IEEE MTT-V053-I05 (2005-05) [53, 05 ed.]

Citation preview

MAY 2005

VOLUME 53

NUMBER 5

IETMAB

(ISSN 0018-9480)

PAPERS

Chebyshev Collocation and Newton-Type Optimization Methods for the Inverse Problem on Nonuniform Transmission Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .M. Norgren Four-Port Microwave Networks With Intrinsic Broad-Band Suppression of Common-Mode Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. M. Fathelbab and M. B. Steer Ultra-Sensitive Detection of Protein Thermal Unfolding and Refolding Using Near-Zone Microwaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. M. Taylor and D. W. van der Weide Low-Reflection Subgridding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Kulas and M. Mrozowski Table-Based Nonlinear HEMT Model Extracted From Time-Domain Large-Signal Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. C. Currás-Francos An Advanced Low-Frequency Noise Model of GaInP–GaAs HBT for Accurate Prediction of Phase Noise in Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . .J.-C. Nallatamby, M. Prigent, M. Camiade, A. Sion, C. Gourdon, and J. J. Obregon Appropriate Formulation of the Characteristic Equation for Open Nonreciprocal Layered Waveguides With Different Upper and Lower Half-Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Rodríguez-Berral, F. Mesa, and F. Medina Compact MMIC CPW and Asymmetric CPS Branch-Line Couplers and Wilkinson Dividers Using Shunt and Series Stub Loading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Hettak, G. A. Morin, and M. G. Stubbs Synthesis and Design of In-Line -Order Filters With Real Transmission Zeros by Means of Extracted Poles Implemented in Low-Cost Rectangular -Plane Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. R. Montejo-Garai, J. A. Ruiz-Cruz, J. M. Rebollar, M. J. Padilla-Cruz, A. Oñoro-Navarro, and I. Hidalgo-Carpintero Digital Subband Filtering Predistorter Architecture for Wireless Transmitters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. Hammi, S. Boumaiza, M. Jaïdane-Saïdane, and F. M. Ghannouchi Design Guidelines for Terahertz Mixers and Detectors . . . . . . . . . . . . . . . . . . . P. Focardi, W. R. McGrath, and A. Neto CMOS RF Amplifier and Mixer Circuits Utilizing Complementary Characteristics of Parallel Combined NMOS and PMOS Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Nam, B. Kim, and K. Lee A Low-Power -Band Voltage-Controlled Oscillator Implemented in 200-GHz SiGe HBT Technology . . . . . . . . . . . . . . . . . . . . . . . . . Y.-J. E. Chen, W.-M. L. Kuo, Z. Jin, J. Lee, Y. V. Tretiakov, J. D. Cressler, J. Laskar, and G. Freeman Modeling Distortion in Multichannel Communication Systems . . . . . . . . . . . . . . . . . . K. M. Gharaibeh and M. B. Steer

1561 1569 1576 1587 1593 1601 1613 1624

1636 1643 1653 1662 1672 1682

(Contents Continued on Back Cover)

(Contents Continued from Front Cover) An Analytical Technique for the Synthesis of Cascaded -Tuplets Cross-Coupled Resonators Microwave Filters Using Matrix Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Tamiazzo and G. Macchiarella Effect of Mode-Stirrer Configurations on Dielectric Heating Performance in Multimode Microwave Applicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Plaza-González, J. Monzó-Cabrera, J. M. Catalá-Civera, and D. Sánchez-Hernández Compact Planar and Vialess Composite Low-Pass Filters Using Folded Stepped-Impedance Resonator on Liquid-Crystal-Polymer Substrate . . . . . . . . . . . . S. Pinel, R. Bairavasubramanian, J. Laskar, and J. Papapolymerou Precision Open-Ended Coaxial Probes for In Vivo and Ex Vivo Dielectric Spectroscopy of Biological Tissues at Microwave Frequencies . . . .D. Popovic, L. McCartney, C. Beasley, M. Lazebnik, M. Okoniewski, S. C. Hagness, and J. H. Booske Experimental Class-F Power Amplifier Design Using Computationally Efficient and Accurate Large-Signal pHEMT Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Wren and T. J. Brazil Radiometric Millimeter-Wave Detection via Optical Upconversion and Carrier Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. A. Schuetz, J. Murakowski, G. J. Schneider, and D. W. Prather Experimental Validation of Analysis Software for Tunable Microstrip Filters on Magnetized Ferrites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. León, M. J. Freire, R. R. Boix, and F. Medina Systematic Linearity Analysis of RFICs Using a Two-Port Lumped-Nonlinear-Source Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q. Liang, J. M. Andrews, J. D. Cressler, and G. Niu Measurement of the Dielectric Constants of Metallic Nanoparticles Embedded in a Paraffin Rod at Microwave Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y.-S. Yeh, J.-T. Lue, and Z.-R. Zheng Computational Approach Based on a Particle Swarm Optimizer for Microwave Imaging of Two-Dimensional Dielectric Scatterers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Donelli and A. Massa Design of Dielectric-Filled Cavity Filters With Ultrawide Stopband Characteristics . . . . . . . . . . . . . . . . . . C. Rauscher Simplified Analysis Technique for the Initial Design of LTCC Filters With All-Capacitive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Rambabu and J. Bornemann -Band Orthomode Transducer With Waveguide Ports and Balanced Coaxial Probes . . . G. Engargiola and A. Navarrini Optimum Operation of Asymmetrical-Cells-Based Linear Doherty Power Amplifiers—Uneven Power Drive and Power Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Kim, J. Cha, I. Kim, and B. Kim Miniaturized Parallel Coupled-Line Bandpass Filter With Spurious-Response Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Cheong, S.-W. Fok, and K.-W. Tam Periodically Nonuniform Coupled Microstrip-Line Filters With Harmonic Suppression Using Transmission Zero Reallocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Sun and L. Zhu Parallel Coupled Microstrip Filters With Floating Ground-Plane Conductor for Spurious-Band Suppression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. del Castillo Velázquez-Ahumada, J. Martel, and F. Medina

1693 1699 1707 1713 1723 1732 1739 1745 1756 1761 1777 1787 1792 1802 1810 1817 1823

LETTERS

Comments on “On Deembedding of Port Discontinuities in Full-Wave CAD Models of Multiport Circuits” and Related Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Farina Authors’ Reply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. I. Okhmatovski, J. D. Morsey, and A. C. Cangellaris Comments on “Toward Functional Noninvasive Imaging of Excitable Tissues Inside the Human Body Using Focused Microwave Radiometry” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. N. Reznik Authors’ Reply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. S. Karanasiou, N. K. Uzunoglu, and C. C. Papageorgiou Corrections on “Mode Discriminator Based on Mode-Selective Coupling”. . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Wang

1829 1831 1833

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

1834

1829 1829

CALLS FOR PAPERS

14th Topical Meeting on Electrical Performance of Electronic Packaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1835

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 and will receive this TRANSACTIONS upon payment of the annual Society membership fee of $14.00 plus an annual subscription fee of $24.00. 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. C. GUPTA, President M. P. DE LISO S. M. EL-GHAZALY M. HARRIS T. ITOH

K. VARIAN, Vice President D. HARVEY J. HAUSNER L. KATEHI

A. MORTAZAWI, Secretary T. LEE D. LOVELACE J. MODELSKI

S. KAWASAKI J. S. KENNEY N. KOLIAS

Honorary Life Members A. A. OLINER K. TOMIYASU T. S. SAAD L. YOUNG

L. E. DAVIS W. GWAREK W. HEINRICH W. HOEFER

M. HARRIS, Treasurer

V. J. NAIR B. PERLMAN D. RUTLEDGE Distinguished Lecturers T. ITOH B. KIM J. LASKAR J. C. RAUTIO

K. VARIAN R. WEIGEL S. WETENKAMP

W. SHIROMA R. SNYDER R. SORRENTINO D. RYTTING M. SHUR P. SIEGEL R. J. TREW

Past Presidents R. J. TREW (2004) F. SCHINDLER (2003) J. T. BARR IV (2002)

MTT-S Chapter Chairs Albuquerque: G. WOOD Atlanta: J. PAPAPOLYMEROU Austria: R. WEIGEL Baltimore: B. MCCARTHY Beijing: Y.-R. ZHONG Beijing, Nanjing: W.-X. ZHANG Belarus: A. GUSINSKY Benelux: D. V.-JANVIER Buenaventura: L. HAYS Buffalo: M. R. GILLETTE Bulgaria: F. FILIPOV Cedar Rapids/Central Iowa: D. JOHNSON Central New England/Boston: F. SULLIVAN Central & South Italy: R. TIBERIO Central No. Carolina: T. IVANOV Chicago: R. KOLLMAN Cleveland: G. PONCHAK Columbus: J.-F. LEE Croatia: J. BARTOLIC Czech/Slovakia: P. HAZDRA Dallas: P. WINSON Dayton: A. TERZOULI, JR. Denver: K. BOIS East Ukraine: A. KIRILENKO Eastern No. Carolina: D. PALMER Egypt: I. A. SALEM Finland: T. KARTTAAVI Florida West Coast: S. O’BRIEN

Foothill: C. ANTONIAK France: O. PICON Germany: W. HEINRICH Greece: I. XANTHAKIS Hong Kong: W. Y. TAM Houston: J. T. WILLIAMS Houston, College Station: K. MICHALSKI Hungary: T. BERCELI Huntsville: M. A. BARNES India/Calcutta: P. K. SAHA India Council: K. S. CHARI Israel: S. AUSTER Japan Council: H. JUI-PANG Kitchener-Waterloo: R. R. MANSOUR Long Island/New York: M. HANCZOR Los Angeles Council: T. LEE Malaysia: Z. AWANG Melbourne: R. BOTSFORD Milwaukee: S. G. JOSHI Mohawk Valley: P. RATAZZI Montreal: K. WU New Hampshire: T. PERKINS New Jersey Coast: G. N. S. PRASANNA New South Wales: G. TOWN North Italy: G. GHIONE North Jersey: K. DIXIT North Queensland: A. TSAKISSIRIS Northern Nevada: B. S. RAWAT Norway: M. OLAVSBRATEN Orange County: H. J. DE LOS SANTOS Oregon: T. RUTTAN Orlando: T. WU

Editor-In-Chief MICHAEL STEER WOLFGANG MENZEL North Carolina State Univ. Univ. of Ulm Raleigh, NC Germany 27695-7911 USA email: Phone: +1 919 515 5191 [email protected] Fax: +1 919 513 1979 ANDREAS CANGELLARIS email: Univ. of Illinois, Urbana Champaign [email protected] USA email: [email protected] ANTTI RÄISÄNEN Helsinki Univ. of Technol. Finland email: [email protected] M. GUPTA, Editor, IEEE Microwave Magazine

Ottawa: J. E. ROY Philadelphia: J. B. McCORMACK Phoenix: C. WEITZEL Poland: M. P. MROZOWSKI Poland, Lithuania: I. NAIDIONOVA Portugal: V. FERENDES Princeton/Central Jersey: W. CURTICE / A. KATZ Queensland: A. ROBINSON Rio de Janeiro: J. R. BERGMANN Rochester: F. POLO Romania: I. SIMA Russia, Jt. Nizhny: Y. BELOV Russia, Jt. Saint Petersburg: M. SITNIKOVA Russia, Moscow: V. KALOSHIN Russia, Saratov-Penza: N. RYSKIN Saint Louis: D. MACKE San Diego: U. S. DHALIWAL San Fernando Valley: J. C. WEILER, JR Santa Clara Valley/San Francisco: M. SHAKOURI Seattle: T. RASCHKO Seoul Council: H.-Y. LEE Siberia, Jt. Novosibirsk: V. SHUVALOV Siberia, Tomsk: E. GOLOVIN Singapore: O. B. LEONG South Africa: J. JOUBERT South Australia: B. BATES Associate Editors

South Brazil: L. C. KRETLY Southeastern Michigan: L. M. ANNEBERG Southern Alberta: E. C. FEAR Spain: L. DE HARO Springfield: S. C. REISING Sweden: J. F. JOHANSSON Switzerland: J. HESSELBARTH Syracuse: E. ARVAS Taipei: C.-S. LU Thailand: V. VIVEK Toronto: G. V. ELEFTHERIADES Tucson: N. BURGESS / S. MORALES Turkey: O. A. CIVI Twin Cities: M. J. GAWRONSKI UK/RI: A. REZAZADEH Ukraine West: M. I. ANDRIYCHUK Ukraine, Central Kiev: Y. POPLAVKO Ukraine, East: A. KIRILENKO Ukraine, Rep. of Georgia: R. ZARIDZE Ukraine, Vinnitsya: V. DUBOVOY Venezuela: A. N. BIANCHI Victoria: A. MITCHELL Virginia Mountain: D. MILLER Washington DC/Northern Eastern Northern Virginia: E. ADLER Winnipeg: S. NOGHANIAN Yugoslavia: A. MARINCIC

BUMMAN KIM KENJI ITOH ZOYA POPOVIC Pohang Univ. Sci. Technol. Univ. of Colorado at Boulder Mitsubishi Electric Corp. Korea USA Japan email: email: email: [email protected] [email protected] [email protected] AMIR MORTAZAWI DYLAN F. WILLIAMS RUEY-BEEI WU Univ. of Michigan at Ann Arbor NIST National Taiwan Univ. USA Taiwan, R.O.C. USA email: email: email: [email protected] [email protected] [email protected] YOSHIO NIKAWA ALESSANDRO CIDRONALI PETER RUSSER Technische Univ. Muenchen Univ. of Florence Kokushikan Univ. Germany Italy Japan email: email: email: [email protected] [email protected] [email protected] R. VAHLDIECK, Editor, IEEE Microwave and Wireless Component Letters T. LEE,

STEVEN MARSH Midas Consulting U.K. email: [email protected] MANH ANH DO Nanyang Technological Univ. Singapore email: [email protected] VITTORIO RIZZOLI Univ. of Bologna Italy email: [email protected] Web Master

IEEE Offficers LEAH H. JAMIESON, Vice President, Publication Services and Products W. CLEON ANDERSON, President and CEO MARC T. APTER, Vice President, Regional Activities MICHAEL R. LIGHTNER, President-Elect DONALD N. HEIRMAN, President, IEEE Standards Association MOHAMED EL-HAWARY, Secretary JOHN R. VIG, Vice President, Technical Activities JOSEPH V. LILLIE, Treasurer GERARD A. ALPHONSE, President, IEEE-USA ARTHUR W. WINSTON, Past President MOSHE KAM, Vice President, Educational Activities STUART A. LONG, Director, Division IV—Electromagnetics and Radiation

Executive Staff

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 $69.00 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 © 2005 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. Printed in U.S.A.

Digital Object Identifier 10.1109/TMTT.2005.849468

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

1561

Chebyshev Collocation and Newton-Type Optimization Methods for the Inverse Problem on Nonuniform Transmission Lines Martin Norgren, Member, IEEE

Abstract—A frequency-domain inverse problem for the nonunitransmission line is considered. The parameters of form the nonuniform line are interpolated by Chebyshev polynomials, and the Telegraphers equations are solved by a collocation method using the same polynomials. The interpolation coefficients for the unknown parameters are reconstructed by means of Newton-type optimization methods for which the Jacobian matrix has been calculated explicitly. For the reconstruction of one or two parameters, the algorithm is tested on synthetic data, and the necessity to use regularization is discussed. Finally, the algorithm is tested with measured reflection data to reconstruct shunt capacitances with piecewise constant profiles. Index Terms—Collocation, inverse problem, optimization, transmission line.

I. INTRODUCTION

T

HE INVERSE problem of parameter reconstruction on nonuniform transmission lines is of importance in various sensor applications. For example, a medium, e.g., like soil or snow, can be diagnosed using the reflection/transmission data from a submerged transmission line, which has been designed so that the parameters are sensitive to the properties of the surrounding media (see, e.g., [1]). For nonuniform transmission lines, analytical inversion methods have been developed [2], [3], but the existence or tractability of an analytical method depends largely on the particular parameter model used. Hence, many inverse problems for transmission lines have instead been solved by means of optimization methods both in time-domain [4], [5] and frequency-domain [6], [7] settings. Optimization methods exhibit a large versatility for different parameter models and are also easy to implement due to the availability of well-developed numerical software packets. The most frequently used optimization method for reconstructing transmission-line parameters has been the conjugate gradient (CG) method (see, e.g., [1], [4]–[6]). However, CG utilizes information from the first-order derivatives only and is thus likely to be more slowly convergent than, e.g., quasi-Newton (QN) methods, which utilize approximate second-order derivatives. The inferiority of

the CG against the QN method has been confirmed for some classes of inverse problems, in [8] for reconstructing piecewise constant profiles and in [9] for optical tomography. In this paper, we consider the possibility of QN methods as an alternative when reconstructing the line parameters from widebanded measurement data in the frequency domain. The analysis is based on the Chebyshev polynomials as an interpolation basis for both the unknown parameters and the voltage and current, in the transmission-line equations. Chebyshev polynomials together with the collocation (pseudospectral) method [10], [11] have been applied successfully in [12] for analyzing transient wave propagation on nonuniform multiconductor transmission lines. Here, this technique is used to solve the propagation problem in the frequency domain. Of similar reasons as in [12], we chose the collocation method: it is suitable for nonconstant coefficient differential equations and is expected to have rapid convergence for smooth parameter profiles [10]. For inverse problems that have been solved by means of the CG method [1], [4]–[6], the gradient has been calculated explicitly by forming the Fréchet differential of the objective function where the gradient is obtained from as a continuous function that is discretized when implemented numerically. However, that approach does not provide any information about higher order derivatives. Using the collocation method, the necessary truncation results already from the beginning in a finite-dimensional parameter space with a quite low dimension for smooth parameters. The resulting algebraic framework facilitates explicit calculations of derivatives, in principal, of any order. Hence, the direct solver based on Chebyshev collocation can easily be extended to higher order optimization methods like, e.g., full Newton methods. This paper is organized as follows. The collocation method is described in Section II and the calculation of the explicit Jacobian matrix is described in Section III. In Section IV, reconstruction results are presented, from synthetic as well as measured data. Section V contains the conclusions. II. DIRECT PROBLEM

Manuscript received May 28, 2004; revised October 24, 2004. This work was supported by The Fifth Framework Programme of the European Commission under a grant. The author is with the Division of Electromagnetic Theory, Kungliga Tekniska Högskolan, SE-100 44 Stockholm, Sweden. Digital Object Identifier 10.1109/TMTT.2005.847045

We consider a nonuniform transmission line with the series inductance , shunt capacitance , series resistance , and shunt conductance , where is the position along the line. For convenience of the analysis, we assume that . the nonuniform line occupies the region

0018-9480/$20.00 © 2005 IEEE

1562

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

The Telegrapher’s equations for the voltage are rent

and the cur-

(1) (2) where an time dependence has been assumed and suppressed. The nonuniform line is excited at from a uniform line with a characteristic impedance , and is terminated at . The voltages and curwith a load impedance rents at the feeding and load ends are denoted

The collocation points shall be located at the extrema of viz. , (cf. [10]). is omitted since it is where the Note that the point boundary condition (5) is enforced. For evaluation of , , and their derivatives at the collocation points, we introduce an matrix .. .

..

.. .

.

(11)

At the collocation points, we thus obtain

(3) (4)

.. .

.. .

(12)

The direct problem is posed as a one-ended boundary-value problem, where the boundary values at are chosen as (open end) (short-circuited end) otherwise

(5) is the load impedance that terminates the transmiswhere sion line. From the solution of (1), (2), and (5), the reflection coefficient at the feeding end becomes (6)

and analogous expressions for the current . The method requires the values of , , , and at the collocation points only. However, for the purpose of solving the inverse problem, it is convenient to interpolate the values using the same set of Chebyshev polynomials as for the expansion of the voltage and current. Hence, we also introduce the corresponding coefficient , etc. for the parameters. With the vectors and , the enforceabbreviations ments of the Telegrapher’s equations at the collocation points become

A. Collocation Method

(13)

The direct problem is solved by a spectral collocation method, , the similar to the one used in [12]. In the region voltage and current are expanded as (7)

denotes the element-wise product between matrices: , and is an row vector with unit elements. At the load end, (5) and (7) together with imply the property where

(14) where and polynomial, and efficients. The derivatives of as follows:

is the th-order Chebyshev are the expansion coand are expanded similarly

The expansion coefficient vectors from the equation

and are finally determined

(15) (8) where The coefficients in (8) and (7) are related as [11] (16)

(9) where and , . In the numerical implementation, we truncate the expansion after the th-order polynomial. The expansion coefficients are column vectors , collected into etc., whereby the linear relations in (9) can be written as the following matrix multiplications: (10) where the matrix represents the derivative (higher order derivatives follow analogously: ).

(17)

is an column vector

row vector with unit elements and with null elements.

is an

III. OPTIMIZATION APPROACH TO THE INVERSE PROBLEM Using Chebyshev interpolation for the parameters, the inverse problem is to determine a subset of the coefficients

NORGREN: CHEBYSHEV COLLOCATION AND NEWTON-TYPE OPTIMIZATION METHODS

from reflection data measured at different frequencies. For brevity, we introduce the parameter vector , containing all interpolation coefficients

1563

is solved. Introducing the row vector now follows from (21)–(23) that

, it

(24)

Let

denote the calculated reflection coefficient and let denote the measured reflection coefficient, at the angular frequency . If the reflection coefficient has been meaof frequencies, we obtain the folsured over a set lowing vector with nonlinear equations:

.. .

(18)

Note that the computational efficiency when solving (23) can has been LU factorbe increased one order if the matrix ized when (15) was solved. Let the vectors obtained from the derivatives of with respect to the coefficients and , respectively, build up the columns in the and , respectively. It then follows from (16) and matrices and are obtained from the equation (23) that

(25) The corresponding derivative matrices for the reactive parameters are obtained as

The inverse problem is formulated as the minimization of the scalar cost function

(26)

(19)

Now it follows that the row–vector gradient of the reflection coefficient becomes [cf. (24)]

Many available optimization routines incorporate the first-order derivatives and approximate second-order derivatives through the Jacobian matrix . The accuracy and, especially, the computational efficiency, can then be improved considerably if the Jacobian is calculated explicitly rather than by numerical perturbations in the parameters. With the Jacobian available, the gradient of the cost function becomes

(27) whereby we finally obtain the Jacobian matrix

.. .

(20) To build the Jacobian matrix, we need the gradient of the reflection coefficient with respect to the vector . Let denote a single component of , and let us concentrate on the reflection coefficient from one particular frequency (hence, the -dependence is left out for brevity). From (6), it follows that

A. Regularization

, we introduce the vector , when (3) and (7) yield

To suppress erroneous rapid oscillations in the reconstructed parameter profiles, due to model errors and noise-contaminated measurement data, we must add regularization terms to the equation system (18). The conventional Tikhonov regularization, i.e., suppressing the squared norm of the spatial derivative of the parameter, is enforced by extending the vector in (18) with the following vector of equations [cf. (10)]:

(22)

(29)

(21) From the property

(28)

where the extension of the Jacobian is

From (15), we obtain (23)

(30)

where and is the regularization parameter. Note that any desired th-order regularization can be applied by instead using .

1564

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

IV. RECONSTRUCTION RESULTS For the particular model with four nondispersive real-valued parameters , , , and , it has been shown in [13] that at most two parameters can be reconstructed simultaneously. Thus, we will only consider reconstruction of one or two parameters. Mathematically, there are two different cases of one-parameter or , and four different cases of reconstructions, or , two-parameter reconstructions ( , or , or , ).

Fig. 1. Reconstruction of shunt capacitance. The true profile is depicted with a solid line. (a) Without noise:  = 0 dotted line,  = 0:01 dashed line. (b) With noise:  = 0 dotted line,  = 0:1 dashed line.

A. Reconstructions From Synthetic Data In all reconstructions from synthetic data, we consider a line occupies the region scaled problem where the . The characteristic impedance of the feeding line and load impedance are . The frequency range goes from 0.025 to 1.000 in steps of 0.025, i.e., 40 frequency points are used. In the spectral approximation and interpolation , i.e., the highest Chebyshev of the parameters, we use polynomial used is . To avoid the inverse crime, the synthetic data has been generated by numerical integration of the Telegrapher’s equations (1) and (2) (cf. [6]) instead of using the spectral method, which is used as a direct solver in the optimization procedure. As initial guesses of the parameters, we have used their mean values, which may be estimated from low-frequency impedance measurements when the far end of the cable is either short-circuited ( and ) or open-ended ( and ). If a good initial guess is unavailable, one can run the algorithm several times starting from different initial guesses and take the best fit as the reconstruction result. A more systematic approach, which, in practice, has been proven to reduce the problem with local minima even if one starts from a poor initial guess, is to start with a limited set of data from the lower end of the frequency band to obtain a rough reconstruction of the large-scale variations. One then gradually incorporates data from higher frequencies to obtain the small-scale variations (see, e.g., [8]). When evaluating the reconstruction algorithm against noise contaminated data, random noise has been added to the calculated reflection coefficient. The magnitude of the complexvalued noise data has been generated from a Gaussian distribution with a mean value zero and a standard deviation of 0.05. The argument of the noise data has been generated from a uniform distribution in the range from 0 to . 1) Reconstruction of or : When studying one-parameter reconstruction, we chose the following profiles for the parameters:

( denotes the th-order Bessel function). All of the one-parameter reconstructions are from reflection data when exciting . the transmission line at the left-hand side The shunt capacitance was first reconstructed from noisefree data without using any regularization. As can been seen from the dotted line in Fig. 1(a), the reconstruction fails due to superimposed rapid oscillations in the reconstructed profile.

Fig. 2. Reconstruction of shunt conductance. (a) Without noise:  = 0 dotted line,  = 0:005 dashed line. (b) With noise:  = 0 dotted line,  = 0:1 dashed line.

This reconstruction error is a consequence of the numerical difference because a different direct solver has been used when generating the data (if the inverse crime is committed, the reconstruction would be virtually perfect). However, by imposing a small amount of regularization, the effect of this systematic error can be effectively suppressed. The smallest deviation in , which the reconstructed profile was obtained with yields a virtually perfect reconstruction; see the dashed line in Fig. 1(a). With noisy data, we see in Fig. 1(b) that, without regularization, the reconstruction fails even more. To stabilize the reconstruction, we thus need more weight on the regularization term than in the noise-free case. The best reconstruction result [see the dashed line from noisy data was obtained with in Fig. 1(b)]. The results when reconstructing the shunt conductance are similar to the ones when reconstructing . Again, regularization is needed to handle model errors and noise contaminated data (see Fig. 2). 2) Comparison Between Different Optimization Methods: All reconstruction results presented have been obtained using a trust-region (TR) optimization method, provided in the MATLAB numerical software [14]. In the , we compared with other methods: reconstruction of the Gauss–Newton (GN), the Levenberg–Marquardt (LM) method, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, another TR method (denoted TRH) using the exact Hessian matrix, where the Hessian was calculated explicitly, as described in the Appendix. All these methods were available in the numerical software, and for details, we refer to [14] and [15]. Furthermore, we compared them with the commonly used CG method, which was implemented by the author. The results of the comparison are presented below in Table I. The fastest method is TR, while the CG is the slowest. For noise-free data, GN is fast, but slower for noisy data, which is in accordance with the GN being more efficient for problems with small residuals [15]. TRH is rather slow due to the calculation of the Hessian. The large number of cost-function evaluations indicate that the CG has a slow rate of convergence, although

NORGREN: CHEBYSHEV COLLOCATION AND NEWTON-TYPE OPTIMIZATION METHODS

1565

TABLE I COMPARISON BETWEEN DIFFERENT OPTIMIZATION METHODS, WHEN RECONSTRUCTING THE SHUNT CAPACITANCE C (x)

Fig. 5. Reconstruction of series inductance and shunt capacitance from double-sided data (lossless case). Noise-free data with  = 0:006: dotted line. Noisy data with  = 0:09: dashed line.

Fig. 3. Reconstruction of shunt capacitance and shunt conductance from noise-free data. Left-hand-sided data with  = 0:02: dotted line. Right-hand-sided data with  = 0:004: dashed line. Double-sided data with  = 0:005: dashed–dotted line.

Fig. 6. Reconstruction of series inductance and shunt capacitance from double-sided data (lossy case). Noise-free data with  = 0:004: dotted line. Noisy data with  = 0:05: dashed line.

Fig. 7. Reconstruction of series resistance and shunt conductance. Noise-free data with  = 0:003: dotted line. Noisy data with  = 0:04: dashed line. Fig. 4. Reconstruction of shunt capacitance and shunt conductance from noisy data. Left-hand-sided data with  = 0:05: dotted line. Right-hand-sided data with  = 0:03: dashed line. Double-sided data with  = 0:09: dashed–dotted line.

the number can be slightly reduced with an improved line-search procedure. and : Using the 3) Simultaneous Reconstruction of same values of the parameters as when reconstructing one parameter, we next consider the simultaneous reconstruction of and . The previous study in [6] indicated that a two-parameter reconstruction is unstable using only one-sided data. To investigate in more detail about that indication, we test the algorithm with one-sided reflection data, both from the left- and right-hand side, as well as double-sided data. The results using noise-free data are depicted in Fig. 3. Even though regularization has been used to suppress the influence of small numerical errors, the reconstructions from one-sided data clearly exhibit deviations. The reconstructions from double-sided data are, on the other hand, virtually perfect. The results when using noisy data are depicted in Fig. 4, where the reconstructions from one-sided data exhibit much larger deviations than the ones from double-sided data. 4) Simultaneous Reconstruction of and : When reconstructing both reactive parameters, we chose

We first consider the lossless case, when . The reconstructions for and from double-sided data are depicted in Fig. 5. The reconstructions exhibit clear deviations for both noise-free and noisy data. (still ). Next, we introduce losses by setting Compared with the previous case, we see in Fig. 6 that the deviations are smaller, especially when using noise-free data. The above results conform with the uniqueness result reported in [13], in which, in the lossless case, one cannot obtain a unique simultaneous reconstruction of and . 5) Simultaneous Reconstruction of and : When reconstructing both dissipative parameters, we chose

The reconstructions for and from double-sided data are depicted in Fig. 7. The deviations, using noise-free or noisy data, are larger than in the other considered cases. B. Reconstructions From Measurement Data Finally, we test the reconstruction algorithm on measured data. The transmission lines to be investigated are built from the same kind of flat band cable that has been used for soil moisture

1566

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

TABLE II NOMINAL VALUES FOR THE DIELECTRIC MEDIA AROUND THE FLAT BAND CABLE

Fig. 8.

Resistors at the load end.

Fig. 10.

Connector at the feeding end.

Near the load end, we can expect an increased pile up of charges on the conductors, which yields an extra amount of capacitance at the load. The load impedance thus becomes (32)

Fig. 9. Flat band cable sandwiched between two blocks of Plexiglas.

and snow water content determinations at the Institute for Meteorology and Climate research (IMK), Karlsruhe, Germany (see [1] and [5]). The band cable consists of three strip conductors of copper embedded in a plastic band of polyethylene (see, e.g., Fig. 8). The series inductance for the even mode is estimated to nH/m [1]. For a cable submerged into a dielectric medium, experiments at IMK indicate that the shunt capacitance for the even mode can be estimated with the formula (31)

where pF is the stray capacitance at the load end. The measurements were performed with an HP8510C/HP8517A network analyzer system. The feeding coaxial cable from the network analyzer has a . At the feeding end, the characteristic impedance outer conductors of the band cable are attached to the screen of the coaxial connector via two wires, which are approximately 2 cm in length (see Fig. 10). At the feeding point, these wires yields an increased amount of shunt capacitance, which is modeled as a capacitor pF. To compensate for this stray capacitance, we must use the following reflection coefficient as the input to the reconstruction algorithm:

is the relative permittivity in the surrounding where and media, of which properties vary along the cable. , are part capacitances with the values [1] pF/m

pF/m

pF/m.

To obtain a varying , i.e., a nonuniform shunt capacitance, certain sections of the cable are sandwiched between blocks of plastic material (see Fig. 9). The two kinds of plastic materials used in the experiments are Plexiglas and polyethylene. We neither had information from any manufacturer about the dielectric constants for these media, nor did we measured the dielectric constant by any other means. Thus, the only information we compare with in this study is the following nominal values found in [16] and [17] (see Table II). Since the plastic materials have negligible losses, we take the . The series resistance in the copper shunt conductance . conductors is also small and, hence, neglected, i.e., 1) Measurement Setup and Compensation for Stray Capacitances: The length of the band cable was 2.00 m, and it was terminated with two 390- resistors between the central conductor to each of the outer conductors, yielding a load resistance for the even mode (see Fig. 8).

(33) where is the reflection coefficient recorded by the network analyzer. and were estimated Both of the stray capacitances from prior measurements on shorter samples of the cable, which were surrounded by air and either short circuited or resistively loaded at the rear end. 2) Reconstructions of Relative Permittivity From One-Sided Reflection Data: In the first experiment, a 775-mm section of the cable was sandwiched between two blocks of Plexiglas (cf. Fig. 9). The order and lengths of the sections along the cable were air

mm Plexiglas

mm air

mm

Measurement data from 382 evenly spaced frequencies in the range of 45–500 MHz were used for the reconstruction. In the . The inversion alspectral approximation, we used gorithm was tested with constant initial guesses for the shunt capacitance in the range of 20–30 pF/m in steps of 2 pF/m. The smallest value of the cost function and at the same time

NORGREN: CHEBYSHEV COLLOCATION AND NEWTON-TYPE OPTIMIZATION METHODS

Fig. 11. Reconstruction of shunt capacitance along a cable sandwiched between Plexiglas in 0:500 m < x < 1:275 m.

the smallest number of iterations were obtained with the initial guess of 26 pF/m. The result of the reconstruction is depicted in Fig. 11. From the prior knowledge that the permittivity of the surrounding media is piecewise constant, we see that the reconstruction is successful. The boundaries of the blocks of Plexiglas are located correctly and, elsewhere, the profile tends to fluctuate around piecewise constant values. Where the cable is surrounded by air, the shunt capacitance is approximately 18 pF/m, which, according to the experimentally determined mixing forin the surrounding medium (air). mula (31), yields In the Plexiglas region, the shunt capacitance is approximately for the Plexiglas; a value 37 pF/m, which yields slightly below the nominal value in Table II. There are mainly two reasons for this difference: the mixing formula (31) has been determined using media having a large extent outside the cable. Here, the blocks of Plexiglas has a thickness of 24 mm (see Fig. 9) whereby the air region outside the blocks reduces the shunt capacitance. There are also small air gaps between the cable and the blocks of Plexiglas. As can be seen in Fig. 10, the cable has indentations in the insulation near the conductors, where the field is strong. Hence, between the Plexiglas and the insulation, there will be air gaps, which reduce the shunt capacitance. In the second experiment, one section of the cable was sandwiched between the same two blocks of Plexiglas that were used in the first experiment. Another section of the cable was sandwiched between two blocks of polyethylene. The order and lengths of the sections along the cable were air

mm Plexiglas mm air mm polyethylene mm air mm

Measurement data from 633 evenly spaced frequencies in the range of 45–800 MHz were used. In the spectral approximation, . The algorithm was tested with constant initial we used guesses in the range of 26–32 pF/m in steps of 2 pF/m. The smallest value of the cost function and the smallest number of iterations were obtained with the initial guess of 30 pF/m. The result is depicted in Fig. 12. Also in this more difficult case, the reconstruction is successful. The boundaries between the different regions are located correctly and, elsewhere, the profile tend to fluctuate around piecewise constant values. For the regions where the cable is surrounded by air, we obtain slightly different values of the shunt capacitance in the range of approximately –1.11. For the Plexiglas, 18-19 pF/m, which yields we get a reconstructed value around 36 pF/m, which yields

1567

Fig. 12. Reconstruction of shunt capacitance along a cable sandwiched between Plexiglas in 0:250 m < x < 1:025 m and polyethylene in 1:300 m < x < 1:658 m.

, i.e., quite close to the value in the previous case. For the polyethylene, we get a reconstructed value around , i.e., below the nominal 33 pF/m, which yields value in Table II. However, although there are systematic errors due to the air gaps, the proportion between the reconstructed values for Plexiglas and polyethylene are in accordance with the proportion between the nominal values in Table II. V. DISCUSSION AND CONCLUSIONS For the direct problem, the convergence of the Chebyshev collocation method is very fast for smooth parameter profiles. Thus, the optimization-based algorithm is very efficient for reconstructing smooth profiles, requiring a moderate number of expansion functions. It is straightforward to change the model to other parameter dependencies by modifying the matrix in (15). Such dependencies are, e.g., a surface-resistance model for [6] or a dispersion model for the dielectric medium, which determines both and [7]. The simulations on synthetic data show that stable reconstructions of one parameter can be achieved using one-sided reflection data. For two parameters, one must, in practice, use double-sided reflection data and the only cases with reasonable stability are the simultaneous reconstructions of a reactive parameter together with a dissipative parameter. In all circumstances, one has to use regularization to achieve stable reconstructions. The comparison between different optimization methods indicate that Newton-type methods are more efficient than the CG method. The reconstructions from measured reflection data show that the method may also be used for reconstructing piecewise constant parameters, but the convergence then requires a larger number of expansion functions, which slows down the computation. The reconstructions of the permittivities of different materials around the band cable yields consistent results, although the values are slightly too high in the air regions and slightly too low in the regions surrounded by plastic materials. These errors may partly be explained by the air gaps. Also, for the band cable, there are uncertainties in the values of the series inductance and the part capacitances in the mixing formula (31) (see, e.g., [18] where slightly different values have been proposed and used). Another source of error for this kind of open cable is radiation, which introduces dissipation, which cannot be easily included in the model. The introduction of stray capacitances at the endpoints is necessary to account for the geometrical mismatches at the connector and load.

1568

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

APPENDIX I EXPLICIT HESSIAN MATRIX

REFERENCES

The Hessian of the cost function (19) reads (34)

where is the Hessian of the th component in (18). is either the real or imaginary part of the Hessian of the reflection coefficient, which reads

(35) where all first-order derivatives are available from the previous calculation of the Jacobian. What remains is the second-order derivatives of the input voltage and current. Considering the part first, let be any of or , from the equation [cf. (25)] and determine the matrix

(36) where and are known from the calculation of the Jacobian. Constructing the matrix .. .

(37)

the part of the second-order derivative term in (35) with respect to the and coefficients is obtained as (38)

[1] C. Hübner, “Entwicklung hochfrequenter Meßverfahren zur Boden-und Schneefeuchtebestimmung,” Ph.D. dissertation, Inst. Meteorol. und Klimaforschung, Forschungszentrum Karlsruhe, Karlsruhe, Germany, 1999. [2] P. V. Frangos and D. L. Jaggard, “Inverse scattering: solution of coupled Gel’fand–Levitan–Marchenko integral equations using successive kernel approximations,” IEEE Trans. Antennas Propag., vol. 43, pp. 547–552, Jun. 1995. [3] J. Lundstedt and S. Ström, “Simultaneous reconstruction of two parameters from the transient response of a nonuniform LCRG transmission line,” J. Electromagn. Wave Applicat., vol. 10, pp. 19–50, Jan. 1996. [4] J. Lundstedt and S. He, “A time-domain optimization technique for the simultaneous reconstruction of the characteristic impedance, resistance and conductance of a transmission line,” J. Electromagn. Wave Applicat., vol. 10, pp. 581–601, Apr. 1996. [5] S. Schlaeger, “Inversion von TDR-Messungen zur Rekonstruktion räumlich verteilter bodenphysikalischer parameter,” Ph.D. dissertation, Inst. Bodenmech. Felsmechanik, Universität Fridericiana Karlsruhe, Karlsruhe, Germany, 2002. [6] M. Norgren and S. He, “An optimization approach to the frequencydomain inverse problem for a nonuniform LCRG transmission line,” IEEE Trans. Microw. Theory Tech., vol. 44, pp. 1503–1507, Aug. 1996. [7] J. Lundstedt and M. Norgren, “Comparison between frequency domain and time domain methods for parameter reconstruction on nonuniform dispersive transmission lines,” J. Electromagn. Wave Applicat., vol. 17, pp. 1735–1737, Dec. 2003. [8] M. Norgren and S. He, “A gradient-based optimization approach to the inverse problem for multilayered structures,” Int. J. Appl. Electromagn. Mech., vol. 10, pp. 315–335, Aug. 1999. [9] A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomography image reconstruction,” Inverse Problems, vol. 19, pp. 387–409, Apr. 2003. [10] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia, PA: SIAM, 1977. [11] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Berlin, Germany: Springer-Verlag, 1994. [12] O. A. Palusinski and A. Lee, “Analysis of transients in nonuniform and uniform multiconductor transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 37, pp. 127–138, Jan. 1989. [13] J. Lundstedt, “A Time-domain wave-splitting approach to signal restoration, internal source and parameter reconstruction,” Ph.D. dissertation, Detp. Electromagn. Theory, Royal Inst. Technol., Stockholm, Sweden, 1995. [14] Optimization Toolbox, User’s Guide, Version 2, The Math Works Inc., Natick, MA, 2000. [15] J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, NJ: Prentice-Hall, 1983. [16] M. N. Afsar, Y. Wang, and A. Moonshiram, “Measurement of transmittance and permittivity of dielectric material using dispersive Fourier transform spectroscopy,” Microwave Opt. Technol. Lett., vol. 38, pp. 27–30, Jul. 2003. [17] The Electrical Engineering Handbook, R. C. Dorf, Ed., CRC, Boca Raton, FL, 1993. [18] G. Håkansson, “Reconstruction of soil moisture profile using time-domain reflectometer measurements,” M.S. thesis, Dept. Electromagn. Theory, Royal Inst. Technol., Stockholm, Sweden, 1997.

where the complete second-order derivative term becomes (39)

ACKNOWLEDGMENT The author is grateful to Dr. P. Fuks, Kungliga Tekniska Högskolan, Stockholm, Sweden, for valuable discussions and assistance with the measurements.

Martin Norgren (M’97) received the Ph.D. degree in electromagnetic theory from Kungliga Tekniska Högskolan, Stockholm, Sweden, in 1997. He is currently an Associate Professor with the Division of Electromagnetic Theory, Kungliga Tekniska Högskolan. His current research interests are electromagnetic theory in general, with special interests in guided-wave problems and scattering problems involving complicated materials, and the related inverse scattering problems and inverse source problems.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

1569

Four-Port Microwave Networks With Intrinsic Broad-Band Suppression of Common-Mode Signals Wael M. Fathelbab, Member, IEEE, and Michael B. Steer, Fellow, IEEE

Abstract—Two classes of four-port microwave networks are defined with discriminative even- and odd-mode transmission characteristics. Design of the networks is based on filter prototypes with predefined sets of transmission zeros and, consequently, a desired response can be synthesized. A configuration consisting of a pseudodifferential active circuit in cascade with either class results in a sub-system with high common-mode rejection ratio over a broad frequency range. A pseudodifferential circuit implementation demonstrated low common-mode gain over a 3 : 1 operating bandwidth ranging from 250 to 750 MHz. Index Terms—Common-mode signals, four-port networks, matching/filtering, pseudodifferential circuits, transmission zeros.

I. INTRODUCTION

D

IFFERENTIAL circuit design leads to stable, noise-tolerant monolithically integrated analog circuits and is compatible with current-mode biasing. Symmetrical differential circuits are generally less sensitive to component variations and when appropriately designed can reject common-mode signals, particularly coupled noise. However pseudodifferential circuits, as shown, for example, in the dotted box in Fig. 1(a), typically have a unit common-mode rejection ratio (CMRR) when biased by a pair of inductors, but will have dynamic range larger than that of fully differential circuits as the constant current source is mostly eliminated. A new coupled-line based biasing scheme for pseudodifferential RF circuits leading to enhanced CMRR was recently proposed [1]. A central attribute of coupled lines is the ability to discriminate between odd and even modes and, in terms of the subject of this paper, between common- and differential-mode signals. This paper extends the biasing scheme to one based on filter prototypes and circuit synthesis procedures. In particular, two classes of four-port networks are introduced that are suited to biasing differential circuits and obtaining broad-band common-mode signal suppression. A sub-system schematic of a pseudodifferential circuit in cascade with a four-port network connected to single-ended is illustrated in Fig. 1. The four-port network ideally loads presents distinct common- and differential-mode impedances to the active devices and has the following properties: • inherent broad-band isolation or attenuation of commonmode signals, as well as broad-band matching of differential-mode signals; Manuscript received June 8, 2004; revised September 8, 2004. This work was supported by the U.S. Army Research Office as a Multidisciplinary University Research Initiative on Multifunctional Adaptive Radio Radar and Sensors under Grant DAAD19-01-1-0496. The authors are with the Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695-7911 USA. Digital Object Identifier 10.1109/TMTT.2005.847046

Fig. 1. Pseudodifferential circuit connected to a pair of single-ended loads R separated by a four-port inter-stage network. Drain voltage V is applied at relevant points within the inter-stage network to bias the active devices.

• provides external dc biasing to the active devices. From a circuit synthesis perspective, the first requirement necessitates that the network possesses significantly different evenand odd-mode transmission characteristics. This is essentially the rationale behind the solutions introduced in this paper. The second requirement is satisfied by appropriate selection of the set of transmission zeros of the odd-mode sub-network of the four-port inter-stage circuit. In Section II, two classes of microwave networks with the aforementioned characteristics are presented, while in Section III, each network class is, in turn, implemented and cascaded with a pseudodifferential amplifier. Broad-band suppression of common-mode signals is demonstrated. II. FOUR-PORT INTER-STAGE NETWORKS Presentation of the new network classes will now commence.

0018-9480/$20.00 © 2005 IEEE

1570

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

circuit present in the circuit. Thus, the operation of Class-A inter-stage networks in suppressing common-mode signals relies on the isolating property of the microwave section shown in the dashed box in Fig. 2(c) [6], [7]. Class-A networks also possess short-circuited points marked in Fig. 2(a) where dc bias of the active devices can be applied with the use of decoupling capacitors. The square of the modulus of the transmission transfer function of the reciprocal odd-mode network of Fig. 2(d) may now as be defined in terms of its characteristic polynomial (1) replaces or . where With the even-mode network, ideally there is no transmission through the network and, thus, the design objective is (2) replaces or . is the Richards variable [5] defined as , where is the real frequency variable, is the resonant frequency of the transmission-line resand onators. Now the square of the modulus of the CMRR of the four-port inter-stage network may be defined as where In (1) and (2),

(3) and, consequently, the resulting CMRR of the sub-system configuration consisting of the pseudodifferential circuit in cascade with the four-port inter-stage network [the dashed box in Fig. 1(a)] is (4) Fig. 2. Generalized topology of Class-A inter-stage network made of two pairs of coupled lines and matching/filtering networks. (a) Physical topology. (b) Even- and odd-mode sub-networks. (c) Even-mode prototype. (d) Odd-mode prototype with at least one zero at dc, and a single UE.

A. Class-A Network The first class, denoted Class A, is shown in Fig. 2(a) and here comprises a pair of coupled-line sections connected to a pair of matching or filtering networks. The topology of the matching/filtering network is arbitrary and may consist of coupled-line sections, transmission lines, and lumped components dependent on the choice of transmission zeros required to satisfy a particular electrical specification. Application of the modal analysis to this structure results in the sub-networks of Fig. 2(b) with equivalent even- and odd-mode prototypes, as shown in Fig. 2(c) and (d), respectively. These equivalent prototypes result because the plane of symmetry transforms to either an open or short circuit for common- or differential-mode signal excitations, respectively, at Ports 1 and 2. In the odd-mode effective network, differential-mode signals , whereas in the even-mode are passed to single-ended loads equivalent circuit, it is seen that common-mode signals do not pass through the network by virtue of the effective series open

where and are the single-ended common- and differential-mode gains, respectively, of the pseudodifferential cirand are idencuit. For a pseudodifferential circuit, tical [1] and, thus, the sub-system CMRR is dominated by the performance of the inter-stage network. An example of a Class-A inter-stage network is illustrated in Fig. 3. It consists of a th-order network where is the number of resonators in either the even- or odd-mode sub-networks. It is seen that the th resonators of the even-mode prototype [see Fig. 3(b)] are completely decoupled from each other, thus offering optimum isolation of common-mode signals. On the other hand the odd-mode prototype, shown in Fig. 3(c), consists of cascades of short-circuited interdigital coupled-line sections [2]–[5], [7] loaded by open-circuited stubs. The odd-mode network [4] will possess a zero at dc, zeros at infinity, and zeros at leading to the existence of quarter-wave-long transmission lines (also known as unit elements (UEs) [5]). The general design procedure of an inter-stage network starts with the synthesis of an th-order -plane odd-mode prototype corresponding to an -plane response with a specific passband centered at with lower and upper band-edge frequencies ,

FATHELBAB AND STEER: FOUR-PORT MICROWAVE NETWORKS WITH INTRINSIC BROAD-BAND SUPPRESSION OF COMMON-MODE SIGNALS

1571

Fig. 4. Quadrature hybrid coupler. (a) Physical topology. (b) Even- and (c) odd-mode prototypes. Fig. 3. Example of Class-A inter-stage network made of a pair of appropriately connected bandpass interdigital filters. (a) Physical topology. (b) Even- and (c) odd-mode prototypes.

and , respectively, and a commensurate frequency . A set of transmission zeros selected to suit a particular electrical specification with a minimum of one zero at dc and a UE may be chosen to generate the appropriate characteristic polynomial [4], [10], [11] after which the square of modulus of the reflection coefficient of the odd-mode prototype is constructed. Based on the assumption of a lossless system, the square of modulus of the odd-mode reflection transfer function is, from (1), (5) which can be redefined in terms of

as

consider the inhomogeneous transfer admittance of the isolating section [the dashed box in Fig. 2(c)] [7]

(9) and are the even- and odd-mode impedances of where and are the even- and odd-mode the coupled lines and phase lengths, respectively. It is seen from (9) that the numerator is zero only when the even- and odd-mode phase lengths are equal, resulting in the equivalence of Fig. 2(c). Thus, the deterioration in sub-system performance due to the inhomogeneous media will be manifested in lower achievable CMRR since the will have transmission of the even-mode network a finite value. B. Class-B Network

(6) may then be found with the knowledge of (7) From this, the input impedance of the odd-mode prototype can then be derived in a 1- system using

Another four-port circuit with valuable characteristics is the quadrature coupler of Fig. 4(a) with even- and odd-mode prototypes illustrated in Fig. 4(b) and (c). The even- and odd-mode prototypes consist typically of sections of shunt eighth-wavelength open- and short-circuited stubs, respectively, separated by quarter-wavelength-long sections of transmission lines of uniform impedance . Each quarter-wave transmission-line section may then be split into two equal sections and the rematrix of a cascade pair of eighth-wavelength sulting transmission lines is

(8) and then synthesized using standard element extraction. A possible source of CMRR degradation in Class-A networks is the leakage of common-mode energy through the network. This is particularly a concern in inhomogeneous media such as microstrip or coplanar waveguide. To investigate this further,

(10)

1572

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

From (10), it is seen that the resulting matrix is invariant with respect to the transformation (11) except for a change of sign [8]. This leads to a very interesting property. The above implies that the odd-mode prototype of the coupler can be driven from its even mode, and vice versa, by simply replacing the open-circuited stubs by short-circuited stubs [8]. Now in the -plane, the even- and odd-mode prototypes have low-pass and high-pass responses, respectively. In the -plane, the low-pass and high-pass responses become periodic functions with bandstop and bandpass responses, respectively, at the center frequency . Thus, over a frequency band centered at , the even-mode network suppresses commonmode signals, while the odd-mode prototype passes differentialmode signals. Thus, the CMRR of this network has the relationship already defined in (3). In this case, the square of modulus of the even-mode transmission transfer function only dips to zero at the center of the band and, thus, the CMRR will have a varying characteristic with enhanced performance in the middle of the operating band. The quadrature coupler utilized as an inter-stage network has where and are the a maximum bandwidth ratio ( passband edge frequencies) of 3 : 1 over which there is discrimination of common- and differential-mode signals. This condition is satisfied only if the cutoff frequency of the even-mode prototype is chosen to be

Fig. 5. Generalized topology of Class-B inter-stage network. (a) Physical topology. (b) Odd-mode prototype with five zeros at dc and i UEs (i = 4; 6; 8; 10; . . . :).

(12) leading to an upper cutoff frequency of (13) Thus, in principle, the quadrature coupler could be used as an inter-stage network, but in contrast to a Class-A network, it is a low-pass structure. This structure can be modified for the purpose of biasing active devices with the inclusion of a pair of wide-band bias T’s [9] at the input arms. However, by proper selection of the relevant set of transmission zeros, a coupler-like network that supports dc bias can be designed. This leads to the Class-B inter-stage network shown in Fig. 5. The generalized odd-mode -plane prototype of a Class-B circuit is illustrated in Fig. 5(b) and consists of a minimum of five zeros at dc and UEs, where is an even number to maintain physical symmetry [11]. This has the benefit of providing dc bias to other parts of the system if required. It is now intuitive to conclude that the even- and odd-mode sub-networks of the coupler-like network will provide the necessary attenuation and matching of common- and differential-mode signals. This is illustrated in Fig. 6 for a set of synthesized Class-B prototypes with a 3 : 1 passband centered at 500 MHz, for various filter degrees. It is seen that the even- and odd-mode responses are considerably different from each other over the frequency range of interest. The main advantage of a Class-B network is simplicity in design, control of the even-mode attenuation level, and inherent broad-band operation. It is worth pointing out that the effective

Fig. 6. Simulated transmission characteristics of Class-B inter-stage network for different degrees. (a) Odd- and (b) even-mode responses centered at 500 MHz with a 3 : 1 bandwidth.

electrical lengths of the transmission lines forming the branches of the coupler-like section of Class-B networks are a half-wavelength long, whereas the branches of the true quadrature coupler are a quarter-wavelength long. In general, Class-B networks are particularly suited for systems operating at very high frequency, as electrical miniaturization of a Class-B structure is not possible, as it is the case with Class-A networks. III. EXPERIMENTAL RESULTS Experimental implementations of the two classes of interstage networks will now be presented.

FATHELBAB AND STEER: FOUR-PORT MICROWAVE NETWORKS WITH INTRINSIC BROAD-BAND SUPPRESSION OF COMMON-MODE SIGNALS

1573

A. Implementation of a Class-A Inter-Stage Network A pseudodifferential amplifier (HELA-10B)1 was selected and integrated with a Class-A network. The amplifier has a gain of 11 dB and broad-band operation from 50 MHz to 1 GHz in a 50- system. The chip requires a single 12-V dc power supply. An odd-mode -plane bandpass filter prototype was designed corresponding to an -plane response with a center frequency, of 500 MHz, band-edge frequencies ( and ) of 250 and 750 MHz, a commensurate frequency of 2 GHz, and a return loss (RL) of 16 dB. The set of transmission zeros chosen for this filter are three zeros at dc, three zeros at infinity, and two nonredundant UEs. Based on this, the characteristic polynomial of the filter prototype is constructed using a classic procedure [4], [10]. This gives (14)

from which the input reflection coefficient is evaluated using (6) and (7) leading to

(15) was then evaluated using The input impedance (8). The synthesized prototype is shown in Fig. 7(a) in a 50system. Construction of the four-port inter-stage network is now possible after transformation of parts of the odd-mode prototype to a structure made of coupled lines and lumped capacitors. Fig. 7(b) shows the final layout. Implementation on an FR4 board with a substrate thickness of 62 mil (1.57 mm), relative dielectric constant of 4.7, and loss tangent of 0.016 translate coupled-line even- and odd-mode impedances of 276.589 and 48.69 to a pair of transmission line that are 10-mil (0.254 mm) wide and 7-mil (0.177 mm) apart. The coupled lines are 910-mil (23.114 mm) long. These dimensions were obtained using Agilent’s Advanced Design Tool (ADS).2 The even- and odd-mode transmission characteristics of the physical layout of Fig. 7(b) were measured, and are presented in Fig. 7(c). It is clear that there is an appreciable difference between the two responses, as predicted by the theory. In the odd mode, a bandpass response of 3 : 1 bandwidth ratio is observed when the even-mode isolation is at least 22 dB. The finite even-mode isolation level is primarily due to the difference between the even- and odd-mode phase velocities of the microstrip media leading to the transmission of

150-MHz–1-GHz 2ADS,

amplifier, Mini-Circuits. ver. 2003A, Agilent Technol., Palo Alto, CA.

Fig. 7. Class-A inter-stage network in a 50- system. (a) Synthesized odd-mode prototype. (b) Physical layout loaded with lumped capacitors in picofarads. (c) Measured even- and odd-mode transmission characteristics of (b). (d) Measured common- and differential-mode gains of sub-system configuration comprising pseudodifferential amplifier in cascade with the inter-stage network.

common-mode signals through the isolating sections, as discussed in Section II. Another source of energy leakage contributing to this finite isolation level is the parasitic coupling beyond nearest neighboring resonators leading to unwanted couplings throughout the even-mode networks of the inter-stage

1574

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

network. This is particularly true since there is very weak coupling between closely spaced pairs of resonators of isolating sections giving rise to crosscoupling routes. At this point, the pseudodifferential amplifier and inter-stage network were integrated and the overall characteristics measured. The result is shown in Fig. 6(d). Very low common-mode of less than 12 dB was recorded over the operating gain bandwidth while a 3 : 1 bandwidth ratio was achieved with a of approximately 10 dB. The excess differential-mode gain 1 dB of loss is attributed to loss in the inter-stage network. Also, it is observed that the differential-mode gain ripples over the measured passband and rolls off rapidly toward the upper band edge frequency, i.e., 750 MHz. This is primarily due to the impedance-level mismatches since the measured RL of the inter-stage circuit is only 6 dB [see Fig. 7(c)]. The deteriorated RL was due to the restriction imposed by the FR4 board manufacturing process that only allowed a minimum of 7-mil spacing between the coupled lines. To achieve a better RL, e.g., 12 dB, a 5-mil spacing between the lines is required. B. Implementation of a Class-B Inter-Stage Network A Class-B inter-stage network was also designed on an FR4 board with the same specification to that used in Section III-A. In this case, the objective is to highlight the characteristics of the even- and odd-mode networks of a Class-B circuit without integration with the pseudodifferential amplifier. The odd-mode prototype was designed for an -plane response identical to that defined in Section III-A, but with an RL of 36 dB. Here, the commensurate frequency of the -plane prototype coincides with the center frequency since the prototype corresponds to an -plane high pass. The set of transmission zeros of the odd-mode prototype of the coupler-like network are chosen to be five zeros at dc and four UEs. The characteristic polynomial was directly generated using [11] to give

(16) In a manner similar to that used previously, and are constructed using (6)–(8). Direct synthesis of the above transfer function leads to the prototype of Fig. 8(a). Application of the appropriate Kuroda transformations to the circuit of Fig. 8(a) results in the circuit of Fig. 8(b). It is seen from Fig. 8(b) that the characteristic impedance of the inner pair of shunt short-circuited stubs are of value 33.27 . This pair of stubs was forced to have a value of 50 by application of some circuit optimization leading to a degradation of the RL from 36 dB to approximately 15 dB with a slight increase in bandwidth. This is acceptable since there was a lot of RL to factor. The prototype is shown in Fig. 8(c). At this point, the layout of the inter-stage network was constructed from the odd-mode prototype, as illustrated in Fig. 8(d). However, the input and output sections (series open-circuited stubs separated by a UE) are difficult to realize using microstrip technology and equivalence to coupled lines must be made

Fig. 8. Class-B inter-stage network in a 50- system. (a) Synthesized odd-mode prototype. (b) After application of relevant Kuroda transformations. (c) After some circuit optimization. (d) and (e) Physical layout. (f) Measured even- and odd-mode transmission characteristics of the circuit of (e).

leading to the final network of Fig. 8(e). Now the even- and of odd-mode impedances of values 133.741 and 41.306 each coupled-line section were then translated into physical dimensions resulting in a width of 27 mil (0.685 mm), spacing of 7 mil (0.177 mm), and length of 3483 mil (88.468 mm). The width of the other transmission lines of 50, 70.047, and 83.594 were 110 (2.794 mm), 57 (1.447 mm), and 38 mil

FATHELBAB AND STEER: FOUR-PORT MICROWAVE NETWORKS WITH INTRINSIC BROAD-BAND SUPPRESSION OF COMMON-MODE SIGNALS

(0.965 mm), respectively. The lengths of all the transmission lines were also 3483 mil (88.468 mm). The measured even- and odd-mode transmission characteristics of the inter-stage network are shown in Fig. 8(f), showing significant difference, as expected.

1575

[9] B. J. Minnis, “Decade bandwidth bias Ts for MIC applications up to 50 GHz,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 6, pp. 597–600, Jun. 1987. [10] H. J. Orchard and G. C. Temes, “Filter design using transformed variable,” IEEE Trans. Circuit Theory, vol. 15, pp. 385–408, Dec. 1968. [11] M. Horton and R. Wenzel, “General theory and design of optimum quarter-wave TEM filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 5, pp. 316–327, May 1965.

IV. CONCLUSION This paper has presented the derivation and design procedures of two new classes of four-port microwave inter-stage networks that choke common-mode signals over a very broad frequency range. The design procedure presented for the new networks is based on the synthesis of an odd-mode matching/filtering prototype with broad bandwidth and out-of-band rejection levels. Subsequent integration of either class of these networks with pseudodifferential circuits leads to dramatic improvement of the sub-system CMRR over very large bandwidth. This improvement was demonstrated experimentally. Both classes of networks possess fundamentally different even- and odd-mode transmission characteristics and reflect undesired common-mode energy. Class-B networks could be designed to posses a finite attenuation level to reject common-mode signals. This feature does not exist with Class-A networks and basically the rejection level of common-mode signals will depend on how homogeneous the realization media is. Choice of which class to use is dependent on the system application and on the evaluated strengths of undesired common-mode signals. REFERENCES [1] W. M. Fathelbab and M. B. Steer, “Distributed biasing of differential RF circuits,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5, pp. 1565–1572, May 2004. [2] G. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters Impedance-Matching Networks, and Coupling Structures. Norwood, MA: Artech House, 1980. [3] R. Mongia, I. Bahl, and P. Bhartia, RF and Microwave Coupled-Line Circuits. Norwood, MA: Artech House, 1999. [4] R. J. Wenzel, “Synthesis of combline and capacitively loaded interdigital bandpass filters of arbitrary bandwidth,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, no. 8, pp. 678–686, Aug. 1971. [5] I. Hunter, Theory and Design of Microwave Filters. London, U.K.: IEE Press, 2001. [6] H. Ozaki and J. Ishii, “Synthesis of a class of strip-line filters,” IRE Trans. Circuit Theory, vol. CT-5, pp. 104–109, Jun. 1958. [7] G. I. Zysman and A. K. Johnson, “Coupled transmission line networks in an inhomogeneous dielectric medium,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 10, pp. 753–759, Oct. 1969. [8] R. Levy and L. F. Lind, “Synthesis of symmetrical branch-guide directional couplers,” IEEE Trans. Microw. Theory Tech., vol. MTT-16, no. 2, pp. 80–89, Feb. 1968.

Wael M. Fathelbab (M’03) received the Bachelor of Engineering (B.Eng.) and Doctor of Philosophy (Ph.D.) degrees from the University of Bradford, Bradford, U.K., in 1995, and 1999 respectively. From 1999 to 2001, he was an RF Engineer with Filtronic Comtek (U.K.) Ltd., where he was involved in the design and development of filters and multiplexers for various cellular base-station applications. He was subsequently involved with the design of novel RF front-end transceivers for the U.K. market with the Mobile Handset Division, NEC Technologies (U.K.) Ltd. He is currently a Research Associate with the Department of Electrical and Computer Engineering, North Carolina State University, Raleigh. His research interests include network filter theory, synthesis of passive and tunable microwave devices, and the design of broad-band matching circuits.

Michael B. Steer (S’76–M’82–SM’90–F’99) received the B.E. and Ph.D. degrees in electrical engineering from the University of Queensland, Brisbane, Australia, in 1976 and 1983, respectively. He is currently a Professor with the Department of Electrical and Computer Engineering, North Carolina State University, Raleigh. In 1999 and 2000, he was a Professor with the School of Electronic and Electrical Engineering, The University of Leeds, where he held the Chair in microwave and millimeter-wave electronics. He was also Director of the Institute of Microwaves and Photonics, The University of Leeds. He has authored over 260 publications on topics related to RF, microwave and millimeter-wave systems, high-speed digital design, and RF and microwave design methodology and circuit simulation. He coauthored Foundations of Interconnect and Microstrip Design (New York: Wiley, 2000). Prof. Steer is active in the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). In 1997, he was secretary of the IEEE MTT-S. From 1998 to 2000, he was an elected member of its Administrative Committee. He is the Editor-in-Chief of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (2003–2006). He was a 1987 Presidential Young Investigator (USA). In 1994 and 1996, he was the recipient of the Bronze Medallion presented by the Army Research Office for “Outstanding Scientific Accomplishment.” He was also the recipient of the 2003 Alcoa Foundation Distinguished Research Award presented by North Carolina State University.

1576

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Ultra-Sensitive Detection of Protein Thermal Unfolding and Refolding Using Near-Zone Microwaves Kimberly M. Taylor, Student Member, IEEE, and Daniel W. van der Weide, Member, IEEE

Abstract—We use planar slot antennas proximal to proteins in solution to detect changes in conformation. The antennas are attached to fused-quartz or glass sample holders and the cuvette/antenna assembly is placed in the sample holder of an optical spectrophotomer (either UV/VIS or fluorescence polarization), allowing simultaneous dielectric and optical measurements. Return loss is recorded using a vector network analyzer. This system was used to study the equilibrium thermal unfolding and refolding of a small globular protein, as well as the binding of small hormones to a receptor. Good agreement between optical and microwave measurements was obtained for all systems studied. We show that microwave measurements can be made at protein concentrations as low as 0.3 ng/mL (19 pM), several orders of magnitude lower than that required for optical spectroscopy. The results from these experiments demonstrate that resonant slot antennas can be used to detect changes in protein conformation and the presence of microwave radiation does not perturb the system under study. Index Terms—Biomedical applications of electromagnetic radiation, biophysics, proteins, slot antennas.

I. INTRODUCTION

D

IELECTRIC spectroscopy is used to investigate gases, liquids, and solids in the radio, microwave, and terahertz frequency regimes (approximately 10 –10 Hz). The technique has been successfully applied to biological specimens using various methods of application including time-domain reflectometry [2], [3], transmission lines [4], coaxial probes [5], and microfabricated waveguides [6]. While these studies have yielded potentially useful results, many of them suffer from one or more of the following disadvantages. 1) Experiments are performed on dry, but “hydrated” samples [3], [7], yielding results that may not be applicable to biomolecules in solution. 2) Results are qualitative rather than quantitative, i.e., a phenomenon is demonstrated, but its extent is not, or cannot, be further quantified [4]. 3) Results are qualitative, but are not correlated with previously published results obtained with other techniques [8], [9].

Manuscript received April 21, 2004; revised October 31, 2004. This work was supported by the Army Research Office under Grant DAAD19-02-1-0081 and by the Office of Naval Research. The work of K. M. Taylor was supported by the National Institutes of Health under a Molecular Biophysics Training Grant. The authors are with the the Department of Electrical and Computer Engineering, University of Wisconsin–Madison, Madison, WI 53706 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2005.847047

We introduce a new application of dielectric spectroscopy in which proteins or other biomolecules are studied in an aqueous environment and in which optical spectroscopy is simultaneously performed [10]. This method not only allows measurement of proteins and other biomolecules in their “natural” aqueous state, but also permits correlation and comparison with results obtained using an established spectroscopic method. Variations of this technique have been previously described in the literature [11]–[13]. The phenomenon measured by dielectric spectroscopy is the polarization relaxation (also called dispersion) of one or more dielectric species. For biological samples, these relaxations are generally divided into three regimes. The -dispersion is a broad rotational orientation of either the entire macromolecule or of specific groups (e.g., protein sidechains with a high degree of freedom may undergo their own dispersion) [14]. The frequency of the -dispersion is inversely proportional to molecular size, and this type of dispersion generally occurs below 100 MHz. The -dispersion is caused by absorption of bulk water, and is generally seen as a wide peak with a maximum around 19.4 GHz for pure water at room temperature [15], [16]. Water is strongly dielectric, and -dispersion is generally the largest dispersion observed in the radio, microwave, and terahertz regimes. The -dispersion is located between the - and -dispersions, in the high radio to low microwave region (approximately 500 MHz–20 GHz), and is largely caused by reorganization of “bound” water molecules near the surface of the protein or macromolecule [15]. These “bound” waters undergo dispersion at a lower frequency because they are rotationally and translationally hindered, and possess higher and more ordered dipole moments compared with bulk solution [17]. By addressing an appropriate frequency range in which -dispersion is large while the contribution from -dispersion is strong but static, the dispersion of bound water can be used as a reporter for protein conformational change. The bound water shells are modified in response to changes in the conformation of a biological macromolecule such as folding or unfolding, association, and ligand binding. This concept has been used by other researchers: Denisov and Halle used water O and H magnetic resonance dispersion to examine the thermal denaturation of bovine pancreatic ribonuclease (RNase A) and other proteins [18]. In order to exploit -dispersion, we chose to examine a frequency range between 3–20 GHz, and used planar slot antennas driven by a 50- coaxial cable. Planar slot antennas are commonly used at microwave frequencies [19], and consist of a rect-

0018-9480/$20.00 © 2005 IEEE

TAYLOR AND VAN DER WEIDE: ULTRA-SENSITIVE DETECTION OF PROTEIN THERMAL UNFOLDING AND REFOLDING

angular window cut into the ground plane of a metal-clad didevices, electric material. Slot antennas can be treated as where the longest dimension of the slot is equal to approximately half of the principal resonant wavelength [19]. However, the impedance of the slot in this fundamental mode is very high [20], and a lower impedance (resulting in a better match to the coaxial feed) may be obtained by using the second mode, in which the slot length is equal to a full wavelength. The slot antenna is attached to a fused quartz cuvette (for simultaneous measurements in the range of 190–820 mm) or to the sample holder of a fluorescence polarimeter. We chose thermal denaturation of bovine pancreatic ribonuclease (RNase A) as a test system for very large conformational changes. Unfolding of a protein from its native to its denatured state is the largest conformational change a protein experiences under equilibrium conditions and, therefore, should be easily detected using the slot antenna system. RNase A is a highly stable 13.7-kDa globular protein that undergoes reversible two-state unfolding over a wide range of conditions [21], [22]. Three series of experiments were performed. In the first, unfolding and refolding of RNase A was performed over a range of pH from 2.5 to 5.0. In the second series, the sensitivity of the slot antenna system was investigated by performing unfolding/refolding of RNase A over concentrations from 0.26 ng/mL to 9.3 mg/mL (19 pM to 680 M). Finally, the effects of microwave power on the unfolding of RNase A were explored by varying the power delivered by the network analyzer from 35 to 5 dBm (from 18 W to 1.78 mW). A scaled version of the planar slot antenna resonant in the 10–20-GHz range was used to perform simultaneous dielectric and fluorescence polarization measurements on the binding of hormones to human estrogen receptor . Fluorescence polarization (FP) is a technique that exploits differences in the tumbling rate of fluorescently labeled ligands to detect binding events [23]. The protein human recombinant estrogen receptor- (ER- ) [24] was titrated with aliquots of fluormone ES2, a fluorescently labeled estradiol, in order to probe the thermodynamics of binding and obtain a dissociation constant . This titration was repeated with unlabeled estradiol. The binding of unlabeled estradiol was monitored using microwave measurements only since neither estradiol, nor ERresponds fluorescently. Previous measurements of ER- /estradiol binding could only be performed using radiolabeled ligand [25].

1577

Fig. 1. Return loss of antenna/cuvette assembly employed for simultaneous UV/VIS and microwave recordings recorded at room temperature. Solid line is empty (air-filled) cuvette. Solid symbol ( ) is buffer at pH 4.5; open symbol ( ) is 5.5 mg/mL (0.40 mM) RNase A in the same buffer. Inset: dimensions of slot antenna.

NaCl for pH 3.5 and below. The heat capacity of RNase A was treated as a constant 1.15 kcal mol K for all calculations [21]. Human recombinant estrogen receptor , fluormone ES2 -estradiol were (a fluorescently labeled estradiol), and obtained from PanVera, a division of Invitrogen, Carlsbad, CA. Buffer conditions were 100-mM potassium phosphate, 100- g/mL bovine -globulin, 0.02% NaN , pH 7.4. B. Slot-Antenna Fabrication Slot antennas were fabricated using RO-4003C high-frequency laminate from Rogers Corporation, Rogers, CT. The board had permittivity of 3.38, height of 0.813 mm, and was doubly clad with 1 oz of copper. Antennas were cut using either an LPKF ProtoMat (LPKF Laser and Electronics AG, Garbsen, Germany) or a TAIG MicroMill (TAIG Tools, Chandler, AZ). Copper cladding was removed from one face of the antenna after fabrication, and a semirigid coaxial cable was soldered across the center of the slot window. Antenna dimensions for the antenna used for simultaneous UV/VIS spectroscopy and dielectric measurements are shown in Fig. 1.

II. METHODS AND MATERIALS A. Protein Preparation and Experimental Conditions

C. Spectroscopy and Thermal Denaturation

RNase A (Type XII-A) was obtained in lyophilized form from the Sigma Chemical Company, St. Louis, MO, and was used without further purification. All protein solutions were dialyzed exhaustively before use. For concentrations above 1 M, protein concentration was determined from the absorbance at 278 nm, using an extinction coefficient of 0.72 mg mL cm [21]. At lower concentrations, a modified Lowry assay [26] (Pierce Chemical, Rockford, IL) was used to determine concentration. Buffer conditions were: 1) 50-mM sodium acetate/acetic acid, 100-mM NaCl for pH 4–5 and 2) 50-mM glycine/HCl, 100-mM

An HP 8452A diode-array spectrophotometer equipped with a water-jacketed cell holder (Agilent Technologies, Palo Alto, CA) was used for all UV/VIS spectroscopy measurements. UV-transparent fused-quartz cuvettes of 0.5-cm pathlength were obtained from Hellma GmbH and Company, Müllheim, Germany. An RTE-111 circulating water bath (Thermo Neslab, Portsmouth, NH) was used for temperature control. Return loss from the slot antenna was recorded using an HP 8720ES vector network analyzer (VNA) (Agilent Technologies, Palo Alto, CA). Unless otherwise specified, network-analyzer output

1578

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

power was fixed at 5 dBm (1.78 mW). The UV/VIS spectrophotometer, VNA, and water bath were controlled by a PC using LabView 6i software (National Instruments, Austin, TX). All temperature scans were taken under the following conditions: from 20 C to 80 C and back in 2 C intervals with a 5-min equilibration time at each temperature. UV/VIS absorbance was recorded at 288 nm [21]. A volume of 1.6-mL buffer or protein solution was used in all experiments. Generally, three complete cycles from 20 C to 80 C and back were taken for each protein or buffer sample; reported thermodynamic parameters are the average of parameters from each of these scans. Unfolding and refolding were more than 95% reversible. and van’t Hoff enthalpy Midpoint temperature were obtained from unfolding or refolding scans using the method of Marky and Breslauer [27]. Assuming that the protein undergoes a reversible two-state transition from native ( ) to unfolded ( ) states, the equilibrium constant can be defined as

If the ligand is assumed to exist in only two states (bound to the receptor or free in solution), then an equilibrium exists between the free receptor, bound receptor, and complex such that

(4) is the dissociation constant [in units of moles/liter Here, or molar (M)]. The anisotropy should be equal to

(5) and are the fractions of bound and free where and are the anisotropy values for ligand, and is the concentration of the rebound and free ligand. If ceptor–ligand complex, and and are, respectively, the total concentrations of ligand and receptor, then (4) can be rewritten as [29]

(1) Here, is the fraction of protein in the native state, is the gas constant, is the temperature (in degrees Kelvin), and is the midpoint temperature (defined as the temperature at which is equal to 0.50 and is equal to 1). The signal ( is any observable, in this case, either the UV/VIS absorbance or the peak frequency) was equal to (2) and are, respectively, the native and unfolded where baselines, defined as linear functions of temperature. An alternative baseline-free approach [28] was also used in which (3) Here, is a scaling factor. The two methods yielded near-identical thermodynamic parameters, but the baseline-free method . gave better determination of the midpoint temperature D. FP and Ligand Binding A Beacon 2000 fluorescence polarization system (Invitrogen, Carlsbad, CA) equipped with the standard spectrum range (360–700 nm) was used for all FP measurements. A small-scale slot antenna resonant in the range of 10–20 GHz was attached to the Beacon’s sample holder. The temperature of the sample was kept constant using the Beacon’s built-in Peltier temperature control unit. Samples were contained in 50 mm test tubes (Fisher Scientific, borosilicate glass 6 Hampton, NH), and sample volume was maintained at 500 L (for complete coverage of antenna). All instrumentation was computer controlled using the manufacturer’s software for the Beacon system and LabView 6i for the VNA.

(6) Data from either the Beacon instrument or the network analyzer were fit to (5) and (6) using nonlinear regression. III. RESULTS A. Slot Antenna Design and Return-Loss Spectra The antenna dimensions were chosen to yield a principal resonance of approximately 3.5 GHz when loaded with a cuvette filled with aqueous solution. Slot dimensions are shown in Fig. 1. Representative spectra of the empty (air-filled) cuvette and of the same cuvette filled with either buffer (50-mM sodium acetate/acetic acid, 100-mM NaCl, pH 4.5) or a solution of 0.40-mM RNase A in the same buffer are shown in Fig. 1 The slot was expected to yield a single resonant peak, but multiple additional peaks are clearly present. In an ideal slot, the ground plane and dielectric substrate extend to infinity [30]. By contrast, the slot used in these experiments has a large slot and a relatively small ground plane. The unexpected resonances may also result from reflections from the cuvette and/or the cell holder. The shape of the reflection return loss was invariant: a large peak at approximately 3.2–3.4 GHz with multiple smaller side peaks. The return loss for buffer and RNase A are superficially similar, although the protein return loss was larger. We noted that any change in the cuvette/antenna assembly (including emptying or refilling of the cuvette) caused perturbation of the driving coaxial cable and, thus, changes in the return response. The return-loss pattern was very stable if the cuvette assembly was not disturbed. Peak positions within the return-loss pattern were determined by the fitting to a series of Lorentzian peaks. In general, 7–11

TAYLOR AND VAN DER WEIDE: ULTRA-SENSITIVE DETECTION OF PROTEIN THERMAL UNFOLDING AND REFOLDING

1579

Lorentzian peaks were needed to obtain a good fit to the returnloss pattern. B. Antenna Response to Solute Titration As seen in Fig. 1, the return loss of a buffer solution differs from that of a solution of the same buffer containing 0.40-mM RNase A. This dissimilarity is chiefly observed in the magnitude of the return-loss spectra, particularly at the major peak around 3.6 GHz. of the antenna Theoretically, the resonant frequency . Thus, a should depend on permittivity as decrease in the permittivity is expected to increase the antenna’s resonant frequency. The permittivity of a mixture of two dielectric materials can be approximated using a simplified form of the Wiener mixture equation [31] (7) and are the volume fraction of materials and , where and are the permittivities of materials and . It is and assumed in (7) that the two materials are miscible, but do not interact. Equation (7) can also be expressed as

Fig. 2. Variation in peak position with approximately dielectric constant for acetone (closed symbols) and methanol (open symbols). Lines indicate fit to the inverse square root of the permittivity: acetone (solid line; R = 0:994) and methanol (dashed line; R = 0:989).

(8) As a test for the sensitivity of the antenna resonance to permittivity changes, a solution of distilled deionized water was titrated with aliquots of low-permittivity solvent. Chosen for this experiment were methanol ( at 25 C) and aceat 25 C). The real portion of the permittivity tone ( of water at room temperature was taken as 80.4 at 20 C. The variation in peak position with permittivity for acetone and methanol is shown in Fig. 2. Data for both organic solvents vary in a similar manner with permittivity, and both fit well to an exponential model (not shown). Data at low organic volume fraction, or high permittivity, also fitted well to the inverse square root of the calculated permittivity . Fits to the inverse square root formula were not possible for below 60, probably due to uncertainty in the volume of organic solute. Fits for this data are indicated in Fig. 2. Titration of a buffer solution was also performed with two biological macromolecules: RNA (polycytidylic acid, or polyC) and the protein RNase A (shown in Fig. 3). Due to solubility concerns, concentrations of both macromolecules could not be measured above approximately 220 M. Permittivity values for biological macromolecules vary, but many published accounts set at only 2–4 for both RNA and protein [32]. Approximations such as in the derivation of (7) or (8) cannot be applied to aqueous solutions of most biological macromolecules because there are significant interactions between solvent water and the macromolecules. However, peak positions for both polyC and RNase A vary linearly with protein concentration over this limited concentration range. Thus, our technique is sensitive to the concentration of aqueous solutes, including biological macromolecules, in a linear fashion.

Fig. 3. Variation in peak position with concentration of the biological macromolecules RNA (polycytidylic acid, or polyC, closed symbols) and RNase A (open symbols). Peaks were derived from Lorentzian fit of the complete return-loss spectrum (data not shown). Lines are linear fits to: RNA (solid line; R = 0:803) and RNase A (dashed line; R = 0:963).

C. Protein Unfolding and Refolding Although the buffer and RNase A return loss scans as described above were superficially similar, their temperature-dependent behaviors were quite different. Fig. 4 shows the behavior of RNase A return loss at selected temperatures between 20.43 C–76.56 C. The amplitude of the major resonant peak decreases with increasing temperature, and the position of this peak shifts to lower frequency. The center frequencies of all peaks were determined by fitting the return-loss scan at each temperature to a series of Lorentzian peaks (Fig. 5). A fixed number of peaks (7–11) were used to fit all return-loss scans in an unfolding/refolding series. The behavior of individual peaks could be assessed by plotting peak position versus temperature. The behavior of a peak

1580

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Fig. 4. Variation of return loss of 5.6 mg/mL (0.41 mM) RNase A at pH 4.5 with temperature. From bottom to top, temperatures are 20.43 C, 30.10 C, 38.99 C, 48.72 C, 57.96 C, 67.28 C, and 76.56 C.

Fig. 5. Fitting of return loss of RNase A at 20.43 C with nine Lorentzian peaks. This data is from the same experiment featured in Fig. 4. Solid symbols ( ) are raw data, the solid line is the fit. Dotted lines indicate individual Lorentzians; the peak at approximately 3.1 GHz plotted in Fig. 6 is shown as open symbols ( ).

at approximately 3.1 GHz from the RNase A solution is shown in Fig. 6. The temperature-dependent variation of this peak frequency has a sigmoidal appearance characteristic of a cooperative phenomenon such as protein unfolding, and this data fitted well to a two-state unfolding model. After an initial increase in the peak position, the frequency of the corresponding peak from the buffer return loss scans decreased monotonically with increasing temperature, and could not be fit to a two-state model (data not shown). Fig. 6 also displays the fraction of protein in the native state derived from fits to the peak at approximately 3.1 GHz from the network-analyzer data and to the UV/VIS absorbance at 288 nm. Midpoint temperatures for the two data sets were similar (59.18 C for the network-analyzer data versus 62.46 C for the UV/VIS data), as was unfolding enthalpy (59.2 and 63.7 kcal/mol, respectively). This result demonstrates that

Fig. 6. Temperature-dependent behavior of a single peak (highlighted in Fig. 5) from Lorentzian analysis of return loss. Top panel shows variation of peak position with temperature (solid symbols) and fit to a two-state unfolding model (solid line). Bottom panel shows fraction native calculated from peak position (solid symbols and line) and from simultaneous UV/VIS absorbance (open symbols and dotted line).

the return loss data can be used to detect changes in protein conformational change. Generally, one or more peaks from protein unfolding or refolding data fitted to a two-state model with similar midpoint to that obtained from UV/VIS data. Due to varitemperature ability of the return-loss pattern, peaks did always respond idenreported tically to protein conformational change. Values of was often are averages of all fits to peak position, and error in high due to variation in peak position fits. Determination of unwas more problematic. In most cases, folding enthalpy reasonable values of enthalpy were obtained, as in the example above, but errors in enthalpy were very large, exceeding 20%. will not be reported for the other experiments Therefore, in this paper. These problems in midpoint temperature and enthalpy determinations, and possible solutions, will be revisited in the Section IV. D. pH Series To ensure that the results obtained at pH 4.5 were not an anomaly related to the behavior of the cuvette/assembly around 60 C, a series of unfolding/refolding assays at pH values between 2.5–5 were performed. Values of and decrease with decreasing pH in this pH range as the protein stability diminishes with decreasing charge. At each pH value, two unfolding/refolding assays were performed, which are: 1) UV/VIS absorbance alone and 2) simultaneous UV/VIS absorbance and dielectric measurements. Values of midpoint temperature from these experiments are summarized in Table I. The pH series results from UV/VIS and network-analyzer peak fitting are summarized in Fig. 7. Also included in this figure are published midpoint temperatures obtained from differential scanning calorimetry [22]. derived from UV/VIS is assumed to Note that the error in be 0.50 C based on estimates of the accuracy of the water bath

TAYLOR AND VAN DER WEIDE: ULTRA-SENSITIVE DETECTION OF PROTEIN THERMAL UNFOLDING AND REFOLDING

TABLE I VARIATION IN MIDPOINT TEMPERATURE WITH pH

1581

TABLE II VARIATION IN MIDPOINT TEMPERATURE WITH PROTEIN CONCENTRATION

UV/VIS absorbance was recorded simultaneously with return loss data at all concentrations, but absorbance was too low for analysis of UV/VIS data below 8.8

M. Error in

midpoint temperature determined from UV/VIS absorbance at 8.8 M is higher than usual due to low signal.

E. Concentration Series

Fig. 7. pH dependence of midpoint temperature T from UV/VIS absorbance , from UV/VIS absorbance recorded in the presence of recorded alone , and from peak position analysis of return loss data . microwave power Also included are published midpoint temperatures derived from differential scanning calorimetry [22].

() ( ) (r)

(1)

and thermometer; fitting error in all cases is actually consideris the standard deably lower. Reported error in the VNA viation of the average of all midpoint temperatures determined from peak fitting. Midpoint temperature recorded using UV/VIS absorbance in the presence of microwave power appears to be up to 1.6 C greater than that obtained using the same method in the absence of microwave power. The average increase is 0.42 C, slightly smaller than experimental error. Could microwave power at these low levels stabilize the protein, particularly at high pH? The premise is intriguing, but cannot be proven at this time. Based on published reports [33], it was expected that the microwave radiation might enhance the protein’s folding rate, but would ultimately destabilize the protein due to non-Joule heating effects. No microwave-dependent destabilization has been observed in our experiments. Another interesting result is the lower midpoint temperatures determined from network-analyzer peak fits. With the exception determined from network-analyzer of the data at pH 2.5, data is an average of 3.27 C lower than that determined from simultaneously obtained UV/VIS absorbance. As evidenced by values from the sample unfolding at pH 4.5 above, lower network-analyzer data are a standard result.

A series of unfolding/refolding assays at decreasing concentrations was performed in order to determine the limits of sensitivity of the slot antenna system and to test for evidence of destabilization at low protein concentrations. Protein concentration began at 680 M and was serially decreased to 19 pM. Simultaneous UV/VIS and network-analyzer measurements were made at each concentration. A separate UV/VIS absorbance measurement in the absence of microwave power was also made for concentrations above 8.8 M. Below 8.8 M, UV/VIS absorbance at 288 nm was lower than experimental error and data could be analyzed. Midpoint temperatures determined from UV/VIS absorbance and VNA peak fitting at all concentrations are summarized in Table II. The apparent decrease in midpoint temperature determined from UV/VIS absorbance with decreasing temperature is attributed to the decreasing signal-to-noise ratio at lower protein concentrations. This effect is unlikely to be due to multimerization (formation or unfolding of dimers, trimers, or other higher order structures) since RNase A is a monomer and has been shown to undergo two-state unfolding under these conditions [21], [22], [33]. Protein solutions displayed no evidence of aggregation and unfolding was highly reversible. Once again, the midpoint temperature derived from UV/VIS absorbance in the presence of microwave radiation exceeds the midpoint temperature determined from UV/VIS absorbance alone by 0.31 C–1.97 C. For concentrations of 78 M or greater, this temperature difference exceeds the experimental error and may be statistically significant. At 8.8 M, the signal can only be determined to a precistrength was so low that sion of approximately 1.0 C, and the temperature difference may be attributed to experimental error. Two facts may be gleaned from the concentration-dependent behavior of the midpoint temperature derived from peak fitting. derived from network-analyzer measurements is First, derived from simultaneous UV/VIS again lower than the

1582

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

TABLE III MINIMUM PROTEIN CONCENTRATION FOR POPULAR PROTEIN THERMODYNAMICS TECHNIQUES

TABLE IV VARIATION IN MIDPOINT TEMPERATURE WITH NETWORK ANALYZER POWER

TABLE V BINDING OF FLUORMONE AND ESTRADIOL TO ESTROGEN RECEPTOR

absorbance. The magnitude of this decrease is difficult to determination at low gauge due to the difficulty in accurate determined from concentrations. Second, while values of network-analyzer measurements showed a wide variation with concentration, from 46.66 C to 56.09 C, no clear concentration-dependent trend was evident. Contrary to expectation, no decrease in stability with at low concentrations was observed. The average midpoint temperature for all concentrations was 51.76 C 3.08 C. These concentration dependence studies establish an approximate operating range for the slot antenna system. A wide concentration range (nearly eight orders of magnitude) was used in these experiments. Midpoint temperature could be determined with reasonable accuracy at protein concentrations as low as became unac19 pM; at lower concentrations uncertainty in ceptably high as the signal strength decreased. The upper concentration limit was not determined in this study, but will probably be limited by protein solubility. Minimum protein concentrations for several popular techniques for measurement of unfolding/refolding thermodynamics are summarized in Table III. With the exception of fluorescence spectroscopy, all of these techniques require at least 20 g/mL of protein or greater (1.5- M RNase A) to obtain good results. Good fluorescent spectroscopy results were obtained at concentrations as low as 0.8 nM of fluorescein-labeled RNase A (approximately 10 ng/mL) [1]. This study establishes that thermodynamic data can be obtained from as little as 0.3 ng/mL (19 pM) of RNase A, more than 40 times lower than the minimum needed for fluorescence spectroscopy. F. Power Series In an attempt to understand the apparent microwave-induced stability increase observed in results from UV/VIS absorbance, a series of unfolding/refolding assays at increasingly low power levels was attempted. If the stability increase was indeed caused by microwave power, then this increase should decrease

or even reverse at sufficiently low power levels. An initial unfolding/refolding UV/VIS absorbance assay was obtained in the absence of microwave power. The results of this assay were used as a baseline against which all other assays were compared. Unfolding/refolding assays were performed at five power levels evenly spaced from 35 to 5 dBm (from 18 W to 1.78 mW). All experiments were performed at a concentration of 0.36 mg/mL (26.2 M) in a pH 3.5 buffer. Results of the power series are summarized in Table IV. Midpoint temperature determined from UV/VIS absorbance alone was 50.66 C 0.50 C. For unknown reasons, this is approximately 5 C lower than that previously determined. However, midpoint temperatures determined from network-analyzer data were at the expected values. Stability increase in the presence of microwave power as ascertained from UV/VIS absorbance ranged from 0.11 C to 1.11 C with an average increase of 0.63 C. No clear dependence of midpoint temperature on test port power is observed for either UV/VIS or return response data. G. Ligand Binding Experiments Due to the use a long coaxial cable and lack of impedance matching, the return-loss pattern of the scaled-down slot antenna employed for fluorescence polarization experiments displayed a number of peaks in the 10–20-GHz range. A typical return-loss pattern is shown in Fig. 8. In contrast to the return loss of the larger antenna described above (see Fig. 1), peak position varied little. Instead, peak magnitude was sensitive to a number of factors, including temperature, protein concentration, and ligand concentration. Fig. 9 shows the variation of return-loss magnitude with estrogen receptor (ER- ) concentration for titration of 0.24-nM fluormone with ER- at 25 C. Return loss is high at low ERconcentration (around 0.1 nM), decreases in the range from 0.7 to 10 nM, then increases with increasing ER- concentration. The first region corresponds to primarily free fluormone and the third region corresponds to primarily bound fluormone, with the inner region representing a mixture of bound and free ligand.

TAYLOR AND VAN DER WEIDE: ULTRA-SENSITIVE DETECTION OF PROTEIN THERMAL UNFOLDING AND REFOLDING

1583

tive species were used. As with the fluormone titration experiments, peak position shifted little with concentration, although peak magnitudes varied. Several peaks fitted well to (5) and (6) varying from 1.12 to 2.68 nM and an average with of 1.60 0.62 nM (data not shown). The dissociation constant for this system measured by gel filtration analysis was 0.09 nM determined from flu[25]. Values of dissociation constant orescence polarization and return-loss measurements are summarized in Table V.

IV. DISCUSSION

Fig. 8. Representative return loss for scaled slot antenna employed for fluorescence polarization experiments. Sample is 38.8-nM estrogen receptor with 0.24 nM in buffer. The grey circle indicates the peak considered in more detail in Fig. 9.

Fig. 9. Variation of return loss at 11.75 GHz with concentration of estrogen receptor. Concentration of fluormone was fixed at 0.24 nM. Return-loss data (closed symbols) was fitted to (5) and (6); fits are indicated by solid lines. Fraction ligand bound (open symbols and dashed line) from Beacon data are included in the lower graph. Dissociation constant K was determined to be 6.17 4.44 nM for return-loss data and 5.71 nM for Beacon data.

6

6

The data fits well to (5) and (6) with dissociation constant equal to 6.17 4.44 nM. Several other peaks were identified that yielded good fits to a single-site binding model; values of varied from 1.94 to 8.78 nM, with average 5.40 2.59 nM. Fluorescence data taken simultaneously yielded a value of 5.71 0.47 nM. Both of these values compare well to the published dissociation constant (5.53 nM) for a similar fluorescent estradiol [34]. We also performed label-free measurements of the binding of estradiol and ER- using return loss alone. FP data were not recorded for these experiments since no fluorescently ac-

Our coaxial-fed slot antenna has been used successfully for detection of protein conformational changes in the antenna’s near zone. We have demonstrated that both larger (unfolding/refolding) and smaller (ligand binding) conformational changes can be monitored. In all cases, similar thermodynamic parameters (for either unfolding/refolding or binding) were obtained with both the slot antenna and conventional optical means. The combined cuvette/antenna system allows simultaneous measurements of UV/VIS absorbance and return reflection loss, enabling correlation of antenna measurements with an established method. Fitting the return reflection loss to a series of Lorentzian peaks allows examination of the temperature-dependent behavior of individual peaks. Selected peaks, usually 1–2 peaks per return-loss pattern, could be fitted to a two-state unfolding model, and frequency shifts of these peaks appear to correlate with changes in protein conformation. Midpoint temperatures determined from these frequency shifts are compavalues derived from UV/VIS absorbance, and rable to the exhibit similar pH-dependent behavior. Thermodynamic parameters were obtained at concentrations as low as 19 pM, far lower than the minimum detectable concentration from UV/VIS abmeasured by UV/VIS sorbance. The midpoint temperature absorbance in the absence of microwave power did not decrease when simultaneous UV/VIS and dielectric measurements were performed, implying that the protein is not destabilized by microwave power. An advantage of this slot antenna approach over other approaches is that proteins need not be labeled or otherwise altered. The titration experiments discussed above in Section III-B demonstrate that any dielectric material, including proteins and nucleic acids, in aqueous solution can be examined by this technique. Position of peaks in the return-loss spectra were shown to vary with calculated permittivity for titration of acetone and methanol into water; data at permittivities below 60 fitted well to the expected inverse square root of the permittivity (see Fig. 2). Peak shifts were large, up to 200 MHz, for these experiments because of the large changes in solution permittivity. Peaks were shown to vary linearly with concentration of either RNase or RNA (see Fig. 3). Peak shifts for the biological macromolecule titrations were much smaller than the organic solvents, probably due to the low concentration of macromolecule (200 M or less) and the higher water concentration (55.6 M). These results suggest that the antenna resonance responds to nonspecific changes in permittivity, meaning that a resonant slot antenna can be used to observe any activity involving a permittivity change.

1584

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Unfolding/refolding of RNase A was measured at concentrations as low as 19 pM. Of the techniques currently used for similar biological studies, only fluorescence spectroscopy can be used to obtain measurements at protein concentrations in the low picomolar to nanomolar range, but this technique requires that most biomolecules be labeled with a fluorescent entity (see Table III). Other common methods such as UV/VIS or circular dichroism spectroscopy require much higher protein concentrations. Several unresolved issues need to be addressed. First, although midpoint temperature may be ascertained to reasonable accuracy from frequency shifts of network-analyzer peaks, dehas been far termination of unfolding/refolding enthalpy from network-analyzer less satisfactory. The values of peak fitting varied widely among different data sets. In some cases, as in the RNase A unfolding at pH 4.5 above, values of that are lower than those from UV/VIS absorbance, but values still reasonable, were obtained. In most cases, from network-analyzer data are small and extremely high in error. These low values of enthalpy are unlikely to be due to protein destabilization since unfolding/refolding enthalpy values from UV/VIS absorbance from this work were very similar to UV/VIS and calorimetric values from the literature (data not shown). There are two explanations for these small and error-prone enthalpy values. First, accurate enthalpy determination is rendered impossible by error in the native and unfolded baselines. Generally, return response peak frequency shifts correspond to a linear model at temperatures at which the protein is either folded or unfolded, but baseline fitting to network-analyzer data was usually not as accurate as was fitting to UV/VIS absorbance. Uncertainty in baseline determination does not entirely explain from the baseline-free fitthis problem since values of ting method were similar in magnitude and error to those obtained from the baseline-dependent method. These errors may be due to a low- antenna; improvements in the antenna system and . may facilitate more accurate determination of On the other hand, these values of unfolding or refolding enthalpy may indeed be accurate. Data from UV/VIS absorbance and VNA return loss reflect very different phenomena. UV/VIS absorbance assesses changes in the absorbance of aromatic residues (in this case, tyrosine only, since wild-type or naturally occurring RNase A possesses no tryptophan) and disulfide bonds [35]. Aromatic residues such as tyrosine tend to be located in the hydrophobic core at the protein interior. Of the six tyrosines in RNase A, two (Tyr 76 and 115) are completely solvent exposed; the remaining four are partially or completely buried in the hydrophobic exterior [36]. Dielectric spectroscopy in the observed frequency range focuses on changes in water permittivity, particularly that of bound water [16]. Since RNase A contains no internal water molecules, the waters affected by the microwave radiation must be either on the protein surface or in bulk solution. It is possible that the waters on the protein surface respond to protein unfolding at a lower temperature than do the hydrophobic molecules of the core. Perhaps the entropy of the hydrophilic surface residues increases long before the unfolding of the core, and this is the phenomenon observed by may the network-analyzer data. The lower values of

also be explained if these changes in the entropy of surface residues extend over a wider temperature range than does unfolding of the hydrophobic core. Additional experiments will be necessary to resolve these enthalpy determination issues. The low levels of microwave power used in these experiments did not destabilize the protein. On the contrary, midpoint temperatures measured by UV/VIS absorbance were higher in the presence of microwave power. Our results support those of Bohr and Bohr [33], in which the folding rate of a protein was enhanced by microwave power at 2.45 GHz. If the radiation did not affect the protein’s unfolding rate, then the presence of microwave power would, therefore, increase protein stability. This increase was expected to disappear at low power levels, but we observed no power-dependent effects. There are several possible explanations for this negative result. First, the apparent increase of stability with microwave power may not be statistically significant since the magnitude of the increase is approximately the size of the experimental error The power output from the network analyzer may not be sufficiently low to eliminate the stability increase. These experiments were performed over a fairly small power range, only two orders of magnitude. Any change in stability increase over this power range may be smaller than the experimental error. Significantly lower power outputs may be needed to eliminate or reverse the stability increase. Ligand-binding experiments with estrogen receptor demonstrate that the slot antenna system’s usefulness is not limited to unfolding/refolding experiments. It is expected that the same phenomenon is observed by all slot antenna measurements: changes in water permittivity, especially changes due to reorganization of the water shells surrounding proteins or small molecules in solution. Although estrogen receptors, like other nuclear receptors, undergo considerable structural change upon ligand binding [37], this conformational change is not expected to be as large as that associated with protein unfolding/refolding. The smaller conformational change may be one reason for error in dissociation constant determination. Unlike experiments with larger-scale antennas used for unfolding/refolding experiments, peak positions did not vary, although peak magnitude appeared to reflect conformational changes. Another cause for error was the mechanical instability of the system. As with unfolding/refolding experiments, the coaxial cable feed was highly sensitive to perturbation. In addition, the position of the test tube within the sample holder had a large effect on return loss. Nonetheless, very similar dissociation constants were obtained from data taken by both fluorescence polarization and by return loss (5.71 nM versus 5.40 nM, respectively). We also demonstrated that the system could be used alone to detect binding events inaccessible to fluorescence polarization by measuring the binding of unlabeled estradiol to the same was somewhat larger receptor. Although our value of than that measured by other means (1.60 nM versus 0.09 nM [34]), it is very promising that our technique can be used to measure quantities previously accessible only through binding of radiolabeled ligands. Overall, the slot antenna system performed very well. Temperature-dependent changes in RNase A could be measured, and

TAYLOR AND VAN DER WEIDE: ULTRA-SENSITIVE DETECTION OF PROTEIN THERMAL UNFOLDING AND REFOLDING

thermodynamic parameters from these measurements were similar to those derived from UV/VIS absorbance. Binding of a fluorescent ligand to estrogen receptor was recorded using both fluorescence polarization and return-loss measurement; similar were found from both values of the dissociation constant sets of data. Measurement of the binding of unlabeled estradiol was monitored using only return-loss measurements; such measurements cannot be performed using fluorescence polarization. Future designs will include a more mechanically robust and higher antenna design. Also, other types of experiments such as enzyme kinetics, chemical denaturation, and ligand binding will be performed.

ACKNOWLEDGMENT The authors thank to A. Bettermann, Dr. D. McCaslin, and Dr. C. Paulson, all of the University of Wisconsin–Madison, L. Palmer, Epic Systems, Madison, WI, and Dr. J. Peck, Stanford Research Systems, Sunnyvale, CA.

REFERENCES [1] R. L. Abel, M. R. Haigis, C. Park, and R. T. Raines, “Fluorescence assay for the binding of ribonuclease A to the ribonuclease inhibitor protein,” Anal. Biochem., vol. 306, pp. 100–107, Jul. 2002. [2] I. Ermolina, H. Morgan, N. G. Green, J. J. Milner, and Y. Feldman, “Dielectric spectroscopy of tobacco mosaic virus,” Biochem. Biophys. Acta, vol. 1622, pp. 57–63, Apr. 2003. [3] A. Markelz, S. Whitmire, J. Hillebrecht, and R. Birge, “THz time domain spectroscopy of biomolecular conformational modes,” Phys. Med. Biol., vol. 47, pp. 3797–3805, Nov. 2002. [4] J. Hefti, A. Pan, and A. Kumar, “Sensitive detection method of dielectric dispersions in aqueous-based, surface-bound macromolecular structures using microwave spectroscopy,” Appl. Phys. Lett., vol. 75, pp. 1802–1804, Sep. 1999. [5] M. Suzuki, J. Shigematsu, and T. Kodama, “Hydration study of proteins in solution by microwave dielectric analysis,” J. Phys. Chem., vol. 100, pp. 7279–7282, Apr. 1996. [6] G. R. Facer, D. A. Notterman, and L. L. Sohn, “Dielectric spectroscopy for bioanalysis: From 40 Hz to 26.5 GHz in a microfabricated wave guide,” Appl. Phys. Lett., vol. 78, pp. 996–998, Feb. 2001. [7] S. E. Whitmire, D. Woldpert, A. G. Markelz, J. R. Hillebrecht, J. Galan, and R. R. Birge, “Protein flexibility and conformational state: A comparison of collective vibrational modes of wild-type and D96N bacteriorhodopsin,” Biophys. J., vol. 85, pp. 1269–1277, Aug. 2003. [8] Y. D. Feldman and V. D. Fedotov, “Dielectric relaxation, rotational diffusion and the heat denaturation transition in aqueous solutions of RNase A,” Chem. Phys. Lett., vol. 143, pp. 309–312, Jan. 1988. [9] A. Bonincontro, A. De Francesco, and G. Onori, “Temperature-induced conformational changes of native lysozyme in aqueous solution studied by dielectric spectroscopy,” Chem. Phys. Lett., vol. 301, pp. 189–192, Feb. 1999. [10] D. W. van der Weide and K. M. Taylor, “Microwave dielectric spectroscopy method and apparatus,” U.S. Patent 6 801 029, Oct. 11, 2002. [11] M. K. Choi, K. Taylor, A. Bettermann, and D. W. van der Weide, “Broadband 10–300 GHz stimulus-response sensing for chemical and biological entities,” Phys. Med. Biol., vol. 47, pp. 3777–3787, Nov. 2002. [12] K. M. Taylor and D. W. van der Weide, “Sensing folding of solution proteins with resonant antennas,” presented at the 9th Int. Terahertz Electronics Conf., vol. 9, Charlottesville, VA, 2001. [13] , “Microwave assay for detecting protein conformation in solution,” in Instrumentation for Air Pollution and Global Atmospheric Monitoring, R. L. Spellicy, Ed. Boston, MA: SPIE Press, 2002, vol. 4574, Proceedings of SPIE, pp. 137–143. [14] S. Bone and B. Zaba, Bioelectronics. Chichester, U.K.: Wiley, 1992. [15] E. H. Grant, R. J. Sheppard, and G. P. South, Dielectric Behavior of Biological Molecules in Solution. Oxford, U.K.: Oxford Univ. Press, 1978.

1585

[16] R. Pethig, “Protein–water interactions determined by dielectric methods,” Annu. Rev. Phys. Chem., vol. 43, pp. 177–205, 1992. [17] R. H. Henchman and J. A. McCammon, “Structural and dynamic properties of water around acetylcholinesterase,” Protein Sci., vol. 11, pp. 2080–2090, Nov. 2002. [18] V. P. Denisov and B. Halle, “Thermal denaturation of ribonuclease A characterized by water 17O and 2H magnetic relaxation dispersion,” Biochemistry, vol. 37, pp. 9595–9604, Jun. 1998. [19] H. G. Akhavan and D. Mirshekar-Syahkal, “Slot antennas for measurement of properties of dielectrics at microwave frequencies,” presented at the Nat. Antennas Propagation Conf., 1999. [20] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [21] T. A. Klink, K. J. Woycechowsky, K. M. Taylor, and R. T. Raines, “Contribution of disulfide bonds to the conformational stability and catalytic activity of ribonuclease A,” Eur. J. Biochem., vol. 267, pp. 566–572, Jan. 2000. [22] C. N. Pace, G. R. Grimsley, S. T. Thomas, and G. I. Makhatadze, “Heat capacity change for ribonuclease A folding,” Protein Sci., vol. 8, pp. 1500–1504, Jul. 1999. [23] G. J. Parker, T. L. Law, F. J. Lenoch, and R. E. Bolger, “Development of high throughput screening assays using fluorescence polarization: Nuclear receptor–ligand–binding and kinase/phosphatase assays,” J. Biomol. Screening, vol. 5, pp. 77–88, 2000. [24] K. Pettersson and J.-A. Gustafsson, “Role of estrogen receptor beta in estrogen action,” Annu. Rev. Physiol., vol. 63, pp. 165–192, 2001. [25] G. G. J. M. Kuiper, J. G. Lemmen, B. Carlsson, J. C. Corton, S. H. Safe, P. T. van der Saag, B. van der Burg, and J.-A. Gustafsson, “Interaction of estrogenic chemicals and phytoestrogens with estrogen receptor b,” Endocrinology, vol. 139, pp. 4252–4263, Oct. 1998. [26] O. H. Lowry, N. J. Rosebrough, A. L. Farr, and R. J. Randall, “Protein measurement with the Folin phenol reagent,” J. Biol. Chem., vol. 193, pp. 265–275, Nov. 1951. [27] L. A. Marky and K. J. Breslauer, “Calculating thermodynamic data for transitions of any molecularity from equilibrium melting curves,” Biopolymers, vol. 26, pp. 1601–1620, Sep. 1987. [28] D. M. John and K. M. Weeks, “van’t Hoff enthalpies without baselines,” Protein Sci., vol. 9, pp. 1416–1419, Jul. 2000. [29] Fluorescence Polarization Technical Resource Guide, 3rd ed., PanVera Corporation, Madison, WI, 2002. [30] M. Kahrizi, T. K. Sarkar, and Z. A. Maricevic, “Analysis of a wide radiating slot in the ground plane of a microstrip line,” IEEE Trans. Microw. Theory Tech., vol. 41, pp. 29–37, Jan. 1993. [31] R. Pethig, Dielectric and Electronic Properties of Biological Materials. Chichester, U.K.: Wiley, 1979. [32] M. A. Olson, “Calculations of free-energy contributions to protein-RNA complex stabilization,” Biophys J., vol. 81, pp. 1841–1853, Oct. 2001. [33] H. Bohr and J. Bohr, “Microwave-enhanced folding and denaturation of globular proteins,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 61, pp. 4310–4314, Apr. 2000. [34] S. Suzuki, K.-I. Ohno, T. Santa, and K. Imai, “Study on interactions of endocrine disrupters with estrogen receptor-b using fluorescence polarization,” Anal. Sci., vol. 19, pp. 1103–1108, Aug. 2003. [35] S. C. Gill and P. H. von Hippel, “Calculation of protein extinction coefficients from amino acid sequence data,” Anal. Biochem, vol. 182, pp. 319–326, Nov. 1989. [36] M. H. Dung and J. A. Bell, “Structure of crystal form IX of bovine pancreatic ribonuclease A,” Acta Cryst., vol. D53, pp. 419–425, Jul. 1997. [37] D. Moras and H. Gronemeyer, “The nuclear receptor ligand-binding domain: Structure and function,” Curr. Op. Cell Biol., vol. 10, pp. 384–391, Jun. 1998.

Kimberly M. Taylor (S’04) received the B.S. degree in physics from the University of Virginia, Charlottesville, in 1994, the M.A. degree in biophysics from The Johns Hopkins University, Baltimore, MD, in 1997, and is currently working toward the Ph.D. degree in biophysics at the University of Wisconsin–Madison. She was a National Institutes of Health (NIH) Molecular Biophysics Trainee. Her research interests include protein function and thermodynamics and the applications of RF, microwave, and terahertz techniques to problems of biomedical and biochemical interest. Ms. Taylor was the recipient of a Howard Hughes Predoctoral Fellowship.

1586

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Daniel W. van der Weide (S’86–M’86) received the B.S.E.E. degree from the University of Iowa, Iowa City, in 1987, and the Master’s and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1989 and 1993, respectively. He held summer positions with the Lawrence-Livermore National Laboratory and Hewlett-Packard, and full-time positions with Motorola as an Engineer and the Watkins-Johnson Company as a Member of the Technical Staff. From 1993 to 1995, he was a Post-Doctoral Researcher with the Max-Planck-Institut für Festkörperforschung (Solid State Research), Stuttgart, Germany, after which he joined the Department of Electrical and Computer Engineering, University of Delaware, as an Assistant and Associate Professor and Director of the Center for Nanomachined Surfaces. In 1999, he joined the Department of Electrical and Computer Engineering, University of Wisconsin–Madison, as an Associate Professor. From 2002 to 2004, he was a University of Wisconsin Vilas Associate. He was the Principal Investigator on a 2003 Air Force Office of Scientific Research (AFOSR) Multiuniversity Research Initiative (MURI) overseen by Lt. Col. G. Pomrenke entitled, “Nanoprobe Tools for Molecular Spectroscopy and Control.” His current research involves ultrafast electronics, one-dimensional (1-D) electron systems, and the application of high-frequency techniques in biotechnology. Dr. van der Weide was the recipient of the National Science Foundation (NSF) CAREER and PECASE Awards in 1997 and the Office of Naval Research (ONR) Young Investigator Program Award in 1998.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

1587

Low-Reflection Subgridding Lukasz Kulas, Student Member, IEEE, and Michal Mrozowski, Senior Member, IEEE

Abstract—The paper presents a novel three-dimensional subgridding scheme applicable to the finite-difference technique in the time and frequency domains. Transfer of fields between a main grid and a refined volume is performed using a simple linear interpolation. Very low-reflection levels from the main to local grid interface are obtained by co-location of fields used in the interpolation process. The technique allows material traverse without any special boundary treatment. The accuracy of the scheme is verified in numerical tests showing excellent performance even for high refinement factors. Index Terms—Finite difference frequency domain (FDFD), finite difference time domain (FDTD), subgridding technique.

I. INTRODUCTION

T

HE ACCURACY of the finite-difference time-domain (FDTD) method can be increased without significantly increasing numerical cost by using subgridding in a restricted volume of the computational space where higher spatial resolution is required [1]–[6]. Subgridding consists in refining the mesh locally. The equations inside the volume are updated more often than in the rest of the grid and both grids have to be coupled in a way that ensures continuous information flow. While subgridding is usually associated with the FDTD method, it can also be applied in the frequency domain [7], in which case, one is only concerned with the spatial relationship between fields at the grid interface. Recent developments have shown that subgridding becomes even more computationally attractive if it is combined with model-order reduction techniques [8]–[12]. In this case, one may use time steps in the FDTD scheme exceeding the fine-grid Courant time-step limit [8], [9], [12], [13], while finite-difference frequency-domain (FDFD) formulations show faster convergence of iterative solvers [10]. The quality of subgridding can be measured by observing the level of reflections from the dense to coarse mesh interface. For low-reflection levels, the accuracy of the subgridding is the highest. Subgridding schemes presented in the literature differ in reflection levels, subgridded volume’s refinement factors, and possibility of traversing meshes’ interface by different materials and metallic walls. To give a rough estimate of reflection levels in published subgridding algorithms, we have compared calculations conducted for a similar setup (subgridding volume fills all or almost all space in the transverse waveguide direction) . and reflections at the frequency corresponding to The most accurate three-dimensional (3-D) subgridding scheme developed thus far was presented in [1], where the authors used and sophisticated the mesh refinement factor of two

Manuscript received April 26, 2004; revised September 9, 2004. This work was supported by the Foundation for Polish Science under the Senior Scholar Grants Scheme. The authors are with the Department of Electronics, Telecommunications, and Informatics, The Gdan´sk University of Technology, 80-952 Gdan´sk, Poland. Digital Object Identifier 10.1109/TMTT.2005.847048

spline interpolation. Reflection levels reported were around 65 and 50 dB in a broad frequency range for a subgridded region placed centrally inside a parallel-plate waveguide (PPW) and rectangular waveguide (RW), respectively. When faster linear interpolation was used, the results were above 40 dB in both cases. In [2], linear interpolation was used, which brought the (RW, one reflection level to approximately 34 dB for frequency point), while in [3], the linear interpolation method resulted in PPW reflections around 40 dB (a broad frequency range). In a more practical 3-D scheme that allows material traverse and an odd integer refinement factor, the reflection , in an RW are more than 44 dB in levels, tested for the broad frequency range [4]. A similar traversing scheme was also proposed in [5], but reflection levels calculated for increasing from 2 to 11 at the single frequency point in an RW were considerably higher ( 20 dB). The interpolation scheme is only one factor that affects the reflection level. The other is the choice of the field samples within the refined volume that are used to provide boundary conditions for the main grid (MG). Various schemes presented in the literature differ in this choice and the way the fine grid fields are processed. As a result, the reflection levels for the techniques employing linear interpolation are different. In this paper, we present a 3-D subgridding scheme, which is based on linear interpolation, allowsforthematerialtraverseand,yet,showsverylow-reflection levels. Low reflections from the interface between grids are obtained by coupling the base mesh and subgridded volumes using collocated fields [6], [8]. Fields collocation is a technique where fine grid fields taking part in the grid computing process are selected in such a way that the interpolation and averaging do not cause dislocation of the fields evaluated at the interface between meshes. In this paper, we demonstrate how to use this approach in 3-D in both the time and frequency domains. The new algorithm is validated in numerical tests, which show that a low-reflection level holds for high-refinement factors. Additionally, since the dislocation errors are absent, subgridded volume may be traversed by metal walls without any special treatment, as demonstrated on one of the examples involving a resonator with an iris. II. LOW-REFLECTION SUBGRIDDING SCHEME USING LINEAR INTERPOLATION In finite differences, a computational domain, meshed with the uniform space discretization step , is represented by a set of points forming a uniform grid. Maxwell’s equations discretized on Yee’s mesh result in a set of linear equations, which can be written in the Laplace -domain in the following operator form:

0018-9480/$20.00 © 2005 IEEE

(1)

1588

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Fig. 1. Local mesh refinement for the refinement factor of 3. Solid lines are electric-field grid lines, while the dual mesh (magnetic-field grid lines) is shifted by = in each direction (dashed lines).

12

Fig. 3. MG magnetic boundary fields evaluated from corresponding LG fields. Note that to achieve fields collocation, the magnetic-field boundary plane is shifted by = with respect to the MG–LG interface.

12

Fig. 2. Position of MG electric-field boundary providing boundary conditions for the LG. LG electric fields in the MG–LG interface plane are calculated from corresponding MG vectors. Shaded area is an electric-field boundary plane, whose position coincides with the MG–LG interface.

where , and , are discrete curl operators and diagonal material matrices, respectively. Vectors and contain -, -, and -field components collocated with electric and magnetic field grids, in each direction. which are shifted by In order to subgrid a mesh locally, a local subvolume in the discrete domain described by the MG has to be chosen. This subvolume is discretized more finely. If the mesh refinement factor is an odd number, some local grid (LG) lines are collocated with the MG, as presented in Fig. 1, where all MG electric- and magnetic-field lines have its collocated counterpart in the LG volume. To provide boundary conditions for the refined volume, LG electric fields, which lie in the plane constituting the MG–LG interface, are interpolated from MG electric fields (see Fig. 2). Similarly,asinsidetheLG therearenoMGfieldcomponents,magnetic , fields, which provide boundary conditions for the MG (e.g., , , and in Fig. 3) have to be evaluated from magnetic and fields inside the LG. Normal field components (e.g., in Fig. 2 and 3) do not require any treatment. To obtain low-reflection transition from one grid to the other, one has to carefully choose the magnetic fields within the LG that serve as boundary conditions for magnetic field in the MG. Fig. 3 shows two planes within the LG: one, nearest to the

Fig. 4. Coupling between MG and LG electrical fields (see text for explanation).

MG–LG interface (nearest plane), and the other, shifted by with respect to the MG–LG interface and from the plane in MG containing fields , , , and (collocated plane). The latter plane should be chosen to evaluate the missing magnetic fields. This way, dislocation errors will not appear, resulting in low reflections from the grid interfaces [6]. A. Linear Interpolation Scheme The subgridding scheme is based upon simple linear interpolation and integral interpretation of fields at the MG–LG interface. LG electric fields are interpolated from MG fields in transverse and averaged [6] in longitudinal directions. This process - and -field component and the is illustrated in Fig. 4 for . LG fields are evaluated from refinement factor and in the following way [see Fig. 4(a)]:

(2)

KULAS AND MROZOWSKI: LOW-REFLECTION SUBGRIDDING

1589

[7], [13]. For both grids, Maxwell’s equations take the following form:

(4)

(5) where and are the LG electric fields interpolated from MG and MG magnetic fields interpolated from LG, respectively. and may be written in a convenient Boundary fields matrix form using MG electric and LG magnetic fields Fig. 5. Coupling between MG and LG magnetic fields (see text for explanation).

It can be noted that the sum of LG fields , , and is equal to the MG field . Factors 1/9, 2/9, and 3/9 in (2) may be regarded as coefficients, which are used to couple LG boundary electric fields to one or two MG boundary fields. The interpolaand other interface planes is analogous. tion procedure for As for magnetic fields, integral interpretation [6] of magnetic flux leads to evaluation of MG magnetic fields from surrounding LG magnetic fields included in the collocated boundary plane. For the boundary MG magnetic field presented in Fig. 5, the averaging equation is

(3) To deal with corner boundary MG magnetic fields, this scheme has to be applied to each of refined volume’s faces separately. This means that magnetic fields included in the collocated boundary plane (see Fig. 5) are used to calculate only those of -plane. This alMG electric fields that are tangential to the lows one to use the same interpolation for all magnetic fields. The extension of the interpolation algorithm to an arbitrary refinement factor is straightforward. From each, two LG boundary fields boundary MG electric fields are interpolated— in the longitudinal direction [ in Fig. 4(a)] in the transverse one [ in Fig. 4(a)]. Corresponding and . Similarly, coupling factors are equal LG fields, MG magnetic fields are calculated from a sum of in each direction [see Fig. 5 and (3)] with factor 3 in (3) replaced by . B. Grid Coupling in an Operator Form For the FDTD formulation, the subgridding scheme described above can be presented in a usual update formula for individual field components. However, for generality, which allows the scheme to also be used in the frequency domain and in reduced-order models [12], it is more convenient to cast the equations for coupling LG and MG grids in the matrix form. This form is also convenient to see if the coupling scheme satisfies stability conditions [7], [13], [14]. To this end, one has to define matrix operators at each grid and field coupling matrices

(6) where and are and matrices, which choose MG electric and LG magnetic fields tangential to the and are the number of MG elecMG–LG boundary. tric and LG magnetic fields, respectively, while and denote the number of fields that are used in interpolation at the MG–LG interface. The selection of proper fields is achieved and corresponding to by inserting ones at columns of the chosen fields (one element in each row). Similar operations and of size and . are performed by These matrices correspond to Maxwell’s grid operators applied to boundary fields and perform coupling of interpolated fields to the Maxwell’s equations by placing them in the correct row with the correct sign in (4) and (5). This results in 1 and 1 at and . The remaining and certain positions inside of are and interpolation matrices, which contain by (2) and (3). coupling factors given for The coupling matrices given by (6) provide boundary conditions for both grids and can be substituted into (4) and (5). This results in a compact form of the global operator describing field behavior in the entire domain as follows: (7) (8) The above relations can be used for both time- and frequencydomain analysis. For the former case, the substitution in (7) and (8) is made, and upon introducing the time discretization, one obtains a leapfrog iterative scheme, which may be implemented in a form of a classical FDTD method. The frequency-domain formulation is obtained after substitution and imposing appropriate excitation and termination conditions. III. LOW-REFLECTION SUBGRIDDING SCHEME IN THE TIME DOMAIN Since two grids are used, the iterative scheme may be defined in two ways. The first approach is to use the same time step in both grids [7]. This implies that the time step in the entire computational space has to be reduced by a factor of . In other ( and are the words, Courant number and maximal wave velocity, respectively). It is,

1590

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

however, more computationally efficient to use different time steps for the MG and LG. LG equations are iterated times calculated in the MG field update is for one MG cycle, and kept constant for the entire MG cycle. After the last iteration of are updated and the MG LG equations, MG magnetic fields iteration is performed [2]. , the staWhen both grids work with the same time step bility analysis can be performed using the theory presented in [14] and [13]. From (7) and (8), one may easily obtain the canonical form of a second-order system [13], which works stable if (9) Unfortunately, for the proposed low-reflection subgridding scheme, (9) does not hold, which means that long-time stability cannot be guaranteed. To investigate if this is a severe limitation, we have conducted several numerical tests to verify if long-time instabilities appear and how they are related to various conditions. The results are presented in Section IV. In the case of different time steps used in MG and LG volumes, the stability analysis is much more complicated and we are unaware of any theory concerning that type of subgridding algorithms. Usually, for stable operation, the algorithm requires to be properly adjusted from its maximum value ( in 3-D for the uniform grid) to lower values. As the refinement factor increases, it is more difficult to couple MG and LG meshes1 [15], so even when a subgridding algorithm satisfies condition (9) (as a scheme proposed in [7]), its practical 3-D implementations based on separate time updates in the LG and MG may suffer from late-time instabilities. It has been observed that late time instabilities often appear in subgridding schemes [1], [4], [15], but in most cases, the period of stable operation is long enough for the subgridding to be used in practice. This is also the case for the scheme proposed in this paper, as shown in Section IV. IV. NUMERICAL RESULTS To verify the accuracy and stability of the proposed subgridding scheme, three tests were conducted. For two tests, we have chosen the same structures and meshing as those investigated in [1]. Following [1], reflections from the subgridded region were tested. To this end, a 8 4 10 mm centrally placed subgridded volume was introduced in a 15 10 mm section of a PPW. The space discretization step of the MG was 1 mm in the transverse mm in the wave propagation direction, which plane and at 15 GHz. To avoid reflections corresponds to from imperfect absorbing boundary conditions, two runs of the FDTD analysis were performed (with and without the subgrided volume). The results presented in Fig. 6 indicate that our ap, proach gives 60-dB reflections up to 15 GHz which is close to the results obtained using spline interpolation in [1]. Reflections are even much better at lower and frequencies and, moreover, do not practically increase with the refinement factor. 1In [5], authors propose a subgridding scheme working for maximal Courant number C C with increasing k, but the scheme shows high 20-dB reflections from subgridded volume in an empty RW.

=

0

Fig. 6. Reflections from a subgridded volume placed in a PPW for different mesh refinement factors k .

Fig. 7. Reflections from a subgridded volume placed in an RW for different mesh refinement factors k .

Next, we repeated the waveguide test reported in [1] for the same computing and meshing conditions. The subgridded 16 3 40 mm volume was placed in a 20 7 mm RW with the same discretization as in the previous test.2 In that case, the reflections are lower than 47 dB (see Fig. 7) for the frequency range up to 19 GHz (approximately ) and . Results do not deteriorate significantly for an increasing refinement factor, but are worse than in the PPW case. In a RW, there is a standing wave across the waveguide. The angle of incidence on the MG–LG interface varies from 90 at cutoff to zero at high frequencies. For an oblique wave, incidence subgridding schemes show higher reflection levels [16]. As the incidence angle grows for lower frequencies, the reflections in an RW do not decrease for lower frequencies as the do in the PPW case. Reflection zeros in Fig. 7 are related to the subgridded volume’s internal resonances. 2Although in [1] space discretization of the RW is not given, the authors have provided the necessary data.

KULAS AND MROZOWSKI: LOW-REFLECTION SUBGRIDDING

TABLE I NUMBER OF ITERATIONS, AFTER WHICH LATE TIME INSTABILITY ) VERSUS THE REFINEMENT FACTOR k OBSERVABLE (n

TABLE II COURANT NUMBER REDUCTION WITH NUMBER OF ITERATIONS n TESTED PPW AND RW

1591

IS

FOR

2

2

Fig. 8. 10.16 22.86 36.83 mm rectangular resonator used in numerical tests and subgridding volumes’ placement (k = 3). TABLE III COMPARISON OF RELATIVE ERROR AND SIMULATION TIME IN STANDARD AND SUBGRIDDED FDTD FORMULATIONS (FOR SUBGRIDDING COURANT NUMBER C IS ADDITIONALLY INCLUDED) AND SUBGRIDDED FDFD

In both tests, instability was observed within a time window required to obtain the reflection-coefficient curves shown in Figs. 6 and 7 (1600 and 2000 coarse grid iterations for the PPW and RW cases, respectively). To investigate when the late time instability occurs, we extended the simulation time until the onset of instability. The results are collected in Tables I and II. Table I shows the data for the scheme employing the same shortened time step in the whole computational space. It is seen that the problem is not severe as the numbers are high. The results for the scheme operating with different time steps are given in Table II. The data is given for various refinement factors and the time step (the Courant number) reduction factor . As discussed in Section III, the time-step reduction factor plays an important role here. For higher refinement factors, the time step has to be approximately halved for the late time instability not to interfere with the simulations. The third test verifies the practical usefulness of the presented scheme in dealing with structures, where the subgridded volume is traversed by metal. In [4], it was shown that tests with metal walls are the most critical, as metal introduces high reflections and is a source of serious phase errors. For this reason, in our test, we have used a structure containing a resonant metal iris presented in Fig. 8. The resonator was discretized using mm and four volumes surrounding the iris edge were covered with a fine mesh. The size of the subgridded volumes was for the vertical edges and for horizontal ones. This arrangement is very challenging, as metal walls traverse MG–LG interfaces many times, thus, only highly accurate subgridding is able to provide acceptable results. In this setup, using four subgridded volumes along the edges instead of one is recommended especially for large windows. In that case, only the edges are covered with a high-density mesh and not the volume between them. As a result, fewer variables are processed at each iteration. A resonator structure is also a very good stability test of a time scheme. The structure was analyzed with three different methods, which are: 1) the standard FDTD; 2) the FDTD with subgridding and different time steps in the MG and LG; and 3) the

FDFD with subgridding. For the standard FDTD scheme, the mesh was refined in the whole resonator. The results for different refinement factors are presented in Table III. The reference value of 9.4794 GHz was calculated using the mode-matching technique for a sufficiently high number of modes. As the refinement factor grows, the relative error decreases, which indicates that MG–LG interfaces of subgridded volumes are transparent for traversing metal walls and do not introduce phase errors to the boundary fields. The Courant number used in numerical tests had to be decreased with an increasing refinement factor, which is the price for coupling grids with different densities and using different time steps in the MG and LG. Even then, the analysis is conducted much faster than in the standard FDTD approach (see Table III). The last column show the results obtained using the eigenfrequency problem formulation (7) and (8) with respect to magnetic fields using an iterative Jacobi–Davidson eigensolver [17]. It is seen that error levels in subgridded FDTD and subgridded FDFD schemes are almost identical, which proves that the method developed in this paper can be used in both the time and frequency domains. V. CONCLUSIONS A new low-reflection 3-D subgridding scheme has been presented. The scheme can be used in the time or frequency domains. The technique is not suitable for nonuniform grids, as field collocation has to be achieved. It can, however, be combined with local schemes [18] or locally conformal meshes [19] for better resolution of the structure’s shape. The efficiency of the scheme can be further enhanced by combining it with the 3-D model-order reduction technique proposed in [12]. Boundary fields at the refined volume interface

1592

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

are evaluated using simple linear interpolation. High accuracy was achieved by choosing collocated fields for the interpolation and averaging process. The algorithm gives the accuracy comparable only to those obtained in [1] for the most sophisticated interpolation scheme and the refinement factor of two. Late time instabilities appear in the scheme, but their presence did not influence the results. ACKNOWLEDGMENT The authors would like to thank Prof. M. Okoniewski, University of Calgary, Calgary, AB, Canada, for providing figures used as a reference in numerical tests. REFERENCES [1] M. Okoniewski, E. Okoniewska, and M. A. Stuchly, “Three-dimensional subgridding algorithm for FDTD,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 422–428, Mar. 1997. [2] P. Thoma and T. Weiland, “A consistent subgridding scheme for the finite difference time domain method,” Int. J. Numer. Modeling, vol. 9, pp. 359–374, Sep. 1996. [3] K. M. Krishnaiah and C. J. Railton, “A stable subgridding algorithm and its application to eigenvalue problems,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 5, pp. 620–628, May 1999. [4] M. W. Chevalier, R. J. Luebbers, and V. P. Cable, “FDTD local grid with material traverse,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 411–421, Mar. 1997. [5] M. J. White, Z. Yun, and M. F. Iskander, “A new 3-D FDTD multigrid technique with dielectric traverse capabilities,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 422–430, Mar. 2001. [6] L. Kulas and M. Mrozowski, “A simple high-accuracy subgridding scheme,” in 33rd Eur. Microwave Conf., Munich, Germany, Oct. 2003, pp. 347–350. [7] O. Podebrad, M. Clemens, and T. Weiland, “New flexible subgridding scheme for the finite integration technique,” IEEE Trans. Magn., vol. 39, no. 5, pp. 1662–1665, May 2003. [8] B. Denecker, F. Olyslager, L. Knockaert, and D. De Zutter, “Generation of FDTD subcell equations by means of reduced order modeling,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1806–1817, Aug. 2003. [9] L. Kulas and M. Mrozowski, “Reduced order models of refined Yee’s cells,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 4, pp. 164–166, Apr. 2003. [10] , “Macromodels in the frequency domain analysis of microwave resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 3, pp. 94–96, Mar. 2004. , “Multilevel model order reduction,” IEEE Microw. Wireless [11] Compon. Lett., vol. 14, no. 4, pp. 165–167, Apr. 2004. [12] , “A fast high-resolution 3-D finite-difference time-domain scheme with macromodels,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2330–2335, Sep. 2004. [13] , “Stability of the FDTD scheme containing macromodels,” IEEE Microw. Wireless Compon. Lett., to be published.

[14] M. Mrozowski, “Stability condition for the explicit algorithms of the time domain analysis of Maxwell’s equations,” IEEE Microw. Guided Wave Lett., vol. 4, no. 8, pp. 279–281, Aug. 1994. [15] M. J. White, M. F. Iskander, and Z. Huang, “Development of a multigrid FDTD code for three-dimensional applications,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 10, pp. 1512–1517, Oct. 1997. [16] M. Celuch-Marcysiak and J. Rudnicki, “A study of numerical reflections caused by FDTD mesh refinements in 1-D and 2-D,” in 15th Int. Microwaves, Radar and Wireless Communications Conf., vol. 2, May 2004, pp. 626–629. [17] Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Eds., SIAM, Philadelphia, PA, 2000. [18] P. Przybyszewski and M. Mrozowski, “A conductive wedge in Yee’s mesh,” IEEE Microw. Wireless Compon. Lett., vol. 8, no. 2, pp. 66–68, Feb. 1998. [19] W. K. Gwarek, “Analysis of an arbitrarily-shaped planar circuit—A time-domain approach,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 10, pp. 1067–1072, Oct. 1985.

Lukasz Kulas (S’02) was born in Ketrzyn, Poland, in 1977. He received the M.S.E.E. degree (with honors) from The Gdan´ sk University of Technology (GUT), Gdan´ sk, Poland, in 2001, and is currently working toward the Ph.D. degree at The GUT. He is currently a Research and Teaching Assistant with The GUT. His research interests include model-order reduction (MOR) methods and their use in the FDTD method, FDFD method, finite-element method (FEM), and other numerical methods in electromagnetics. Mr. Kulas was the recipient of the 2004 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Graduate Fellowship.

Michal Mrozowski (M’96–SM’02) received the M.S.E.E., Ph.D., and D.Sc. degrees (with honors) in microwave engineering from The Gdan´ sk University of Technology (GUT), Gdan´ sk, Poland, in 1983, 1990 and 1994, respectively. Since 2001, he has been a Full Professor with The GUT. His interests are computational electromagnetics and field theory. He has authored or coauthored over 50 reviewed journal papers and two monographs on computational electromagnetics and guided electromagnetic waves. Dr. Mrozowski is a member of the Electromagnetics Academy. He is chairman of the Polish joint Aerospace and Electronic Systems (AES)/Antennas and Propagation (AP)/Microwave Theory and Techniques (MTT) Chapter.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

1593

Table-Based Nonlinear HEMT Model Extracted From Time-Domain Large-Signal Measurements M. Carmen Currás-Francos

Abstract—This paper presents an empirical table-based nonlinear HEMT model fully extracted from time-domain large-signal measurements. A simple and direct extraction procedure, based on a vector nonlinear network analyzer measurement system with load–pull facilities, demonstrates by experimental results on microwave transistors how a very reduced number of measurements is enough to obtain the current and charge generators to fill a lookup model. Table-based model extraction, implementation, and validation are described in this paper. Index Terms—Field-effect MODFETs.

transistors

(FETs),

modeling, Fig. 1. Intrinsic quasi-static nonlinear HEMT model.

I. INTRODUCTION

E

MPIRICAL table-based models have been developed to reproduce the complex nonlinear behavior of semiconductors from measured data tables as an alternative to nonlinear analytical models. Lookup models presented in the literature [1]–[6] have a common nexus: the small-signal measurement data used to generate the nonlinear functions that fill the tables. This is an indirect extraction method based on the integration of small-signal quantities to predict the nonlinear properties of devices. Main drawbacks of this approach are the: 1) high number of required measurements; 2) bias-path-dependent integrals due to low-frequency dispersion issues; 3) uncertainties associated with the use of different systems for generation and validation; 4) extraction under hot thermal conditions; and 4) impossibility to characterize breakdown and maximum dissipation regions unless pulsed systems are available. Recent development of vector nonlinear network analyzer (VNNA) systems, able to obtain magnitude and phase information of the devices behavior under large-signal excitation [7]–[10], has opened new direct model extraction strategies, which should improve the above-mentioned problems. Different modeling techniques related to large-signal-based nonlinear model generation have been proposed. Most approaches have extracted only the main nonlinearities from large-signal measurements using small-signal data to complete nonlinear models. Bandler et al. [11] was one of the first authors in pointing out the use of harmonics for large-signal model extraction. His method uses the large-signal data to extend the optimization process involved in the extraction of the coefficients of a nonlinear analytical model. Werthof et al. [12] presented a method where VNNA measurements are

Manuscript received July 2, 2004; revised September 20, 2004. The author is with the Departamento de Tecnología Electrónica, University of Vigo, 36200 Vigo, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.847049

used to obtain some of the nonlinear relations, obtaining the remaining ones from small-signal measurements at different bias points. Demmler et al. [13] has used explicitly VNNA measurements to remove the delay between the input and output of the device, extracting the output current function for an analytical nonlinear model. The remainder of the nonlinear functions are obtained from small-signal data. Wei et al. [14] has also used the VNNA measurements to extract the nonlinear current generators of an HBT device considering a symmetrical contribution of the displacement currents. Schreurs et al. [15] has shown how it is possible to obtain all nonlinear functions from large-signal measurements by injecting signals in the input and in the output of the device. Schreurs et al. [16] has also shown how the parameters of nonlinear empirical or artificial neural-network models can be fully estimated by optimization using only vectorial large-signal measurements. The goal of this paper is to present a table-based nonlinear model fully extracted from time-domain large-signal measurements with a method similar to the one established in [15], but based on a load–pull characterization, which enables an easier and systematic extraction procedure and a valid nonlinear model generation tool. II. NONLINEAR MODEL EXTRACTION An equivalent circuit that can be separated out as an extrinsic and intrinsic part commonly represents microwave HEMTs. The extrinsic part is related to device layout and modeled with linear elements. The intrinsic part is often described in terms of state functions: nonlinear current and charge generators representing the conduction and displacement currents, respectively. Here, it is considered the quasi-static intrinsic device model, presented schematically in Fig. 1, consisting of a parallel combination and a voltage-conof a voltage-controlled current source at the input (gate) and output (drain) of trolled charge source at port can be the device. The current terminal behavior

0018-9480/$20.00 © 2005 IEEE

1594

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

expressed as a sum of nonlinear conduction and displacement terms

(1) By defining the nonlinear capacitances as

(2) Equation (1) can then be rewritten in the following way:

(3) The following approach is followed in this study: the output dependence on the input voltage is exvoltage ploited and applied to (3), resulting in (4) as follows: (4) A new capacitance element has been defined [17] as follows: (5)

Fig. 2. Typical dynamic voltage transfer characteristic measured with a nominal 50- measurement system. Fundamental frequency (f ) = 4 GHz.

Consequently, with only one large-signal measurement, it is possible to extract the nonlinear and functions for all , ) measured values by solving the set of the ( , ) value in the time-domain equations for each ( waveforms. Fig. 3(b) shows the extraction result of the output current nonlinear function using only one large-signal time-domain measurement that fulfills the required condition between and , as Fig. 3(a) illustrates. functions along the given ( , ) contour The just by can be directly calculated [20] from the extracted using (9) as follows:1

Therefore, (4) can be rewritten as follows: (9) (6) Due to the two-dimensional dependence of the functions, the solution of (6) is not straightforward. Nevertheless, the proper selection of the load presented to the device can overcome this problem and the equation can exactly be solved [17]–[22]. In order to exactly solve (6), it is necessary to fulfill the condition and for two time instants . To meet this requirement, a new transfer characteristic versus . has to be considered: the RF versus characteristic is a The looping in the function of the load presented to the device, among other factors related to the transistor itself. Using a 50- measurement system, this dynamic voltage transfer characteristic presents looping, as shown in Fig. 2. A proper selection of the output load presented to the device can remove this looping, i.e., versus becomes a single-valued function. Once the previous condition has been accomplished, (6) can be solved since, for each port , it is possible to write two equations with two unknowns ( and ), where and for as follows:

(7)

(8)

Therefore, the use of a single large-signal measurement is enough for the extraction of both current and charge functions along the ( , ) contour defined in the gate and drain waveforms. The full state-functions plane can be obtained with this procedure just by changing the bias point of the large-signal waveforms used for the extractions, and selecting the appropriate input power. III. THEORETICAL VERIFICATIONS The first analysis that has been carried out is the comparison of the extracted state functions following two approaches, i.e., the classical approach based on the integration of small-signal parameters at multibias points [1] and the new direct extraction approach based on the use of several large-signal waveforms. In order to avoid possible measurement errors, a theoretical verification has initially been made by using a well-known analytical large-signal model for an HEMT [23]. This analytical model has been used to generate the data necessary to follow both previous approaches, i.e., 51 different gate bias and 51 different drain bias to cover a wide area in the device plane, generating 2601 small-signal -parameter data points for the first approach, and large-signal waveforms at only 51 different bias points for the 1Integration boundaries: t location of maximum C dV

= location of minimum C dV =dt value.

=dt value, t

=

CURRÁS-FRANCOS: TABLE-BASED NONLINEAR HEMT MODEL

1595

Fig. 3. (a) RF voltage characteristic fulfilling the nonlooping condition. (b) RF nonlinear output current extracted using only one large-signal measurement. Fundamental frequency (f ) = 2:5 GHz. Load impedance 0 (f ) = 0:32 21 .

Fig. 4.

Comparison of: (a) extracted nonlinear current and (b) charge functions using large-signal waveforms (lines) and small-signal S -parameters (dots).

second approach. As can be observed, the number of required measurements is dramatically reduced for the second case. With the data obtained from both approaches, the nonlinear functions have been extracted. The comparison of both types of extractions is presented in Fig. 4. This figure clearly shows a very good agreement between both extraction methods with the advantage of the large-signal approach being that the number of required measurements is reduced from to . This theoretical verification demonstrates the new technique is mathematically valid.

IV. EXPERIMENTAL VERIFICATIONS A. Model Extraction The necessary nonlooping dynamic voltage transfer characteristic has to be analyzed in the intrinsic plane of the device. As a consequence, a previous deembedding of the extrinsic linear elements needs to be performed to transform the extrinsic current and voltage measured waveforms into the intrinsic ones. The linear elements are determined from small-signal measurements, following well-known [24] cold field-effect transistor

1596

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

(FET) techniques, measuring FETs with different gatewidths to obtain accurate parasitic capacitances, and taking into account channel resistance by using forward bias conditions. Once parasitics are known, a deembedding process is performed using large-signal measurements, as explained in Appendix I. The VNNA system used in this study [10], based on the Agilent Microwave Transition Analyzer, includes a calibration routine that permits the connection of any load to the device-undertest. This fact allows connecting the optimal load to the transistor in order to achieve the desired condition. For that purpose, a variable load consisting of a phase shifter and an attenuator, ended with a coaxial termination, is used to obtain the required load impedance that removes the looping in the intrinsic versus transfer. The nonlinear and functions are extracted using an automated VNNA measurement system where it is only necessary to choose the adequate fundamental frequency, input power, bias points, and complex output load. The fundamental frequency chosen for the extractions is GHz . In order to cover all the voltage space, the amplitude of the waveforms has to be controlled by the input power applied to the device. With this parameter, one can select how introduced are the large-signal waveforms in the breakdown, pinchoff, knee, and forward regions. The bias points place the large-signal measurements in the device plane. Class-A bias points have been selected in order to center the waveforms to get a symmetrical clipping. To obtain the full state-functions plane, a procedure has been implemented where the gate bias is kept constant and the drain bias is swept from an initial to an end value. This can be observed in Fig. 5, where the extractions have been performed by sweeping the drain bias voltage from 0 to 5.75 V with 0.25-V steps and V as constant gate bias. This results in 24 using large-signal measurements to extract the nonlinear model. Note that the density of the extracted state functions becomes dependent only on the steps of one bias voltage, the drain voltage bias, and not on both. The output load impedance controls the slope and looping of the dynamic voltage transfer characteristic. The VNNA system used does not have an automatic control of the load. Nevertheless, after experimental investigation, the complex load that reversus characmoves the looping in the transfer teristic was observed to be almost bias independent. Therefore, was used for the same load impedance the extraction of the state functions in the full device plane. It can be observed in Fig. 5 that the looping in that intrinsic plot is nearly removed in all bias points used. An additional large-signal measurement, which crosses all large-signal measurements in Fig. 5, is used in order to RF link all the extractions, avoiding the immediate dc link used in [13], which can shift the result. Following the procedure explained in Section II, the state functions have been obtained and a table-based model has been constructed and implemented into the Agilent commercial harmonic-balance software, where the measurements in the table can be spline interpolated by the simulator.

Fig. 5. Large-signal measurements fulfilling the nonlooping condition obtained using a complex load impedance 0 (f ) = 0:38 175 . Fundamental frequency (f ) = 8 GHz. Bias points: V = 0:25 V, V = 0 V to 5.75 V.

0

Fig. 6. Extracted intrinsic nonlinear output current at different input powers. = 7 V. Fundamental frequency = 8 GHz. Bias: V = 0:25 V, V

0

The extracted nonlinear functions correspond to a quasi-static formulation, which is a good first-order model approximation, but does not cover all device operation conditions. For this reason, a low-frequency dispersion modeling has been included to extend the bandwidth of the model from the used extraction frequency down to dc. For that purpose, the measured dc characteristics have also been incorporated to the table-based model in a similar way as that presented in [1]. The use of quasi-static charges also limits the prediction of model at higher frequencies beyond the one used for extractions. Nonquasi-static charges have also been introduced in the model by means of two constant delays to the gate and drain quasi-static charges, as explained in [6], in order to increase model bandwidth. B. Model Consistency Model consistency checks that data accomplish the necessary conditions for the existence and uniqueness of the constitutive relations. This implies the possibility of constructing unique current and charge relations for a given device. In order to confirm the validity of the technique and the uniqueness of the extracted RF surfaces, several extraction studies have been undertaken covering changes in the input power level, dc bias, frequency, and output load impedance. Fig. 6 presents a good overlapping, in common voltage regions, in the extracted intrinsic characteristic, showing there is no dependence on the input power in the extractions.

CURRÁS-FRANCOS: TABLE-BASED NONLINEAR HEMT MODEL

Fig. 7. Extracted intrinsic nonlinear output current at different bias points. Fundamental frequency = 8 GHz. Input Power = 16 dBm. Drain bias: V = 7 V.

1597

Fig. 9. Extracted intrinsic nonlinear output current with two different output load impedances. Fundamental frequency = 8 GHz.

Fig. 8. Extracted intrinsic nonlinear output current at different fundamental = 3 V. frequencies. Bias: V = 0:25 V, V

0

Another proof of the validity of this process is to check whether the extracted functions can also be obtained independently of the dc-bias point. Fig. 7 illustrates how changing the bias points gets the extracted characteristics in different voltage regions, but with good overlapping in common voltage regions. Besides the independence of the extractions on the input power levels and on the dc bias used, it has also been verified how the extracted intrinsic functions are independent of the fundamental frequency of the waveforms, as shown in Fig. 8. This is verified for the whole bandwidth in which the quasi-static model has better predictions. The last experiment shows in Fig. 9 how two different output load impedances that guarantee the nonlooping condition obtain the same extracted characteristic. These four analyses are a proof that unique RF functions have been found. C. Model Validation The model has been extracted and validated with a 0.25- m gate length and a 4 60 m gatewidth GaAs HEMT device from Marconi Caswell Technology, Towcester, U.K. Model extraction and validation have been performed using the same measurement system. Measurements under large- and small-signal conditions have been obtained from the VNNA system for model validation purposes. Fig. 10 compares measured and simulated -parameters in the 0.5–40-GHz range. The model extracted at 8-GHz fundamental frequency achieves a good bandwidth response because of the nonquasi-static charge correction.

Fig. 10. Measured (dots) and simulated (lines) small-signal S -parameters in the 0.5–40-GHz range at V = 0:25 V and V = 2 V.

0

Fig. 11 compares large-signal simulations with measurements showing the output fundamental and harmonic powers versus the input power for a fundamental frequency GHz , second GHz , third GHz , GHz , and fifth harmonic GHz . fourth Fig. 12 compares measured and simulated output current and voltage time-domain waveforms at a fundamental frequency of 4 GHz. By plotting the output current versus the output voltage, the dynamic load lines for three different input powers are also obtained as shown in Fig. 13. A complex load that provides a big loop in this output plane has been used for this last verification. The observed discrepancies between measurements and simulations are more related with external factors affecting the measurements than with the model itself, i.e., remaining noise in the measurement system and residual looping in the dynamic transfer voltage plots not considered when using a constant output load impedance in all the extraction process. V. CONCLUSION An empirical table-based nonlinear model extracted from time-domain large-signal measurements has been presented. A fast and easy model generation procedure has been described using an automated measurement system with load–pull facilities. The extraction process based on large-signal VNNA measurements has been shown and also compared with the

1598

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Fig. 13. Measured (discontinuous) and simulated (lines) dynamic load = 0 :2 V , V = 2:9 V, and lines for three input powers at V fundamental frequency = 4 GHz.

0

APPENDIX I PARASITICS DEEMBEDDING WITH LARGE-SIGNAL MEASUREMENTS

Fig. 11. Measured and simulated output power versus input power at V 0:12 V, V = 2:5 V, and fundamental frequency = 8 GHz.

0

=

In order to have access to the intrinsic values of voltages and currents in an HEMT, a transformation of the measured voltage and current waveforms from the outer to the inner reference planes needs to be performed. The procedure developed to deembed the parasitics network from the available extrinsic voltage and current waveforms measurements solves, in the frequency domain, a 3 3 equation system for each value of harmonic content in the signals. The used parasitics cells are shown in Fig. 14. , ) in the dc For dc, there are two-port resistances ( path, at the input and at the output of the device with For RF,

(10)

placed in the dc path do not affect, therefore, (11)

In both cases (dc and RF), (12) (13)

Fig. 12. V

Measured (discontinuous) and simulated (lines) output waveforms at = 2:9 V, and fundamental frequency = 4 GHz.

= 00:2 V, V

classical extraction process based on small-signal measurements. Comparison of both approaches leaded to very similar extraction results, showing the validity of the proposed technique, with the advantage of the large-signal approach in the big reduction of needed measurement data. A table-based nonlinear model generated from large-signal measurements has been experimentally validated using small- and large-signal measurements.

(14) (15)

(16)

CURRÁS-FRANCOS: TABLE-BASED NONLINEAR HEMT MODEL

Fig. 14.

1599

Typical parasitics cell for an HEMT at microwave frequencies.

The known data to solve these equations are the linear elements values and the measured extrinsic voltages and currents , ). Consequently, a 3 3 system (16) has to be ( , , solved for the unknowns ( , , ). This system needs to be solved for the harmonics present in the measurements, where ( harmonics).

(17) (18) (19) Once ( , , and ) have been calculated from the 3 system, the equation that obtains the intrinsic voltages is

3

(20)

(

In this way, it is possible to have the values of the voltages , ) and currents ( , ) in the intrinsic planes. ACKNOWLEDGMENT

The author thanks Prof. P. J. Tasker, Cardiff University, Wales, U.K., for his invaluable help with this study. REFERENCES [1] D. E. Root, S. Fan, and J. Meyer, “Technology independent large-signal FET models: A measurement-based approach to active device modeling,” in Proc. 15th ARMMS Conf., Sep. 1991, pp. 1–21. [2] R. R. Daniels, F. P. Harrang, and A. T. Yang, “A nonquasistatic large signal FET model derived from small signal S -parameters,” in Proc. Int. Semiconductor Device Research Symp., 1991, pp. 601–604.

[3] M. C. Foisy, P. E. Jeroma, and G. H. Martin, “Large signal relaxation-time mode for HEMT’s and MESFET’s,” in IEEE MTT-S Int. Microwave Symp. Dig., 1992, pp. 251–254. [4] F. Filicori, G. Vannini, and V. A. Monaco, “A nonlinear integral model of electron devices for HB circuit analysis,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 7, pp. 1456–1465, Jul. 1992. [5] I. Corbella, J. M. Legido, and G. Naval, “Instantaneous model of a MESFET for use in linear and nonlinear circuit simulations,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 7, pp. 1410–1421, Jul. 1992. [6] M. Fernández-Barciela, P. J. Tasker, Y. Campos-Roca, H. Massler, E. Sánchez, M. C. Currás-Francos, and M. Schlechtweg, “A simplified broad-band large-signal non quasi-static table-based FET model,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 3, pp. 395–405, Mar. 2000. [7] F. van Raay and G. Kompa, “A new on-wafer large-signal waveform measurement system with 40 GHz harmonic bandwidth,” in IEEE MTT-S Int. Microwave Symp. Dig., 1992, pp. 1435–1438. [8] T. van de Broeck and J. Verspecht, “Calibrated vectorial nonlinear network analyzers,” in IEEE MTT-S Int. Microwave Symp. Dig., 1994, pp. 1069–1072. [9] M. Demmler, P. J. Tasker, and M. Schlechtweg, “A vector corrected high power on-wafer measurement system with a frequency range to the higher harmonics up to 40 GHz,” in Proc. 24th Eur. Microwave Conf., 1994, pp. 1367–1372. [10] P. J. Tasker, S. S. O’Keefe, G. D. Edwards, W. A. Philips, M. Demmler, M. C. Currás-Francos, and M. Fernández-Barciela, “Vector corrected nonlinear transistor characterization,” in Proc 5th Eur. Gallium Arsenide and Related III–V Compounds Applications Symp., Bologna, Italy, Sep. 1997, pp. 91–94. [11] Q. Z. J. W. Bandler, S. Ye, and S. H. Chen, “Efficient large-signal FET parameter extraction using harmonics,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1099–2108, Dec. 1989. [12] A. Werthof, F. van Raay, and G. Kompa, “Direct nonlinear FET parameter extraction using large-signal waveform measurements,” IEEE Microw. Guided Wave Lett., vol. 3, pp. 130–132, May 1993. [13] M. Demmler, P. J. Tasker, M. Schlechtweg, and A. Hülsmann, “Direct extraction of nonlinear intrinsic transistor behavior from large signal waveform measurement data,” in Proc. 26th Eur. Microwave Conf., Prague, Czech Republic, 1996, pp. 256–259. [14] C. J. Wei, Y. E. Lan, J. C. M. Hwang, and W. J. Ho, “Waveform-based modeling and characterization of microwave power heterojunction bipolar transistor,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 12, pp. 2899–2905, Dec. 1995.

1600

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

[15] D. Schreurs, J. Verspecht, B. Nauwelaers, A. V. D. Capelle, and M. V. Rossum, “Direct extraction of the nonlinear model for two-port devices from vectorial nonlinear network analyzer measurements,” in Proc. 27th Eur. Microwave Conf., Jerusalem, Israel, 1997, pp. 921–926. [16] D. Schreurs, J. Verspecht, S. Vandenberghe, and E. Vandamme, “Straightforward and accurate nonlinear model parameter-estimation method based on vectorial large-signal measurements,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2315–2319, Oct. 2002. [17] M. C. Currás-Francos, P. J. Tasker, M. Fernández-Barciela, S. S. O’Keefe, Y. Campos-Roca, and E. Sánchez, “Direct extraction of nonlinear FET I–V functions from time domain large signal measurements,” Electron. Lett., vol. 34, pp. 1993–1994, Oct. 1998. [18] M. C. Currás-Francos, P. J. Tasker, M. Fernández-Barciela, Y. CamposRoca, and E. Sánchez, “Direct extraction of nonlinear FET C–V functions from time domain large signal measurements,” Electron. Lett., vol. 35, pp. 1789–1791, Oct. 1999. [19] , “Extraction of transistor large signal models from vector nonlinear network analyzers,” in Proc. 55th Automatic RF Techniques Group Conf., Boston, MA, 2000, pp. 9–13. , “Direct extraction of nonlinear FET – functions from time [20] domain large signal measurements,” IEEE Microw. Guided Wave Lett., vol. 10, no. 12, pp. 531–533, Dec. 2000. [21] M. C. Currás-Francos, “Microwave FET large signal modeling and experimental characterization using a vector nonlinear network analyzer,” Ph.D. dissertation, Dept. Tecnologías Comun., Univ. Vigo, Vigo, Spain, 2000.

QV

[22]

, “Comparison of HEMT nonlinear model extraction approaches based on small signal and on large signal measurements,” Int. J. Numer. Modeling, vol. 16, pp. 41–51, Jan.–Feb. 2003. [23] R. Osorio, M. Berroth, W. Marsetz, L. Verweyen, M. Demmler, H. Massler, M. Neumann, and M. Schlechtweg, “Analytical charge conservative large signal model for MODFET’s validated up to mm-range,” in IEEE MTT-S Int. Microwave Symp. Dig., 1998, pp. 595–598. [24] G. Dambrine, A. Cappy, F. Heliodore, and E. Playez, “A new method for determining the FET small-signal equivalent circuit,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 7, pp. 1151–1159, Jul. 1988.

M. Carmen Currás-Francos received the Ingeniera de Telecomunicación and Ph.D. degrees (with honors) from the Universidad de Vigo, Vigo, Spain, in 1994 and 2000, respectively. In 1994 she joined the Telecommunication School, Universidad de Vigo, as a Research Collaborator. In 2000, became an Assistant Professor with the Universidad de Vigo. Her main research interest is the nonlinear real-time characterization and modeling of semiconductor devices and circuits for microwave applications.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

1601

An Advanced Low-Frequency Noise Model of GaInP–GaAs HBT for Accurate Prediction of Phase Noise in Oscillators Jean-Christophe Nallatamby, Michel Prigent, Member, IEEE, Marc Camiade, Arnaud Sion, Cyril Gourdon, and Juan J. Obregon, Senior Member, IEEE

Abstract—We present a new low-frequency noise model of a GaInP–GaAs HBT and the associated extraction process from measurements. Specific measurements enable us to locate the two dominant low-frequency noise sources. Their spectral densities extraction as a function of the emitter bias current is then performed and a normalized scalable model is deduced. The cyclostationarity of the low-frequency noise sources is justified. The whole noise model including the shot noise source is implemented in the nonlinear HBT model used in the United Monolithic Semiconductors foundry. In order to verify the validity of the scalable noise model, several voltage-controlled oscillators with different center frequencies and tuning bandwidth have been designed and processed. Comparisons between the predicted performances and experimental results show an excellent agreement and validate the proposed low-frequency noise modeling of multifinger HBTs. Index Terms—Cyclostationary noise sources, HBT models, lowfrequency noise modeling, monolithic microwave integrated circuit (MMIC) voltage-controlled oscillator (VCO) phase noise, physical noise sources in HBTs.

I. INTRODUCTION

T

HE simulation accuracy of the phase-noise spectrum of oscillators used in microwave applications [1], [2] principally relies, on the one hand, on the ability of the simulation algorithm to handle any kind of circuit architecture and noise source accurately and, on the other hand, on the accuracy of active devices and passive element models and their associated noise sources. Today, the simulation methods of phase noise in oscillators are well established [3]. Modern software packages allow to handle linear and cyclostationary noise sources. The phase-noise spectrum is accurately computed with the same formalism far from and near to the carrier frequency [4]: the simulation, which is no longer quasi-static, now takes into account the low-frequency dynamic effects, i.e., the dynamic interaction between the low-frequency bias circuit and the RF circuit properly so called [5].

Manuscript received July 16, 2004; revised October 4, 2004. This work was supported by SIAE microelettronica, Cologno Monzese, Milan, Italy. J.-C. Nallatamby, M. Prigent, and J. J. Obregon are with the Institut de Recherche en Communications Optiques et Microondes, Institut Universitaire de Technologie Génie Electrique et Informatique Industrielle, Universite de Limoges, 19100 Brive, France (e-mail: [email protected]; [email protected]; [email protected]). M. Camiade, A. Sion, and C. Gourdon are with United Monolithic Semiconductors, 91401 Orsay, France (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2005.847050

Besides, the linear and nonlinear models of the active devices and passive elements are generally fair and accurate up to millimeter wavelengths [6]–[8]. With regard to the noise models, one must distinguish, on the one hand, between the white noise sources: the diffusion noise and correlatively the shot noise, which are fairly well modeled [9]–[12] and, on the other hand, the colored low-frequency noise noise sources, which include – noise and fundamental whose models are still an open matter of investigation [13]–[19]. Phase noise far from carrier is mainly generated by the noise sources centered near the oscillation frequency and its harmonics [20]. Near carrier (up to 100-kHz offset from carrier) phase noise is directly related to the low-frequency noise sources, which are up-converted around the carrier. In this paper, we present a new low-frequency noise model of a GaInP–GaAs HBT and the associated extraction process from measurements. This paper is organized as follows. First, possible locations of the low-frequency noise sources associated to their physical origins are listed. A new low-frequency noise model is proposed. The cyclostationary nature of the low-frequency noise sources in large-signal operating conditions is then justified. The extraction of the noise model from measurements is then presented. A first measurement allows to eliminate some of the possible noise sources. Further measurements allow to locate the two dominant low-frequency noise sources. The extraction of their spectral densities as a function of the emitter bias current is then performed and a normalized scalable model is deduced. The results obtained with this scalable model match the lowfrequency noise measurements performed on transistors with different emitter sizes: from 2 20 m to 2 40 m , and from 1 to 6 fingers, validating the model scalability. The noise sources are implemented into the transistor nonlinear model of the United Monolithic Semiconductors (UMS), Orsay, France, HB20P foundry process, which will be used in the design process. The predicted phase-noise spectra of voltage-controlled oscillators (VCOs), designed at different microwave oscillation frequencies, are presented. Comparisons between the predicted performances and experimental results obtained show an excellent agreement and validate the proposed low-frequency noise modeling of HBTs.

0018-9480/$20.00 © 2005 IEEE

1602

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Fig. 1. Possible localizations of the low-frequency noise current sources in the HBT.

A conclusion showing possible improvements of the model and extensions to other technologies closes this paper. II. LOW-FREQUENCY NOISE HBT MODEL A. Possible Low-Frequency Noise Sources and Their Localization Note that, in order to extract the two main low-frequency noise sources, a T model of an HBT is used [21], [22]. However, the T and Pi models are exactly equivalent by taking into account the appropriate frequency dependence of the collector current source. Microscopic physical noise sources lead to random fluctuations of conductive currents. It then follows that, in a macroscopic electrical representation, every equivalent Norton noise current source must be associated to the conductive element where it originates. Fig. 1 shows all the possible localizations of the low-frequency noise current sources in the HBT. Note that all the noise sources shown in this figure are uncorrelated. Represents the and noise sources generated in the base–emitter junction. Represents the and noise sources of the recombination current principally present at the periphery of the base–emitter junction. and Represent the and noise sources and resistances. generated in the represents the and noise sources generated in the base–collector junction. Represents the and noise sources of the recombination current of the base–collector junction.

B. First Important Observation is the total current flowing in the base–emitter In Fig. 1, junction. This total current may be written as the sum of a curand a low-frequency Norton noise current source . rent Since is part of , it induces a full correlated noise

Fig. 2. HBT small-signal differential model with possible localization of the low-frequency noise current sources.

controlled-current source between base and collector: in the same way as current induces the controlled-current source . It results in a total controlled-current source rep. resented in Figs. 1 and 2 by It must be noted that the low-frequency noise current source of the base–emitter junction generates a partition noise similar to that of shot noise, which is similarly injected from the base in the reverse-biased base–collector junction, creating a correlated current noise source in the electrical model [9], [12]. The main difference is the physical mechanism of noise generation and the resulting spectral density, which is colored for the former and white for the latter. Moreover, strictly speaking, may be different according to its associated the coefficient emitter current: the deterministic current due to the input signal, the shot noise, or the low-frequency noise currents generated in may be the junction. Nevertheless, in a first approximation, taken as identical for input deterministic signals, the shot noise, and the low-frequency noise generated in the base–emitter juncwill be written as tion. (1) where is principally due to the transit time in the reverse-biased base–collector junction. At low frequencies, is neglected. As previously noted, the noise is always generated in a “physical” conductive element. Accordingly, a resulting noise controlled current source in the macroscopic electrical model does not generate noise by itself. A controlled current source is no more than a consequence of the current path into the device. Thus, in order to clarify the location of the physical noise generation, the term “noise source” will be reserved in the HBT T-models (Figs. 1 and 2) for the actual physical location of noise, and not for the resulting controlled sources. Obviously this notation does not change the final results in any way. III. LOW-FREQUENCY NOISE SOURCES AND THEIR ELECTRICAL BEHAVIOR IN LARGE-SIGNAL TIME-VARYING OPERATING CONDITIONS Noise modeling oriented to computer-aided design (CAD) of nonlinear circuits leads to noise representation by macroscopic

NALLATAMBY et al.: ADVANCED LOW-FREQUENCY NOISE MODEL OF GaInP–GaAs HBT

Norton noise current sources associated in the nonlinear electrical models of the semiconductor devices to the conductive elements where they originate. Each Norton noise current has its own electrical behavior [5], which depends: 1) firstly, on the physical process leading to carrier or velocity random fluctuations; 2) secondly, on the conductive current component (diffusion or drift current) prevailing locally in the region where the generation takes place; and finally 3) on the transfer from this local current source to the macroscopic terminal of the conductive element where it originates. Here, we describe the electrical characteristics of the main low-frequency physical noise generation processes and the associated noise current sources, including the noise induced by random fluctuations of carrier number, recombination velocity, noise due to random fluctuations of and the fundamental mobility [5], [13]–[19], [22]–[26]. 1) Carrier’s Random Generation Due to Traps: From the master equation, using the Langevin approach, it may be shown that the – noise process due to traps is a low-pass process; this characteristic leads to a spectral density of carrier generation, , confined to low frequencies. We obtain a carrier namely, spectral density, which is a function of frequency, of the form or

(2)

where , , , and depend on the nature of the traps present in the semiconductor, and are functions of the carrier’s energy. In turn, this energy is a function of the time-dependent applied signal. However, due to the low-pass characteristic of , , , and may be practically the generation process, assumed to be functions of the dc component of the time-dependent applied signal. Once the random carriers have been generated, they give rise to a local noise current, which can be diffusion or drift current. The nature of this resulting conductive current depends on the conditions prevailing in the region where noise generation takes place. It has been shown theoretically [27]–[31] that, in homogeneous resistors and in space charge regions of P–N junctions operating in large signal conditions, the resulting macroscopic Norton noise current source follows the time variations of the periodic applied signal. Hence, it is cyclostationary. 2) Noise Due to Random Fluctuations of Surface Recombination Velocity: The surface recombination velocity is assumed to be modulated by the fluctuating occupancy of surface traps. This noise generating process is a low-pass process by nature: fluctuations of the recombination velocity are confined to low frequencies. Nevertheless, these velocity fluctuations give rise to a local recombination current proportional to the deterministic large-signal excess minority carrier density at the interface [31], [34]. In the presence of a periodic applied signal, this excess minority carrier density becomes periodically time varying, giving rise to a cyclostationary noise current source of velocity recombination. Noise: noise is due to random 3) Fundamental fluctuations of the low field mobility. This process is low pass by nature. Nevertheless, in the presence of an applied periodic electric field, the product of the low-frequency random fluctuations of mobility by the periodically time-dependent electric

Fig. 3.

1603

Basic noise measurement setup.

field gives rise to noise current sources located as sidebands around the dc and RF components of the periodic electric field [5]. It follows that the Norton noise current sources associated noise are cyclostationary. to fundamental IV. EXTRACTION FROM MEASUREMENTS OF THE LOW-FREQUENCY NOISE MODEL A. Determination of the HBT Low-Frequency Noise Model Fig. 2 shows the HBT small-signal differential model deduced from Fig. 1. and are exponential The differential resistances functions of the intrinsic bias voltage ; and of . is a function of and the intrinsic bias voltage voltages. and can be neIn a normal mode of operation, glected, however, for other bias conditions, i.e., for dc/ac opencircuit collector, they must be preserved in the model in order to close up the collector current sources. The principle of the low-frequency measurement setup is shown Fig. 3. This new setup, developed at the Institut de Recherche en Communications Optiqus et Microondes (IRCOM), Brive, France, is an improved version of the one described in [3]. The spectral densities of the input and output low-frequency and are measured for many transisnoise currents tors of the UMS HB20P process with different emitter sizes is inserted in the and finger number. A variable resistance HBT base loop. The principle of the calibration procedure of this setup is explained in [3]. It must be noted that the transistor shot noise can be neglected at low frequencies. Nevertheless, this noise source will be included in the final transistor model in order to accurately simulate the phase noise for all offset frequencies from carrier. Straightforward calculations using the model of Fig. 2 and in the base loop give including

(3)

1604

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

and (4), shown at the bottom of this page, with

Due to the great number of possible low-frequency noise sources, the localization of the main sources is a difficult task is added to the even if the cross-spectral density previous equations. In order to remove this indetermination, we have performed complementary noise measurements on reverse-biased varactors processed on the basis of the HBT base–collector junctions. The results obtained show that spectral densities of and are lower by more than one order of magnitude as compared to the results obtained on the transistors. Thus, these and results enable us to eliminate the noise sources as main contributors to the low-frequency noise of the transistors. This conclusion was expected because the base–collector junction of the HBT is reverse biased in the normal mode of operation. It constitutes a drift region where the low-frequency carrier generation is very low. On the other hand, in this region, noise a high electric field prevails and the fundamental issued from random fluctuations of the carrier mobility at a low field is low. Four low-frequency noise sources remain as main candidates. It must be noted that measurements do not permit to separate of and . In fact, the noise sources associated to and are multiplied by the same coefficients in (3) and (4). In order to discriminate between, on the one hand, the resistance noise sources described together as a single noise voltage source of spectral density (5) and, on the other hand, the noise sources and , according [22], we assume that the HBT low-frequency noise behavior can be mainly described by a combination of two independent sources. We will successively try the couples and and and Since the main noise sources are not known a priori, a first general extraction is performed over more than one decade of for the three bias current by solving (3) and (4) with couples. It results in the extracted spectral densities from the

Fig. 4.

Final low-frequency HBT model.

three couples always showing positive values. Nevertheless, and are linear resistances, at least at the first order, since should show a quadratic evolution with bias current. shows a weak bias current deHowever, the extracted pendence over the bias current range. This dependence is not physically acceptable for linear semiconductor resistors, which can be considered as homogeneous ohmic elements [16], [23]. By contrast, the log–log representation of the spectral and extracted from the couple densities fairly follows straight lines with a downright slope over the whole range of the bias current. We may conclude that the HBT low-frequency noise behavior may be attributed with an accurate assumption to the sources and generated by fluctuations of minority carriers in the base–emitter junction (periphery and volume). Note that the region we denote “volume” includes the interfaces of the emitter–base junction, and the emitter–base space–charge region. B. Accurate Extraction of the Two Main Low-Frequency Noise Sources Fig. 4 represents the low-frequency HBT model retained. is a function of the current flowing in the diode is a function of the current flowing in the diode. and In order to accurately extract these two spectral densities, many dedicated extraction strategies can be worked up [22], and can be extracted from [23], [26]. For example, , and from (4) for large; the results obtained (3) for or by comparing the can then be verified for other values of computed and measured cross-spectral density Other,

(4)

NALLATAMBY et al.: ADVANCED LOW-FREQUENCY NOISE MODEL OF GaInP–GaAs HBT

1605

more or less sophisticated strategies are possible. A comparison between them will be presented and discussed elsewhere. Nevertheless, we propose a simple nonconventional extracand whose accuracy has proven tion procedure of excellent by comparison with more sophisticated strategies. and can be performed An accurate extraction of using two bias/load transistor-operating modes: 1) First Mode of Operation: Normal Mode: Measurement of output noise current spectral density with large. The measured output noise current spectral density can be written as

(6)

Fig. 5.

S

Fig. 6.

S

extracted from S

and S

measurements.

with

Note that for

large, one obtains (7)

may be considered as “large” when becomes prac. In our measurements, has been tically independent of chosen greater than 10 k . Measurement of input noise current spectral density with . Besides the measured input noise current spectral density can be written as

(8) For a null , it becomes (9), shown at the bottom of this page. 2) Second Mode of Operation: Collector AC/DC Open Circuited1: Measurement of input noise current spectral density with . The measured input spectral density of the noise current can be written as

(10) 1It must be noted that the operating mode with the collector ac/dc open circuited has been known for a long time as a possible operating mode for the extraction of R emitter resistance in bipolar transistors [24].

extracted from S

and S

measurements.

3) Extraction Procedure: In order to accurately determine , , the two dominant low-frequency noise sources the following procedure has been chosen. • From (6) and (10), we extract the two spectral densities. • The accuracy of the extraction method is then verified by introducing these in (9), which has not been used for the is then computed and compared with the extraction; measured one. The excellent agreement obtained justifies the proposed method of extraction a posteriori. As an example of extraction results, Fig. 5 and 6 show the results obtained for an HBT of one emitter finger with 2 30 m size as a function of frequency and for different emitter currents. The verification of the accuracy of the extracted spectral densities is then performed by applying (9). For the sake of brevity,

(9)

1606

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

detailed results of this verification will be presented in Section IV-D on the final scalable model. C. Scalable Noise Model For the determination of the scalable model, the length unit is the micrometer and the current unit is the ampere. 1) Surface Recombination Velocity Noise Source: According to [15], [23], [25] and [26], for a fixed frequency, the theoretical spectral density of the surface recombination velocity noise source for homogeneous generation along the emitter periphery can be written as (11) where where the section area writes as finger number; emitter width; emitter length. However, the actual devices require a more detailed description, for example, in [27] and [28], it is pointed out that if noise generation is not homogeneously distributed, but principally located at the edge of each emitter finger, this expression is no more valid: the noise spectral density is then explicitly a function of the finger number . A more general expression is of the current needed for an accurate representation of devices; it can be described by

S

L N

) computed from (16) for

f = 1 kHz.

S

L N

) computed from (16) for

f = 10 kHz.

Fig. 9. 10 log(S

L N

) computed from (16) for

f = 100 kHz.

Fig. 7.

10 log(

Fig. 8.

10 log(

(12) Besides, the noise spectral densities are extracted as a function of the current . By taking into account the relation and , as proposed in [28], may be between . For the measured transistors at written as a function of K, we get the following relation: (13) . with For a fixed frequency, one can then write the general expression (14) All the measured transistors of this study have the same width m, then can be written (15) To find a generic scalable model, the spectral density must be described as a function of the current density . , can By taking into account finally written as (16) (16) is fitted by taking into account all the measured transistors from 1 2 20 to 6 2 40 m size. For the ten measured is plotted versus . transistors,

Fig. 7–9 show the results for three frequencies, i.e., 1, 10, and 100 kHz. A very good fit is obtained in the full range of bias current density from 0.05 to 0.3 mA m for all the transistors but

NALLATAMBY et al.: ADVANCED LOW-FREQUENCY NOISE MODEL OF GaInP–GaAs HBT

FITTED COEFFICIENTS OF S

1607

TABLE I FOR

THREE NOISE FREQUENCIES

one: the plot relative to the transistor with one 2 20 m size finger deviates from the others. This problem is often encountered with small-size transistors whose actual dimensions must be precisely known in order to obtain accurate scalability. Table I shows the fitted coefficients for the three noise frequencies for the HB20P UMS process. 2) Volume – Noise Source: Concerning – noise generated in the emitter–base junction, a theoretical modeling of the noise spectral density for one homogeneously distributed trap predicts the following expression for a fixed noise frequency [26]:

10 log(S

=A

) computed from (16) for

f

= 1 kHz.

Fig. 11. 10 log(S

=A

) computed from (16) for

f

= 10 kHz.

10 log(S

=A

) computed from (16) for

f

= 100 kHz.

Fig. 10.

(17) A more realistic expression must take into account many different nonhomogeneously distributed traps. A generalized expression will take the form (18) As for the source, we can express . of the current density Let , (15) can be written as

as a function

(19) . This expression constitutes an accurate model with of low-frequency noise generated in the emitter–base junction for the HB20P process. is plotted For the ten measured transistors, . Fig. 10–12 show the results for three frequenversus cies, i.e., 1, 10, and 100 kHz. We note an excellent agreement of the scalable model of the . low-frequency noise source Table II shows the fitted coefficients for the three noise frequencies for the HB20P UMS process. Finally, experimental results show that, in the HBTs under noise can be neglected as compared to the analysis, “pure” – noise in the frequency range under consideration. D. Verification of the Scalable Model In order to verify the extraction procedure, (9) is now used. is computed using (16) and (19). Results are compared with the corresponding measured data. Figs. 13–15 show this comparison for three transistors of emitter size 2 30 m and with one, two, and four fingers at 1-, 10-, 100-kHz frequencies. and of Note that the weights of present in (9) are of the same order of magnitude, leading to

Fig. 12.

an accurate verification of both and . The agreement obtained is excellent for all the transistors. This verification confirms the validity of the extraction and the assumption and are the two main sources of low-frethat quency noise in the HBTs under analysis.

1608

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

TABLE II FITTED COEFFICIENTS OF S FOR THREE NOISE FREQUENCIES

Fig. 15. Comparison of S computed using (15) and (18) with measured data at 1, 10, and 100 kHz for a transistor of four fingers of 2 30 m .

2

Fig. 13. Comparison of S computed using (15) and (18) with measured data at 1, 10, and 100 kHz for a transistor of one finger of 2 30 m .

2

2) Low-Frequency Noise Sources: In dc operating conditions, the spectral densities of the low-frequency noise sources and are given by (16) and (19). A modulation theory of low-frequency noise sources operating in large-signal conditions is still under development [5], [18], [19], [29]–[33]: the actual modulation functions to be applied to real transistors including many nonhomogeneously distributed traps, cannot yet be determined by theoretical calculations. Thus, physics-based empirical expressions will be used. From [32] and [33], it may be inferred with an accurate assumption that the modulation function of the spectral density of the low-frequency noise current generated in the volume is proportional to the square of the dynamic current . Besides, according to [32] and [36], the same assumption may be inferred for the modulation function of the spectral density of the low-frequency noise current generated by the surface recomwill then be taken as proportional to bination velocity. . the square of the dynamic recombination current density From (16) and (19), by also taking into account (13), we obtain the final expressions of the scalable cyclostationary low-frequency noise sources

Fig. 14. Comparison of S computed using (15) and (18) with measured data at 1, 10, and 100 kHz for a transistor of two fingers of 2 30 m .

(22)

E. Modeling of the Cyclostationary Noise Sources of the HBT

(23)

2

1) Shot Noise Source: Contrary to the – noise, the shot noise process in semiconductor junctions is an all-pass process by nature (white noise); it is associated to the diffusion noise of minority carriers in the quasi-neutral regions of junctions. In dc operating conditions, the spectral density of the shot noise source of the emitter–base junction, located in parallel , is given by with (20) In dynamic operating conditions, the shot noise source, which is a white noise source, is fully modulated [34], [35]; it results proportional from (20) a cyclostationary spectral density . Then to the dynamic current density (21) Note that shot noise due to

is negligible.

and where Tables I and II).

are functions of the frequency

(see

V. EXPERIMENTAL VERIFICATION: APPLICATION TO MICROWAVE VCOs In order to verify the validity of the scalable noise model, several VCOs have been processed by the UMS foundry. The noise model was included in the UMS electrical model of the HB20P process. Note that the cyclostationary noise sources are straightforwardly introduces as current-controlled current sources. The thermal noise sources associated to the linear resistances in the whole VCO circuit are also taken into account in the noise simulations. Two push–push oscillators are presented: one with a fundamental output frequency centered at 2 GHz. The other with the output frequency at the second harmonic centered at 13 GHz.

NALLATAMBY et al.: ADVANCED LOW-FREQUENCY NOISE MODEL OF GaInP–GaAs HBT

Fig. 17. voltage.

1609

Simulated and measured oscillation frequencies versus tuning

Fig. 16. Electrical schematic principle of the designed VCOs.

For measurement purposes, the chips were mounted on sockets with microstrip lines and coaxial connectors. The simulations of VCOs have been performed with a harmonic-balance simulator. Phase noise was simulated with an accurate fast method based on conversion matrices and noise correlation matrices formalism. This method of noise simulation is naturally related to the harmonic-balance formalism [3], [5]. The electrical schematic principle of the two VCOs is presented in Fig. 16: the oscillator circuit has a push–push structure. Two complementary output ports are available in the 2-GHz fundamental frequency oscillator. In the 13-GHz oscillator, the ) and ( ) are connected together to the input output ports ( of a 6-dB attenuator followed by a two-stage single-ended buffer amplifier. This output circuit includes its own stationary and cyclostationary noise sources for phase-noise simulations. Detailed architectures and simulations of the different oscillator circuits will be presented elsewhere. A. 2-GHz VCO The center oscillation frequency is 2 GHz. Transistors have 18 emitter fingers of 2 60 m size. Fig. 17 shows the predicted and measured oscillation frequencies versus tuning voltage. The tuning bandwidth is 360 MHz. The output power is 0 dBm 1.5 dB in the bandwidth Fig. 18 shows the predicted phase noise and a comparison with the measurement results @ V. Fig. 19 shows the predicted phase noise @ 10-kHz offset frequency versus tuning voltage and a comparison with the measurement results

Fig. 18. Spectral phase-noise density for a tuning voltage @ 4 V versus frequency offset from carrier.

Fig. 19.

Phase-noise spectrum @ 10 kHz: predicted and measured results.

It must be noted that the comparison of the predicted and measured results obtained with this VCO validates the extension of the scalable noise model up to 18-emitter-finger transistors. B. 6.5

2 GHz VCO

The fundamental oscillation frequency is 6.5 GHz. The circuit design was oriented to obtaining a high-output power @ 13 GHz and wide tuning bandwidth. Transistors used in the oscillator circuit have six emitter fingers of 2 30 m size.

1610

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Fig. 22. Measured phase noise for a tuning voltage @ 3 V versus frequency offset from carrier of the whole chip. Fig. 20. voltage.

Simulated and measured oscillation frequencies versus tuning

Fig. 23. Phase-noise spectrum @ 100 kHz: predicted and measured results of the whole chip. Fig. 21. Simulated and measured output power versus tuning voltage.

Transistors used in the buffer amplifier have two emitter fingers of 2 30 m size. Figs. 20 and 21 show the predicted and measured oscillation frequencies and output power versus tuning voltage. Fig. 22 shows the predicted phase noise and a comparison V. with the measurement results @ Fig. 23 shows the predicted phase noise @ 100-kHz offset frequency versus tuning voltage and a comparison with the measurement results. We have noted that the low-frequency noise sources of the reverse-biased varactors do not significantly increase the phase noise. The very good agreement obtained for the complete sources validates the extraction procedure of the scalable low-frequency noise model for the HB20P process. It must be noted that the resulting characteristics of the processed circuits are excellent for the two VCOs. In terms of phase noise/tuning bandwidth, the characteristics of the VCO centered at 2 GHz represent the state-of-the-art. VI. CONCLUSION An accurate scalable model of low-frequency noise of GaInP–GaAs HBT has been proposed. The extraction of the spectral density of noise sources from measurements has been

detailed. Measurement results of the spectral density of noise sources obtained on transistors of different emitter sizes (from 2 20 m to 2 40 m ) and finger numbers (from 1 to 6) show an excellent reproducibility. The cyclostationary nature of the low-frequency noise sources has been justified and introduced in the model used in the circuit design CAD process of the UMS foundry. Monolithic microwave integrated circuit (MMIC) oscillator circuits have been designed, and the experimental results have been compared with the predicted one. Excellent agreement has been found, validating the proposed low-frequency noise scalable model for multifinger HBTs. It must be noted that the comparison of the predicted and measured results obtained with the 2-GHz VCO validates the extension of the scalable noise model up to 18-emitter-finger transistors. In terms of phase noise/tuning bandwidth, the characteristics of the MMIC VCO centered at 2 GHz represent the state-of-the-art. To our knowledge, it is the first time that phase-noise results obtained on complete microwave sources, designed in an industrial environment, have been compared with the predicted ones using full scalable models of transistors including the low-frequency noise sources. The match between the predicted and measured phase noise of different oscillator circuits designed with different objectives, at different operating frequencies, is excellent without any retrofitting.

NALLATAMBY et al.: ADVANCED LOW-FREQUENCY NOISE MODEL OF GaInP–GaAs HBT

Further improvements of the noise model will lead to extend its validity to different widths of emitter fingers and to extract bias voltage. models, which are also functions of the The proposed extraction methodology can be extended to other technologies by considering two possible noise sources at a time, and extracting their noise spectral densities as a function of frequency and bias current; physical considerations then allow to eliminate some candidates and finally enable to find the dominant noise sources and their associated behavior. ACKNOWLEDGMENT The authors wish to acknowledge P. Auxemery, and H. Blanck, both of United Monolithic Semiconductors, Orsay, France, and S. Delage and D. Floriot, both of Thales TRT, Orsay, France, for valuable and fruitful technical discussions. REFERENCES [1] M. Camiade et al., “Fully MMIC-based front end for FMCW automotive radar at 77 GHz,” in Eur. Microwave Conf., vol. 1, Paris, France, 2000, pp. 9–12. [2] “Session TU2D: Integrated circuits for 40 Gb/s fiber systems,” in IEEE MTT-S Int. Microwave Symp. Dig, Seattle, WA, May 2002, pp. 75–91. [3] J. Obregon and J. C. Nallatamby et al., “High-frequency oscillator circuit design,” in RF and Microwave Oscillator Design, M. Odyniec et al., Ed. Norwood, MA: Artech House, 2002. [4] P. Bolcato et al., “A unified approach of PM noise calculation in large RF multitone autonomous circuit,” in IEEE MTT-S Int. Microwave Symp. Dig, Boston, MA, Jun. 2000, pp. 417–420. [5] J. C. Nallatamby et al., “Semiconductor device and noise sources modeling, design methods and tools, oriented to non linear H.F. oscillator CAD,” presented at the SPIE Int. Symp. Conf., Maspalomas, Gran Canaria, Spain, May 2004. [6] Design Manuals of the MMIC Foundries, United Monolithic Semiconduct., Orsay, France, 2000. [7] T. Peyretaillade et al., “A pulsed measurement based electro thermal model of HBT with thermal stability prediction capability,” in IEEE MTT-S Int. Microwave Symp. Dig, Denver, CO, Jun. 1997, pp. 1515–1518. [8] T. Dhaene, J. De Geest, and D. De Zutter, “Constrained EM-based modeling of passive components,” in IEEE MTT-S Int. Microwave Symp. Dig, vol. 3, Seattle, WA, May 2002, pp. 2113–2116. [9] J. G. Tartarin, L. Escotte, and J. Graffeuil, “Small signal model extraction technique dedicated to noise behavior of microwave HBTs,” in GaAs Symp., Amsterdam, The Netherlands, 1998, pp. 301–306. [10] A. Van der Ziel and E. R. Chenette, “Noise in solid state devices,” in Advances in Electronics and Electron Physics, S. L. Marton and C. Marton, Eds. New York: Academic, 1978, vol. 46. [11] G. Niu et al., “A unified approach to RF and microwave noise parameter modeling in bipolar transistors,” IEEE Trans. Electron Devices, vol. 48, no. 11, pp. 2568–2574, Nov. 2001. [12] M. Rudolph et al., “An HBT noise model valid up to transit frequency,” IEEE Electron Device Lett., vol. 20, no. 1, pp. 24–26, Jan. 1999. [13] S. Mohammadi and D. Pavlidis, “A non fundamental theory of low frequency noise in semiconductor devices,” IEEE Trans. Electron Devices, vol. 47, no. 11, pp. 2009–2017, Nov. 2000. [14] P. Heymann et al., “Modeling of low-frequency noise in GaInP/GaAs hetero-bipolar transistors,” in IEEE MTT-S Int. Microwave Symp. Dig, vol. 3, Boston, MA, Jun. 2001, pp. 1967–1970. [15] A. Penarier et al., “Low frequency noise in III–V high speed devices,” Proc. Inst. Elect. Eng.—Circuits, Devices, Syst., vol. 49, no. 1, pp. 59–67, Feb. 2002. [16] Y. S. Kim et al., “Influence of magnetic field on 1=f noise and thermal noise in multiterminal homogeneous semiconductor resistors and discrimination between the number fluctuation model and the mobility fluctuation model for 1=f noise in bulk semiconductors,” Solid-State Electron., vol. 48, pp. 641–654, 2004. [17] R. Plana et al., “Excess noise in microwave GaInP/GaAs heterojunction bipolar transistors,” in Proc. Eur. GaAs and Related III–V Compounds Applications Symp., Turin, Italy, Apr. 1994, pp. 131–134.

1611

[18] A. Cappy et al., “Noise analysis in devices under nonlinear operation,” Solid State Electron., vol. 43, pp. 21–26, 1999. [19] F. Bonani, S. D. Guerrieri, and G. Ghione, “Noise source modeling for cyclostationary noise analysis in large-signal device operation,” IEEE Trans. Electron Devices, vol. 49, no. 9, pp. 1640–1647, Sep. 2002. [20] J. C. Nallatamby, M. Prigent, and J. Obregon, “On the role of additive and converted noise in the generation of phase noise in nonlinear oscillators,” IEEE Trans. Microw. Theory Tech.. [21] B. Li, S. Prasad, L.-W. Yiang, and S. C. Wang, “Large signal characterization of AlGaAs/GaAs HBT’s,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1743–1746, Sep. 1999. [22] D. R. Pehlke et al., “Direct parametric extraction of 1=f noise source magnitude and physical location from baseband spectra in HBT’s,” in IEEE MTT-S Int. Microwave Symp. Dig, vol. 3, Jun. 1996, pp. 1305–1308. [23] T. G. M. Kleinpenning, “Low-frequency noise in modern bipolar transistors: Impact of intrinsic transistor and parasitic series resistances,” IEEE Trans. Electron Devices, vol. 41, no. 11, pp. 1981–1991, Nov. 1994. [24] G. Kulke and S. L. Miller, “Accurate measurement of emitter and collector series resistances in transistors,” Proc. IRE, vol. 45, pp. 171–175, Jan. 1962. [25] Y. Takanashi and H. Fukano, “Low-frequency noise of InP/InGaAs heterojunction bipolar transistors,” IEEE Trans. Electron Devices, vol. 45, no. 12, pp. 2400–2406, Dec. 1998. [26] A. Pénarier et al., “Low-frequency noise of InP/InGaAS heterojunction bipolar transistors,” Jpn. J. Appl. Phys., vol. 40, pp. 525–529, 2001. [27] E. Zhao et al., “Temperature dependence of 1=f noise in polysilicon–emitter bipolar transistors,” IEEE Trans. Electron Devices, vol. 49, no. 12, pp. 2230–2236, Dec. 2002. [28] E. Hayama and K. Honjo, “Emitter size effect on current gain in fully self-aligned AlGaAs/GaAs HBT’s with AlGaAs surface passivation layer,” IEEE Electron Device Lett., vol. 11, no. 9, pp. 388–390, Sep. 1990. [29] S. Perez et al., “Microscopic analysis of generation-recombination noise in semiconductors under dc and time varying electric fields,” J. Appl. Phys., vol. 88, no. 2, Jul., 15 2000. , “Monte Carlo analysis of the influence of DC conditions on the up [30] conversion of generation-recombination noise in semi-conductors,” in Revue Semiconductor Science and Technology, No. 16. London, U.K.: IOP, 2001, pp. L8–L11. [31] J. E. Sanchez, G. Bosman, and M. E. Law, “Simulation of GR noise of resistor and junctions under periodic large signal steady state condition,” in 16th Int. Noise in Physical System and 1=f Noise Conf., Gainesville, FL, Oct. 2001, pp. 645–648. , “Two-dimensional semiconductor device simulation of trap-as[32] sisted generation-recombination noise under periodic large-signal conditions and its use for developing cyclostationary circuit simulation models,” IEEE Trans. Electron Devices, vol. 50, no. 5, pp. 1353–1362, May 2003. [33] J. E. Sanchez, “Semiconductor device simulation of low-frequency noise under period large-signal conditions,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Univ. Florida, Gainesville, FL, 2000. [34] C. Dragone, “Analysis of thermal and shot noise in pumped resistive diodes,” Bell Syst. Tech. J., pp. 1883–1902, Nov. 1968. [35] A. R. Kerr, “Noise and loss in balanced and sub harmonically pumped mixers: Part I—Theory,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 12, pp. 938–943, Dec. 1979. [36] O. Jantsch, “Flicker (1=f ) noise generated by random walk of electrons in interfaces,” IEEE Trans. Electron Devices, vol. ED-34, pp. 1100–1115, May 1987.

Jean-Christophe Nallatamby received the D.E.A. degree in microwave and optical communications and Ph.D. degree in electronics from the Universite de Limoges, Brive, France, in 1988 and 1992, respectively. He is currently a Lecturer with the Institut Universitaire de Technologie Génie Electrique et Informatique Industrielle, Universite de Limoges. His research interests are nonlinear noise analysis of nonlinear microwave circuits, the design of the low phase-noise oscillators, and the noise characterization of microwave devices.

1612

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Michel Prigent (M’93) received the Ph.D. degree from the Universite de Limoges, Brive, France, in 1987. He is currently a Professor with the Universite de Limoges. His field of interest are the design of microwave and millimeter-wave oscillator circuits. He is also involved in characterization and modeling of nonlinear active components (field-effect transistors (FETs), pseudomorphic high electron-mobility transistors (pHEMTs), HBTs, etc.) with a particular emphasis on low-frequency noise measurement and modeling for the use in monolithic microwave integrated circuit (MMIC) computer-aided design (CAD).

Arnaud Sion received the Ph.D. degree in high-frequency electronic from the Universite de Limoges, Brive, France, in 2002. Since 1999, he has been with United Monolithic Semiconductors, Orsay, France, where he is currently a Research and Development Monolithic Microwave Integrated Circuit (MMIC) Designer. His area of expertise includes microwave and millimeter-wave active circuits with emphasis on automotive front ends.

Cyril Gourdon received the Master degree from the Universite de Limoges, Brive, France, in 2002 in Radio Frequencies and Optical Communications. Since 2003, he has been with United Monolithic Semiconductors, Orsay, France, where he is currently working toward the Ph.D. degree from the Universite de Limoges. His research work addresses the analysis and design of advanced MMICs/digital non linear circuits.

Marc Camiade was born in France, in 1958. He received the Dpl.Eng. degree in physics and electronic engineering from the Institut National des Sciences Appliquées, Toulouse, France, in 1981. In 1982, he joined Thomson-CSF as a Design Engineer of hybrid circuits, during which time he participated in a variety of microwave and millimeterwave circuits. Since 1988, he has been an Application Group Manager in charge of new product development based on microwave integrated circuit (MIC) and MMIC components. In 1996, he joined United Monolithic Semiconductors, Orsay, France, where he is currently in charge of the development of components for defense and automotive applications. He is also currently and mainly involved in all the functions for radar front-ends from L- to W -bands.

Juan J. Obregon (SM’91) received the E.E. degree from the Conservatoire National des Arts et Métiers (CNAM), Paris, France, in 1967, and the Ph.D. degree from the Universite de Limoges, Brive, France, in 1980. He then joined the Radar Division, Thomson-CSF, where he contributed to the development of parametric amplifiers for radar front-ends. He then joined RTC Laboratories, where he performed experimental and theoretical research on Gunn oscillators. In 1970, he joined the DMH Division, Thomson-CSF, and became a Research Team Manager. In 1981, he was appointed Professor at the Universite de Limoges. He is currently Professor Emeritus with the Universite de Limoges. His fields of interest are the modeling, analysis, and optimization of nonlinear microwave circuits, including noisy networks.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

1613

Appropriate Formulation of the Characteristic Equation for Open Nonreciprocal Layered Waveguides With Different Upper and Lower Half-Spaces Raúl Rodríguez-Berral, Francisco Mesa, Member, IEEE, and Francisco Medina, Senior Member, IEEE

Abstract—This paper will study the most convenient way of formulating the characteristic equation for shielded, parallel-plate, grounded, and open reciprocal/nonreciprocal planar layered waveguides, including the possibility of different upper and lower half-spaces. A detailed study of the suitable mapping for each kind of waveguide will lead to the formulation of analytic characteristic equations for shielded/parallel-plate/grounded reciprocal/nonreciprocal waveguides and also for open reciprocal ones. Although no mapping has been found to remove all the branch points of the characteristic equation for open nonreciprocal waveguides with different half-spaces, a robust approach will be proposed to overcome the main drawbacks caused by the multivalued nature of this problem. The combination of the suitable formulation of the characteristic equation with a systematic integral-nature root-searching strategy bears a reliable and efficient method. Some novel numerical results will be presented for open anisotropic reciprocal waveguides and for open magnetized ferrite slabs to illustrate the performance of the present proposal. Index Terms—Leaky waves, nonreciprocal wave propagation, planar waveguides, root loci.

I. INTRODUCTION

P

LANAR layered waveguides are of great interest in microwave engineering and optics because of their use as the background structure for integrated circuitry. Hence, from early on, much effort has been devoted to study the propagation characteristics of the modal spectrum of these waveguides [1]–[27]. Additionally, that study is key to determine the characteristics of the Green’s function involved in the solution of scattering and propagation problems for antennas and high-frequency printed circuits by means of the integral-equation method. Specifically, the propagation constants of the waveguide determine the singularities of the spectral-domain Green’s function [9], and are also required in obtaining accurate closed-form expressions for the space-domain Green’s functions [28], [29]. Thus, numerous

Manuscript received July 15, 2004. This work was supported by the Spanish Ministry of Education and Science and by FEDER funds under Project TEC2004-03214. R. Rodríguez-Berral and F. Mesa are with the Grupo de Microondas, Departamento de Física Aplicada 1, ETS de Ingeniería Informática, Universidad de Sevilla, 41012-Seville, Spain (e-mail: [email protected]). F. Medina is with the Grupo de Microondas, Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla, 41012-Seville, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.847051

and diverse approaches have been proposed to deal with different kinds of planar waveguides: shielded, laterally open, and grounded reciprocal/nonreciprocal waveguides [2]–[12], [15], [16], [19]–[24], [27]. Planar waveguides opened to both (and eventually different) upper and lower half-spaces, henceforth, open planar waveguides, have also been analyzed, e.g., in [13], [17], [18], [20], [23], [25], [26]. Nevertheless, to the authors’ knowledge, previous studies have not considered the case of open waveguides whose layered substrate includes nonreciprocal materials (the case of open reciprocal anisotropic waveguides treated in [26] was restricted to anisotropic dielectric slabs in free space). The rich phenomenology provided by gyrotropic (e.g., magnetized ferrites) substrates in an open environment could be of potential practical interest to design, e.g., new types of radiators [30]–[32]. The great variety of approaches reported in the literature to study each specific type of planar waveguide can make it difficult to draw a general view of the problem under consideration, and to achieve a clear understanding of the similarities and differences existing between the application of a given approach to each kind of waveguide. Hence, this study will attempt to collect all of the possible different cases of planar layered waveguides, including the open nonreciprocal case, with the purpose of giving a general perspective of them within a congruent and efficient approach. The problem of finding the modal propagation constants of open planar layered waveguides will be reduced to a root-searching procedure in a similar way as was done in [19] for grounded structures. That paper posed a corresponding eigenvalue problem whose characteristic equation provided a formulation of the waveguide dispersion relation as the complex zeros of a complex function free of poles and branch points. The treatment of this eigenvalue problem will be extended here to deal with the case of open waveguides with different upper and lower half-spaces. Moreover, this paper will also study how the peculiarities of each kind of waveguide demands to pose its corresponding characteristic equation in a different and more appropriate mathematical form. Specifically, the further removal of the possible branch-point singularities of the characteristic function will be key to build a robust and reliable algorithm based on root-searching procedures. As is well known, these latter procedures do not work properly in those regions of the complex plane where the function is not

0018-9480/$20.00 © 2005 IEEE

1614

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

, and the characteristics matrix that depends only on of the layered substrate. The introduction of the corresponding upper and lower impedance/admittance boundary conditions allows us to write the following characteristic equation [19]: (2) where the (2

is given by (3)

N z

Fig. 1. Cross section of the planar -layers waveguide under study. The structure is infinite in both the - and -directions, and can be bounded by electric/magnetic walls or different lower and upper half-spaces.

x

2) matrix

analytic (it should be reminded that only analytic functions are complex differentiable at every point of the region). In this sense, a general discussion on the mappings that could succeed in removing the possible branch points of the characteristic function will be carried out for each kind of waveguides. As a result, the characteristic functions proposed in this study will be analytic for any kind of structure, except for the case of open nonreciprocal waveguides. Nevertheless a new robust approach will also be proposed for this latter case in spite of not having found a mapping to unfold the associated Riemann surface. The numerical efficiency of this proposal is also conveniently enhanced by the use of a integral-nature root-searching procedure [33], [34] following the systematic strategy reported in [27]. This paper is organized as follows. Section II will briefly outline the mathematical formulation of the problem. Section III will expose the scheme used to deal with reciprocal waveguides. This section will distinguish between the treatment for parallel-plate and/or grounded waveguides and that for open waveguides. Finally, the study of nonreciprocal structures will be presented in Section IV, also differentiating the grounded and open cases. II. FORMULATION OF THE PROBLEM The planar waveguide under consideration (see Fig. 1) is layers having the following general linear composed of constitutive relations: (1) where and are the (3 3) complex tensors accounting, respectively, for the electric permittivity, magnetic permeability, and optical activity. The upper and bottom boundaries of the waveguide can be electric and/or magnetic walls, and free-space and/or dielectric half-spaces. (The theory proposed here will also be able to deal with any type of boundary condition susceptible to be implemented in the spectral domain by means of impedance/admittance dyadics. This would also include, for example, the cases of lossy conductors under the approximation of the surface impedance, periodic boundary conditions, etc.) Assuming a common dependence of the fields of the type , and following [7], [9], [19], and [35]–[37], the transverse-to- fields at the and ) can be related by upper and lower interfaces ( , where , and is a (4 4)

being the (2 2) submatrices of matrix and the upper and lower impedance matrices. (Other equivalent formulations are possible, also involving the admittance matrices instead of the impedance ones.) In the following, and without loss of generality, the - and -axes orientation will be chosen such that propagation always takes place along (for periodic or laterally the -direction assuming should be taken so as to satisfy the corshielded structures, responding lateral boundary conditions). For isotropic and/or anisotropic/gyrotropic materials with their main axis along the -direction, the above assumption is irrelevant since the and -directions are completely equivalent. For more general cases, a change in - and -axes orientation would only imply a rotation of the characteristic tensors around the -axis. with

III. RECIPROCAL WAVEGUIDES Before discussing the formulation of the dispersion equation for reciprocal planar waveguides, it is convenient to note that the reciprocity of these structures causes the elements of mato be invariant under inversion of the propagation ditrix rection; namely, they are even functions of . In addition, the uniqueness theorem assures that the fields at any point within the multilayer are uniquely determined by the transverse fields at the lower interface. As a consequence, the elements in maare single-valued functions of and, therefore, the trix branch points of the characteristic function (if any) will arise from the impedance matrices. Finally, it should be recalled that do not have poles for any specific the elements of matrix form of the characteristic tensors [19], whereas the elements of for half-space the impedance matrices are even functions of boundary conditions [38]. A. Parallel-Plate and Grounded Reciprocal Waveguides Since parallel-plate waveguides have null upper and lower impedance matrices, the characteristic function for this case can be written as (4) which is fully analytic in the complex -plane. Thus, there is no need for a mapping in order to use other computational variables (i.e., the independent variable in which the characteristic equation is finally posed) than —the square of the longitudinal wavenumber . The root-searching strategy presented in [27] can then be straightforwardly applied to the characteristic function in (4). (The choice for the computational variable reported in [27] would not make sense for anisotropic reciprocal

RODRÍGUEZ-BERRAL et al.: FORMULATION OF CHARACTERISTIC EQUATION FOR OPEN NONRECIPROCAL LAYERED WAVEGUIDES

waveguides because of the presence of ordinary and extraordinary rays.) does For grounded structures, the determinant of matrix present a branch point in the complex plane at [38], with and being the permittivity and where permeability of the upper half-space, respectively (grounded waveguides will be always considered to have null impedance at the lower interface). The two-sheeted nature of the Riemann surplane comes from the two posface defined in the complex sible choices in the sign of the vertical wavenumber in the upper for each value of . Taking half-space, into account the even nature of the elements in matrices and , it can be asserted that the mapping (5) will remove the branch points for reciprocal grounded waveguides. Thus, the following analytic characteristic function is proposed: (6) has been introduced to rewhere the multiplicative factor arising from the upper half-space move the pole at impedance matrix. B. Open Reciprocal Waveguides For open reciprocal waveguides, has two branch and points in the complex -plane located at [13], [17], [18], [20], where with and being the lower half-space permittivity and permeability. The reason for the new branch point comes from the ambiguity in the sign of the vertical wavenumber in the lower half-space . Hence, the complex -plane is now a Riemann surface that comprises four sheets, one for each comand . In order to find a mapping bination of the signs of that unfolds this Riemann surface, the new variable should be and so that it contains all chosen as a combination of the information concerning the signs of these two variables. and Looking for simple combinations, the product of must be discarded because it takes the same value for different . The same combination of signs, e.g., can be adduced for , which, in addition, has a singular . Therefore, the following linear combination point at of these two variables will be next considered and examined:

1615

. In consewhich is a second-order complex polynomial in corresponding to each value quence, there are two values of . Thus, considering the latter relation beof unless tween and , mapping (7) must have the form (9) thus leading to the following single-valued expression of a function of :

as

(10) It should be noted that the sign in (10) does not indicate any ambiguity in the sign of , but is simply the sign choice in (9). Looking at (10), it can be concluded that an important feature to is that this function must not have be satisfied by function would diverge bezeros for any finite value of (otherwise, cause of the appearance of a singular point in the mapping). The can be also equivalently previous analysis involving variable carried out using variable , leading to the same requirements and . The expression of as a function of is for found to be (11) It can then be concluded that any mapping having the form being an analytical function without given in (9), with zeros, is regular and removes the branch points of the characteristic function for open reciprocal waveguides. For example, the family of mappings proposed in [20] is a particular case of the general mapping in (9). Our specific choice will be (12) which is actually very similar to that proposed in [18]. Nevertheless, it should be pointed out that our proposal has followed a different rationale that can complement the previous discussions on this subject. Using (12), the expressions relating and to are (13) (14) (15) which would lead to the following analytic expression of the characteristic function for these structures:

(7)

(16)

is a given analytic function and are two comwhere plex constants. For this mapping to unfold the Riemann surand corface, there must be an unique value of both and responding to each value of . In other words, both must be single-valued functions of . Taking into account that , where , mapping (7) can be written as

where the determinant of matrix has been multiplied by and in order to remove the poles at and arising from the upper and lower impedance matrices. The basic reason for our proposal (12) is that it leads to a straightforward mapping of the different sheets of the original Riemann surface onto the complex plane (see Fig. 2). Since the complex logarithm is a multivalued function with an infinite number of branches, there will be an infinite number of values and [18], of corresponding to each pair of values of

(8)

1616

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

Fig. 3. Normalized phase constants of the first modes of an open anisotropic planar slab, previously analyzed in [26, Fig. 2] with h = 2 = 2:25" ; " = 4" , and the optical axis oriented with cm, "  = 45 ;  = 60 . Solid lines: our results. Points: data in [26]. Black color lines and points stand for proper modes, grey color for improper modes. Fig. 2. Schematic representation of the mapping in (12) for: (a) k > k and (b) k > k . In each zone of the region of interest of the complex  -plane, the power flow along the vertical direction is indicated by arrows and the exponential behavior of the fields by increasing/decreasing curves.

[20]. However, this fact does not pose any troubles provided that the principal value of the logarithm is always taken, thus, re. The eight zones of stricting the domain to Fig. 2 have been labeled in a manner similar to [18]: the arrows represent the power flow along the vertical direction, whereas the curves represent the exponential behavior of the fields. Note or bethat the mapping depends on whether cause will be respectively real or imaginary (in both cases, the nonzero part of is chosen to be positive). Bound modes and both imag(BMs) in lossless waveguides must have and and, therefore, they will inary with segment of the be located on the complex -plane for and on for . As losses increase, BMs will move from their original location to the right entering the and zones, respectively. The particular case of reciprocal waveguides having identical upper and lower half-spaces must be treated separately since and when mapping (9) would lead to . This drawback can be overcome by choosing as the computational variable, which would yield a Riemann surface for the characteristic function comprising two unconnected sheets [13]. This can be understood as an splitting of the characteristic function into the two following functions: (17)

both of them being analytic in the entire complex -plane. , accounts for BMs and for improper modes The first one, whose corresponding fields grow both upwards in the upper half-space and downwards in the lower half-space. On the other correspond to improper modes whose hand, the zeroes of related fields diverge only in one of the two half-spaces. Once the fully analytic characteristic functions (4), (6), (16), and (17) have been formulated, the root-searching strategy reported in [27] can be now applied to these functions to compute the propagation constant for the modes of reciprocal waveguides. First, our results will be checked versus some previous results reported in the literature. A first comparison has been carried out with the results presented in [25, Table I] (which, in turn, compares with [14]) for the propagation constant of several leaky modes in a four-layer open isotropic waveguide with different upper and lower half-spaces. An excellent agreement of more than eight identical digits has been always found between our results and those in [25]. Next, a comparison is shown in Fig. 3 for the reciprocal anisotropic dielectric slab surrounded by free space previously studied in [26, Fig. 2]. The slab shows uniaxial anisotropy along the optical axis and along any with permittivity direction perpendicular to the optical axis (the orientation of the optical axis is shown in Fig. 3). An excellent agreement has been also found in this case between our numerical results and those reported in [26] for both the proper and improper modes. This figure includes an additional fundamental mode that was not considered in [26]. As a new case study, the proposed method will be now used to compute the complete dispersion relation for the BMs of an anisotropic dielectric layered waveguide with different lower

RODRÍGUEZ-BERRAL et al.: FORMULATION OF CHARACTERISTIC EQUATION FOR OPEN NONRECIPROCAL LAYERED WAVEGUIDES

1617

Fig. 4. Normalized propagation constants of the BMs of an open anisotropic reciprocal dielectric waveguide with h = h = h = h = 2 mm, " = = 4" ; " = 9:8" ; " = 2:2" ; " = " ; " = 4" ; " = 2:25" ; " 12" ;  =  = 45 .

and upper half-spaces. Thus, the layered substrate considered in Fig. 4 has four layers, three of them being isotropic and one presenting uniaxial anisotropy with a tilted optical axis (the orientation of the optical axis has been chosen arbitrarily as ). The data in Fig. 4 have been computed by applying the root-searching algorithm presented in [27] to the analytic characteristic function (16) along the segment of the (which corresponds to complex -plane between ) and (i.e., ). It should be pointed out that all the data corresponding to the 22 modes plotted in Fig. 4 have been obtained automatically without any interaction with the program during execution. Moreover, the algorithm is reliable even in the zones where two or more modes are very close, and no BM is lost in the considered frequency range. In addition, our strategy turns out to be remarkably efficient: the computation of the approximately 9500 values of the propagation constant plotted in Fig. 4 took approximately 11 min on a Pentium IV platform at 2 GHz. An interesting feature of the dispersion curves presented in Fig. 4 is that no BMs are found for frequencies below 2.6 GHz, although the layers of the analyzed structure have permittivities considerably higher than those for and the lower and upper half-spaces ( versus and ). The above fact can be better appreciated in Fig. 5(a), which shows the evolution as frequency decreases from 15 to 1 GHz of the imaginary and real parts of for the first three BMs and an improper mode of the structure considered in Fig. 4. It can be observed that the second and third modes that are present and modes, respectively) couple toat 15 GHz (the gether for a frequency range under 13 GHz. For lower frequenmode as a consecies, the third mode (which is now the

Fig. 5. Frequency evolution of: (a) the imaginary and real parts of  and (b) the normalized phase constant of the HE , EH , and HE modes of the open reciprocal waveguide previously analyzed in Fig. 4. Black solid lines stand for proper mode, black dotted lines for IRM, and grey lines for ICM.

quence of the coupling) keeps on being a proper mode until, at approximately 12.1 GHz, it crosses the real axis of the complex plane to enter the zone [see Fig. 2(b)] where it turns into an improper real mode (IRM). This improper mode meets another IRM at 11.9 GHz to give rise to a pair of improper and oppocomplex modes (ICMs) with the same value of site values of , namely, they are complex conjugate in the plane. As frequency decreases down to 1 GHz, this pair of ICMs stays in the same zone of the complex plane without any further change in the signs of the real and imaginary parts and . It can also be observed that the and of modes transition into IRMs at 8.1 and 2.6 GHz, respectively. It should be highlighted that the computation of the propagation constant in the neighborhood of a BM-to-IRM modal transition has not raised any troubles because of the appropriate choice of the independent variable. Otherwise, these transitions would have implied the excursion of the modal solution into a different Riemann sheet with the consequent complication in the tracking procedure. This fact is illustrated in Fig. 5(b), which shows the frequency evolution of the normalized phase constants of the modes plotted in Fig. 5(a). This figure shows how the BM’s , where they cross the cortransition into IRMs at responding branch cut in the complex -plane to go into the improper sheet. In a similar fashion, it has been the specific root-searching strategy employed here that has allowed for a

1618

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 5, MAY 2005

any composition of the layered slab. Thus, and according to the above rationale, it can be concluded that there are not fundamental modes in any kind of layered slab when the upper and lower half-spaces are different. IV. NONRECIPROCAL WAVEGUIDES

Fig. 6. Normalized propagation constant