fig. 2(a) and (b), respectively. Both structures were simulated using the three-dimensional (3-D) EM commercial software, CST Microwave Studio [13], using the following dimensions: is 0.5/4/0.5 mm, where is the width of the center line and is the gap of the CPW, is 0.5 mm, is 1 mm, is 6 mm, and is 5 mm. The substrate height is 1.6 mm and the dielectric constant is 4.6. Both structures were fabricated on FR4 whose parameters are similar to those used in the simulation. A picture of the fabricated M-DGS is shown in Fig. 2(c). Measured data are plotted on Fig. 2(d), which also shows simulation results. Conventional DGS and M-DGS have the same shape with a slight shift in the central frequency, which confirms the theoretical prediction. Measurements are in a good agreement with simulations. Some discrepancies appear at frequencies higher than the resonance frequency, which is due to the radiation loss.

Fig. 2. Current distribution on the upper ground of: (a) the conventional dumbbell-shaped DGS and (b) the modified dumbbell-shaped DGS. (c) Fabricated structure. (d) Simulation results of the two structures versus measurements.

III. SYMMETRY STUDY OF THE M-DGS The M-DGS, as shown in Fig. 2(b), has two axes of symmetry and . The insertion of any elements may disturb the current in both dimensions and, hence, the equivalent circuit may not be as a simple resonance circuit as in the case of the conventional DGS. Therefore, three different cases of the M-DGS have been studied: firstly, four short circuits have been inserted so that the 2-D symmetry is preserved, secondly, two short circuits have been inserted so that one-dimensional (1-D) (the ) symmetry is preserved while the second symmetry ( ) is violated, thirdly, one short circuit has been inserted so that the 2-D symmetries are violated. In each case, the structure was simulated using the 3-D EM simulator, the equivalent circuit was developed, and the simulation results were confirmed by the experimental data. A. M-DGS With Four Short Circuits Fig. 3(a) shows the M-DGS with four short circuits, two at each side. They are parallel to the center line and they connect the patch to the CPW ground. The width of each short circuit, i.e., , is 1 mm. All other dimensions are kept the same. This configuration preserves the symmetry in the two dimensions.

The structure was also fabricated on FR4. The fabricated structure is shown in Fig. 3(b). Fig. 3(c) shows the EM simulated versus the measured . Full agreement was achieved. The obtained behavior is similar to the case of the conventional DGS and/or the M-DGS with the rejected frequency shifted to a higher frequency, 7.6 GHz instead of 4.7 GHz as in the case of the M-DGS. This can be explained by the fact that the combination of the current path, which is shorter in this case, and the presence of the defect in the regions of high EM fields, lead to a smaller equivalent inductance and/or capacitance and, thus, a higher rejected frequency. The equivalent circuit of the M-DGS with four short circuits can simply be presented by a parallel , and , as shown in Fig. 3(d), which is similar to the presentation of the conventional DGS and/or M-DGS. The extraction of these parameters is straightforward [1]. , , and are found to be equal to 1530 , 1.27 nH and 0.35 pF, respectively. The simulation results of the equivalent circuit, which consists of the equivalent circuit of the defect and two transmission lines each of length equal to the half of the structure length corresponding to the two CPW lines at both sides, is also shown in Fig. 3(c). Full agreement with the EM simulation and measurement results can be observed.

SAFWAT et al.: TUNABLE BANDSTOP DGS RESONATOR USING RECONFIGURABLE DUMBBELL-SHAPED CPW

Fig. 3. (a) 2-D symmetrical four-short M-DGS. The black arrow shows the direction of the current. (b) Fabricated structure (c) EM, circuit simulated, and measured S -parameters. (d) Equivalent-circuit model.

B. M-DGS With Two Short Circuits The M-DGS with two short circuits is similar to the previous configuration, except that two short circuits have been removed. The resulting structure preserves the symmetry around the axis, while it violates the symmetry. The schematic of the M-DGS with two shorts is shown in Fig. 4(a), while Fig. 4(b) shows the picture of the fabricated structure on FR4. Fig. 4(c) shows the EM simulated data and the measurement results as well. Full agreement can be observed. These data present a very interesting feature, which is the appearance of two resonance frequencies. This is similar to the case presented in [12], in which an asymmetric DGS relative to the center line had been studied. However, now we can conclude that the presence of a single resonance in M-DGS necessitates the preservation of the symmetry not only in 1-D, the center line, but in 2-D, as shown in Fig. 2(b). The presence of two resonance frequencies can be presented by the equivalent-circuit model shown in Fig. 4(d). It consists of

3561

Fig. 4. (a) 1-D symmetrical two-short M-DGS. The black arrow shows the direction of the current. (b) Fabricated structure (c) EM, circuit simulated, and measured S -parameters. (d) Equivalent-circuit model.

two resonance circuits; each has a parallel and . These parameters were extracted from the EM simulated data and their values were found to be as follows: is 1330 , is 0.87 nH, is 0.47 pF, is 878 , is 2.08 nH, and is 1.15 pF. The of the circuit simulation is also included in Fig. 4(c). The full agreement with the EM and measured data validates the proposed equivalent-circuit model of the defect. C. M-DGS With One Short Circuit The M-DGS with one short circuit, shown in Fig. 5(a), violates the 2-D symmetry. This configuration, similar to those previously described, was studied using the 3-D EM simulation. The structure was also fabricated on FR4, and its picture is shown in Fig. 5(b). The EM and the measured results, shown in Fig. 5(c), suggest that the M-DGS with one short circuit is a superposition of the M-DGS and the M-DGS with two short circuits and hence three resonance frequencies appear in . This new configuration

3562

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 6. (a) Layout of the M-DGS with two short circuits positioned at the top. (b) EM simulated data.

Fig. 5. (a) Aymmetrical one-short M-DGS. (b) Fabricated structure. (c) EM, circuit simulated, and measured S -parameters. (d) Equivalent-circuit model.

may open the door for wideband stop filters with very compact defect size, which can be achieved by the proper choice of the defect dimensions and the position of the short. Hence, the equivalent circuit of this configuration is three cascaded parallel resonance circuits, as shown in Fig. 5(d). The parameter extraction is straightforward as well. In this case, the extracted parameters were found to have the following values: is 1069 , is 1.03 nH, is 2.02 pF, is 714 , is 0.72 nH, is 1.52 pF, is 988 , is 0.46 nH, and is 1.04 pF. The of the circuit simulation is also included in Fig. 5(c), and full agreement with the EM and measured data is achieved. D. Discussion In M-DGS, the multiple resonance behavior could be due to the current path, as shown in Figs. 3 and 4, i.e., when there is one resonance the current paths in the upper and lower DGS are identical and have the form of a loop. For two resonances,

the current paths have the form of two loops inside each other, which produces two resonances, and for three resonances, there are three loops, one in the upper half of the DGS and two in the lower half. The question now is: do these configurations converge? And to insert a variable device, where can it be placed? To answer these questions, the structure shown in Fig. 6(a) was EM simulated. It has two short circuits placed at the top of the defect. It can be seen that it is the limit at which the M-DGS with four short circuits and the M-DGS with two short circuits converge since it preserves the 2-D symmetry. Moreover, from the EM simulation, shown in Fig. 6(b), this structure has one resonance frequency, which coincides with the resonance frequency of the M-DGS. Hence, it is the limit of the M-DGS as well. Therefore, placing a variable device in this position will not affect its operation. IV. TUNABLE BANDSTOP DGS RESONATOR The tunability of the M-DGS can be achieved if short circuits are replaced by varactors. However, in this case, the resonance frequency will not be determined by the DGS only, but by the combination of the equivalent circuit of the DGS and the equivalent circuit of the varactors. The position of the short circuits studied previously serves in determining the location of the inserted varactors. These, in turn, are inserted at approximately the same position as the short circuits in the case of the M-DGS with two short circuits, as shown

SAFWAT et al.: TUNABLE BANDSTOP DGS RESONATOR USING RECONFIGURABLE DUMBBELL-SHAPED CPW

3563

Fig. 7. (a) Schematic of the tunable bandstop filter. (b) Fabricated M-DGS with the capacitors. (c) Measured, equivalent-circuit model, and EM simulation data. In this case, C = 2:5 pF.

in Fig. 7(a). Bias was provided through a wire bonding that connects the bias pad to the patch. As an alternative, bias can also be provided by connecting the patch to a thin metal strip, which can be realized on the bottom plane of the substrate and carries the bias through vias. Fig. 7(b) shows the fabricated structure. This configuration has been chosen since it gives a low resonance frequency and it requires less varactors. Fig. 7(c) shows the measured data. The equivalent-circuit model is similar to the M-DGS with two short circuits. The simulated circuit response, as well as the EM simulation, are also depicted in Fig. 7(c). Excellent agreement between the circuit model and measured data is observed. Little discrepancy between EM simulation and measurements is due to the capacitor parasitic effects. The value of the capacitor used in Fig. 7(a) is 2.5 pF and they are two; one is connected in the upper ground plane and the second is connected to the lower ground plane. The values of the equivalent-circuit model are as follows: , , and equal 1.25 pF, 1.7 nH, and 300 , respectively. These three determine the shape and magnitude of the first resonance, while , , and , which equal 0.4 pF, 1.4 nH, and 940 , respectively, determine the shape and magnitude of the second resonance. The variation of the resonance frequency is shown in Fig. 8(a) and (b), where measured and are depicted for three different values of the varactors, i.e., 0.4, 0.7, and 2.5 pF, respec-

Fig. 8. (a) S and (b) S at three capacitance values: 0.4, 0.7, and 2.5 pF. (c) The variation of the two resonance frequencies as a function of the inserted capacitances.

tively. These values were realized using the commercial M/A COM MA46H071-1088 varactor diode. Other alternatives may provide a wider range of variation. Fig. 8(c) shows the variation of the two resonance frequencies as a function of the inserted capacitances both in EM simulation and measurement. Two tuning ranges are obtained, the first is 700 MHz centered at 3.7 GHz, and the second is 1.4 GHz centered at 7.4 GHz corresponding to 19%.

3564

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

For different values of the varactor, the parameters of the equivalent circuit were extracted. In each case, a full agreement as was shown in Fig. 7(c) was achieved. Clearly the presence of the varactor disturbs the field distribution and, hence, varies the values of the equivalent-circuit parameters. V. CONCLUSION A square patch has been inserted inside the conventional dumbbell-shaped DGS to modify its frequency response and to allow the control of the rejected frequency. Low, high, or even multiple frequencies can now be rejected by the proper choice of the positions of the short circuits that can be placed along the circumference of the square patch. The M-DGS possesses 2-D symmetry. The performance of the structure in terms of the symmetry preservation has been carefully studied. The relation between the symmetry, the number of resonance frequencies, and the equivalent-circuit model has been concluded. M-DGS also allows the insertion of variable devices between the square patch and the CPW ground, and bias can always be provided. This led to a tuning capability using varactors. A tuning range that extends to 19% centered at 3.7 and 7.4 GHz was achieved using commercially available diode varactors. REFERENCES [1] D. Ahn, J. S. Park, C. S. Kim, Y. Qian, and T. Itoh, “A design of the low-pass filter using the novel microstrip defected ground structure,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 86–93, Jan. 2001. [2] Y.-T. Lee, J.-S. Lim, S. Kim, J. Lee, S. Nam, K.-S. Seo, and D. Ahn, “Application of CPW based spiral-shaped defected ground structure to the reduction of phase noise in V -band MMIC oscillator,” in IEEE MTT-S Int. Symp. Dig., 2003, pp. 2253–2256. [3] K. T. Chan, A. Chin, M.-F. Li, D.-L. Kwong, S. P. McAlister, D. S. Duh, W. J. Lin, and C. Y. Chang, “High-performance microwave coplanar bandpass and bandstop filters on Si substrates,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 9, pp. 2036–2040, Sep. 2003. [4] J.-S. Yoon, J.-G. Kim, J.-S. Park, C.-S. Park, J.-B. Lim, H.-G. Cho, and K.-Y. Kang, “A new DGS resonator and its application to bandpass filter design,” in IEEE MTT-S Int. Symp. Dig., 2004, pp. 1605–1608. [5] M.-L. Her, C.-M. Chang, Y.-Z. Wang, F.-H. Kung, and Y.-C. Chiou, “Improved coplanar waveguide (CPW) bandstop filter with photonic bandgap (PBG) structure,” Microw. Opt. Technol. Lett., vol. 38, no. 4, pp. 274–277, Aug. 2003. [6] E. K. I. Hamad, A. M. E. Safwat, and A. Omar, “L-shaped defected ground structure for coplanar waveguide,” in IEEE AP-S/USNC/URSI Nat. Radio Sci. Int. Symp., Washington, DC, Jul. 3–8, 2005. [7] ——, “2D periodic defected ground structure for coplanar waveguide,” in Germany Microw. Conf., Ulm, Germany, Apr. 2005, pp. 25–28. [8] J.-S. Lim, Y.-T. Lee, C.-S. Kim, D. Ahn, and S. Nam, “A vertically periodic defected ground structure and its application in reducing the size of microwave circuits,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 12, p. 479, Dec. 2002. [9] T.-Y. Yun and K. Chang, “Uniplanar one-dimensional photonic-bandgap structures and resonators,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 86–93, Mar. 2001. [10] Q. Xue, K. M. Shum, and C. H. Chan, “Novel 1-D microstrip PBG cells,” IEEE Microw. Guided Wave Lett., vol. 10, pp. 403–405, Oct. 2000. [11] E. K. I. Hamad, A. M. E. Safwat, and A. Omar, “Controlled capacitance and inductance behavior of L-shaped defected ground structure for coplanar waveguide,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 152, no. 5, pp. 299–304, Oct. 2005. [12] D. Woo and T. Lee, “Suppression of harmonics in Wilkinson power divider using dual-band rejection by asymmetric DGS,” IEEE Trans. Microw. Theory Tech., vol. MTT-53, no. 6, p. 2139, Jun. 2005. [13] “Microwave Studio User’s Guide,” CST, Wellesley, MA, 2003.

Amr M. E. Safwat (S’91–M’01) was born in Cairo, Egypt, in May 1970. He received the B.Sc. and M.Sc. degrees from Ain Shams University, Cairo, Egypt, in 1993 and 1996, respectively, and the Ph.D. degree from the University of Maryland at College Park, in 2001, all in electrical engineering. From 1993 to 1997, he was a Research and Teaching Assistant with Ain Shams University. From 1997 to 2001, he was a Research Assistant with the Laboratory for Physical Sciences, University of Maryland at College Park. From August 2001 to August 2002, he was with Cascade Microtech Inc., where he codeveloped infinity probes and on-wafer differential calibration standards. In August 2002, he joined the Electronics and Communications Engineering Department, Ain Shams University, as an Assistant Professor. In 2004, he was a Visiting Professor with Otto-Von-Guericke University, Magdeburg, Germany. In 2005, he was with the Institut National Polytechnique de Grenoble (INPG), Grenoble, France. His research interests include on-wafer probing, microwave passive planar structures, and microwave photonics.

Florence Podevin was born in Arras, France, on August 18, 1976. She received the M.Sc degree in electronics and microelectronics engineering and Ph.D. degree in microelectronics from the Université des Sciences et Technologies de Lille (USTL), Lille, France, in 1998 and 2001, respectively. In 2001, she joined the Institut de Microélectronique, Electromagnétisme et Photonique (IMEP), Grenoble, France, as an Assistant Professor, where she is involved with the optical control of microwave devices and periodic structures filtering.

Philippe Ferrari was born in Ugine, France, in 1966. He received the B.Sc. degree in electrical engineering and Ph.D. degree from the Institut National Polytechnique de Grenoble (INPG), Grenoble, France, in 1988 and 1992, respectively. In 1992, he joined the Laboratory of Microwaves and Characterization, University of Savoy, Savoy, France, as an Assistant Professor of electrical engineering. From 1998 to 2004, he was the Head of the laboratory project on nonlinear transmission lines and tunable devices. Since September, 2004, he has been a Professor with the University Joseph Fourier of Grenoble, France, and continues his research with the Institut de Microélectronique, Electromagnétisme et Photonique (IMEP), INPG. His main research interest is the conception and realization of tunable and miniaturized devices such as filters, phase shifters, and power dividers, and also new circuits based on periodical structures such as filters or phase shifters. He is also involved in the development of time-domain techniques for the measurement of passive microwave devices and soil moisture content.

Anne Vilcot (M’93) received the grade of Engineer in Electronics from the Ecole Nationale Supérieure d’Electronique et de Radioélectricité de Grenoble (ENSERG), Institut National Polytechnique de Grenoble (INPG), Grenoble, France, in 1989, and the Ph.D. degree in microwaves from the Laboratory of Electromagnetism Microwaves and Optoelectronics (LEMO), Grenoble, France, in 1992. In 1989, she joined LEMO. Since then, she has been involved with the optical control of microwave devices. She is currently a Professor with the Institut National Polytechnique de Grenoble (INPG), Grenoble, France, and the ViceDirector of the Institut de Microélectronique, Electromagnétisme et Photonique (IMEP). Her research interest is the field of microwaves photonics.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3565

Adaptive Nonuniform-Grid (NG) Algorithm for Fast Capacitance Extraction Amir Boag, Senior Member, IEEE, and Boris Livshitz, Student Member, IEEE

Abstract—A novel method for computing the capacitance matrices of arbitrary shaped three-dimensional geometries is presented. The proposed approach combines a novel nonuniform-grid (NG) algorithm for fast evaluation of potentials due to given source distributions with an iterative solution of the pertinent integral equations. The NG algorithm is based on the observation that locally the potential produced by a finite size source can be interpolated from its samples at a small number of points of a nonuniform spherical grid. This observation leads to a multilevel algorithm comprising interpolation and aggregation of potentials. The resulting hierarchical algorithm attains an ( ) asymptotic complexity and memory requirements. The computational efficiency is further improved for quasi-planar geometries by the use of adaptive grids. Index Terms—Capacitance extraction, circuit simulation, fast algorithm, iterative solver, multilevel algorithm.

I. INTRODUCTION APACITANCE extraction in complicated interconnect geometries is often effected by a numerical solution of the integral equation, which is based on a representation of the electrostatic potential in terms of unknown surface charge densities. Discretization of the integral equation using the conventional method of moments (MoM) [1], also referred as the boundaryelement method, leads to generation of a system of linear equations. The resulting number of unknowns is typically quite large due to the geometrical complexity of the structures of practical interest. To that end, the computational complexity of direct MoM solvers makes them virtually impractical and necessitates the use of iterative schemes. When solving the integral equations iteratively, each iteration requires an evaluation of the electric potential due to a given surface charge. The classical evaluation of this potential at points by surface integration using quadrature points (an equivalent of the direct MoM matrix–vector product) requires operations. This high computational burden underscores the need for fast evaluation techniques such as the fast multipole method (FMM) [2]–[5], the multiscale (multigrid) methods [6], [7], and the singular value decomposition (SVD) compression approach [8], all of which evaluate the potential in operations as opposed to cost of the direct computation. However,

C

Manuscript received November 28, 2005; revised April 15, 2006. This work was supported by the Israel Science Foundation under Grant 224/03. A. Boag is with the School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected]). B. Livshitz was with the School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel. He is now with the Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA 92037 USA. Digital Object Identifier 10.1109/TMTT.2006.879776

complexity the large constant multipliers hidden in the and memory requirements of the above methods remain to be improved. In this paper, we present a novel static version of the recently proposed nonuniform grid (NG) approach [9]–[11], which facilitates a numerically efficient evaluation of potentials produced by given charge distributions. The approach is based on the observation that the potential at a sufficient distance away from a finite size source can be interpolated from its samples on a spherical NG. Furthermore, the number of points in the NG depends only on the distance of the observation domain from the source domain normalized to the size of latter. With this in mind, the NG algorithm relies on a multilevel domain decomposition of the sources. The potential is computed directly for the NGs of the smallest subdomains at the finest level of decomposition. The near-field potentials are also computed directly. Subsequently, the potentials of progressively larger subdomains on the corresponding NGs are computed by interpolation and aggregation. Finally, the potential at the observation points is computed by multilevel interpolation employing a hierarchy of local grids. The resulting hierarchical algorithm attains an asymptotic complexity of similar to that of the FMM. The NG approach can be considered a spatial counterpart of the FMM, which relies on spectral (multipole) representation. Also note that the static version of the NG approach is closely related to the NG-based frequency- and time-domain MoM analysis schemes [12]–[14], as well as the fast physical optics algorithms for high-frequency scattering [15]. Indeed, the direct spatial sampling of the fields and potentials allows seamless transition from static all the way to the high-frequency–dynamic-frequency regime. The numerical results support the performance characteristics of the proposed method. Being an essentially spatial technique (as opposed to the spectral nature of the FMM), this approach can also be easily adapted to quasi-planar [two-andone-half dimensional (2.5-D)], elongated, and other special geometries. In those specific cases, which are common in modern interconnects and packaging, the algorithm attains significant reduction in complexity and use of memory. Since the NG approach uses spatial sampling of the conventional Green’s function, it is also easy to adapt as an enhancement to existing MoM codes. The outline of this paper is as follows. In Section II, we formulate the problem of capacitance extraction and describe the standard integral-equation approach leading to matrix equation generation. In Section III, the NG algorithm is then formulated and analyzed based on the NG sampling criteria derived in the Appendix. In Section IV, we describe some numerical

0018-9480/$20.00 © 2006 IEEE

3566

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

experiments demonstrating the performance of the multilevel NG algorithm coupled with an iterative solver for solving the capacitance extraction problem. Finally, conclusions are given in Section V. II. PROBLEM SPECIFICATION Consider the capacitance extraction problem for a three-dimensional (3-D) complex structure comprising conductors embedded in a uniform dielectric medium characterized by permittivity . Extension of the current formulation to piecewise homogeneous dielectric media can be effected by introducing surface polarization charges and will be discussed elsewhere. The capacitance matrix is defined by relation (1) is the total charge on the th conductor and is the where voltage on conductor . The th element of the capacitance matrix can be found by calculating charge induced on conductor by enforcing a potential of 1 V on conductor while grounding the rest. The total charge on conductor , is found by integrating the surface charge density over the surface of the conductor

Fig. 1. NG for a source box. (Color version available online at http://ieeexplore. ieee.org.)

entail that the potentials can be computed with a prescribed accuracy using a proper interpolation rule. The far-field and local interpolation representations to be employed in the algorithm are the spatial counterparts of the spectral entities of the FMM, namely, the multipole and local expansions, respectively. The numerical efficacy of the sampling representations stems from our ability to construct optimal sampling grids comprising minimal numbers of points and appropriate interpolation rules. To that end, we construct spherical NGs for representing the far-field and local potentials. Specifically, the far-field potential of a charge distribution contained in a sphere of radius centered at point can be computed with a prescribed error as

(4) (2) is the surface of conductor . Therefore, the main where goal is to determine the surface charge distributions due to known applied potentials. Let denote the combined surface of all conductors. The charge density is then a solution of the equation

(3) is the known impressed potential on conductor where surfaces and is the electrostatic Green’s function. When solving (3), each iteration requires evaluation of the integral operator on the right-hand side of (3), which is akin to computation of the potential due to a given surface charge distribution. Straightforward evaluation of the potential at points by surface integration involving summation of terms amounts to operations. Such direct evaluation is equivalent to a straightforward matrix–vector multiplication for the MoM discretized representation of (3). This high computational burden underscores the need for using fast potential evaluation techniques. To that end, in Sections III and IV, we concentrate on accelerating the evaluation of potentials due to given charge distributions. III. FAST ALGORITHM The proposed algorithm for fast evaluation of the integral operator in (3) relies on numerically efficient representations of potentials in terms of their spatial samples. Such representations

where

are the interpolation coefficients and the set of

is the spherical NG depicted in Fig. 1 and satispoints fying the sampling criteria derived in Appendix. In (4), also denotes the neighborhood of the observation point such within this neighborhood contribute to the that grid points local interpolation. The far-field condition is determined by parameter . We show in the Appendix that the number of grid points is independent of and depends only on the desired accuracy and the value of . In fact, thanks to the use of local interpolation, the grid is constructed only where necessary for the interpolation depending on problem geometry. The potential at the grid points can be evaluated directly by numerical quadrature. Alternatively, these values can be obtained by interpolation from other NGs, as will be discussed below. In order to optimally combine the two ways of computing the potentials on NGs, the proposed technique uses a hierarchical decomposition of the problem domain into an incomplete octal tree of boxes. As the first step of such decomposition, the whole geometry is enclosed in a single box referred as level . Subsequent multilevel decomposition entails that each “parent” box of level is subdivided into eight “children” boxes of level , where . The subdivision hierarchy forms a tree, where each box of level has up to eight pointers to nonempty smaller boxes of level . The th box of level , its center, and the radius of the minimal circumscribing sphere are denoted , , and , respectively. The boxes of level are the smallest boxes not subdivided any further and assumed to contain unknowns (quadrature

BOAG AND LIVSHITZ: ADAPTIVE NG ALGORITHM FOR FAST CAPACITANCE EXTRACTION

points), i.e., a number independent of the problem size. Therefore, on the finest level of decomposition, we compute the potential directly. Samples of the far-field potential on the corresponding NG

due to charges in box

are computed as

(5) . Note that on level , the numerical where quadrature in (5) requires operations. Subsequently, the far-field potentials for all boxes on their respective grids for all other levels can be obtained recursively by interpolation and aggregation of contributions from child boxes to their parents. Thus, the far-field potential of box is obtained by adding up contributions from its child boxes

(6)

Note that in (6), potentials

of child boxes

terpolated as in (4) from their respective NGs NG

of their parent box

are into the new

, while

denote the corresponding interpolation coefficients. Direct transition from the far-field potentials to the final values of the total potential leads to an algorithm. Achieving an complexity entails transition from the far field to local representations. Let denote the local potential within box due to distant charges. We also define a local grid allowing computation of the potential in box via interpolation by analogy with (4). The potential at a local grid point of box comprises far-field contributions from boxes in the interaction list of the same level, as well as the local potential of its parent box (computed by interpolation), i.e.,

(7) where

is the parent box of , i.e., and denotes local interpolation coefficients. The interaction , denoted , is a set boxes of the same level such list of

3567

, while respective parents that they are well separated from on level are not. Here, we say that box is well separated from box if the distance between their centers satisfies . Equation (7) can be applied recursively like in the case of (6) (but in the opposite direction of increasing with the initial conditions ) to obtain the local potentials on the local grids of all boxes. Finally, at the finest level , the total potential at an arbitrary point of box is obtained by interpolation of the local potential with additional near-field contributions from the charges residing in and its near-neighbor boxes. We have

(8) is a list of all where near-neighbor boxes of including itself. Using (8) for all observation points in all boxes of level concludes the evaluation process. Now let us estimate the computational complexity of all the steps combined. To that end, we recall that the number of points in every NG is of , i.e., it is bounded and the bound is independent of the large parameter , as well as the box or problem size. On level , potential computations for boxes using (5) require operations. Complexity of performing a local interpolation for observation points in (8) is also of . The computation of the near-neighbor contribution is performed directly by numerical quadrature over the near-neighbor surfaces and its complexity is also of . Transitions between levels and using (6) and (7) require operations, which add up to for all the levels combined. Therefore, the total complexity of the algorithm is of . It can be shown that the total storage requirements are of as well. The multiplicative factors implicit in the above asymptotic complexity and storage estimates depend on the NG parameters. When local interpolation is employed, the NG related complexity and storage are defined by the order of the grids and is proportional to (see the Appendix), contrary to respective and computational cost and storage needs of the FMM. Note that the complexity and memory of the NG approach are reduced for the cases of planar and elongated geometries by using adaptive grids to and , respectively, while those for the conventional FMM remain unchanged. Additional massive computations and data structures are related to point-to-point near-field interactions. This suggests choosing a minimal coefficient for the “well separated” criterion. On the other hand, lowering leads to an increase in , which, in turn, causes a growth of NG sizes. Thus, the total complexity and storage requirements of the algorithm are minimized by striking the balance between the near-field and the grid sizes. IV. COMPUTATIONAL RESULTS The NG algorithm has been implemented and tested on 3-D geometries of bus structures. An example of crossing buses with

3568

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

2

Fig. 2. 4 4 conductor bus crossing structure. Potential of 1 V is applied to the lower right conductor. The shades of gray represent the surface charges.

Fig. 3. Quasi-planar bus structure with 16 conductors.

TABLE I COMPARISON OF EXTRACTION METHODS FOR THE 4 4 CONDUCTORS PROBLEM

2

Fig. 4. Number of operations per iteration as a function of the problem size.

4 4 conductors is shown in Fig. 2. In this example, all conductors have 1 1 9 m size with 1-m gap. The discretization is completed by dividing each surface section into square panels. For this application, the NG for the fast potential evaluation was coupled with the general minimal residual (GMRES) iterative scheme (with a tolerance of 0.01). The NG parameters are determined according to the rules defined in the Appendix. In our examples, parameter is set to 1.6. Typically the number of points in NGs is less than 20 on the finest level and goes down as we go to coarser levels. For optimal performance, the number of sources in the boxes of the finest level of decompositions should be roughly the same as the number of points in the NG. The algorithm uses linear interpolation. To examine the effect of the NG order on the accuracy, in Table I, we compare the resulting capacitances computed for the 4 4 conductor problem by a direct solver, GMRES iterative solver with direct matrix–vector multiplication, the multipole accelerated generalized conjugate residual (MGCR) [4], and the GMRES–NG method for orders 0 and 1. Each column represents the capacitance associated with one pair of conductors. Taking the direct method results as a reference, the results

indicate that our algorithm with zero-order expansion is comparable in terms of accuracy with the FMM of the first order. Overall, for , the charge evaluation attains better than 3% rms accuracy. The comparison is especially favorable for the smallest coupling capacitance, which is two orders of magnitude smaller than the diagonal entries. To explore the geometric adaptivity of the proposed technique, we consider a quasi-planar (2.5-D) geometry of a 16-line bus shown in Fig. 3. The numbers of multiply operations required for one matrix–vector product while using the regular and quasi-planar versions of the NG algorithm, as well as the FMM for this case, are plotted in Fig. 4 as functions of the problem size. Also, Fig. 5 presents the memory requirements of the regular and adaptive NG scheme and FMM versus the number of unknowns. The direct multiplication and a linear -dependence lines are also shown for reference in both figures. The operations count and memory usage results exemplify the linear dependence of the computational cost versus the number unknowns for the proposed method and a quadratic one for the direct computation. As can be seen, the NG algorithm requires 450- and 200-N operations per iteration for 3-D and 2.5-D geometries, respectively. One can observe that the regular version of the NG

BOAG AND LIVSHITZ: ADAPTIVE NG ALGORITHM FOR FAST CAPACITANCE EXTRACTION

where

3569

, are the expansion coefficients and denotes the angular harmonics. Furthermore,

expression

(A2)

describes the order of expansion in (A1) as a function of the normalized radius and the prescribed bound for the relative error incurred in (A1), . Towards designing the sampling criteria, we examine the be-

Fig. 5. Storage requirements of various algorithms versus the number of unknowns.

algorithm is comparable with the FMM in terms of both complexity and memory requirements. On the other hand, the NG algorithm employing adaptive grids appears to be superior for such 2.5-D geometries. V. CONCLUSION The NG algorithm proposed in this paper facilitates a numerically efficient evaluation of potentials produced by given charge distributions such as encountered in iterative solution of the capacitance extraction problems. As compared to other techniques, this method promises additional flexibility in treating various geometries such as quasi-planar or elongated structures. With the NG approach, the transition from the static to the time- or frequency-domain analysis also appears to be almost straightforward. We surmise that the NG can be extended to a variety of environments where the asymptotic behavior of the pertinent Green’s function is known. For example, fields of sources embedded in stratified media can be computed by approximating the far-zone behavior of the Green’s function with a finite number of terms (representing, e.g., surface waves, complex images, and so on), applying (a slight modification of) the NG kernel to each term separately, and summing the results [17]. APPENDIX In order to justify the proposed algorithm, we have to prove that the NGs can, indeed, be constructed. Here, we concentrate on the sampling criteria for the far-field NGs, whereas construction of local grids is somewhat simpler and can be justified in a similar fashion. For a charge distribution confined to a sphere of radius centered at the origin, the potential at an observation point can be approximately expressed using a truncated ( -term) multipole expansion [2] as

(A1)

havior of in (A1) versus each of the spherical coordinates separately. For fixed and , the expression enclosed in the curly brackets in (A1) can be viewed as a polynomial of dein a new variable with the cogree efficients delimited by the expressions in the square brackets. Note that, in (A1), the relative distance from the source region . Therefore, the use of polynomial interpolation [16] of order is natural for the domain . Here, is the maximum dimension of the problem geometry. Clearly, such -sampling and interpolation uses a highly nonuniform sampling grid versus , . Note that the factor in (A1) can be accounted for analytically. Turning to angular sampling on a sphere of radius , we note that, in (A1), the order of expansion can be reduced for larger values of based on (A2). For fixed and , taking into account that angular harmonics comprise the associated Legendre functions with , the funccan also be considered as a signal consisting of tion harmonics with integer frequencies in the range . Consequently, the minimal number of -sampling points is , and trigonometric interpolation [16] enables the potential computation with the precision like the multipole expansion. Similar considerations apply to sampling, and the minimal number of -sampling points is given by . Based on these sampling criteria, the potential can be sampled on a spherical NG with angular and radial densities decreasing with the distance away from the source (see Fig. 1). Note that the bound on the number of points in the NGs is of and it depends only on and . The combination of polynomial radial interpolation and trigonometric angular interpolation constitute a global scheme involving all NG points in each computation. Thus, global interpolation between different NGs can reach complexity of . Such behavior is especially unfavorable for high accuracy/order computations. To that end, we turn to a local interpolation. In this case, the interpolation involves only the NG points surrounding the target point, though at the price of certain over-sampling. The local interpolation provides an additional advantage as compared to the global one, namely, the NGs need to be constructed only around the target surface. Thus, the algorithm becomes geometrically adaptive. For example, quasi-planar and other essentially surface geometries require NGs comprising rather than the regular points.

3570

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

REFERENCES [1] R. F. Harrington, Field Computation by Moment Methods. Malabar, FL: Krieger, 1982. [2] L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems. Cambridge, MA: MIT Press, 1988. [3] L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., vol. 73, pp. 325–348, 1987. [4] K. Nabors and J. White, “FastCap: A multipole accelerated 3-D capacitance extraction program,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 10, no. 11, pp. 1447–1459, Nov. 1991. [5] K. Nabors, S. Kim, and J. White, “Fast capacitance extraction of general three-dimensional structures,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 7, pp. 1496–1506, Jul. 1992. [6] A. Brandt, “Multilevel computations of integral transforms and particle interactions with oscillatory kernels,” Comput. Phys. Commun., vol. 65, pp. 24–38, 1991. [7] J. Tausch and J. White, “A multiscale method for fast capacitance extraction,” in Proc. 36th Design Autom. Conf., 1999, pp. 537–542. [8] S. Kapur and D. E. Long, “IES3: Efficient electrostatic and electromagnetic simulation,” IEEE Comput. Sci. Eng., vol. 5, no. 4, pp. 60–67, Oct.–Dec. 1998. [9] A. Boag and B. Livshitz, “Domain decomposition and non-uniform spherical grid interpolation (NSGI) algorithm for fast solution of potential problems,” in Proc. IEEE 22nd Conf. of Electrical and Electronics Engineers in Israel, Tel Aviv, Israel, Dec. 2002, pp. 86–88. [10] ——, “Non-uniform grid algorithm for fast capacitance extraction,” in IEEE Signal Propag. on Interconnects Workshop, Heidelberg, Germany, May 2004, pp. 109–112. [11] ——, “Non-uniform grid (NG) algorithm for fast potential evaluation,” in URSI Radio Sci. Meeting, Monterey, CA, Jun. 2004, p. 343. [12] A. Boag, E. Michielssen, and A. Brandt, “Non-uniform polar grid algorithm for fast field evaluation,” IEEE Antennas Wireless Propag. Lett., vol. 1, no. 7, pp. 142–145, 2002. [13] A. Boag, E. Michielssen, V. Lomakin, and E. Heyman, “Fast evaluation of time domain fields by domain decomposition and non-uniform spherical grid interpolation,” in Proc. URSI Radio Sci. Meeting, Columbus, OH, Jun. 2003, p. 159.2. [14] A. Boag, V. Lomakin, and E. Michielssen, “Non-uniform grid time domain (NGTD) algorithm for fast evaluation of transient wave fields,” IEEE Trans. Antennas Propag., to be published. [15] A. Boag, “A fast iterative physical optics (FIPO) algorithm based on non-uniform polar grid interpolation,” Microw. Opt. Technol. Lett., vol. 35, no. 3, pp. 240–244, Nov. 2002. [16] A. Papoulis, Signal Analysis. New York: McGraw-Hill, 1977.

[17] V. Lomakin and A. Boag, “A non-uniform grid algorithm for rapid evaluation of fields radiated in layered media,” in Proc. URSI Radio Sci. Meeting, Albuquerque, NM, Jul. 2006, p. 307.

Amir Boag (S’89–M’91–SM’96) received the B.Sc. degree in electrical engineering (summa cum laude) and B.A. degree in physics (summa cum laude), M.Sc. degree in electrical engineering, and Ph.D. degree in electrical engineering from the Technion—Israel Institute of Technology, Haifa, Israel, in 1983, 1985, and 1991, respectively. From 1991 to 1992, he was a faculty member with the Department of Electrical Engineering, Technion. From 1992 to 1994, he was a Visiting Assistant Professor with the Electromagnetic Communication Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign. In 1994, he joined Israel Aircraft Industries, as a Research Engineer and became a manager of the Electromagnetics Department in 1997. Since 1999, he has been with the Physical Electronics Department, School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel, where he is currently an Associate Professor. He has authored or coauthored over 60 journal papers and presented over 100 conference papers on electromagnetics and acoustics. His interests are electromagnetic theory, wave scattering, imaging, and design of antennas and optical devices.

Boris Livshitz (S’04) received the M.Sc. degree in physics and electrical engineering from St.-Petersburg State Technical University (formerly the Polytechnic Institute), St.-Petersburg, Russia, in 1995, and the Ph.D. degree in electrical engineering from Tel Aviv University, Tel Aviv, Israel, in 2005. In 2006, he joined the Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, as a Post-Doctoral Associate. His research interests include computational electromagnetics, algorithms for analysis, simulation, and modeling of electromagnetic fields in complex configurations, and fast numerical methods for solutions of integral equations.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3571

Inductively Compensated Parallel Coupled Microstrip Lines and Their Applications Ravee Phromloungsri, Mitchai Chongcheawchamnan, Member, IEEE, and Ian D. Robertson, Senior Member, IEEE

Abstract—A simple method using lumped inductors to compensate unequal even- and odd-mode phase velocities in parallel coupled microstrip lines is presented. The singly and doubly compensated cases are analyzed to enable the optimum inductor values and the electrical lengths of the compensated coupled lines to be calculated from closed-form expressions. The technique proposed not only improves the performance, but also yields a more compact design. To demonstrate the technique’s broad range of applicability, the compensated coupled-line structure is used to enhance the performance of a 900-MHz Lange coupler, a 1-GHz multisection 10-dB coupler, a 900-MHz planar Marchand balun, and a 1.8-GHz parallel coupled bandpass filter. Index Terms—Coupled-line resonator, Marchand balun, microstrip, parallel coupled filter, parallel coupled lines.

I. INTRODUCTION ARALLEL-COUPLED lines are extensively used in microwave and millimeter-wave circuits for filters, impedance-matching networks, directional couplers, baluns, and combiners [1], [2]. Since microstrip is easily incorporated in hybrid and monolithic microwave integrated circuits [3], it is the most common technology for microwave and millimeter-wave circuits. As microstrip is an inhomogeneous medium, parallel coupled microstrip lines exhibit poor directivity [4] resulting from the inequality of even- and odd-mode wave phase velocities [5], [6]. The unequal phase velocities in parallel coupled microstrip lines not only cause poor directivity in couplers, but also significantly deteriorate the performance of other circuit components circuits. For example, it is well known that the parallel coupled microstrip bandpass filter has an asymmetrical passband response and spurious responses at harmonics of the filter passband [7]. It was recently reported [8] that degradation in amplitude/phase balance of the microstrip Marchand balun partly stems from the unequal phase velocities. Over the past decades, the notorious problem of unequal phase velocities in parallel coupled microstrip lines has been tackled by several previously proposed techniques. The techniques can be classified into two main categories, which are distributed and lumped compensation approaches.

P

Manuscript received December 27, 2005; revised April 26, 2006. R. Phromloungsri and M. Chongcheawchamnan are with the Research Center of Electromagnetic-Wave Applications and Telecommunication Department, Mahanakorn University of Technology, Bangkok, Thailand. I. D. Robertson is with the School of Electronic and Electrical Engineering, The University of Leeds, Leeds LS2 9JT, U.K. Color versions of Figs. 10, 13, 16, and 20 are available online at http:// ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2006.881026

The methodology based on the distributed approach is to modify either the parallel coupled-line structure [9], [10], dielectric layer [11], or ground plane [12] such that the phase velocities of both modes are equalized. No external components or extra space are needed for this approach. The main disadvantage of this approach is the lack of closed-form design equations, meaning the design task heavily relies on the electromagnetic (EM) simulation stage, which, in turn, consumes much effort and computing time. Additionally, techniques based on this approach are often not suitable for some standard fabrication processes, thus more cost demand is unavoidably required. The lumped compensation approach [1], [13], [14] involves connecting external reactive components between or shunted with the parallel coupled-lines’ ports. Based on the reactive types, this approach can be categorized into two techniques, which are capacitive [1], [13], [14] and inductive compensation techniques [13]. The size of the lumped-compensated parallel coupled lines is about the same as the original size (the uncompensated coupled lines) since the length of compensated parallel coupled lines is shorter than that of the uncompensated coupled lines. Another distinct advantage of the lumped compensation technique is its simple design procedure because the closed-form design equations can be derived. The disadvantages of the technique are, from a practical point-of-view, the lumped components’ parasitics and difficulty in layout [1], [13]. In this paper, we present a simple, yet effective inductive compensation technique to improve the directivity of the parallel coupled microstrip lines. The technique can achieve high isolation and, hence, high-directivity coupled microstrip lines by connecting small inductors in series with the coupled-lines’ ports. This paper is organized as follows. Section II presents two proposed inductive compensation techniques, referred to as the singly and doubly compensated cases. Analysis of the techniques will be performed and closed-form equations for determining the optimum values of inductor for the singly and doubly compensated cases are provided. The derived closed-form equations are validated by comparison with analysis results of the uncompensated and compensated coupled lines. Applications of the inductively compensated coupled-line structure to a Lange coupler, a three-section 10-dB coupler, a planar Marchand balun, and a parallel coupled microstrip bandpass filter will be demonstrated in Section III. The design equations for these circuits are also given. Design examples of the circuits based on the inductively compensated coupled-line section as well as the EM simulated and measured results will be presented in Section IV. This paper is then concluded in Section V.

0018-9480/$20.00 © 2006 IEEE

3572

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 1. Schematics of: (a) singly and (b) doubly compensated parallel coupled lines.

II. INDUCTIVELY COMPENSATED COUPLED LINES Due to the different phase velocities associated with the evenand odd-mode wave propagation, parallel coupled microstrip lines cannot easily achieve directivity values better than 12 dB [3]. Here, we propose an inductive compensation technique to equalize phase velocities in coupled microstrip lines, which, in turn, leads to a high isolation and, hence, high-directivity coupled microstrip lines. Both a singly and doubly compensated technique can be used. A. Singly Compensated Case Fig. 1(a) shows the schematic of parallel coupled lines using a single inductor for compensation. The parallel coupled-line section has the characteristic impedance of and coupling coefficient of . Both and are related to the even- and odd-mode characteristic impedances ( and ) of the parallel coupled lines [3]. The compensated inductor is connected in series with the coupled port (port 2), giving what we define as port 2’, the coupled port of the compensated coupled-line section. As the compensated inductor is connected in series with the coupled port of the uncompensated parallel coupled lines, open-circuited impedance parameters ( -parameter) are more appropriate to analyze this problem. Assuming the uncompensated coupled lines are symmetrical, the -parameters of the coupled lines can be characterized completely by just four elements of the uncompensated parallel coupled lines’ -parameter matrix denoted by and . Applying the two-port network theory to the circuit of Fig. 1(a) [15], the -parameters of the singly compensated parallel coupled lines in matrix form can be written as

(1)

Terminating the singly compensated coupled lines shown in Fig. 1(a) with at all ports and then applying the -parameter definitions, the coupling and isolation coefficients of the singly compensated coupled lines can be obtained from

. Similarly, from (2b) is the ratio between is denoted by the terminal voltage at port 1 and the voltage source injected at the isolated port (port 3), which is denoted by . The directivity of the singly compensated coupled-line section is immediately obtained by dividing from (2a) by from (2b). The compensating inductor can improve considerably with a small change in . Thus, the optimum value of with such a condition can be solved. For our analysis, we apply the -parameters obtained from (1) into (2) to solve for the optimum value of . At the operating frequency of , this must provide the minimum magnitude of and minimum difference between and , the original coupling coefficient. After some mathematical manipulation, we obtain the optimum value of shown as follows [16]:

(3a) where

denotes an imaginary part of

and

where: 1) is the even-mode effective dielectric constant; 2) is the odd-mode effective dielectric constant; 3) is the even-mode electrical length of the uncompensated coupled lines; and 4) is the odd-mode electrical length of the uncompensated coupled lines. Note that the frequency where maximum directivity is obtained for the singly compensated technique does shift slightly below the original . Therefore, the electrical length of the singly compensated coupled lines must be modified as follows [16]: (3b) B. Doubly Compensated Case The doubly compensated technique applied to parallel coupled microstrip lines is shown in Fig. 1(b). Here, two identical compensating inductors are connected in series with the coupled (port 2) and direct ports (port 4) of the uncompensated parallel coupled microstrip lines. We assume that the uncompensated coupled-line section shown in Fig. 1(b) is symmetrical and the impedance parameters are and . The coupled and direct ports of the compensated coupled lines are now port 2’ and 4’, respectively. The analysis procedure for this structure is similar to the procedure applied to a singly inductive compensation case, which was already described in Section II-A. From Fig. 1(b), the -parameters of the doubly inductive-compensated coupled lines in matrix form can be written as

(2a) (2b) is the ratio between the voltage at port 1 From (2a), and the voltage source injected at coupled port (port 2’), which

(4)

PHROMLOUNGSRI et al.: INDUCTIVELY COMPENSATED PARALLEL COUPLED MICROSTRIP LINES AND THEIR APPLICATIONS

3573

With similar conditions and similar approach applied in Section II-A, the optimum value of connecting in series at ports 2’ and 4’ can be determined as follows [16]: (5) where

(6a) and (6b) Again, slightly shifts from the original . Hence, to shift where maximum directivity occurs back to the original, the electrical length of the doubly compensated coupled lines must be shortened as follows [16]: (7) C. Analysis Results and Performances The validity of the inductive compensation technique is proven by applying the technique to a microstrip 10-dB coupler operating at 1.8 GHz . The substrate used is RF60–0600 from Taconic Inc., Petersburgh, NY ( mm, and ). Based on these parameters, and calculated from (3a) and (5) are 1.24 and 1.31 nH, respectively. In all cases, is 69.37 and is 36.03 . The degree of directivity and isolation improvement of the inductive compensation technique is compared with two capacitive compensation techniques [1], [4]. The compensated capacitors are 0.185 [1] and 0.187 pF [4]. Simulations of all of these coupled-line topologies are performed and the results compared. We assume that the inductive and capacitive lumped components used are ideal. Fig. 2(a) shows the simulation results of directivity and isolation performances of the capacitive compensation techniques [1], [4] in comparison with the uncompensated 10-dB parallel coupled microstrip lines. From 1.5 to 2.1 GHz, the directivity performances of both techniques are at least 12 [1] and 14 [4] dB more than that of the uncompensated design. Fig. 2(b) shows the simulation of directivity and isolation performance of singly and doubly compensated cases. The directivity and isolation performances from the singly and doubly compensated techniques are 6 and 11 dB better, respectively, than those of the uncompensated coupled lines from 0.7 to 2.7 GHz. At the of 1.8 GHz, the doubly compensated design provides isolation and directivity performance approximately 25 dB better than the uncompensated coupled lines, while the singly compensated design achieves around 8-dB improvement. Comparing the performances between inductive and capacitive compensation techniques, we find that the doubly compensated technique provides best directivity improvement for a narrowband design. Although the performance of the capacitive compensation techniques is better for wideband design, such relatively small capacitors needed in the design examples may

Fig. 2. Simulation results of directivity and isolation performances of the: (a) previous works and (b) singly (- - -) and doubly (—) compensated 10-dB coupled lines compared with the uncompensated case (1 1 1).

cause practical limitations for some applications. In addition, layout of the capacitive compensation technique is more difficult than the proposed technique. For the RF60–0600 substrate, gains of the capacitive-compensated [1], [4] and the proposed compensated coupled microstrip lines’ directivities over the uncompensated coupled-lines’ directivities at for each coupling factor are shown in Fig. 3 (top). At least 7.5- and 10-dB directivity gain is achieved from the singly and doubly compensated cases over the coupling factor range of 1 to 15 dB. From 3 to 11-dB coupling factor, the doubly inductive and capacitive compensation techniques exhibit a similar degree of directivity gain. For very tight coupling, only the singly compensated case provides the directivity gain. The effect of deviation in optimum compensated components on the directivity (described in terms of percentage of directivity variation) of both inductively and capacitively compensated coupled lines has also been studied. Fig. 3 (bottom) shows the analysis results of the percentage of directivity variation when optimum inductors or capacitors are deviated within 10%. The directivity sensitivity of the singly compensated case is constant around 20% from over 1- to 15-dB coupling factor. It is shown in Fig. 3 that the directivity of the doubly compensated technique is not very sensitive to , especially for tight coupling. Maximum sensitivity result from the effect of component variation is owned by the capacitive-compensated design proposed in [4]. However, it should be noted that

3574

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 4. Singly compensated Lange coupler.

Fig. 3. (top) Directivity gain over the uncompensated design at f and (bottom) its variation over 10% change of optimum compensated inductors on the RF600600 substrate.

such deviations are solely specified to parallel coupled lines designed on the specific substrate, not for any substrate and not necessarily coupled-line-based circuits. The effect of deviation in compensating inductor values on the performance of other coupled-line based circuits should be carefully considered for each design case.

Fig. 5. Schematics of: (a) uncompensated and (b) inductive-compensated three-section coupled lines.

B. Multisection Parallel Coupled Lines III. APPLICATION CIRCUITS In order to demonstrate the usefulness of the inductive compensation technique, it is applied to four frequently used parallel coupled microstrip circuits: the Lange coupler, a three-section coupler, the planar Marchand balun and, finally, the parallel coupled bandpass filter. A. Lange Coupler The Lange coupler is a popular tight quadrature coupler used in microwave circuit design. It has been well known that the tight coupling in the Lange coupler is achieved by increasing the odd-mode capacitance with the numbers of parallel line because the parallel-line structure in a Lange coupler causes poor directivity, especially at the high-frequency band edge when realizing a Lange coupler in microstrip technology. To solve this problem, either the singly or doubly compensated technique can be applied to the Lange coupler. For the sake of brevity, here we demonstrate only the design detail of the singly compensated Lange coupler. For the doubly compensated Lange coupler, the design procedure is similar. Fig. 4 shows the singly compensated technique applied to a Lange coupler. The oddand even-mode characteristic impedances are and , respectively. The compensated inductor is connected at the coupled port and the electrical length is . With these parameters, the singly compensated Lange coupler can be fully characterized by (1). Consequently, and , shown in Fig. 4, are calculated from (3a) and (3b) for designing the singly compensated Lange coupler.

Parallel coupled lines with multioctave bandwidth can be realized using a multisection topology. However, the poor directivity of each parallel coupled-line section deteriorates the isolation performance and degrades the coupling bandwidth. The proposed inductive compensation technique applied to each parallel coupled-line section preserves the desired coupling bandwidth, as well as improves the isolation performance of the multisection parallel coupled lines. Since there are various parallel coupled lines in the multisection topology, numerous forms of inductive-compensation-based multisection parallel coupled lines can be obtained by applying each coupled-line section with different combinations of singly and/or doubly compensated techniques. In this paper, we apply a combination of singly and doubly compensated designs to a three-section 10-dB coupler. Fig. 5(a) and (b) shows the schematics of the uncompensated and inductively compensated three-section coupled lines, respectively. As shown in Fig. 5(b), the singly and doubly compensated techniques are applied to the outer and center coupled lines, respectively. The design procedure of the compensated multisection coupled lines follows that of the uncompensated coupled lines [17], [18]. After obtaining the parameters of each parallel coupled-line section, all the uncompensated parallel coupled lines will be replaced with the inductive-compensated and parallel coupled lines. The compensated inductors are calculated from (3a) and (5). The electrical lengths of are calculated from (3b) and each coupled-line section (7). Finally, based on these electrical parameters, the physical parameters of each coupled-line section are synthesized.

PHROMLOUNGSRI et al.: INDUCTIVELY COMPENSATED PARALLEL COUPLED MICROSTRIP LINES AND THEIR APPLICATIONS

Fig. 6. Schematics of: (a) uncompensated and (b) singly compensated coupledline-based Marchand balun.

C. Marchand Balun Having been extensively used in a large variety of microwave circuits, the planar Marchand balun is basically formed by two parallel coupled-line sections connected in back-to-back configuration, as shown in Fig. 6(a). The coupled-line section has coupling coefficient of , characteristic impedance of , and electrical length of . This conventional (uncompensated) Marchand balun can transform unbalanced port impedance to balanced port impedance if one selects as follows [8]: (8)

It has been reported that the Marchand balun exhibits poor amplitude/phase balance when the circuit is realized in inhomogeneous medium such as microstrip. The imbalance partly comes from the poor directivity of the parallel coupled microstrip lines [8], hence, an approach to improve the directivity of the coupled lines can enhance the amplitude/phase balance of the Marchand balun. For simplicity’s sake, the singly compensated technique is applied to the Marchand balun. Fig. 6(b) shows the proposed technique based on the singly compensated technique. The optimum value of can be calculated from (3a). From Fig. 6(b), the driving-point impedance at unbalanced port (port 1) at , denoted by , is calculated from the -parameter of the singly compensated coupled line. As is already known, each -parameter of the compensated coupled-line section can be obtained from (1). This -parameter will be used as basis parameter to calculate . For our analysis, the electrical length of each compensated coupled-line section is obtained by applying the following condition:

(9) Hence, (10)

3575

Fig. 7. Schematic of: (a) uncompensated, (b) singly, and (c) doubly compensated coupled-line resonators.

Designing the proposed balun with and calculated from (3) and (10), the real part of is nearly equal to , while the imaginary part of is inductive. This inductive part must be cancelled out to give the compensated Marchand balun good matching at the unbalanced port. The inductive part can be simply eliminated by a series capacitor . The value of can be determined by (11) has little effect on the ampliIt should be noted that tude/phase balance of the balun. It significantly affects to only the return loss at the unbalanced port. The design procedure of the balun based on the singly compensated coupled lines starts by determining from (8). With the known substrate and uncompensated coupled-line parameters, all electrical parameters and can be calculated. Subsequently, is calculated from (3a). Finally, and are obtained from (10) and (11), respectively. With this design procedure, a balun design with good amplitude/phase balance and good return loss at the unbalanced port across a large bandwidth can be obtained. D. Parallel Coupled Filter In an inhomogeneous medium such as microstrip, each coupled-line resonator in the parallel coupled filter cooperatively contributes a spurious response at twice the center frequency and beyond. Since the poor directivity is an outcome of phase-velocity inequality, the inductively compensated coupled-line resonator with high directivity can suppress the spurious response of the filter effectively [19]. The resonators based on uncompensated and compensated coupled lines, both singly and doubly compensated cases, are depicted in Fig. 7(a)–(c), respectively. The optimum values of compensating inductors and in Fig. 7(b) and (c) can be determined from (3a) and (5). To preserve the original filter response, the transmission response of the compensated resonator would be preserved or minimally change from that of the uncompensated resonator. Hence, this condition will be applied for extracting the electrical lengths of each compensated coupler. For tight coupling

3576

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 8. Analysis results of frequency responses of the singly (- 1 -) and doubly (—) compensated coupled-line resonators compared with the uncompensated case (- - -). Fig. 10. PCB photographs of: (a) conventional and (b) singly compensated Lange couplers.

Fig. 9. Simulated frequency responses of the uncompensated (- 1 -) and singly compensated (—) Lange couplers.

( dB), the electrical length of the singly inductive-compensated coupled-line resonator can be determined by (12a) and for loose coupling ( 23 dB

dB),

(12b) For the doubly compensated coupled-line resonator, its electrical length can be calculated from (13) To explicitly show the spurious-suppression performances of both singly and doubly compensated coupled-line resonators, two resonators based on the proposed techniques were designed, and their frequency responses are compared with the results of the uncompensated parallel coupled resonator. We start with the

Fig. 11. Measured results of the singly (—) compensated and uncompensated (- - -) Lange couplers.

uncompensated coupled-line resonator. The resonator was designed from 8.6-dB parallel coupled lines operating at 1.0 GHz on an RF60–0600 microwave substrate from Taconic Inc. The required value of is found to be 78.84 and is 36.48 . The values of and for singly and doubly compensated coupled-line resonators were calculated from (3a), (5), (12), and (13). The values of and are found to be 2.13 nH, 2.07 nH, , and , respectively. With these parameters, the frequency responses of three resonators are plotted in Fig. 8. As shown in Fig. 8, the magnitude of of the uncompensated coupled-line resonator (shown as the dotted line) is around 7 dB at the first harmonic of the desired passband response . Comparing the responses obtained from the singly and doubly compensated resonators with that obtained from the uncompensated resonator, the magnitudes of around are nearly equal, while the responses around 2 and beyond are distinctly different. The suppression performances obtained from the compensated coupled-line resonators at odd and even harmonics of are considerably better than the uncompensated coupled-line resonator. For the singly compensated case, the degree of suppression at and are approximately 14, 7, and 7 dB, respectively, while the

PHROMLOUNGSRI et al.: INDUCTIVELY COMPENSATED PARALLEL COUPLED MICROSTRIP LINES AND THEIR APPLICATIONS

3577

TABLE I PARAMETERS OF THE THREE-SECTION COUPLERS @ 1.0 GHz

Fig. 12. Simulated frequency responses of the uncompensated (- 1 -) and compensated (—) three-section couplers.

Fig. 14. Measured results of the: (a) uncompensated (- - -) and (b) compensated (—) three-section coupler.

TABLE II PARAMETERS OF THE BALUNS @ 0.9 GHz

Fig. 13. PCB photographs of: (a) uncompensated and (b) compensated threesection couplers.

doubly compensated case gives 50, 15, and 30 dB, respectively. To apply the compensated coupled-line resonators to bandpass filter design, the uncompensated coupled-line resonators are initially synthesized. Each coupled-line resonator is then replaced with the singly or doubly compensated coupled-line resonator. All electrical parameters of the compensated coupled-line resonators are calculated from (3a), (5), (12), and (13) according to the singly or doubly compensation case. Finally, the dimensions of each proposed resonator are synthesized from the derived electrical parameters. Noted that the doubly inductive compensated technique proposed can be applied to suppress considerable spurious response for narrowband design due to the technique itself being narrowband. IV. DESIGN AND EXPERIMENTAL RESULTS

of the coupling, isolation, and directivity performances of the uncompensated and singly compensated 10-dB Lange couplers are shown in Fig. 9. The printed circuit board (PCB) photographs of the uncompensated and singly compensated Lange coupler are shown in Fig. 10(a) and (b), respectively. The sizes are 10 60 mm and 10 55 mm . The measurement was performed with an HP8720C vector network analyzer test system. Fig. 11 shows the measured isolation, coupling, and directivity performances of the singly compensated and uncompensated Lange couplers. The measured results are in good agreement with the simulated results shown in Fig. 9. The directivity of the singly compensated one is improved by around 8 dB at 0.9 GHz compared to the uncompensated one. Moreover, the singly compensated Lange coupler provides directivity that is 6 dB better than that of the uncompensated Lange coupler across a 500-MHz bandwidth.

A. Lange Coupler An example of the singly compensated Lange coupler is demonstrated with a 10-dB coupler operating at 0.9 GHz on an RF35–0600 microwave substrate from Taconic Inc. ( mm, ). It is noted that such a coupling factor is chosen for implementation sake due to the tolerance limitation of our in-house fabrication process. The required values of the singly compensated design are 69.37 and 36.03 . and calculated from (3a) and (3b) are 2.05 nH and , respectively. The simulated results

B. Three-Section Parallel Coupled Lines Next, a three-section 10-dB maximally flat coupler operating at 1.0 GHz on an RF60–0600 substrate was designed and simulated for a 50- system. Followed from the procedure of the conventional design [17], [18], 27.2 and 8.2-dB coupledline sections are needed for the outer and center sections shown in Fig. 5(b). From these required coupling coefficients, we obtain and . These uncompensated parallel coupled

3578

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 15. Analysis results of: (a) amplitude and (b) phase imbalance of the 50–150- compensated Marchand balun with various values of the compensating inductors.

Fig. 17. Measured amplitude response of: (a) uncompensated and (b) singly compensated Marchand balun.

Fig. 16. PCB photographs of the fabricated: (a) uncompensated and (b) singly compensated Marchand balun.

lines are then converted to singly and doubly compensated coupled-line sections. The values of and needed for these compensated coupled lines calculated from (3a), (5), (3b), and (7) are 4.14 nH, 1.88 nH, , and , respectively. The simulated results of the coupling, isolation, and directivity performances of the compensated and uncompensated threesection parallel coupled lines are shown in Fig. 12. The PCB photographs of the uncompensated and compensated three-section couplers are shown in Fig. 13(a), and (b), respectively. The electrical and physical parameters of the uncompensated and the compensated three-section parallel coupled lines are summarized in Table I. The size of the compensated three-section

Fig. 18. Comparison of the measured (upper) magnitude and (lower) phase responses of the uncompensated (1 1 1) and singly compensated (—) Marchand balun.

parallel coupled lines is 10 76 mm , which is approximately 70% of the size of the uncompensated design (10 107 mm ). The measured results of the proposed and the uncompensated three-section coupled lines are shown in Fig. 14. Clearly, the coupling bandwidth of the compensated coupler is better than that of the uncompensated coupler, which is degraded by poor

PHROMLOUNGSRI et al.: INDUCTIVELY COMPENSATED PARALLEL COUPLED MICROSTRIP LINES AND THEIR APPLICATIONS

3579

Fig. 19. Schematics of the third-order: (a) uncompensated, (b) singly, and (c) doubly compensated parallel coupled filters.

directivity in each uncompensated section. From the measured results, the directivity of the compensated coupler is 6.5 dB more than that of the uncompensated coupled lines over 80% of the operating bandwidth. C. Marchand Balun To prove the validity of the technique for the Marchand balun, 900-MHz microstrip 50–150- impedance transformation Marchand baluns based on the uncompensated and the singly compensated coupled lines were designed on an FR4 substrate mm, ). The and of the coupled lines are 74.42 and 33.59 , respectively. The parameters calculated from (3a), (10), and (11) for these Marchand balun are shown in Table II. The sensitivity to of the amplitude/phase imbalance is investigated for the 50–150- Marchand balun. Based on the analysis results, variations of amplitude and phase balance resulting from different compensated inductors are shown in Fig. 15(a) and (b), respectively. The amplitude and phase balance of the singly compensated Marchand balun is excellent across the operation bandwidth. The proposed technique is rather practical for the balun since the balance performance is not very sensitive

to the optimum value of the compensating inductor. As shown in Fig. 15(a) and (b), with the 20% change of compensating inductors, the amplitude imbalance of the balun is maintained within 0.3 dB and the phase difference is 180 2 over a 30% bandwidth. The physical dimensions of the two baluns were synthesized from the parameters listed in Table II. Fig. 16(a) and (b) shows the PCB photographs of the designs. The measured frequency responses of the uncompensated Marchand balun are shown in Fig. 17(a). The uncompensated Marchand balun achieved a 3.5-dB transmission coefficient and less than 13-dB return loss at the unbalanced port. The amplitude balance is good only at 900 MHz, while at other frequencies, especially in the high-frequency band edge, it is very poor. Fig. 17(b) shows the measured results of the singly compensated Marchand balun. At 900 MHz, the transmission coefficient is around 3.7 dB and the return loss of the unbalanced port is better than 25 dB. The proposed technique exhibits only 0.2 dB more loss than the uncompensated balun, which is due to loss from the series capacitor. Comparing the measured 10-dB return loss bandwidth, the bandwidth of the proposed technique is 170 MHz larger than the uncompensated balun.

3580

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

TABLE III PARAMETERS OF THE FILTER DESIGNS

Fig. 18 shows the amplitude/phase balance of the uncompensated (shown as the dotted line) and the compensated Marchand balun (shown as the solid line). Compared with the uncompensated balun, the amplitude/phase balance of the singly compensated balun is considerably enhanced. The amplitude and phase balance of the proposed technique is excellent, with 0.1 dB and 180 1 tracking from 700 MHz to 1.1 GHz. D. Parallel Coupled Filter The effectiveness of the inductively compensated parallel coupled filter is proven with three filter designs. These three filters are an uncompensated, a singly, and a doubly compensated design. The filter prototype is a third-order Chebyshev bandpass of 1.8 GHz, fractional filter designed at center frequency of 10%, and passband ripple of 0.1 dB. The bandwidth circuits were designed and fabricated on the RF60–0600 substrate from Taconic Inc. The schematics of three filters are depicted in Fig. 19(a)–(c), respectively. With the design procedure mentioned in Section III, the parameters of three filters were derived and are shown in Table III. The physical dimensions of all three filters are synthesized from the parameters in Section II. Fig. 20(a)–(c) shows the PCB photographs of three filters designed with the uncompensated, singly, and doubly compensated coupled-line section. In our design, the compensation inductors were implemented by shorted stubs. The total sizes of the singly and doubly compensated parallel coupled filters shown in Fig. 20(b) and (c) are 17 70 mm and 18 65 mm , which are approximately 85.4% and 84% of the uncompensated parallel coupled filter’s size. Fig. 21 presents the EM simulated results of the uncompensated and the compensated parallel coupled filters. The frequency response of the uncompensated parallel coupled filter (dash line) evidently shows spurious response more . More than 45- and 28-dB suppression than 12 dB at and are obtained from of the spurious response at the singly (dotted line) and doubly (thick line) compensated parallel coupled filters. The measurement was performed with an HP8720C vector network analyzer calibrated from 0.1 to 10 GHz. HPVEE6.0 software was used to collect the experimental data via a general-purpose interface bus (GPIB) card. Sonnet-Lite and MATLAB were used for simulation, data processing, and display. The measured results of the microstrip filter designed with the uncompensated and the singly compensated coupled-line resonators are shown in Fig. 22(a). The

Fig. 20. PCB photographs of: (a) conventional, (b) singly, and (c) the doubly compensated parallel coupled filters.

Fig. 21. Comparisons of EM simulated results of the singly (—) and doubly (—) compensated compared with the uncompensated case (- - -) parallel coupled filters.

measured insertion and both input/output return losses of the uncompensated filter are 1.2 dB and better than 14 dB, while the singly compensated filter’s insertion and input/output

PHROMLOUNGSRI et al.: INDUCTIVELY COMPENSATED PARALLEL COUPLED MICROSTRIP LINES AND THEIR APPLICATIONS

3581

and parallel coupled bandpass filter. Design procedures for these circuits with the compensation technique have been described. The closed-form expressions for determining the compensation inductor values and coupled-line parameters simplify the design task. Since there are various microwave communication circuits whose structures consist of parallel coupled lines, it is believed that the technique is highly applicable and suitable for modern wireless communications. ACKNOWLEDGMENT The authors are grateful to Taconic Inc., Petersburgh, NY, for supplying the Taconic RF60-0600 microwave substrate for this research. The authors also thank the anonymous reviewers for their valuable comments and suggestions. REFERENCES

Fig. 22. Comparisons of measured results of: (a) singly (—) and (b) doubly (—) compensated compared with the uncompensated case (- - -) parallel coupled filters.

return losses are 1.4 dB and better than 12 dB, which is in good agreement with the EM simulated results. The measured spurious response obtained from the uncompensated parallel coupled filter (dashed line) is around 13 dB . More than 39- and 42-dB suppression of spurious reat and 3 are obtained from the singly compensponse at sated parallel coupled filter. Fig. 22(b) shows the measured results of the doubly compensated parallel coupled filter (thick and are more than line). The harmonic suppressions at 49 and 34 dB, respectively. Over the operating bandwidth, insertion losses are less than 1.5 dB, while both input and output return losses are better than 12 dB. V. CONCLUSIONS Based on the simple inductive compensation technique proposed, we have presented two new methods to achieve high-directivity parallel coupled lines in microstrip. The compensating inductor connected in series with either coupled or direct port of the coupled-line structure equalizes phase velocity, leading to a high-directivity coupled-line design. Compared with previously proposed lumped compensation techniques, the inductive compensation technique provides better directivity performance for a 10% fractional bandwidth. The optimum value of compensation inductor is relatively small, so the technique can be implemented practically in microwave and millimeter-wave applications. The inductive compensation technique has been demonstrated in four microwave coupled-line-based microstrip circuits, which are the Lange coupler, multisection coupler, planar Marchand balun,

[1] M. Dydyk, “Accurate design of microstrip directional couplers with capacitive compensation,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1990, pp. 581–584. [2] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave ImpedanceMatching Network and Coupling Structures. New York: McGrawHill, 1964, pp. 583–593. [3] T. Edward, Foundation for Microstrip Circuit Design. West Sussex, U.K.: Wiley, 1992, pp. 173–228. [4] S. L. March, “Phase velocity compensation in parallel-coupled microstrip,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1982, pp. 581–584. [5] A. Riddle, “High performance parallel coupled microstrip filter,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1988, pp. 427–430. [6] S. M. Wang, C. H. Chen, and C. Y. Chang, “A study of meandered microstrip coupler with high directivity,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 63–66. [7] I. J. Bahl, “Capacitively compensated performance parallel coupled microstrip filter,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1989, pp. 679–682. [8] C. Y. Ng, M. Chongcheawchamnan, and I. D. Robertson, “Analysis and design of a high-performance planar Marchand balun,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, pp. 113–116. [9] A. Podell, “A high directivity microstrip coupled lines technique,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1970, pp. 33–56. [10] S. Uysal and H. Aghvami, “Synthesis, design, and construction of ultrawide-band nonuniform quadrature directional couplers in inhomogeneous media,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 6, pp. 969–976, Jun. 1989. [11] C. S. Kim, Y. T. Kim, S. C. S. Kim, Y. T. Kim, S. H. Song, W. S. Jung, K. Y. Kang, J. S. Park, and D. Ahn, “A design of microstrip directional coupler for high directivity and tight coupling,” Eur. Gallium Arsenide and Other Semiconduct. Applicat. Symp., pp. 126–129, Sep. 2001. [12] F. R. Yang, Y. Qian, and T. Itoh, “A novel uniplanar compact structure for filter and mixer applications,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 6, pp. 969–976, Jun. 1989. [13] M. Dydyk, “Microstrip directional couplers with ideal performance via single-element compensation,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 6, pp. 956–964, Jun. 1999. [14] S. F. Chang, J. J. Chen, Y. H. Jeng, and C. T. Wu, “New high-directivity coupler design with coupled spurlines,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 2, pp. 65–67, Feb. 2004. [15] C. A. Desoer, Basic Circuit Theory. New York: McGraw-Hill, 1966, pp. 409–469. [16] R. Phromloungsri and M. Chongcheawchamnan, “A high directivity design using an inductive compensation technique,” in Asia–Pacific Microw. Conf., Dec. 2005, pp. 2840–2843. [17] E. H. Fook and R. A. Zakarevicius, Microwave Engineering Using Microstrip Circuits. New York: Prentice-Hall, 1990. [18] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [19] R. Phromloungsri, S. Patisang, K. Srisathit, and M. Chongcheawchamnan, “A harmonic-suppression microwave bandpass filter based on an inductively compensated microstrip coupler,” in Asia–Pacific Microw. Conf., Dec. 2005, pp. 2836–2839.

3582

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Ravee Phromloungsri was born in Khon Kaen, Thailand. He received the B.Sc. degree in applied physics in solid-state electronics from the King Mongkut Institute of Technology, Ladkrabang (KMITL), Thailand, in 1992, the M.Eng. degree in electrical engineering (telecommunication) from Mahanakorn University of Technology (MUT), Bangkok, Thailand, in 2000, and is currently working toward the D.Eng. degree in electrical engineering in MUT. Since 1992, he has been a Lecturer with the Department of Telecommunication Engineering, MUT. He is a member of the Research Center of Electromagnetic Waves Applications (RCEWs). His research and teaching interests include microwave passive/active and RF circuits design.

Mitchai Chongcheawchamnan (M’96) was born in Trang, Thailand. He received the B.Eng. degree in telecommunication engineering from the King Mongkut Institute of Technology, Ladkrabang (KMITL), Thailand, in 1992, the M.Sc. degree in communication and signal processing from Imperial College, University of London, London, U.K., in 1995 and the Ph.D. degree in electrical engineering from the University of Surrey, Surrey, U.K., in 2001. He is currently a Director of the Research Center of Electromagnetic-Wave Applications and Assistant

Professor with the Department of Telecommunication Engineering, Mahankorn University of Technology, Bangkok, Thailand. His research and teaching interests include RF and microwave passive and active circuits. Dr. Chongcheawchamnan is a member of the Institution of Electrical Engineers (IEE), U.K.

Ian D. Robertson (M’96–SM’05) received the B.Sc. (Eng.) and Ph.D. degrees from King’s College London, London, U.K., in 1984 and 1990, respectively. From 1984 to 1986, he was with the Monolithic Microwave Integrated Circuit (MMIC) Research Group, Plessey Research (Caswell) Ltd. Since then, he has held academic posts with King’s College London and the University of Surrey. In June 2004, he became the Centenary Chair in Microwave and Millimeter-Wave Circuits with The University of Leeds, Leeds, U.K. He is currently an Honorary Editor of the IEE Proceedings-Microwave, Antennas & Propagation. He edited MMIC Design (IEEE, 1995) and coedited RFIC and MMIC Design and Technology (IEE, 2001, 2nd ed.). He has authored or coauthored over papers in the area of microwave integrated circuit (MIC) and MMIC design. Dr. Robertson has organized numerous colloquia, workshops, and short courses for both the Institution of Electrical Engineers (IEE), U.K., and the IEEE.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

3583

Broadband Space Conservative On-Wafer Network Analyzer Calibrations With More Complex Load and Thru Models Sathya Padmanabhan, Member, IEEE, Lawrence Dunleavy, Senior Member, IEEE, John E. Daniel, Member, IEEE, Alberto Rodríguez, Member, IEEE, and Peter L. Kirby, Member, IEEE

Abstract—An improved vector network analyzer (VNA) calibration approach is demonstrated that utilizes planar lumped shortopen-load-thru standards and achieves accuracy comparable to thru-reflect-line (TRL) at high frequency, without the commonly occurring errors in TRL at low frequency. The approach relies on complex load and thru models for coplanar waveguide and microstrip standards that are not currently available in typical VNA firmware. It is shown that the RF performance changes due to variations in fabrication of load can be addressed by “calibrating” or adjusting the load model with the measured dc resistance for a particular load. Good results are shown for a wide range of substrates (GaAs, alumina, and FR-4) and frequencies to 110 GHz. Index Terms—Calibration, error correction, millimeter-wave measurements, scattering parameters, standards.

I. INTRODUCTION CHIEVING accurate broadband on-wafer calibrations with a minimum set of common footprint calibration standards decreases the cost of real estate on planar substrates and wafers. It also reduces calibration time. The goal of this study is to establish a methodology for characterizing custom standards with sufficient accuracy for broadband calibrations with a manageable set of standard definitions. We also sought a technique that would account for potential fabrication variability of on-wafer (or on-board) loads. Most on-wafer lumped standard calibration techniques use an ideal lossless thru-line model. More appropriate modeling of the thru line, as well as the load standard, enables the proposed method to achieve accurate calibrations from very low to high frequencies (e.g., dc to

A

Manuscript received February 1, 2006; revised May 12, 2006. This work was supported in part under grants by Anritsu, M/A-Com Companies, and Modelithics Inc. S. Padmanabhan was with the Center for Wireless and Microwave Information Systems, Department of Electrical Engineering, University of South Florida, Tampa FL 33620 USA. She is now with Semflex Inc., Mesa, AZ 85215 USA. L. Dunleavy and A. Rodríguez are with the Center for Wireless and Microwave Information Systems, Department of Electrical Engineering, University of South Florida, Tampa, FL 33620 USA (e-mail: [email protected]). J. E. Daniel was with the Center for Wireless and Microwave Information Systems, Department of Electrical Engineering, University of South Florida, Tampa, FL 33620 USA. He is now with the Insyte Corporation, Palm Harbor, FL 34683 USA. P. L. Kirby is with the School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. Color versions of Figs. 1 and 3–27 are available online at http://ieeexplore. ieee.org. Digital Object Identifier 10.1109/TMTT.2006.881027

110 GHz) using a minimal set of compact short-open-load-thru (SOLT) standards, including cases where the reference plane is not at the center of thru. The work described herein follows from a series of developments for on-wafer calibration improvement [1]–[3]. A method for using measured lumped standard definitions in place of the more common equivalent-circuit models (mSOLT) was shown to be capable of transferring the accuracy of thru-reflect-line (TRL) calibration to a compact set of lumped standards. Since mSOLT suffers the low-frequency inaccuracies commonly associated with TRL calibrations, a broadband accurate calibration is not established. A multiline TRL calibration can extend the classical TRL band to lower frequencies, but it requires different probe-to-probe spacing with each line, and this is a major disadvantage with semiautomated calibrations. mSOLT, as well as the technique of this paper, solves this problem. Although the technique presented depends on TRL for the initial model parameters, it will be clear later on that the accuracy of the parameters is not affected by the low-frequency issues of TRL. A careful study of multiple microstrip and coplanar calibration standards on different substrates showed that the -parameter variations in the load standards were the most significant. This variation in RF performance was verified to be directly correlated with the dc resistance [2]. An attempt to fit existing VNA load models to broadband TRL-calibrated load measurements revealed that a more complex load model is needed to represent planar coplanar waveguide (CPW) and microstrip loads [3], [4]. This paper demonstrates a modified physical model that fits measured data well and has a resistive part directly tied to the measured dc resistance of the load. An established approach that has some of the advantages as of the proposed method is the line-reflect-match (LRM) method [5], [6]. The main drawback to the original method is that it is assumed that the -parameter measurements are referenced to the impedance of the match standard. In our study, we found that, for some load standards, their impedance are complex with a real part that varies with the standard’s dc resistance. The reference planes are also established in the middle of the thru-line standard, and are often moved to the probe tips (for commercial substrate calibrations) using a lossless thru-line assumption. The line-reflect-reflect-match (LRRM) method [7] is an improvement to LRM that addresses the complex nature of the load through determination of a series inductance–resistance (R–L) model for the load standard. Another method also

0018-9480/$20.00 © 2006 IEEE

3584

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

accounts for imperfections in LRM loads [8] using a -determined TRL (typically) to first measure the complex impedance of the load and then translate the -parameters to a desired (e.g., 50 ) reference impedance. However, in some cases, as illustrated in this paper, where the load varies significantly with frequency, the technique can fail because its equation-based model cannot track the measured load. One drawback to LRM and TRL methods, in general, is that one-port calibrations cannot be directly accomplished. This paper demonstrates an alternate practically useful calibration approach, called the complex SOLT (cSOLT) that achieves TRL level accuracy at high frequency and avoids commonly occurring low-frequency TRL calibration issues, which have been briefly introduced in [9]. The method is implemented using LabVIEW software, allowing for differing definitions on each port when necessary. The method has been demonstrated to produce excellent results using the Anritsu “Lightning” 37397C and Wiltron 360B vector network analyzers (VNAs) in a number of different calibration experiments conducted on a wide range of substrates (GaAs, alumina, and FR-4) and at frequencies extending to 110 GHz.

Fig. 1. Microphotograph of a typical GaAs microstrip load.

Fig. 2. Complex load model for CPW and MS loads, which accounts for dc resistance variation.

B. Loads II. CSOLT METHODOLOGY The SOLT calibration method is a well-established method for VNA calibration [10]. In many situations, it has been supplanted by the generally more accurate TRL method [11], [12], however, there are situations where a SOLT calibration can be advantageous. In the current case, the interest is in producing a solution that can lead to minimizing the space occupied by on-wafer standards, while maximizing broadband (e.g., dc to over 110 GHz) calibration accuracy. The methodology used can achieve accurate broadband calibrations. Modeling of the load is accomplished by combining dc data with -corrected TRL calibrated data. The thru model is modeled by fitting a lossy propagation constant equation that accounts for attenuation as a function of frequency to a thru-line standard measured with a probe-tip referenced TRL calibration. Once the standards have been modeled, the cSOLT carries out subsequent SOLT calibrations. III. STANDARD MODELS AND MEASURED DATA This study emphasizes the use of improved models for the load and thru standards, while utilizing conventional models for the open and short standards. In all the cases, standard models are derived from -corrected 50- TRL measurements enabled with the National Institute of Standard’s (NIST)’s MultiCal software [12], [13] for each of the standards, combined with dc measurements of the loads. A. Shorts/Opens We found no improvement necessary to the conventional approach to modeling calibration opens and shorts. Hence, the measured open/short were used to fit the respective capacitance or inductance models.

Fig. 1 shows a typical microstrip load after fabrication. The existing load model from NIST [4] accounts for the dc resistance of the load , series inductance ( ), and the capacitance to ground ( ) (Fig. 2). However, with the results shown further here, it is seen that with an increase in frequency, the basic model cannot track the variations of the load well. Thus, the proposed load model includes elements that help the model fit the high-frequency measured loads. It includes a gap capacitance ( ) that exists between the signal line to via pad and inductance ( ) to ground through the via. When a CPW load is of interest, the via inductance is set to zero in the model. The load models are derived such that they fit high-frequency TRL data, but smoothly transition to the dc resistance at low frequencies. The model is also used to fit each of the ports separately since the dc resistances of the ports may vary independently. The fitting procedure is important because, though most of the commercial substrates are laser trimmed and well fabricated (hence, negligible variation in load impedance with frequency), there are many cases where the load is not so well behaved. It is seen in Figs. 3 and 4 that the proposed load model tracks well with the dc resistance of the load at lower frequencies and follows the high-frequency measured data through the entire frequency range when compared to the existing models. The physical significance of the model is validated by comparing the simulation-based optimized model parameters to those calculated using RF transmission-line theory. Table I gives the simulated values of the parameters used in the model for microstrip loads contained in a calibration set designed by Imparato, Tampa, FL [1] and fabricated by ITT GaAs Tek (now M/A-Com), Lowell, MA. The inductance of the load and capacitance to ground were calculated with LINPAR [17] (calculates matrix parameters for transmission lines). The parameters were calculated treating the load as a transmission line. The load inductance and capacitance to ground are found to be 48 pH and 18 fF, respectively. The gap capacitance is a low value, as

PADMANABHAN et al.: BROADBAND SPACE CONSERVATIVE ON-WAFER NETWORK ANALYZER CALIBRATIONS

Fig. 3. Real part of load impedance of center of thru referenced GaAs microstrip load measured (R = 50:256 ) versus modeled load impedances for different models.

3585

Fig. 5. Real part of load impedance of probe tip referenced GGB CS5 CPW load (R = 49:9 ) measured versus modeled load impedance.

Fig. 4. Imaginary part of load impedance of center of thru referenced GaAs microstrip load measured (R = 50:256 ) versus modeled load impedances for different models.

Fig. 6. Imaginary part of load impedance of probe tip referenced GGB CS5 CPW load (R = 49:9 ) measured versus modeled load impedance.

TABLE I TABULATION OF LOAD MODEL PARAMETERS AND THEIR VALUES FROM SIMULATION

fabricated and, hence, the impedance does not vary widely with increase in frequency. The proposed model is also verified with surface-mount resistors soldered on 14-mil FR-4 substrates with microstrip lines, as shown in Figs. 7 and 8. The significance of modeling such loads is to illustrate that broadband SOLT calibrations can be performed on printed circuit boards (PCBs) with chip resistors as loads. A point not to be overlooked is that due to TRL measurement inaccuracies at low frequencies, the model (which trends to the dc resistance) is a more correct representation of the load than the measured data, especially at low frequencies. Measurement of an ITT GaAs load was made to 110 GHz, shown in Figs. 9 and 10. The broadband measurement data in Figs. 9 and 10 was obtained by combining data taken from two separate VNAs; the Anritsu “Lightning” 37397C (40 MHz–65 GHz) and the Wiltron 360B (65–110 GHz). The model parameters were obtained from the 40-MHz–65-GHz data. Expanding the frequency range to 110 GHz, the load model’s performance holds with no additional tuning required. The degradation in data stability near 65 GHz is due to the two sets reaching their performance limits.

expected. This capacitance represents the combined coupling through the air to the pad on top of the ground via, as well as coupling through the substrate to the conical shaped via below the load (which is more significant). The same model topology fits CPW loads very well with an analogous physical motivation. In addition to the ITT GaAs microstrip load, the complex load model is further validated with CPW and microstrip loads measured on other substrates. Figs. 5 and 6 show the real and imaginary impedance of a load measured on a commercially available substrate (GGB CS5). As stated earlier, the loads are well

3586

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 7. Real part of load impedance of center of thru FR-4 load (surface mount) (R = 49:75 ) measured versus modeled load impedance.

Fig. 8. Imaginary part of load impedance of center of thru FR-4 load (surface mount) (R = 49:75 ) measured versus modeled load impedance.

Fig. 9. Real part of load impedance of center of thru referenced GaAs microstrip load measured (R = 49:9799 ) versus proposed model load impedance.

Fig. 10. Imaginary part of load impedance of center of thru referenced GaAs microstrip load measured (R = 49:9799 ) versus proposed model load impedance.

C. Thru VNA calibration algorithms like SOLT or LRM mostly assume an ideal lossless transmission-line model for the thru-line standard. In reality, the thru line is lossy and the attenuation loss varies with respect to the substrate and conductor properties. Thus, it becomes necessary that the thru line be well modeled in terms of the propagation constant over the design frequency range. Even though most VNA firmware allows incorporating for loss in thru lines, it is very common for practitioners to neglect loss during calibration since it is not mandatory. In this paper, the need for an accurate thru model is emphasized and has been implemented independent of the network analyzer. The thru-line equations [14] treat the transmission line as a lossy line and, thus, compensates for the losses. It is known that the propagation constant of a transmission line is represented as

either not too significant or cannot be modeled directly. The attenuation constant equation was thus designed to be dependent on the aforementioned losses. The conductor loss of a transmission line is represented by the sheet resistivity of the line and is given by

(2) where is frequency (in hertz), is the permeability, and the conductivity of the material. The dielectric loss is represented by the equation

(3)

(1) where is the attenuation constant (nepers/unit length) of the transmission line. The dielectric loss of the substrate and conductor loss of the metal are the two main factors that are generally significant in a thru-line measurement. The other losses are

is

where

(4)

PADMANABHAN et al.: BROADBAND SPACE CONSERVATIVE ON-WAFER NETWORK ANALYZER CALIBRATIONS

Fig. 11. Magnitude of crostrip substrate.

S

3587

for the 8390-m delay line on the ITT GaAs mi-

Fig. 13. Reflection coefficients of loads for the ten GaAs loads used, with varying dc resistances from 49.5 to 52.5 on the Smith chart.

ITT GaAs microstrip substrate. Using (6), the equation-based data traces well with the measured -parameters of the transmission line through the frequency range concerned.

IV. Fig. 12. Magnitude of crostrip substrate.

S

for the 8390-m delay line on the ITT GaAs mi-

where is the effective dielectric constant, is the relative permittivity, is the loss tangent of the conductor, and is the speed of light in vacuum. The phase constant of the line is given by

(5) and the total attenuation constant lowing equation:

is represented by the fol-

(6) The above equation includes a fitting factor , which is multiplied with the total attenuation. This is because the equation uses simple approximations to estimate the losses and, hence, the total loss is underestimated. The thru equation suggested above is a good approximation for typical transmission lines and substrates. Figs. 11 and 12 show the thru equation fit with an 8390- m delay line measured, after a TRL calibration, on the

VARIATION WITH RF PERFORMANCE OF LOAD

The fact that the dc resistance can vary between loads was verified with the availability of a whole wafer (ITT GaAs microstrip) with multiple identical dies. Fig. 13 is a plot of the reflection coefficients of ten different loads measured across the wafer after a corrected center of thru TRL calibration. It is clear from the Smith chart that the load impedances vary from die to die on the wafer. This can be attributed to fabrication issues like film resistivity and thickness tolerances. The advantage of the proposed load model when compared to existing models is its ability to track the dc resistance of the load at the lower end and at the same time follow the load behavior at higher frequencies more accurately. It also lends itself to a better physical justification than that proposed in [3]. For validation, 14 loads of varying dc resistance were measured. The measured loads are fit with the load model shown in Fig. 2, adjusting for the respective dc resistances, but making no changes to the other model parameters. Example results presented here correspond to loads with dc resistance of 52.045 and 49.8 . Figs. 14 and 15 show the load impedances measured and matched with the compensated model. These data comparisons show the load model can track the RF performance of the load at both dc and higher frequency ranges. As a reminder, the model parameters are values obtained when optimizing for a 50.3- load. It is seen that the model can fit well with 52- and

3588

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 14. Real and imaginary parts of load impedance of measured load versus load model with adjusted R of 52.045 at center of thru reference. Fig. 17. Comparison of vector error difference between the measured and model with and without compensation for varying R .

Fig. 15. Real and imaginary parts of load impedance of measured load versus load model with adjusted R of 49.8 at center of thru reference.

equivalent model for the load standard. For Fig. 16, the load model parameters were obtained by optimizing each of the three different load models for a single load measurement. In order to plot the vector error difference between the model and 14 measured loads, the model’s component was then the only parameter that was adjusted to correspond to the measured dc resistance. It is observed from the graph that the series RL model had the maximum error because the model is basically a constant real impedance and a linear imaginary impedance with little accommodation for complex impedance variations with frequency. The NIST model is seen to present a large improvement and can model varying dc resistance, however, the vector error difference is approximately 14 at 65 GHz and the error through most of the bandwidth is approximately 8 . The proposed load model has an average error of less than 4 throughout the frequency range. The importance of adjusting for is highlighted again in Fig. 17. The plot shows the vector error differences for model versus measured data for varying and nonadjusted cases. It is clear from the graph that if the load is assumed to be 50 with no variation die to die, then the error between the measured and model is much more significant. V. CSOLT IMPLEMENTATION

Fig. 16. Average vector magnitude error of 14 loads comparing the different load model conditions.

49.8- loads and are at most 1.5 different from the simulated load at the higher frequencies. The average vector error difference between the 14 measured loads and three different load models is plotted in Fig. 16 for a complete analysis. The comparison is made between the proposed complex load model, the NIST model, and series RL model that is available in the commercial VNA firmware as the

A LabVIEW program has been implemented to perform the SOLT calibration with the complex load and thru models. The calibration software can presently be used on Anritsu Lightning and Wiltron 360B VNAs. The program accounts for the small variations that can occur between ports by providing independent model options for the two ports of the calibration standards. The program also allows using measured data files for standard definitions as in the case of the mSOLT calibration method. The results illustrated herein are obtained from an Anristu Lightning (0.04–65 GHz), as well as the Wiltron 360B (65–110 GHz) VNA. The improvement in accuracy is verified using a calibration comparison program also implemented for the purpose of this study. The method as presented by Marks et al. in [15] compares the error coefficients of two calibrations and, thus, plots

PADMANABHAN et al.: BROADBAND SPACE CONSERVATIVE ON-WAFER NETWORK ANALYZER CALIBRATIONS

Fig. 18. S and S of a 0.03-pF capacitor with respect to TRL and cSOLT calibrations referenced to center of thru on a GaAs microstrip substrate.

3589

Fig. 19. Vector error difference between S from TRL and cSOLT calibrations referenced to center of thru on a GaAs microstrip substrate.

the maximum upper error bound. The comparison provides a quick and meaningful comparison and establishes the maximum possible error between two calibrations. The program compares two one-tier calibrations directly from the network analyzer or reads from the error coefficients that are saved in a file. The significance of the implementation of this comparison method where the benchmark calibration is not TRL, as described in [16], will be presented in Section VII. VI. CALIBRATION AND DEVICE MEASUREMENTS The SOLT calibration standards on the ITT GaAs and FR-4 substrates were measured after the center of thru TRL calibration and modeled as discussed earlier. A cSOLT calibration was performed with the standard definitions generated with the models fit to TRL data along with the dc resistance measurements. The accuracy of the calibration is verified with -parameter measurements of devices that were available. The devices were also measured with respect to a TRL calibration for comparison purposes. A 0.03-pF capacitor was measured on the GaAs microstrip substrate with respect to cSOLT and TRL calibrations. Fig. 18 shows and of the measured capacitor. It is observed from the graphs that the -parameters of the two calibrations are very close. The worst case difference between the measurements with respect to the calibrations is plotted in Fig. 19. The vector difference for is less than 0.035 throughout most of the frequency range, which is a relatively small difference at 65 GHz. The results from calibrating a 14-mil-thick FR-4 substrate are also illustrated in this paper to verify the broadband accuracy of cSOLT calibrations with surface-mount components. A 0.2-pF chip capacitor was measured with respect to cSOLT and TRL calibrations. It is clear from the graph in Fig. 20 that the -parameter measurements match very well with each other. The vector error difference between the two calibrations was less than 0.02 throughout the frequency range. VII. CSOLT—ACCURACY VERIFICATION In Section VI, capacitors were measured with respect to cSOLT and TRL calibrations, and the difference in the measured -parameters was analyzed for comparison. However,

Fig. 20. S and S of 0.2-pF capacitor with respect to cSOLT and TRL calibrations referenced to center of thru on a 14-mil FR-4 microstrip substrate.

the difference between two calibrations is best predicted with upper error bound graphs. This graph shows the worst possible measurement differences between passive components, which are calculated directly from the error coefficients. The proposed cSOLT is compared with TRL, mSOLT, and front panel SOLT calibrations performed on the same substrate. The comparison graphs presented here are generated from the LabVIEW program implemented for the study. Figs. 21 and 22 show the upper error bound between the calibrations performed on the ITT GaAs microstrip substrate. The benchmark calibrations for the two cases are TRL and cSOLT, respectively. The fact that the cSOLT method is good at lower frequencies and behaves close to the TRL at higher frequencies is highlighted when all the calibrations are compared with cSOLT as the reference calibration. Note in Fig. 22 that the cSOLT compares with SOLT at low frequencies (zoomed-in part of the graph) as desired, and has a maximum upper error bound of 0.05 at 65 GHz when compared to TRL. When the frequency range is expanded to 110 GHz, cSOLT retains its TRL-like performance, as shown in Fig. 23. In contrast, the low-frequency error of the TRL reference calibration in Fig. 21 causes all the calibrations to show a high upper error bound at low frequencies. The absolute value of this rise in error at low frequencies varies depending on various

3590

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 21. Upper error bound between TRL, cSOLT, mSOLT, and SOLT calibrations with respect to TRL as reference on ITT GaAs microstrip substrate with center of thru reference.

Fig. 23. Upper error bound between TRL, cSOLT, and SOLT calibrations with respect to TRL on ITT GaAs microstrip substrate with center of thru reference.

Fig. 24. Upper error bound between cSOLT, mSOLT, and SOLT with respect to TRL calibration on 14-mil FR-4 microstrip substrate. Fig. 22. Upper error bound difference between TRL, cSOLT, mSOLT, and SOLT calibrations with respect to cSOLT on ITT GaAs microstrip substrate with center of thru reference.

factors impacting the quality of the TRL calibration standards (skin depth of metals used, length of longest delay line, etc.). Since the mSOLT technique depends on the TRL measured data files for its definitions of standards, the accuracy of the calibration follows with the accuracy of TRL, which fails at the lower frequency end. The front panel SOLT calibration compared to TRL has a maximum upper bound that increases with frequency because the standards are not defined adequately. However, with the cSOLT method where the standards are accurately defined, the error is reduced to a great extent. cSOLT has minimum error at the lower frequencies when compared to the TRL and mSOLT calibrations and closely follows TRL at high frequency, as desired for a broadband calibration. The compatibility of the load model with surface-mount chip resistors and the increase in accuracy of cSOLT calibrations is illustrated with the FR-4 copper clad substrate. The same comparison as in the GaAs case is also performed with this case. Figs. 24 and 25 show that cSOLT is comparable to the TRL at higher frequencies (maximum upper bound of 0.02 at 18 GHz) and retains low-frequency accuracy of the SOLT.

Fig. 25. Upper error bound between TRL, mSOLT, and SOLT with respect to cSOLT calibration on 14-mil FR-4 microstrip substrate.

VIII. CALIBRATING OVER MULTIPLE DIE WITH CSOLT A best practice strategy for calibrating over multiple die on a whole wafer is suggested in this paper. The short, open, and complex load with thru equation models are established from one of the dies based on measured data after -corrected TRL

PADMANABHAN et al.: BROADBAND SPACE CONSERVATIVE ON-WAFER NETWORK ANALYZER CALIBRATIONS

3591

Fig. 26. Upper error bound difference for the cSOLT calibration versus TRL on multiple die from an ITT GaAs microstrip substrate using cSOLT as the reference calibration.

Fig. 27. Upper error bound between cSOLT (equal-length SOLT standards) and TRL calibration referenced at probe tips on the M/A-Com GaAs microstrip substrate.

calibration. It has been verified that the load standard varies the most from port to port and die to die. However, with the RF performance (load impedance) directly dependent on the of the load, a complex load model with variable dc resistance that accounts for variations in its performance has been illustrated. Thus, the calibration process can be simplified by adjusting for the value of in the load model and perform SOLT calibrations on the multiple dies available. This eliminates the need to model standards after -corrected TRL calibrations for each new die in the substrate without compromising the accuracy of the calibrations. To illustrate the proposed strategy, a cSOLT calibration is performed on the reference substrate whose dc resistance of the load is 49.97 on port 1 and 49.93 on port 2. The model parameters calculated from the reference substrate is retained for the die under test, except for the dc resistance. Another cSOLT is performed on a different die whose on either port are 51.8 and 51.35 . The calibration thus performed is compared with that of a TRL performed on the same substrate. It is also compared with the cSOLT versus TRL comparison data from the reference die. From Fig. 26, it is clear that the error bound shows very good agreement with the upper bound data from the reference substrate up to approximately 45 GHz. It is also observed that the difference between the two error bounds is just approximately 0.03 after 45 GHz.

calibration substrate (100- m-thick microstrip GaAs) and fabricated by M/A-Com. From the design, it was noted that the TRL standards occupied approximately 4.2 mm 9 mm, while the SOLT standards with the same footprint size occupied just approximately 1.7 mm 1.4 mm. Since the cSOLT calibration requires a set of TRL standards to model the SOLT standards, one reference substrate with the required TRL can be fabricated. The other die on the wafer can just have the SOLT standards. It has been illustrated that the loads measured on the other die can be modeled accurately by adjusting the dc resistance of the respective loads. It was observed that the dc load resistance variation was very well controlled for the M/A-Com standards fabricated for the study, as compared to the other GaAs substrate. The equal footprint SOLT standards were measured and modeled after a probe-tip TRL calibration. It is very important that a calibration comparison be performed with these standards, as this is instrumental in proving that the equal-footprint SOLT standards can be substituted for TRL calibration when modeled accurately. Fig. 27 shows the maximum upper bound when TRL and cSOLT calibrations referenced at the probe tip plane are compared with each other. The results as seen are very close to the TRL repeatability data up to approximately 45 GHz. The error rises to 0.055 at 50 GHz and this is because the open and short were resonating at around 47 GHz. The resonance could be because of coupling from the adjacent standards. Nevertheless, the equal-footprint SOLT standards are validated to predict a very low upper error bound when compared to the TRL repeatability data at higher frequencies while still retaining accuracy at lower frequencies.

IX. SPACE CONSERVATIVE CSOLT The importance of the equal length (same footprint) SOLT standards is highlighted in cases where space is an issue in the fabrication of bulk wafers with active and or passive devices. In device wafers, the area allocated for calibration standards might be of concern if it is a significant number. The other advantages to the proposed methodology are that with the compact set of space conservative standards, probing is very easy and less time consuming and still highly accurate. Availability of a semiautomatic probing system ensures more repeatable measurements with the equal footprint standards. To illustrate the same, TRL and SOLT (half the thru and equal footprint) standards (dc–110 GHz) were designed on a custom

X. CONCLUSION An improved approach to on-wafer SOLT calibrations utilizing models currently unavailable in the firmware of network analyzers has been presented. Complex load and thru models have been implemented with external calibration programs in order to account for variations with frequency due to fabrication imperfections and parasitic effects. The use of more complex models allow for a reduction in the error normally incurred in an SOLT calibration as frequency increases, achieving TRL

3592

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

accuracy, while eliminating the low-frequency problems associated with a TRL calibration. The relation between the dc resistance and RF performance of the load is utilized to demonstrate the fact that SOLT calibrations can be performed on multiple die without having to model the load variation on each individual die or pair of load standards. A significant advantage is that cSOLT utilizes a compact set of space-conservative calibration standards in comparison to the long delay lines needed to achieve broadband TRL. The calibration algorithm can also be used for one-port calibrations, which is not possible with TRL or LRM calibration techniques. Thus, broadband accuracy can be achieved when compared to other methods where the compromise is either at the lower or higher frequency ranges.

[12] R. Marks, “A multiline method of network analyzer calibration,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1205–1215, Jul. 1991. [13] R. B. Marks and D. F. Williams, “Characteristic impedance determination using propagation constant measurement,” IEEE Trans. Microw. Guided Wave Lett., vol. 1, no. 6, pp. 141–143, Jun. 1991. [14] B. C. Wadell, “Generalized transmission lines,” in Transmission Line Design Handbook. Norwood, MA: Artech House, 1991, ch. 2, pp. 15–27. [15] R. B. Marks, J. A. Jargon, and J. R. Juroshek, “Calibration comparison method for vector network analyzers,” in 48th Automat. RF Technol. Group Conf. Dig., Clearwater, FL, Dec. 1996, pp. 38–45. [16] D. F. Williams, R. B. Marks, and A. Davidson, “Comparison of on-wafer calibrations,” in 38th Automat. RF Technol. Group Conf. Dig., San Diego, CA, Dec. 1991, pp. 68–81. [17] A. R. Djordjevic, M. B. Bazdar, T. K. Sarkar, and R. F. Harrington, LINPAR for Windows: Matrix Parameters for Multiconductor Transmission Lines. Norwood, MA: Artech House, 1995 [Online]. Available: http://www.artechhouse.com

ACKNOWLEDGMENT The authors would like to thank Dr. D. Williams, National Institute of Standards and Technology (NIST), Boulder, CO, and J. Martens, Anritsu, Morgan Hill, CA, and E. Daw (formerly with Anritsu) for helpful discussions and encouragement, Dr. J.-P. Lanteri and Dr. T. Büber, both with M/A-Com, Lowell, MA, for fabrication of custom GaAs calibration standards, J. Capwell and M. Laps, both with Modelithics Inc., Tampa, FL, for providing surface-mount FR4 calibration boards, and G. Boll, GGB Industries, Naples, FL, for providing some of the probes and commercial calibration standard substrates used. Author L. Dunleavy would also like to credit Dr. D. Metzger, a consultant, as the author of some of the software algorithms used in the early phases of this paper’s research. REFERENCES [1] M. Imparato, T. Weller, and L. Dunleavy, “On-wafer calibration using space-conservative (SOLT) standards,” in IEEE MTT-S Int. Microw. Symp. Dig., Anaheim, CA, Jun. 1999, pp. 1643–1646. [2] P. Kirby, L. Dunleavy, and T. Weller, “The effect of load variations on on-wafer lumped element based calibrations,” in 54th Automa. RF Technol. Group Conf. Dig., Atlanta, GA, Dec. 1999, pp. 81–90. [3] ——, “Load models for CPW and microstrip SOLT standards on GaAs,” in 56th Automat. RF Technol. Group Conf. Dig., Boulder, CO, Dec. 2000, pp. 93–103. [4] D. K. Walker, D. F. Williams, and J. M. Morgan, “Planar resistors for probe station calibration,” in 40th Automat. RF Technol. Group Conf. Dig., Orlando, FL, Dec. 1992, pp. 1–9. [5] H. J. Eul and B. Schiek, “Thru-match-reflect: One result of a rigorous theory for de-embedding and network analyzer calibration,” in Proc. 18th Eur. Microw. Conf., Stockholm, Sweden, Sep. 1988, pp. 909–914. [6] S. Lautzenhiser, A. Davidson, and K. Jones, “Improve accuracy of on-wafer tests via LRM calibration,” Microw. RF, vol. 29, no. 1, pp. 105–109, Jan. 1990. [7] A. Davidson, K. Jones, and E. Strid, “LRM and LRRM calibrations with automatic determination of load inductance,” in 36th Automat. RF Technol. Group Conf. Dig., Monterrey, CA, Nov. 1990, pp. 57–63. [8] D. F. Williams and R. B. Marks, “LRM probe-tip calibrations using non-ideal standards,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 2, pp. 466–469, Feb. 1995. [9] S. Padmanabhan, P. Kirby, J. Daniel, and L. Dunleavy, “Accurate broadband on-wafer SOLT calibrations with more complex load and thru models,” in 61st Automat. RF Technol. Group Conf. Dig., Philadelphia, PA, Jun. 2003, pp. 5–10. [10] J. Fitzpatrick, “Error models for system measurements,” Microw. J., vol. 21, no. 5, pp. 63–66, May 1978. [11] G. F. Engen and C. A. Hoer, “ ‘Thru-reflect-line’: An improved technique for calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 12, pp. 987–993, Dec. 1979.

Sathya Padmanabhan (M’01) received the B.E. degree in electronics and communications from the University of Madras, Madras, India, in 2001 and the M.S.E.E. degree from the University of South Florida (USF), Tampa, in 2004. From October 2001 to May 2004, she was a Research Assistant with the Center for Wireless and Microwave Information Systems, where she was involved with the improvement of the accuracy of calibration techniques over a broadband frequency range. She is currently a Microwave Engineer with Semflex Inc., Mesa, AZ. Her responsibilities include design, development, and testing of high-performance microwave and millimeter-wave connectors and cable assemblies.

Lawrence Dunleavy (S’80–M’82–SM’96) received the B.S.E.E. degree from Michigan Technological University, Houghton, in 1982, and the M.S.E.E. and Ph.D. degrees from The University of Michigan at Ann Arbor, in 1984 and 1988, respectively. Along with four faculty colleagues, he established the Center for Center for Wireless and Microwave Information Systems, University of South Florida (USF), Tampa. In 2001, he cofounded Modelithics Inc., a USF spinoff company to provide a practical commercial outlet for developed modeling solutions and microwave measurement services. He has been involved in industry for E-Systems (1982–1983) and Hughes Aircraft Company (1984–1990), and was a Howard Hughes Doctoral Fellow (1984–1988). In 1990, he joined the Department of Electrical Engineering, USF, where he is currently a Professor. He guides a team of graduate students in various research projects related to microwave and millimeter-wave device, circuit, and system characterization and modeling. From 1997 to 1998, he spent a sabbatical year with the Noise Metrology Laboratory, National Institute of Standards and Technology (NIST), Boulder, CO. He has authored or coauthored over 75 technical papers. Dr. Dunleavy is very active in the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and the Automatic RF Techniques Group (ARFTG).

John E. Daniel (M’00) was born in St. Petersburg, FL, in 1981. He received both the B.Sc. and M.Sc. degrees electrical engineering from the University of South Florida (USF), Tampa, in 2005. While with USF, he was a Research Assistant. He was also an RF Measurement Technician with Modelithics Inc., Tampa, FL. In December 2005, he joined the Innovative Systems and Technology Corporation (Insyte Corporation), Palm Harbor, FL, as a Lead RF Test Engineer. In March 2006, the Insyte Corporation was acquired by the Aerospace and Communications Division, ITT, and has retained the same position with ITT-ACD at the facility in Palm Harbor, FL.

PADMANABHAN et al.: BROADBAND SPACE CONSERVATIVE ON-WAFER NETWORK ANALYZER CALIBRATIONS

Alberto Rodríguez (S’00–M’06) received the B.S. and M.S. degrees in electrical engineering from the University of South Florida (USF), Tampa, in 1997 and 2003, respectively, and is currently working toward the Ph.D. degree in electrical engineering at USF. His current research interests are in the computer-aided design (CAD) and characterization of microwave and millimeter-wave balanced and multimode circuits and devices.

3593

Peter L. Kirby (S’97–M’05) received the B.S. and M.S. degrees in electrical engineering from the University of South Florida (USF), Tampa, in 1999 and 2002, respectively, and is currently working toward the Ph.D. degree in electrical engineering at the Georgia Institute of Technology, Atlanta. His current research interests are monolithicmicrowave integrated-circuit (MMIC) design in the millimeter-wave frequency range and development of alternative terahertz waveguide structures for use in terahertz multiplier applications.

3594

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Scale-Changing Technique for the Electromagnetic Modeling of MEMS-Controlled Planar Phase Shifters Etienne Perret, Student Member, IEEE, Hervé Aubert, Senior Member, IEEE, and Hervé Legay

Abstract—A scale changing approach is proposed for the electromagnetic modeling of phase-shifter elements used in reconfigurable microelectromechanical system (MEMS)-controlled reflectarrays. Based on the partition of the discontinuity plane in planar sub-domains with various scale levels, this technique allows the computation of the phase shift from the simple cascade of networks, each network describing the electromagnetic coupling between two scale levels. The high flexibility of the approach associated with the advantages of the integral equations formulations renders this original approach powerful and rapid. The scale-changing technique allows quasi-instantaneous computing of the 1024 phase shifts achieved by ten RF-MEMS switches distributed on the phase-shifter surface. Moreover, the proposed approach is much better than the finite-element-method-based software in time costing. Experimental data are given for validation purposes. Index Terms—Multiscale structures, planar phase shifter, reflectarrays, RF microelectromechanical systems (MEMS).

I. INTRODUCTION EFLECTARRAY consists of a feeding antenna illuminating a planar microstrip array, which is designed to scatter a planar phase surface in front of the aperture [1], [2]. The introduction of a specific small phase shift for reconstituting a planar phase surface in the desired direction may be achieved by using microstrip patches with passive delay lines [3]–[5], by adjusting the patch size [6], [7], or by tuning the substrate height [8]. Reflectarray antenna with RF microelectromechanical (RF MEMS) switches is an emerging technology for reconfigurable and scanning antennas. Recently, circular [9] and linear [10] polarization reflectarrays controlled by RF MEMS have been designed. The linearly polarized concept was developed in the frame of a research project funded by the European Space Agency (ESA) Noordwijk, The Netherlands [10], [11]. This MEMS-based reflectarray element was derived from a similar passive concept [12]. It is selected for two

R

Manuscript received December 3, 2005; revised March 23, 2006. E. Perret was with the Ecole Nationale Supérieure d’Electrotechnique, d’Electronique, d’Informatique, d’Hydraulique et des Télécommunications, 31071 Toulouse, France. He is now with the Institut d’Electronique Fondamentale, 91405 Orsay, France. H. Aubert is with the Laboratoire d’Analyse et d’Architecture des Systèmes, Centre National de la Recherche Scientifique, 31077 Toulouse, France, and also with the Ecole Nationale Supérieure d’Electrotechnique, d’Electronique, d’Informatique, d’Hydraulique et des Télécommunications, 31071 Toulouse, France (e-mail: [email protected]). H. Legay is with the Research Department, Alcatel Alenia Space, 31037 Toulouse, France. Digital Object Identifier 10.1109/TMTT.2006.879777

applications where MEMS technology offers interesting capabilities, which are: 1) for -band transmission of high flow between small satellites, observation, and scientific expeditions of nanosatellites constellation and 2) for -band missions GEO-telecom requiring a reconfigurable satellite cover. In such a reflectarray antenna, the phase-shift variation is controlled by the UP/DOWN state of a finite number of RF-MEMS switches: switches, phase for a phase-shifter element containing shifts are available. This paper focuses on the electromagnetic modeling of this latter concept. Its full characterization with existing softwares, i.e., the method of moment or the finite-element method, appeared to be very time consuming as the number of switches increases. Moreover, the wide diversity of scales—in practice, the ratio between the largest and smallest dimensions in a single phase-shifter element is higher than 100—may generated convergence issues, probably related to the ill-conditioned matrices in the numerical treatment of the boundary value problem. The integral-equation formulation (IEF) with entire domain trial functions [13] allows a reduction in the number of unknowns, but suffers from low flexibility. An original approach, called the scale-changing technique, was then proposed and developed for handling the multiscale nature of the structure. It is presented in this paper. Using a partition of the discontinuity plane in multiple planar sub-domains of various scale levels, the scale-changing technique allows the computation of the phase-shift variation generated by the MEMS-controlled phase shifter from the simple cascade of networks, each network describing the electromagnetic coupling between two scale levels. Very recently, this approach has been applied with success to the computation of the input impedance of planar antennas [14]. The application of the scale-changing technique to MEMS-controlled planar phase shifters is more complex because it requires handling many scale levels. A large collaborative research study has been reported in [10] including multiple industrial-oriented considerations relative to the design, the technology, and the manufacturing of MEMS-controlled reflectarrays: this paper is not focused on the numerical technique and does not give concrete numerical results for evaluating the key advantages (low computation time, high flexibility) of the scale-changing technique compared with classical numerical techniques. Finally, the research reported in [15] is focused on the derivation of the equivalent network of a single RF-MEMS switch and is not concerned with the electromagnetic modeling of multiscale planar circuits: in [15], the scale-changing technique could be viewed as a special case of the mode-matching technique. In this paper, on the one hand, multiple scale levels are taken into account by an original cascade of more than one scale-changing

0018-9480/$20.00 © 2006 IEEE

PERRET et al.: SCALE-CHANGING TECHNIQUE FOR ELECTROMAGNETIC MODELING OF MEMS-CONTROLLED PLANAR PHASE SHIFTERS

Ku

Fig. 1. Planar phase shifter used in -band MEMS-controlled reflectarrays. The structure has been manufactured on an alumina substrate (relative permittivity: 9.8; thickness: 0.254 mm). One side of the substrate is shown in this figure. The opposite side of the alumina substrate is completed by adding an air layer of thickness = 2 mm followed by a metallic plane (short circuit). For experimental purposes, this planar structure is inserted in the cross section of a metallic waveguide [12].

h

networks and, on the other hand, current application is distinct from the authors’ previous research. This paper is organized as follows. In Section II, the scalechanging technique is applied to the electromagnetic modeling of a planar phase shifter used in MEMS-controlled reflectarrays and key general characteristics of the proposed method are given. The computational results and experimental validations are presented in Section III. The 1024 phase shifts obtained from a phase-shifter element with ten RF-MEMS switches are calculated and discussed. Finally, the ratio between the DOWN- and UP-state capacitances providing a range of 360 phase shift is determined.

3595

, planar regions or domains may be each scale level defined as follows. • At scale level , the waveguide cross section defines the rectangular domain . • At scale level , the rectangular domain of surface gathers together the three patches, the two slits, and the ten RF-MEMS switches. • At scale level , the slot domain (with ) of surface is particularized. • At scale level , the domain of RF-MEMS switches (with and ) of surface is identified. • Finally, at the smallest scale level , the movable part of the RF-MEMS switch is defined. As indicated in Fig. 2, the domain is bounded by perfect magnetic conditions while , , and are enclosed by perfect magnetic and electric conditions. These boundary conditions are imposed at the contour of the various domains and are assumed to not greatly perturb the electromagnetic field in the structure. Due to the formulation of such (artificial) boundary conditions, the scale-changing technique is an approximate approach and not an exact method. Note that the natural basis for expanding the current density on the domain (i.e., on the metallic patch) is the set of modes in a rectangular waveguide of cross section and is bounded by magnetic walls. However, as far as the numerical convergence is reached, we have numerically observed that the set of modes in a rectangular waveguide of cross section and bounded by electric walls provides to also reach an accurate solution for the phase shift, but with a high number of modes. Consequently, the choice between magnetic and electric boundary conditions does not seem to be critical. The electromagnetic field in each domain (with , , , and ) can be expanded on the set of propagating and evanescent modes in an artificial waveguide of cross section . As reported in Section II-B, from such field representation, scale-changing networks can then be derived for the modeling of the electromagnetic coupling between two successive scale levels and .

II. SCALE-CHANGING TECHNIQUE A. MEMS-Controlled Planar Phase Shifters

B. Scale-Changing Network

For the sake of clarity in the theoretical developments, let us consider planar phase shifters composed of three metallic patches and ten RF-MEMS switches (see Fig. 1). Note that the approach can be applied to a planar phase shifter with arbitrary numbers of patches and RF-MEMS switches. Such phase shifters have been advantageously used as the cells of reconfigurable MEMS-controlled reflectarrays [10]–[12]: the UP/DOWN states and positions of the switches allow several operating modes and interesting discrete tuning of the slit length. Experimental characterizations are generally carried out by placing the planar phase shifter in the cross section of a metallic rectangular waveguide and by considering a incident mode [12]. Here, the scale-changing technique is applied for predicting the phase shift introduced by the phase shifter on the mode. Computational results are compared with measurements in Section II-B. As illustrated in Fig. 2, at

The network representation of the electromagnetic coupling between two successive scale levels is now derived. As sketched in Fig. 3(a), consider the domain at scale level as a discontinuity plane composed of the sub-domain (at scale level ) and the complementary perfect electric or magnetic domain . By adopting the set of propagating and evanescent modes in the two artificial waveguides of cross sections and , the impedance or admittance matrix of the discontinuity plane can be derived from a multimodal variational technique [16]. High-order evanescent modes are shorted by their (pure imaginary) impedance and are said to be passive, and propagating and low-order evanescent modes—or active modes—are used to model the electromagnetic coupling between two successive scale levels. The number of active and passive modes is determined a posteriori from the numerical convergence of the phase shift. Active modes are

3596

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 2. Multiscale view of the planar phase shifter located in the cross section of a metallic waveguide.

Fig. 3. (a) Discontinuity plane considered as a building block in the scalechanging technique. (b) Its equivalent network, called here the scale-changing network.

symbolized by ports in the network representation of the discontinuity plane given in Fig. 3(b). This network, called here the scale-changing network, is then characterized by its impedance or admittance matrix such that

Fig. 4. (a) RF-MEMS switch used in the phase shifter at the smallest scale. (b) Its equivalent network.

C. Surface Impedance Matrix for RF-MEMS Switches or (1) denote, respectively, the voltage and current where magnitudes of active modes at scale level .

Fig. 4(a) displays the geometry of RF-MEMS switch in the domain ( and ). As reported in [15], the set of propagating and evanescent modes in an artificial waveguide of cross section allows the derivation of the multiport network modeling the RF-MEMS switch. Note that if only one active mode, i.e., the TEM mode, is adopted in the domain , the network is equivalent to the surface impedance

PERRET et al.: SCALE-CHANGING TECHNIQUE FOR ELECTROMAGNETIC MODELING OF MEMS-CONTROLLED PLANAR PHASE SHIFTERS

3597

Fig. 5. Equivalent network of the phase shifter as the cascade of scale-changing networks shunted by the equivalent networks of the RF-MEMS switches.

, where the capacitance

is given

deduced from the following relationship:

by (3)

(2)

with

and

designates the free-space wavenumber.

D. Formulation of the Scale-Changing Technique As shown in Fig. 5, the equivalent network of the MEMScontrolled planar phase shifter is obtained from the cascade of four scale-changing networks. Each network models the electromagnetic coupling between two successive scale levels. The cascade is shunted by ten multiport networks, each modeling an RF-MEMS switch. Following Sections II-B and C, all the networks are computed separately. The analytical expressions of their impedance or admittance matrices are reported in the Appendix. The input impedance of the cascade is then computed and the phase of the reflection coefficient is finally

where designates the impedance of the mode. The number of active modes in all impedance or admittance matrices is such that the numerical convergence of the phase is reached. Before presenting the computational results and experimental validations, let us point out key characteristics of the proposed scale-changing technique. As introduced in Section II-A, this technique is based on the partition of the discontinuity plane in multiple domains of surface (with ). In order to eliminate numerical problems due to the treatment of ill-conditioned matrices, the partition can be chosen in order to avoid critical aspect ratios: two successive scale levels and may be such that, for instance, . Moreover, at each scale level , the electromagnetic field can be described as precisely as wished by taking an appropriate number of modes in the corresponding domain. Finally, since the computation of all the networks can be performed separately, a modification of the phase-shifter geometry at scale requires the recalculation of only two scale-changing networks. In other words, the partition of the discontinuity plane in multiple domains makes the approach modular (Lego approach). Note that the computation of the phase shift resulting from the only modification of the switches (UP/DOWN) state is instantaneous because it does not require the recalculation of the scale-changing networks.

3598

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

Fig. 6. Measurement cell [10], [11].

Fig. 7. Phase shift versus frequency for four various UP/DOWN state configurations of the ten RF-MEMS switches: configuration A: [00000 00000], configuration B: [11111 00000], configuration C: [11011 11111], configuration D: [11001 00000]; where the first five digits designate the UP/DOWN states of switches in one slot (the digit is 0 when the state in DOWN) and the last five digits describe the states of the switches in the other slot . (—) Measurements, ( ) scale-changing technique. Dimensions are (see Fig. 1, unit: millime= 0:1, ters): a = 15, b = 15, a = 12, b = 9, b = b = 0:75, a = 0:1, x = 2:4, x = 5:3, x = 8:7, x = 11:1, x = 0:45, b = 1:35, x = 4:25, x = 9:65, and x = 11:15. x

222

Boundary conditions are artificially introduced in the formulation of the scale-changing technique. These boundary conditions enclose the various scale-dependent domains and, consequently, the derivation of the scale-changing network reduces to the analysis of the cascade of planar discontinuity planes. For unbounded or nonplanar structures such an approach requires additional approximations and, consequently, is less attractive than in case of planar and bounded circuits. III. COMPUTATIONAL RESULTS The phase shifter shown in Fig. 1 has been characterized in the -band frequency range [10], [11]. As in [12], the planar phase shifter is characterized in a waveguide configuration. The environment of the phase-shifter element is, in fact, slightly different to the real environment of a large reflectarray since electric walls are imposed at the periphery of the element. This, however, does not modify its behavior. The authors refer to [17], which compares the various boundary conditions to be applied for large reflectarrays. The planar phase shifter is located in the cross section of a metallic square waveguide. Fig. 6 displays the elements of the measurement cell. The numerical and experimental data are reported in Fig. 7 for four various MEMS configurations (see the figure caption of Fig. 7). A very good agreement is observed between results obtained from the scale-changing technique and measurements in the whole frequency band. Fig. 8 displays the phase-shift variation versus the discrete accessible states of the ten RF-MEMS switches. We observe a phase-shift range of approximately 360 and the maximum phase shift between two successive configurations is less than 10 . The proposed scale-changing approach is much better than the finite-element method (FEM) software in time costing. Fig. 9 displays the computation time for calculating the phase shift in a given MEMS switches configuration. Electromagnetic simulations are carried out on a PC with 1-GB RAM and 1.8-GHz

Fig. 8. Phase shift versus the configurations of the RF-MEMS switches (1024 configurations are accessible with ten RF-MEMS switches in UP or DOWN state) at 11.7 GHz.

clock frequency. The number of passive modes at the largest scale ( domain) is tuned from 1000 to 4150 (with a step of 50). The number of modes in the intermediate domains is chosen so that their number per is constant. For comparative purposes, the number of tetrahedrons used in the FEM-based software is tuned form 1799 to 106.460 corresponding to 1–16 passes with 35% of mesh refinement per pass. The initial mesh in the FEM-based software is set so that most element lengths are approximately one-quarter wavelength. Fig. 9 indicates that the convergence is reached in 470 s by adopting 3350 active modes in the scale-changing technique (with an error equals to 0.16%), while 1300 s are required with the commercial software for obtaining a result with an error equals to 7.2%. The CPU time

PERRET et al.: SCALE-CHANGING TECHNIQUE FOR ELECTROMAGNETIC MODELING OF MEMS-CONTROLLED PLANAR PHASE SHIFTERS

Fig. 9. Phase shift versus the computation time for configuration A (see Fig. 7). Computation time varies with the number of tetrahedrons (in FEM-based software) or with the number of modes (in the scale-changing technique). It is shown that the convergence of the computational results is reached more rapidly in the scale-changing technique than in the FEM-based software. Moreover, when the convergence is reached, a very good agreement is obtained between results obtained from the scale-changing technique and measurements. (—) Measure) scaling changing technique. ( ) FEM-based software. ments. (

222

for the calculation of the phase shift is 2.5 times less than that of the FEM-based software. Moreover, the computation by the scale-changing technique of all the 1024 available phase shifts is quasi-instantaneous thanks to a post processing, which successively varies the values of each MEMS impedance. This is a major advantage as compared to commercial software, which requires launching an overall simulation when a MEMS state is changed. Now let us consider the computation of the phase shift for the 1024 accessible RF-MEMS configurations. The admittance matrix obtained from the cascade of the four scale-changing networks allows modeling the electromagnetic coupling between the largest scale and the smallest scale . This matrix does not depend on the state of the MEMS switches and its size is 11 11. The 2 configurations associated with the UP/DOWN states of the ten MEMS switches are modeled by ten shunt impedances. Once the admittance matrix is calculated, only 10 s of computation time are required to sweep the 1024 possible configurations of the ten switches. For technological reasons, the UP-state capacitance of RF-MEMS switches is set to 15 fF. Let us find the DOWN capacitance allowing a phase dynamics close to 360 . This problem of a major practical importance cannot be solved easily with the FEM-based software of the high computation time [10], [11]. However, it can be efficiently solved in using the scale-changing technique. Fig. 10 represents the obtained variation of the phase shift versus the RF-MEMS configurations for various ratios and allows easily choosing the DOWN-state capacitance. For achieving a phase dynamics close to 360 , one may choose . IV. CONCLUSION A scale-changing technique has been reported and applied with success to the electromagnetic modeling of a MEMS-controlled planar phase shifter. Very good agreement has been

3599

Fig. 10. Phase shift versus the configurations of the RF-MEMS switches for various ratios C =C (C = 15 fF) at 11.7 GHz.

observed between computational results and measurements in the whole frequency band. Very good performances in terms of accuracy and CPU time have been obtained. The application of the scale-changing technique for the electromagnetic modeling of reflectarrays composed of a finite number of MEMS-controlled planar phase shifters is under way. The scale-changing technique is a generic approach and is not only applicable to RF-MEMS-based reflectarray antennas. It can be advantageously applied to microwave or millimeter-wave circuits with high (pathological) aspect ratios and to planar multiscale (or fractal) structures. APPENDIX EXPRESSION OF

IN

FIG. 5

(4) (5) (6) (7) where

is the transpose of the matrix of the matrix where , is given by

. The element , ,

denotes the TE and TM modes in the domain and , is the inner product. The element of has the same expression as , but with .

3600

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 9, SEPTEMBER 2006

The element

,

, and is given by

of

where the indices

and

where

. The element of , , ), (

(

,

) where , ,( ) is given by

,

,

refer to the notations of Fig. 2, and

( , domain.

the

) denotes the TE and TM modes in

and the element where represents the complex propagation constant of the guided modes on the waveguide of cross section (see Fig. 2). in (4) is given by

of

,

, , but with

has the same expression as The element

,

. ,

,

,

.

of

EXPRESSION OF

where where

IN

FIG. 5

. The element , ,

, and

of , is given by

denotes the TE and TM modes in the and the element of

domain.

is

given by

has the same expression as

where

is related to

EXPRESSION OF

, but with

,

.

,

IN

FIG. 5

. The element

, of

, and is given

by

where

is related to EXPRESSION OF

,

. IN

FIG. 5

The element

(

the

,

of , , ), (

,

,

where ,

,

, ) is given by

( , ) denotes the TE and TM modes in domain. The element of , ,

PERRET et al.: SCALE-CHANGING TECHNIQUE FOR ELECTROMAGNETIC MODELING OF MEMS-CONTROLLED PLANAR PHASE SHIFTERS

has the same expression as

, but with .

,

The element

,

,

, of

where

[13] M. Nadarassin, H. Aubert, and H. Baudrand, “Analysis of planar structures by an integral approach using entire domain trial functions,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 10, pp. 2492–2495, Oct. 1995. [14] E. Perret and H. Aubert, “Scale-changing technique for the computation of the input impedance of active patch antennas,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 326–328, 2005. [15] E. Perret, H. Aubert, and R. Plana, “ -port network for the electromagnetic modeling of MEMS switches,” Microw. Opt. Technol. Lett., vol. 45, no. 1, pp. 46–49, Apr. 2005. [16] J. W. Tao and H. Baudrand, “Multimodal variational analysis of uniaxial waveguide discontinuities,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 506–516, Mar. 1991. [17] M.-A. Milon, R. Gillard, D. Cadoret, and H. Legay, “Comparison between the infinite array approach and the surrounded-element approach for the simulation of reflectarray antennas,” presented at the IEEE AP-S Int. Symp., 2006.

N

is given by

is related to

3601

.

Etienne Perret (S’02) was born in Albertville, France, in October 1979. He received the Eng. Dipl. and Ph.D. degrees in electrical engineering from the Polytechnique National Institute of Toulouse, Toulouse, France, in 2002 and 2005, respectively. In 2005, he joined the Institut d’Electronique Fondamentale (IEF), Orsay, France. His research activities concern electromagnetic modeling of multiscale structures. His current research interests are optical material characterization, modelization, and design of passive device for millimeter and submillimeter-

ACKNOWLEDGMENT The authors wish to thank Dr. E. Girard, Alcatel Alenia Space, Toulouse, France, for providing the measurements of the planar phase shifter. REFERENCES [1] D. G. Berry, R. G. Malech, and W. A. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propag., vol. AP-11, no. 11, pp. 645–651, Nov. 1963. [2] K. Y. Sze and L. Shafai, “Phase properties of single-layer patch arrays with applications to line-source-fed microstrip reflectarrays,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 149, no. 1, pp. 64–70, Feb. 2002. [3] J. Huang, “Microstrip reflectarray,” in IEEE AP-S Int. Symp., 1991, pp. 612–615. [4] ——, “Analysis of a microstrip reflectarray antenna for microspacecraft applications,” Jet Propulsion Lab., Pasadena, CA, JPL TDA Progr. Rep. 42–120, 1995, pp. 153–172. [5] R. D. Javor, X.-D. Wu, and K. Chang, “Design and performance of a microstrip reflectarray antenna,” Microw. Opt. Technol. Lett., vol. 7, no. 7, pp. 322–324, May 1994. [6] S. D. Targonski and D. M. Pozar, “Analysis and design of a microstrip reflectarray using patches of variable size,” in IEEE AP-S Int. Symp., Jun. 20–24, 1994, vol. 3, pp. 1820–1823. [7] J. Encinar, L. Datashvili, H. Baier, M. Arrebola, M. Sierra-Castaner, J. L. Besada, H. Legay, and G. Toso, “Breadboard of a three layer printed reflectarray for dual polarization and dual coverage,” in 28th ESA Antenna Technol. Space Antenna Syst. Technol. Workshop, Noordwijk, The Netherlands, Jun. 3, 2005, pp. 443–448. [8] J. P. Gianvittorio and Y. Rahmat-Samii, “Reconfigurable reflectarray with variable height patch elements: Design and fabrication,” in IEEE AP-S Int. Symp., Jun. 20-25, 2004, vol. 2, pp. 1800–1803. [9] H. Legay, B. Pinte, M. Charrier, A. Ziaei, E. Girard, and R. Gillard, “A steerable reflectarray antenna with MEMS controls,” in IEEE Int. Phased Array Syst. Technol. Symp. , Oct. 14-17, 2003, pp. 494–499. [10] H. Legay, G. Caille, P. Pons, E. Perret, H. Aubert, J. Pollizzi, A. Laisne, R. Gillard, and M. Van Der Worst, “MEMS controlled phase-shift elements for a linear polarised reflectarray,” in 28th ESA Antenna Technol. Space Antenna Syst. Technol. Workshop, Noordwijk, The Netherlands, Jun. 3, 2005, pp. 449–454. [11] H. Legay, G. Caille, E. Girard, P. Pons, H. Aubert, E. Perret, P. Calmon, J. P. Polizzi, J.-P. Ghesquiers, D. Cadoret, and R. Gillard, “MEMS controlled linearly polarised reflect array elements,” presented at the 12th Int. Antenna Technol. Appl. Electromagn. Symp., Montréal, QC, Canada, Jul. 16–19, 2006. [12] D. Cadoret, A. Laisne, R. Gillard, and H. Legay, “New reflectarray cell using coupled microstrip patches loaded with slots,” Microw. Opt. Technol. Lett., vol. 44, no. 3, pp. 270–273, Feb. 2005.

wave applications. Hervé Aubert (M’94–SM’99) was born in Toulouse, France, in July 1966. He received the Eng. Dipl. degree and Ph.D. degree (with high honors) in electrical engineering from the Polytechnique National Institute of Toulouse, Toulouse, France, in 1989 and 1993, respectively. In 1993, he joined the faculty of the Ecole Nationale Supérieure d’Electrotechnique, Toulouse, France, as an Assistant Professor, and then became an Associate Professor in 1997 and Professor in 2001. From 2001 to 2005, he was the Associate Chairman of the Electronics Laboratory, Ecole Nationale Supérieure d’Electrotechnique, d’Electronique, d’Informatique, d’Hydraulique et des Télécommunications. From 2002 to 2005, he was also the Head of the Electromagnetism Research Group of this same laboratory. In February 2006, he joined the Laboratory for Analysis and Architecture of Systems, National Center for Scientific Research, Toulouse, France, where his research activities involve the electromagnetic modeling of multiscale structures and fractal objects. He has authored or coauthored one book, three book chapters, and 26 papers in refereed journals. Prof. Aubert is the vice-chairman of the IEEE Antennas and Propagation French Chapter and was the secretary of this Chapter from 2001 to 2004. He is a member of URSI Commission B. He was the recipient of the 1994 Leopold Escande Prize for his doctoral dissertation. Hervé Legay was born in 1965. He received the Electrical Engineering degree and Ph.D. degree from the National Institute of Applied Sciences (INSA), Rennes, France, in 1988 and 1991, respectively. For two years, he was a Post-Doctoral Fellow with the University of Manitoba, Winnipeg, MB, Canada, where he developed innovating planar antennas. In 1994, he joined Alcatel Alenia Space, Toulouse, France. He initially conducted studies in the areas of military telecommunication antennas and antenna processing. He currently leads research projects in integrated front ends and reflectarray antennas and coordinates the collaborations with academic and research partners in the area of antennas. He is a member of the Alcatel Technical Academy.

INFORMATION FOR AUTHORS Submitting Manuscripts for Publication Authors need to visit the website http://www.mtt.org/publications/Transactions/for_authors_transactions.htm for the author instructions and checklist to make sure their manuscript is ready for submission. Please make the pdf file with a word-processing program or by converting a printer file, rather than by scanning. The maximum file size is 1MB. All fonts must be embedded in the file. Each manuscript must include an abstract of 200 words or less. The manuscript should be formatted so that is easy to read on either a computer screen or on paper printed out in black and white. The manuscript must be single-spaced and double-column, with figures and captions at the appropriate place within the text, rather than at the end. IEEE templates must be used (see http://www.ieee.org/organizations/pubs/transactions/information.htm). A manuscript should look as it will appear when printed. Figures should be large enough to be easily viewed on a computer screen. Label axes carefully and use large lettering. Instrument screen dumps and screen photos should be redrawn to make them clear. Plot theory and experiment on the same graph so that they can be compared easily. Shading and color must generally be avoided. TeX and LaTeX users need to use scalable fonts rather than bitmapped fonts, so that the pdf file can be read easily on a computer screen. TeX and LaTeX are the preferred word processing languages. If you use Microsoft Word all symbols in the document must be entered using the built-in equation editor or using Mathtype (i.e. do not use font changes to enter symbols). Manuscripts are considered as regular papers or letters. Letters are limited to errata and comments on previously published material. Short papers of three pages or less should be sent to the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. Manuscripts that describe simulations of circuits without validating measurements may be returned without reviews. All papers with theoretical contributions must have independent verification with measurement-based validation strongly preferred. The author must notify the editor at the time of submission if the manuscript contains material that has been previously submitted for publication. We will consider papers that are related to material published in conference digests only if the paper is substantially expanded. Please attach pdf files of the previous papers. Page Charges Papers will be reviewed for their technical merit, and decisions to publish will be made independently of an author’s ability to pay page charges. Page charges of $110 (U.S.) per printed page will be requested on papers of six printed pages or less. Overlength page charges of $200 per page are mandatory for each page in excess of six pages. If the author’s organization agrees to honor the total page charge, which includes the page charges on the first six pages plus the mandatory overlength charge, the author will receive 100 reprints. If the supporting organization honors only the mandatory charge, no free preprints will be sent. Copyright and Clearance It is the policy of the IEEE to own the copyright to the technical contributions it publishes on behalf of the interests of the IEEE, it authors, and their employees, and to facilitate the appropriate reuse of this material by others. To comply with the U.S. Copyright Law, authors are required to sign an IEEE copyright transfer form (http://www.ieee.org/about/documentation/copyright) before publication. This form returns to authors and their employers full rights to reuse their material for their own purposes. Authors must submit a signed copy of this form with their manuscripts. The IEEE must assume that material submitted to its publications is available for general dissemination. It is the responsibility of the author, not the IEEE, to determine whether disclosure of the material requires prior consent of other parties and, if so, to obtain it. Do not repeat figures that were published in non-IEEE publications including conferences unless you obtain and forward a copyright release. Website The website for this TRANSACTIONS is http://www.mtt.org/publications/Transactions/transactions.htm. This site has information about future Special Issues, advanced Tables of Contents, and other information about this TRANSACTIONS. Digital Object Identifier 10.1109/TMTT.2006.882612

Digital Object Identifier 10.1109/TMTT.2006.883078

Digital Object Identifier 10.1109/TMTT.2006.883079

EDITORIAL BOARD Editors: D. WILLIAMS AND A. MORTAZAWI Associate Editors: A. CANGELLARIS, A. CIDRONALI, M. DO, K. ITOH, J. LIN, D. LINTON, S. MARSH, Y. NIKAWA, J. PEDRO, Z. POPOVIC, S. RAMAN, V. RIZZOLI, R. SNYDER, R. WU, T. WYSOCKI, A. YAKOVLEV REVIEWERS A. Abbaspour-Tamijani D. Abbott M. Abdulla M. Abe M. Abedin M. Abouzahra M. Abramowicz R. Achar E. Ackerman D. Adam E. Adle M. Adlerstein M. Afsar K. Agarwal K. Agawa K. Ahmed D. Ahn H.-R. Ahn M. Aikawa M. Akaike Y. Akaiwa E. Akmansoy S. Aksoy A. Akyurtlu F. Alessandri C. Algani F. Ali M. Ali W. Ali-Ahmad F. Alimenti C. Alippi B. Alpert A. Alphones S. Al-Sarawi A. Altintas A. Alvarez-Melcom S. Amari C. Anastasiou U. Andersson Y. Ando P. Andreani K.-S. Ang I. Angelov S. Anlage O. Anwar I. Aoki R. Aparicio V. Aparin F. Arndt U. Arz M. Asai P. Asbeck H. Ashok H. Ashoka A. Atalar A. Atia N. Audeh S. Auster P. Auxemery I. Awai A. Aydiner K. Aygun R. Azadegan A. Babakhani I. Bahl D. Baillargeat S. Bajpai W. Bakalski J. Baker-Jarvis B. Bakkaloglu K. Balmain Q. Balzano S. Banba J. Bandler R. Bansal F. Bardati I. Bardi S. Barker D. Barlage J. Barr J. Bartolic D. Bates G. Baudoin Q. Balzano B. Beker G. Belenky D. Belot C. Bell P. Bell J. Benedikt J. Bernhard G. Bertin H. Bertoni E. Bertran W. Beyene A. Beyer M. Bialkowski E. Biebl P. Bienstman S. Bila M. Bilakowski A.-L. Billabert F. Bilotti H. Bilzer O. Biro R. Bisiso B. Bisla D. Blackham M. Blank P. Blondy D. Boccoli F. Bohn B. Boeck L. Boglione R. Boix J. Booske N. Borges de Carvalho V. Boria O. Boric-Lubecke A. Borji J. Bornemann W. Bosch R. Bosisio S. Boumaiza M. Bozzi E. Bracken R. Bradley V. Bratman T. Brazil G. Brehm K. Breuer B. Bridges J. Brinkoff S. Broschat S. Brozovich D. Budimir D. Buechler M. Buff C. Buntschuh J. Bunton J. Burghartz P. Burghignoli Y. Bykov A. Caballero B. Cabon J. Calame

M. Calcatera C. Caloz C. Camacho-Penalosa E. Camargo R. Cameron S. Cammer C. Campbell M. Campovecchio F. Canavero J. Cao J. Capmany F. Capolino G. Carchon R. Carter N. Carvalho F. Casas J. Catala R. Caverly J. Cavers Z. Cendes B. Cetiner R. Chair H. Chaloupka A. Chambarel B. Chambers C.-H. Chan Y.-J. Chan C.-Y. Chang F. Chang G. Chang H.-C. Chang H.-R. Chang K. Chang E. Channabasappa H. Chapell W. Chappell M. Chatras S. Chaudhuri S. Chebolu C.-C. Chen C.-H. Chen H.-H. Chen J. Chen R. Chen W.-K. Chen Y.-J. Chen K.-K. Cheng Y.-C. Cheng W.-C. Chew C.-Y. Chi Y.-C. Chiang C.-F. Chiasserini I.-T. Chiang J. C. Chiao I. Chiba D. Chigrin A. Chin C.-C. Chiu Y. Cho C. Choi J. Choi M.-J. Choi C.-K. Chou Y.-H. Chou D. Choudhury K. Choumei Y. Chow C. Christodoulou C. Christopoulos H.-R. Chuang Y. Chung B. Chye R. Cicchetti C. Cismaru D. Citrin P. Civalleri A. Ciubotaru T. Clark R. Clarke J. Cloete E. Cohen F. Colomb B. Colpitts M. Condon D. Consonni J. Corral A. Constanzo I. Corbella E. Costamagna A. Coustou J. Craninckx J. Crescenzi S. Cripps D. Cros T. Crowe M. Cryan J. Culver C. Curry W. Curtice M. da Cunha W.-L. Dai T. Dahm G. Dambrine B. Danly F. Danneville N. Das M. Davidovich A. Davis C. Davis L. Davis H. Dayal F. De Flaviis H. De Los Santos A. De Lustrac P. De Maagt J. de Mingo R. De Roo L. de Vreede D. De Zutter B. Deal A. Dearn P. Debicki J. Deen A. Deleniv M. DeLisio S. Demir A. Deutsch V. Devabhaktuni Y. Deval A. Diet L. Ding A. Djermoun T. Djordjevic J. Dobrowolski D. Dolfi W. Dou M. Douglas P. Draxler A. Dreher F. Drewniak J. Drewniak D. Dubuc S. Dudorov L. Dunleavy V. Dunn A. Duzdar

S. Dvorak L. Dworsky M. Dydyk M. Edwards R. Ehlers H. Eisele G. Eisenstein G. Eleftheriades M. Elliott T. Ellis A. Elsherbeni R. Emrick N. Engheta A. Enokihara Y. Eo H. Eom C. Ernst M. Esashi L. Escotte I. Eshrah V. Esposti M. Essaaidi K. Esselle H. Estaban J. Esteban C. Fager J. Fan D.-G. Fang M. Farina W. Fathelbab A. Fathy J. Favennec A. Fazal E. Fear M. Feldman A. Fernandez A. Ferrero T. Fickenscher J. Fiedziuszko D. Filipovic A. Fliflet B. Floyd P. Focardi N. Fong K. Foster P. Foster B. Frank C. Free J. Freire M. Freire R. Freund F. Frezza I. Frigyes C. Froehly J. Fu R. Fujimoto T. Fujioka O. Fujiwara H. Fukuyama V. Fusco D. Gabbay N. Gagnon J. Gallego B. Galwas O. Gandhi B.-Q. Gao J. Gao J. Garcia R. Garver A. Gasiewski B. Geelen B. Geller V. Gelnovatch W. Geppert F. Gerecht J. Gering M. Gerken S. Gevorgian R. Geyer O. Ghandi F. Ghannouchi K. Gharaibeh G. Ghione D. Ghodgaonkar F. Giannini J. Gilb A. Glisson M. Goano E. Godshalk M. Goldfarb P. Goldsmith M. Golio N. Gomez X. Gong R. Gonzalo S. Gopalsami A. Gopinath R. Gordon A. Gorur K. Goverdhanam W. Grabherr L. Gragnani J. Grahn G. Grau A. Grebennikov T. Gregorzyk I. Gresham A. Griol D. R. Grischowsky C. Grossman E. Grossman T. Grzegorczyk A. Gupta K. Gupta M. Gupta R. Gutmann W. Gwarek J. Hacker M. Hafizi S. Hadjiloucas S. Hagness D. Haigh P. Hale D. Ham K. Hamaguchi S. Hamedi-Hagh J. Hand K. Hashimoto Q. Han T. Hancock A. Hanke V. Hanna Z. Hao S. Hara L. Harle A. Harish P. Harrison H. Hartnagel J. Haslett G. Hau R. Haupt S. Hay H. Hayashi J. Hayashi L. Hayden J. Heaton

P. Hedekvist W. Heinrich G. Heiter M. Helier R. Henderson F. Henkel J. Herren P. Herczfeld F. Herzel J. Hessler A. Hiatala C. Hicks M. Hieda A. Higgins M. Hikita W. Hioe Y. Hirachi T. Hiraota A. Hirata T. Hiratsuka Y.-C. Ho W. Hoefer K. Hoffmann R. Hoffmann J. Hong J.-S. Hong K. Horiguchi Y. Horii J. Horng J. Horton K. Hosoya R. Howald H. Howe H.-M. Hsu H.-T. Hsu J.-P. Hsu C.-W. Hsue C.-C. Huang C. Huang F. Huang H. Huang H.-C. Huang J. Huang T.-W. Huang P. Huggard H.-T. Hui D. Humphreys A. Hung C.-M. Hung H. Hung J.-J. Hung I. Hunter H.-Y. Hwang T. Idehara S. Iezekiel J.-Y. Ihm Y. Iida H. Iizuka P. Ikalainen Y. Ikeda P. Ikonen K. Ikossi M. Ilic J. Inatani K. Iniewski H. Inokawa A. Inoue M. Ishida A. Ishimaru T. Ishizaki S. Islam Y. Ismail Y. Isota M. Ito T. Itoh Y. Itoh T. Ivanov C. Iversen D. Iverson M. Iwamoto Y. Iyama H. Izumi D. Jachowski C. Jackson D. Jackson R. Jackson M. Jacob S. Jacobsen D. Jaeger B. Jagannathan N. Jain R. Jakoby G. James V. Jandhyala M. Janezic H. Jantunen B. Jarry P. Jarry A. Jastrzbeski E. Jeckein W. Jemison Y. Jeon J. Jeong Y.-H. Jeong G. Jerinic A. Jerng T. Jerse D. Jiao J.-M. Jin J. Joe L. Johansson T. Johnson A. Joseph K. Joshin J. Joubert P. Juodawlkis P. Kabos S.-T. Kahng T. Kaho D. Kajfez T. Kamel Y. Kamimura H. Kamitsuna K. Kamogawa S. Kanamaluru H. Kanaya M. Kanda P. Kangaslahtii V. Kaper M. Kärkkäinen A. Karpov U. Karthaus A. Karwowski T. Kashiwa R. Kaul K. Kawakami A. Kawalec T. Kawanishi S. Kawasaki H. Kayano M. Kazimierczuk R. Keam L. Kempel P. Kenington K. Kenneth S. Kenny

Digital Object Identifier 10.1109/TMTT.2006.883077

A. Kerr A. Khalil A. Khanifar J. Kiang Y.-W. Kiang P.-S. Kildal O. Kilic B. Kim H. Kim I. Kim J.-P. Kim M. Kim W. Kim B. Kimm K. Kimura S. Kimura A. Kirilenko V. Kisel S. Kishimoto A. Kishk T. Kitamura K. Kitayama T. Kitazawa W. Klaus E. Klumprink R. Knerr R. Knöchel L. Knockaert K. Kobayashi Y. Kogami B. Kolner S. Komaki M. Komaru J. Komiak A. Komijani G. Kompa A. Konczykowska Y. Konishi A. Koonen B. Kopp K. Kornegay M. Koshiba T. Kosmanis J. Kot Y. Kotsuka S. Koul V. Kourkoulos A. B. Kozyrev A. Krenitskiy N. Kriplani K. Krishnamurthy V. Krishnamurthy A. Kroenig C. Kromer C. Krowne V. Krozer W. Kruppa R. Kshetrimayum H. Ku H. Kubo E. Kuester Y. Kuga W. Kuhn T. Kuki M. Kumar M. Kunert J. Kuno M. Kunst C.-N. Kuo J.-T. Kuo H. Kurebayashi T. Kuri F. Kuroki S. Kusunoki D. Kuylenstierna M. Kuzuhara I. Kwon Y.-W. Kwon R. Lai Y.-L. Lai P. Lampariello M. Lanagan M. Lancaster P. Lane U. Langmann Z. Lao G. Lapin L. Larson J. Laskar A. Lauer G. Lazzi Y. Le Coz Y. Le Guennec S. Le Maguer B. Lee C. Lee J.-F. Lee J.-W. Lee K. Lee R. Lee S.-G. Lee T. Lee Y.-C. Leong R. Leoni K.-W. Leung P. Leuchtmann G. Leuzzi A. Leven A. Levi R. Levy A. Lewandowski M. Lewis K. Li L.-W. Li X. Li Y. Li Y.-M. Li M. Liberti L. Ligthart S. Lim E. Limiti C. Lin J. Lin Y.-D. Lin Y.-S. Lin L. Lind S. Lindenmeier F. Ling A. Lipparini D. Lippens V. Litvinov C.-P. Liu Q.-H. Liu S.-I. Liu W. Liu O. Llopis D. Lo A. Loayssa R. Loison J. Long K. Lorincz U. Lott J.-H. Loui H.-C. Lu L.-H. Lu S. Lu

W.-T. Lu V. Lubecke G. Lucca S. Lucyszyn R. Luebbers L. Lunardi J. Luy S. Lyshevski J.-G. Ma Z. Ma S. Maas P. Maccarini G. Macchiarella P. Macchiarella J. Machac S. Maci J. Maciel M. Madihian B. Madhavan V. Madrangeas M. Magana S. Mahmoud S. Mahon I. Maio A. Majedi M. Majewski M. Makimoto J. Malherbe D. Malocha T. Manabe G. Manganaro T. Maniwa C. Mann H. Manohara R. Mansour D. Manstretta J. Mao S.-G. Mao S. Marchetti R. Marques J. Martens J. Marti F. Martin E. Martinez K. Maruhashi D. Masotti A. Massa S. Masuda A. Materka B. Matinpour M. Matsuo A. Matsushima A. Matsuzawa S. Matsuzawa G. Matthaei D. Matthews J.-P. Mattia J. Maurer J. Mayock J. Mazierska S. Mazumder G. Mazzarella K. McCarthy T. McKay J. McKinney R. McMillan D. McQuiddy P. Meany F. Medina S. Melle F. Mena C. Meng H.-K. Meng W. Menzel F. Mesa A. Metzger P. Meyer C. Mias K. Michalski G. Michel E. Michielssen A. Mickelson R. Miles D. Miller R. Minasian B. Minnis D. Mirshekar J. Mitchell O. Mitomi R. Mittra M. Miyakawa R. Miyamoto M. Miyazaki K. Mizuno S. Mizushina M. Mohamed S. Mohammadi A. Mohammadian M. Mongiardo J. Morente M. Morgan K. Mori A. Morini N. Morita E. Moros A. Morris J. Morsey H. Mosallaei M. Mrozowski J.-E. Mueller M. Muraguchi K. Murata H. Muthali T. Nagatsuma P. Nagel K. Naishadham T. Nakagawa M. Nakajima N. Nakajima J. Nakayama M. Nakayama M. Nakhla J. Nallatamby S. Nam S. Narahashi A. Natarajan J. Nath B. Nauwelaers J. Navarro I. Nefedovlgor H.-C. Neitzert B. Nelson S. Nelson A. Neri H. Newman D. Ngo E. Ngoya C. Nguyen K. Niclas E. Niehenke P. Nikitin A. Niknejad N. Nikolova T. Nirmalathas K. Nishikawa T. Nishikawa

K. Nishimura T. Nishino K. Nishizawa G. Niu W. Ng S. Nogi K. Noguchi T. Nojima A. Nosich B. Notaros K. Noujeim D. Novak T. Nozokido T. Nurgaliev D. Oates J. Obregon J. O’Callahan M. O’Droma M. Odyneic I. Ogawa M. Ogusu K. Oh M. Ohawa T. Ohira I. Ohta M. Ohtsuka S. Oikawa K. Okada Y. Okano H. Okazaki V. Okhmatovski A. Oki M. Okoniewski A. Oliner J. Olsson F. Olyslager A. Omar M. Omiya K. Onodera B.-L. Ooi I. Oppermann R. Orta S. Ortiz J. Ou T. Owada M. Ozkar J. Page de la Pega W. Palmer G.-W. Pan A. Paolella C. Papanicolopoulos J. Papapolymerou B.-K. Park C.-S. Park W. Park A. Parker D. Parker T. Parker J. Pearce B. Pejcinovic S.-T. Peng R. Pengelly R. Penty J. Pereda B. Perlman L. Perregrini M. Petelin R. Petersen W. Petersen A. Peterson A. Petosa A.-V. Pham J. Phillips H. Pickett M. Pieraccini L. Pierce B. Piernas J. Pierro P. Pieters M. Piket-May L. Pileggi Z.-Y. Ping M. Pirola A. Platzker C. Plett C. Pobanz R. Pogorzelski R. Pokharel R. Pollard G. Ponchak M. Popovic J. Portilla M. Pospieszalski V. Postoyalko A. Pothier S. Prasad D. Prather D. Prescott A. Priou D. Purdy Y. Qian T. Quach C. Quendo R. Quere F. Raab V. Radisic K. Radhakrishnan T. Rahkonen C. Railton A. Raisanen K. Rajab O. Ramahi J. Randa R. Ranson T. Rappaport J. Rathmell C. Rauscher J. Rautio B. Rawat J. Rayas-Sanchez R. Reano G. Rebeiz J. Rebollar B. Redman-White M. Reddy R. Reid H.-M. Rein J. Reinert R. Remis K. Remley C. Rey L. Reynolds A. Rezazadeh E. Rezek A. Riddle B. Riddle J.-S. Rieh E. Rius I. Robertson R. Robertson A. Rodriguez R. Rogers H. Rogier U. Rohde N. Rolland R. Romanofsky

A. Rong Y. Rong D. Root L. Roselli A. Rosen U. Rosenberg L. Roy M. Royer J. Roychowdury T. Rozzi B. Rubin M. Rudolph P. Russer D. Rutledge T. Ruttan A. Rydberg T. Rylander D. Rytting C. Saavedra A. Safavi-Naeini A. Safwat M. Sagawa B. Sahu A. Saitou I. Sakagami K. Sakaguchi K. Sakakibara K. Sakamoto K. Sakoda M. Salazar-Palma C. Samori L. Samoska A. Sanada Y. Sanada M. Sanagi P. Sandhiva U. Sangawa A. Sangster K. Sano K. Sarabandi T. Sarkar C. Sarris H. Sato M. Sato S. Sato H. Sawada H. Sawaya A. Sawicki A. Sayed I. Scherbatko J. Schellenberg G. Schettini F. Schettino B. Schiek M. Schindler E. Schlecht E. Schmidhammer D. Schmitt J. Schneider J. Schoukens A. Schuchinsky R. Schuhmann J. Schultz J. Schutt-Aine A. Seeds Y. Segawa T. Seki S. Selberherr G. Semouchkin E. Semouchkina Y.-K. Seng R. Settaluri J. Sevic O. Sevimli Y. Segawa Z. Shao M. Shapiro A. Sharma S. Sharma T. Shen Z.-X. Shen Y. Shestopalov H. Shigesawa Y.-C. Shih H. Shimasaki S. Shinjo N. Shino N. Shinohara T. Shimozuma W. Shiroma K. Shogen N. Shuley M. Shur D. Sievenpiper A. Sihvola C. Silva M. Silveira M. Silveirinha M. Silveirinhao K. Silvonen G. Simin R. Simons B. Sinha F. Sinnesbichler J. Sinsky J. Sitch H.-J. Siweris R. Sloan A. Smith D. Smith G. Smith P. Smith R. Snyder H. Sobol A. Sochava M. Solano K. Solbach M. Solomon M. Sorolla Ayza R. Sorrentino C. Soukoulis N. Soveiko E. Sovero J. Sowers M. Soyuer R. Sparks P. Staecker D. Staiculescu S. Stapleton J. Staudinger P. Stauffer P. Steenson K. Stephan M. Steyaert S. Stitzer A. Stoehr B. Strassner M. Stubbs M. Stuchly A. Suarez G. Subramanyam R. Sudbury N. Suematsu M. Sugiyama D. Sullivan L. Sundstrom

Y. Suzuki J. Svacina D. Swanson D. Sweeney R. Syms B. Szendrenyi W. Tabbara M. Tabib-Azar A. Taflove M. Taghivand N. Taguchi Y. Tahara G. Tait Y. Tajima T. Takagi K. Takahashi S. Takayama Y. Takayama S. Takeda I. Takenaka M. Taki K. Takizawa S. Talisa N. Talwalkar B.-T. Tan C.-Y. Tan J. Tan C.-W. Tang W.-C. Tang S. Tanaka T. Tanaka Y. Tanaka M. Tani E. Taniguchi H. Tanimoto R. Tascone J. Taub J. Tauritz R. Tayrani D. Teeter F. Teixeira R. Temkin M. Tentzeris K. Thakur H. Thal W. Thiel H.-W. Thim B. Thompson D. Thompson M. Tiebout L. Tiemeijer H. Toda M.-R. Tofighi M. Togashi T. Tokumitsu R. Tomasiunas A. Tombak K. Tomiyasu I. Toyoda S. Tretyakov R. Trew A. Trifiletti C. Trueman A. Truitt C.-M. Tsai E. Tsai L. Tsang H.-Q. Tserng T. Tsiboukis J. Tsui M. Tsuji T. Tsujiguchi T. Tsukahara K. Tsukamoto K. Tsunoda H. Tsurumi S. Tu R. Tucker M. Tur C.-K. Tzuang H. Uchida S. Uebayashi T. Ueda S. Ueno J. Uher F. Uhlmann T. Ulrich T. Umeda Y. Umeda F. Urbani T. Uwano P. Vainikainen P. Valanju F. Van de Water P. van den Berg D. Van der Weide G. Vandenbosch A. Vander Vorst D. Vanhoenacker-Janvie J. Vankka F. Van Straten K. Varian G. Vasilecu A. Vegas-Garcia L. Vegni A. Verma R. Vernon J. Verspecht B. Vidal L. Vietzorreck A. Viitanen A. Vilches C. Vittoria S. Vitusevich D. Viveiros V. Volman K. Wada K. Wakino D. Walker R. Walker M. Wallis C. Walsh C. Wan S. Wane B.-Z. Wang C. Wang D. Wang E. Wang H. Wang J. Wang K.-C. Wang S. Wang T.-H. Wang W. Wang X. Wang K. Warnick P. Warr S. Wartenberg O. Watanabe S. Watanabe R. Waugh D. Webb K. Webb R. Webster S. Wedge C.-J. Wei

J. Weirt R. Weigel G. Weihs R. Weikle C. Weil D. Weile A. Weily S. Weinreb J. Weiss C. Weitzel T. Weller C.-P. Wen M.-H. Weng R.-M. Weng S. Wentworth J. Whelehan L. Whicker J. Whitaker N. Whitbread D. White I. White S. Whiteley A. Whittneben B. Widrow G. Wilkins J. Williams T. Williams A. Williamson B. Willen B. Wilson J. Wiltse T. Winslow J. Winters A. Wittneben M. Wnuk M.-F. Wong S. Wong W. Woo J. Wood R. C. Wood G. Woods D. Woolard B.-L. Wu C. Wu H. Wu K. Wu K.-L. Wu Q. Wu Y.-S. Wu J. Wuerfl M. Wurzer J. Wustenberg G. Xiao C. Xie H. Xin Y.-Z. Xiong J. Xu Y. Xu Q. Xue T. Yakabe K. Yamamo S. Yamamoto S. Yamashita K. Yamauchi F. Yang H.-Y. Yang K. Yang Y. Yang Y.-J. Yang Z. Yang S. Yanagawa F. Yanovsky H. Yao J. Yao J. Yap B. Yarman K. Yashiro H. Yasser K. Yasumoto S. Ye J. Yeo S.-P. Yeo A. Yilmaz W.-Y. Yin S. Yngvesson N. Yoneda T. Yoneyama C.-K. Yong J.-G. Yook J.-B. Yoon R. York I. Yoshida S. Yoshikado L. Young M. Yousefi J.-W. Yu M. Yu P.-K. Yu W. Yu S.-W. Yun P. Yue A. Zaghoul A. Zaghloul A. Zajic K. Zaki P. Zampardi J. Zapata L. Zappelli J. Zehentner L. Zhang Q.-J. Zhang R. Zhang S. Zhang W. Zhang Y. P. Zhang A. Zhao L. Zhao Y. Zhao F. Zhenghe W. Zhou A. Zhu L. Zhu N.-H. Zhu Y.-S. Zhu Z. Zhu R. Zhukavin D. Zimmermann R. Ziolkowski H. Zirath J. Zmuidzinas A. Zozaya