JANUARY 2011 IEEE MTT-V059-I01 (2011-01) [59, 1 ed.]

Citation preview

JANUARY 2011

VOLUME 59

NUMBER 1

IETMAB

(ISSN 0018-9480)

PAPERS

Theory and Numerical Methods A Semianalytical Spectral Element Method for the Analysis of 3-D Layered Structures ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ...... J. Chen, B. Zhu, W. Zhong, and Q. H. Liu A Common Electromagnetic Framework for Carbon Nanotubes and Solid Nanowires—Spatially Dispersive Conductivity, Generalized Ohm’s Law, Distributed Impedance, and Transmission Line Model ..... ......... ........ ... G. W. Hanson Neural-Network Modeling for 3-D Substructures Based on Spatial EM-Field Coupling in Finite-Element Method .... .. .. ........ ......... ......... ........ ......... ......... ........ .... S. Liao, H. Kabir, Y. Cao, J. Xu, Q.-J. Zhang, and J.-G. Ma Adaptive Sampling Algorithm for Macromodeling of Parameterized -Parameter Responses .. ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ... D. Deschrijver, K. Crombecq, H. M. Nguyen, and T. Dhaene A General Approach for Sensitivity Analysis of Distributed Interconnects in the Time Domain ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... N. Nakhla, M. Nakhla, and T. Achar Efficient Implementation for 3-D Laguerre-Based Finite-Difference Time-Domain Method .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ...... Y.-T. Duan, B. Chen, D.-G. Fang, and B.-H. Zhou Passive Components and Circuits Linear Circuit Models for On-Chip Quantum Electrodynamics .. ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ .... A. Mátyás, C. Jirauschek, F. Peretti, P. Lugli, and G. Csaba A Planar Magic-T Structure Using Substrate Integrated Circuits Concept and Its Mixer Applications . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... F. F. He, K. Wu, W. Hong, L. Han, and X. Chen Exact Synthesis and Implementation of New High-Order Wideband Marchand Baluns . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... J.-C. Lu, C.-C. Lin, and C.-Y. Chang Microstrip Branch-Line Couplers for Crossover Application .... ......... ........ ......... ..... J. Yao, C. Lee, and S. P. Yeo Modeling the Effects of Interference Suppression Filters on Ultra-Wideband Pulses .... .. A. E.-C. Tan and K. Rambabu

1 9 21 39 46 56

65 72 80 87 93

(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Hybrid and Monolithic RF Integrated Circuits A Distributed Dual-Band LC Oscillator Based on Mode Switching ...... ........ ......... ......... .... G. Li and E. Afshari Broadband Differential Low-Noise Amplifier for Active Differential Arrays ... ......... ......... ........ ......... ......... .. .. ........ .. O. García-Pérez, D. Segovia-Vargas, L. E. García-Muñoz, J. L. Jiménez-Martín, and V. González-Posadas A 0.18- m Dual-Gate CMOS Device Modeling and Applications for RF Cascode Circuits ..... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ... H.-Y. Chang and K.-H. Liang Six-Port Gigabit Demodulator ... ........ J. Östh, A. Serban, Owais, M. Karlsson, S. Gong, J. Haartsen, and P. Karlsson Differential Amplifier Characterization Using Mixed-Mode Scattering Parameters Obtained From True and Virtual Differential Measurements .... ......... .... ...... ........ . O. Schmitz, S. K. Hampel, H. Rabe, T. Reinecke, and I. Rolfes Efficiency Enhancement of Doherty Amplifier Through Mitigation of the Knee Voltage Effect . ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... J. Moon, J. Kim, J. Kim, I. Kim, and B. Kim Modeling and Design Methodology of High-Efficiency Class-F and Class-F- Power Amplifiers ...... ......... ......... .. .. ........ ......... ......... ........ ......... ......... .... J. H. Kim, G. D. Jo, J. H. Oh, Y. H. Kim, K. C. Lee, and J. H. Jung Design and Performance of a 600–720-GHz Sideband-Separating Receiver Using AlO and AlN SIS Junctions ....... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ....... F. P. Mena, J. W. Kooi, A. M. Baryshev, C. F. J. Lodewijk, T. Zijlstra, R. Hesper, G. Gerlofsma, T. M. Klapwijk, and W. Wild Instrumentation and Measurement Techniques A Calibration Method for RF and Microwave Noise Sources .... ......... ........ ...... .... ......... ........ .... L. Belostotski A Compact Variable-Temperature Broadband Series-Resistor Calibration ...... ......... ......... ........ ......... ......... .. .. .. N. D. Orloff, J. Mateu, A. Lewandowski, E. Rocas, J. King, D. Gu, X. Lu, C. Collado, I. Takeuchi, and J. C. Booth RF Applications and Systems Flicker Noise Effects in Noise Adding Radiometers ..... ......... ......... ........ ......... .... J. J. Lynch and R. G. Nagele

99 108 116 125 132 143 153 166 178 188 196

LETTERS

Corrections to “Editorial” ........ ......... ......... ........ ......... ......... ........ ......... .. D. Williams and A. Mortazawi Corrections to “Wideband IF-Integrated Terahertz HEB Mixers: Modeling and Characterization” ..... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .... F. Rodriguez-Morales, K. S. Yngvesson, and D. Gu Corrections to “Impedance-Transforming Symmetric and Asymmetric DC Blocks” .... ......... ... H.-R. Ahn and T. Itoh

206

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

208

206 207

CALLS FOR PAPERS

Special Issue on Millimeter-Wave Circuits and Systems ......... ......... ........ ......... . ........ ........ ......... ......... .

209

IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY The Microwave Theory and Techniques Society is an organization, within the framework of the IEEE, of members with principal professional interests in the field of microwave theory and techniques. All members of the IEEE are eligible for membership in the Society upon payment of the annual Society membership fee of $17.00, plus an annual subscription fee of $23.00 per year for electronic media only or $46.00 per year for electronic and print media. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. 2010 ADMINISTRATIVE COMMITTEE S. M. EL-GHAZALY, President L. BOGLIONE W. CHAPPELL M. GUPTA

R. SNYDER, President Elect

J. HACKER M. HARRIS J. HAUSNER

K. ITOH S. KOUL J. LASKAR

T. LEE J. LIN

M. MADIHIAN, Secretary A. MORTAZAWI V. J. NAIR

Y. NIKAWA G. PONCHAK

Honorary Life Members T. ITOH A. A. OLINER

T. S. SAAD P. STAECKER

N. KOLIAS, Treasurer D. SCHREURS W. SHIROMA

R. SNYDER B. SZENDRENYI

Distinguished Lecturers K. TOMIYASU L. YOUNG

A. CANGELLARIS S. GEVORGIAN F. ELLINGER F. GHANNOUCHI A. FERRERO S. LUCYSZYN

A. PHAM P. TASKER M. TENTZERIS

R. WEIGEL K. WU

Past Presidents K. WU M. YU

B. PERLMAN (2009) J. MODELSKI (2008) J. S. KENNEY (2007)

2010 MTT-S Chapter Chairs Albuquerque: H. J. WAGNON Atlanta: D. LEATHERWOOD Austria: A. SPRINGER Baltimore: N. BUSHYAGER Bangalore: T. SRINIVAS Beijing: Z. FENG Belarus: A. GUSINSKY Benelux: D. VANHOENACKER-JANVIER Boston: J. MULDAVIN Brasilia: J. DA COSTA/ A. KLAUTAU Buenaventura: M. QUDDUS Buffalo: J. WHALEN Bulgaria: K. ASPARUHOVA Cedar Rapids/Central Iowa: M. ROY Central & South Italy: G. D’INZEO Central No. Carolina: N. S. DOGAN Chengdu: Z. NEI Chicago: H. LIU Cleveland: M. SCARDELLETTI Columbus: F. TEXEIRA Connecticut: C. BLAIR Croatia: Z. SIPUS Czech/Slovakia: P. HAZDRA Dallas: Q. ZHANG Dayton: A. TERZUOLI

Delhi/India: S. KOUL Denver: M. JANEZIC Eastern No. Carolina: T. NICHOLS Egypt: E. HASHISH Finland: A. LUUKANEN Florida West Coast: K. A. O’CONNOR Foothills: F. FREYNE France: P. EUDELINE Germany: K. SOLBACH Greece: R. MAKRI Harbin: Q. WU Hawaii: R. MIYAMOTO Hong Kong: W. S. CHAN Houston: J. T. WILLIAMS Houston, College Station: G. H. HUFF Hungary: T. BERCELI Huntsville: H. G. SCHANTZ Hyderabad: M. CHAKRAVARTI India/Calcutta: D. GUHA India: D. BHATNAGER Indonesia: E. T. RAHARDO Israel: S. AUSTER Japan: K. ARAKI Kansai: T. OHIRA Kitchener-Waterloo: R. R. MANSOUR Lithuania: V. URBANAVICIUS

Editor-In-Chief GEORGE E. PONCHAK NASA Glenn Research Center Cleveland, OH USA Editorial Assistant LINDA GAYDOSH OAI USA

HERBERT ZIRATH Chalmers Univ. Technol. Goteborg, Sweden WENDY VAN MOER Vrije Universiteit Brussel Brussels JAE-SUNG RIEH Korea Univ. Seoul, Korea QUAN XUE City Univ. Hong Kong Hong Kong LEI ZHU Nanyang Technol. Univ. Singapore K. REMLEY, Editor-in-Chief, IEEE Microwave Magazine

Long Island/New York: J. COLOTTI Los Angeles, Coastal: W. DEAL Los Angeles, Metro/San Fernando: F. MAIWALD Malaysia: M. ESA Malaysia, Penang: Y. CHOW Melbourne: K. LAMP Mexico: R. M. RODRIGUES-DAGNINO Milwaukee: S. G. JOSHI Mohawk Valley: E. P. RATAZZI Montreal: K. WU Nanjing: W. X. ZHANG New Hampshire: D. SHERWOOD New Jersey Coast: D. REYNOLDS New South Wales: A. M. SANAGAVARAPU New Zealand: A. WILLIAMSON North Italy: G. VECCHI North Jersey: H. DAYAL/K. DIXIT Northern Australia: M. JACOB Northern Nevada: B. S. RAWAT Norway: Y. THODESEN Orange County: H. J. DE LOS SANTOS Oregon: T. RUTTAN Orlando: X. GONG Ottawa: Q. YE Philadelphia: J. NACHAMKIN Phoenix: S. ROCKWELL Poland: W. J. KRZYSZTOFIK

Portugal: C. PEIXEIRO Princeton/Central Jersey: A. KATZ Queensland: A. RAKIC Rio de Janeiro: J. BERGMANN Rochester: S. CICCARELLI/ J. VENKATARAMAN Romania: G. LOJEWSKI Russia, Moscow: V. A. KALOSHIN Russia, Nizhny: Y. BELOV Russia, Novosibirsk: A. GRIDCHIN Russia, Saint Petersburg: M. SITNIKOVA Russia, Saratov: N. M. RYSKIN Russia, Tomsk: R. V. MESCHERIAKOV Saint Louis: D. MACKE San Diego: G. TWOMEY Santa Clara Valley/San Francisco: M. SAYED Seattle: K. A. POULSON Seoul: S. NAM Serbia and Montenegro: A. MARINCIC Shanghai: J. F. MAO Singapore: A. ALPHONES South Africa: C. VAN NIEKIRK South Australia: H. HANSON South Brazil: R. GARCIA Southeastern Michigan: T. OZDEMIR Southern Alberta: E. FEAR Spain: J. I. ALONSO

Springfield: P. R. SIQUEIRA Sweden: A. RYDBERG Switzerland: M. MATTES Syracuse: E. ARVAS Taegu: Y.-H. JEONG Taipei: F.-T. TSAI Thailand: P. AKKARAEKTHALIN Toronto: G. V. ELEFTHERIADES Tucson: N. BURGESS Turkey: I. TEKIN Twin Cities: M. J. GAWRONSKI UK/RI: A. REZAZADEH Ukraine, Kiev: Y. POPLAVKO Ukraine, East, Kharkov: O. V. SHRAMKOVA Ukraine, East Student Branch Chapter, Kharkov: M. KRUSLOV Ukraine, Rep. of Georgia: D. KAKULIA Ukraine, Vinnitsya: V. DUBOVOY Ukraine, West, Lviv: I. ISAYEV ˇ Venezuela: J. PENA Victoria: K. GHORBANI Virginia Mountain: T. A. WINSLOW Washington DC/Northern Virginia: J. QIU Winnipeg: V. OKHMATOVSKI

Associate Editors KEVIN J. CHEN N. SCOTT BARKER Hong Kong Univ. Sci. Technol. Univ. Virginia Hong Kong Charlottesville, VA USA MING YU COSTAS D. SARRIS COM DEV Univ. Toronto. Cambridge, ON, Canada Toronto, ON, Canada CHIN-WEN TANG CHRISTOPHE FUMEAUX Nat. Chung Cheng Univ. The Univ. Adelaide Taiwan Adelaide, South Australia, Australia BART NAUWELAERS DEUKHYOUN HEO ESAT-TELEMIC Washington State Univ. Belgie, Belgium Pullman, WA USA JOHN PAPAPOLYMEROU Georgia Inst. Technol. Atlanta, GA USA C. TZUANG, Editor-in-Chief, IEEE Microwave and Wireless Component Letters T. LEE, Web Master

IEEE Officers DAVID A. HODGES, Vice President, Publication Services and Products MOSHE KAM, President GORDON W. DAY, President-Elect HOWARD E. MICHEL, Vice President, Member and Geographic Activities ROGER D. POLLARD, Secretary STEVE M. MILLS, President, Standards Association HAROLD FLESCHER, Treasurer DONNA L. HUDSON, Vice President, Technical Activities PEDRO A. RAY, Past President RONALD G. JENSEN, President, IEEE-USA TARIQ S. DURRANI, Vice President, Educational Activities PETER N. CLOUT, Director, Division IV—Electromagnetics and Radiation

IEEE Executive Staff DR. E. JAMES PRENDERGAST, Executive Director & Chief Operating Officer PATRICK MAHONEY, Marketing THOMAS SIEGERT, Business Administration CECELIA JANKOWSKI, Member and Geographic Activities MATTHEW LOEB, Corporate Activities ANTHONY DURNIAK, Publications Activities DOUGLAS GORHAM, Educational Activities JUDITH GORMAN, Standards Activities BETSY DAVIS, SPHR, Human Resources MARY WARD-CALLAN, Technical Activities CHRIS BRANTLEY, IEEE-USA ALEXANDER PASIK, Information Technology IEEE Periodicals Transactions/Journals Department Staff Director: FRAN ZAPPULLA Editorial Director: DAWN MELLEY Production Director: PETER M. TUOHY 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, Piscataway, NJ 08854-4141. NJ Telephone: +1 732 981 0060. Price/Publication Information: Individual copies: IEEE Members $20.00 (first copy only), nonmember $133.00 per copy. (Note: Postage and handling charge not included.) Member and nonmember subscription prices available upon request. 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, Piscataway, NJ 08854-4141. Copyright © 2011 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, Piscataway, NJ 08854-4141. GST Registration No. 125634188. CPC Sales Agreement #40013087. Return undeliverable Canada addresses to: Pitney Bowes IMEX, P.O. Box 4332, Stanton Rd., Toronto, ON M5W 3J4, Canada. IEEE prohibits discrimination, harassment and bullying. For more information visit http://www.ieee.org/nondiscrimination. Printed in U.S.A. Front cover photograph courtesy of LylePhotos.com, Atlanta, GA, used by permission.

Digital Object Identifier 10.1109/TMTT.2010.2103871

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

1

A Semianalytical Spectral Element Method for the Analysis of 3-D Layered Structures Jiefu Chen, Bao Zhu, Wanxie Zhong, and Qing Huo Liu, Fellow, IEEE

Abstract—A semianalytical spectral element method (SEM) is proposed for electromagnetic simulations of 3-D layered structures. 2-D spectral elements are employed to discretize the cross section of a layered structure, and the Legendre transformation is then used to cast the semidiscretized problem from the Lagrangian system into the Hamiltonian system. A Riccati equation-based high precision integration method is utilized to perform integration along the longitudinal direction, which is the undiscretized direction, to generate the stiffness matrix of the whole layered structure. The final system of equations by the semianalytical SEM will take the form of a set of linear equations with a block tri-diagonal matrix, which can be solved efficiently by the block Thomas algorithm. Numerical examples demonstrate the high efficiency and accuracy of the proposed method. Index Terms—Block Thomas algorithm, block tri-diagonal matrix, finite-element method (FEM), Gauss–Lobatto–Legendre (GLL) polynomials, Hamiltonian system, high precision integration (HPI) method, Lagrangian system, layered structure, Riccati equation, semianalytical spectral element method (SEM). Fig. 1. Interconnect problem is a typical layered structure. The structure can be divided into layers. The geometry and material distribution of each layer are homogeneous along the longitudinal direction, but can be arbitrary in the transverse plane.

N

I. INTRODUCTION LECTROMAGNETIC simulations of layered structures are frequently encountered in many areas such as integrated optics, geophysical prospecting, and electronic packaging [1]–[3]. As shown in Fig. 1, the interconnect structure in packaging problems is a typical layered structure. It contains several parallel layers along a specific direction, which is referred to as the longitudinal direction hereinafter. Each layer is homogeneous along the longitudinal direction; while its geometry as well as material distribution can be arbitrary on the transverse plane, i.e., the plane perpendicular to the longitudinal direction. Due to the flexibility in geometric modeling, the finite-element method (FEM) can be employed to perform full wave analysis, and thus to obtain the electrical properties of layered structures. However, as the number of layers and the complexity of each layer increase, directly using the FEM to discretize the whole structure may lead to a huge system of coupled equations, thus making the overall efficiency of conventional FEM very low for the analysis of layered structures.

E

Manuscript received October 15, 2009; accepted September 01, 2010. Date of publication November 15, 2010; date of current version January 12, 2011. J. Chen and Q. H. Liu are with the Department of Electrical and Computer Engineering, Duke University, Durham NC 27708-0291 USA (e-mail: jiefu. [email protected]; [email protected]). B. Zhu and W. Zhong are with the Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2090408

The piecewise homogeneity of layered structures along the longitudinal direction can be exploited to improve the efficiency of the conventional FEM. The numerical mode-matching method (NMM) [4], [5] breaks a whole 3-D (or 2-D) layered regions. After obtaining the eigenmodes of structure into each region by 2-D FEM, the -region wave propagation problem can be solved by introducing generalized reflection and transmission operators. The NMM can reduce a 3-D problem into several 2-D problems, thus greatly decreasing the number of unknowns, and consequently, memory cost and computation time. This method has been applied to various geophysical subsurface sensing problems and open-region waveguide discontinuity problems [6]–[8]. The layered finite element (LAFE) method is another efficient FEM method designed for layered structures [9]–[11]. The LAFE method starts from a 3-D mesh for a 3-D layered structure, and then reduces a 3-D layered system into a 2-D layered system by eliminating volume unknowns, which will be recovered after the 2-D layered system is solved. This method can be used in both time- and frequency-domain simulations. Here, we proposed a semianalytical spectral element method (SEM) for the analysis of layered structures. A piecewise homogeneous 3-D layered structure is first divided into several substructures homogeneous along the longitudinal direction. 2-D scalar and vector spectral elements [12]–[14], which are special types of a higher order FEM, are used to represent longitudinal and transverse unknowns on the cross section of each

0018-9480/$26.00 © 2010 IEEE

2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

substructure, respectively. The semidiscretized system is then transformed from the Lagrangian system into the Hamiltonian system, where a Riccati equation-based high precision integration (HPI) [15]–[17] method is utilized to perform integration along the the longitudinal direction and to generate the stiffness matrix of a substructure. No matter how long a substructure is, HPI for the semidiscretized system can achieve machine precision, i.e., the numerical errors of longitudinal integration by HPI can be as small as the round-off error on a computer. Stiffness matrices of substructures can be directly assembled to a global system matrix taking the form of a block tri-diagonal matrix. A block Thomas algorithm [18] is employed here to solve the final system of equations with high efficiency. In this paper, we will discuss the proposed method as:. Section II governing equation and corresponding functional, Section III 2-D spectral elements for discretization of substructure’s cross section, Section IV semidiscretized system in the Lagrangian system, Section V Hamiltonian system and Riccati equations, Section VI HPI method, Section VII block Thomas algorithm for block tri-diagonal matrix, Section VIII numerical examples, and Section IX conclusion.

Fig. 2. One layer of a layered structure is treated as a substructure and modeled by the semianalytical SEM. This substructure is homogeneous along the z direction, but can be inhomogeneous along the x and y directions.

the cross section and then let the outermost circumference be a PEC or PMC. The one-variable functional corresponding to the above governing equation and boundary conditions is

(4) II. GOVERNING EQUATION AND FUNCTIONAL Consider an -layer piecewise homogeneous structures shown in Fig. 1. Within each layer, the geometry and material distribution are homogeneous along the longitudinal direction, but can be arbitrary on the cross section. For such a structure, layers along the longitudinal we can decompose it into direction and treat each layer as a substructure. Each substructure is modeled by the proposed semianalytical SEM and then be assembled into a global discretized system of the whole structure. Details are elaborated as follows. Consider a substructure as shown in Fig. 2. denotes the plane. cross section of the substructure, which is laid on the The substructure has a finite length along the axis (longitudinal direction). The coordinates of the left and right end are assumed as and , respectively. We first omit the excitation terms, which will be imposed onto the global discretized system after all substructures are assembled together. The governing equation for the substructure is

and right with the unspecified fields on the left end . end Field can be decomposed into transverse and longitudinal components (5) where (6) Operator can also be decomposed into a transverse operator and longitudinal operator (7) where (8)

in

(1) With (5) and (7), the functional (4) can be expressed as

where and denote the relative permittivity and relative permeability, respectively. denotes the wavenumber in the vacuum. Without loss of generality, we can always assume the circumference of the substructure is comprised by a perfect and perfect magnetic conductor electric conductor (PEC) (PMC) , i.e., on on

(2) (3)

For open region problems (with respect to the and direction), we can always apply a perfectly matched layer (PML) to enclose

(9) where

.

CHEN et al.: SEMIANALYTICAL SEM FOR ANALYSIS OF 3-D LAYERED STRUCTURES

3

is the Legendre polynomial of degree of and is its derivative, and is chosen as the roots of . is similar to , but for a different coordinate variable. Within each element, the transverse and longitudinal field components can be expressed as (13) (14)

Fig. 3. Schematics of 2-D spectral elements for semianalytical SEM. The 2-D scalar spectral element is shown in (a), which is used to represent longitudinal field components. The 2-D vector spectral element consists of (b) and (c), which are employed to represent transverse field components. Both scalar and vector spectral elements are defined in a unit square in the reference domain, and they can be curved quadrilaterals in the physical domain after geometric transformais illustrated in this figure. tion.

M =3

III. 2-D SPECTRAL ELEMENTS FOR CROSS SECTION 2-D spectral elements are used to discretize the substructure’s cross section. More specifically, 2-D vector spectral element and scalar spectral element are employed to discretize the transand the longitudinal component , reverse component spectively. The spectral element [12]–[14] is a special type of higher order finite element with sampling points defined as the Gauss–Lobatto–Legendre (GLL) points, which are roots of the derivatives of the GLL polynomials. By choosing GLL points rather than equal-spaced grids as sampling points, the spectral element can avoid the well-known Runge phenomenon [19] and achieve spectral accuracy, which means the numerical results can converge exponentially as the increase of interpolation order of basis functions. As shown in Fig. 3, the 2-D spectral elements are defined . The basis in the reference domain function of the 2-D scalar spectral element with th order of interpolation is defined as (10) and the basis function of the 2-D vector spectral element contains two components

where and denote the degree of freedom of the vector spectral element and scalar spectral element, respectively. Since the spectral elements are constructed in the reference domain, not in the physical domain, co-variant and contra-variant transformations should be employed to the discretized fields and their derivatives. Details are referred to [20] and not elaborated here. IV. SEMIDISCRETIZED SYSTEM AND STIFFNESS MATRIX IN THE LAGRANGIAN SYSTEM Discretizing the cross section of the substructure by the aforementioned spectral elements, we will obtain a semidiscretized functional as

(15) where

(16)

(17) (18) (19) Applying the circumferential boundary conditions and taking the variation for longitudinal fields, we can obtain an expression as of

(11) (20) where

By inserting the above expression into (15) and eliminating the longitudinal components , the original semidiscretized functional can be cast into the Lagrangian system (12)

(21)

4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

where is the vector of the discretized transverse field, and it is renamed here as the generalized displacement following is the Lagrangian function the classic mechanic terms.

where (30)

(22)

The Legendre transformation can also be applied to the discretized functional to generate a new functional and based on variables

(23)

(31)

where

and (24) Since the circumferential boundary conditions are already imposed onto (21), based on the uniqueness theorem [21], all the field values within the substructure are determined if the tangential (viz. the transverse) components on the left and right end are specified (25) Based on the above conclusion, we know that if the longitudinal integration is carried out for (21), the resultant discretized functional should be a quadratic function of and , i.e., (26) , , and are the results of perwhere the matrices forming longitudinal integration for and in the Lagrangian function. After we calculate the values of these three matrices, the global stiffness matrix of this substructure is immediately available (27) Once the stiffness matrices in are obtained, we can assemble the stiffness matrices of all substructures together into a global system matrix. Solving a system of equations with this global matrix, we will obtain the numerical results for the whole layered structure.

By taking variation of the above functional, we will have (32) and (33) From (31)–(33), we know the new functional should be a quadratic function of and , i.e., (34) Based on (26), (31), (33), and (34), a set of mutual transformation relationships can be found between matrices , , , and matrices , ,

(35) (36) Equation (35) shows that instead of directly integrating (21) to obtain the stiffness matrix, we can first calculate matrices , , , , and . and , and then transform them to For a substructure whose distributions of geometry and material are homogeneous along the direction, the matrices , , and are determined by matrices and in the Hamiltonian function, and the length of the substructure (37) It has been proven that the above matrix functions satisfy the Riccati equations [15]

V. HAMILTONIAN SYSTEM AND RICCATI EQUATIONS , and can be obTheoretically, the matrices tained by integrating the Lagrangian function. However, the accuracy cannot be guaranteed if we directly employ numerical integration methods to (21). To calculate the values of the stiffness matrix with high accuracy and construct the so-called semianalytical SEM, we will use a Riccati equation-based HPI method [15], which is performed in the Hamiltonian system. We first introduce a generalized force by Legendre transformation (28) and we will obtain the Hamiltonian function as (29)

(38)

with initial conditions (39) where and denote the zero matrix and identity matrix with the same size as matrices , , and , respectively. The above Riccati equations seem very complex at first glance, however, they can be solved by an algorithm called HPI [15] with numerical errors as small as the rounding error on a computer. After , , and are obtained, the stiffness matrix of the substructure can be calculated based on (35) and it can

CHEN et al.: SEMIANALYTICAL SEM FOR ANALYSIS OF 3-D LAYERED STRUCTURES

be further combined with stiffness matrices of the neighbor substructure to assemble the global system matrix.

5

After repeating (45) for times, the integration interval will , , and be equal to and we can obtain the matrices with very high accuracy.

VI. HPI METHOD The first step of using HPI to solve the Riccati equations (38) algorithm is to divide the integration interval based on the [22] (40) where is a positive integer number. For example, suggested in [15] can achieve machine precision for most cases. means , i.e., even for a substructure as long as 100 wavelengths, a slice with value will be shorter than 1/10000 of a wavelength. It is suitable and accurate to perform a Taylor expansion for matrices , , and within the small interval

(41)

where matrices , , and have the same dimensions as , , and . Since is a very small number, the higher order items by Taylor expansion are usually smaller than or comparable to the minimum quantity defined by the double precision , , and . Omitting these when compared with higher order items will not lead to the loss of any significant digits. Comparing (41) with (38), we can obtain expressions of matrices , , and over the interval

(42)

VII. BLOCK THOMAS ALGORITHM FOR BLOCK TRI-DIAGONAL MATRIX After assembling all the stiffness matrices, applying boundary conditions onto the first and last interfaces of the whole structure, and imposing the excitations, the equations to be solved will take a form of a block tri-diagonal system

.. .

..

.

..

.

..

.

..

.

..

.

.. .

.. . .. .

.. . .. .

(46)

where , , and , and are the discretized generalized displacement and discretized excitation corresponding to the th interface, respectively. The block Thomas algorithm can be used here to accelerate the process of solving (46). The block Thomas algorithm is a generalized version of Thomas algorithm [18], which is designed for a tri-diagonal matrix. The pseudocode of the block Thomas algorithm is as follows:

for

end

(47)

for (43)

end More details, as well as a discussion of efficiency of the block Thomas algorithm can be referred to [18].

(44) VIII. NUMERICAL EXAMPLES AND DISCUSSIONS After the calculation over interval , , , and can be obtained by combining two small together as

(45) It is worth noting that during the combination only the increment of , i.e., , is calculated. That is because this small quantity must be kept away from during computation, otherwise all the significant digits of will be lost due to a much larger quanto the addition of a small quantity tity .

The first example is shown in Fig. 4, where a uniform plane wave is impinging onto a PEC-backed stratified dielectric slab . Along the direction, the stratiwith incident angle cm, and it is evenly dified dielectric slab has thickness vided into ten layers of dielectric with the distribution of permittivity shown in Fig. 5. We simulate this electromagnetic problem by both the conventional FEM and the proposed semianalytical SEM with the same discretization scheme (ten elements per layer along the direction), and then extract the reflection coefficient from the simulated electromagnetic fields. Since this is a 1-D problem and does not require discretization of the cross section, the comparison between the conventional FEM and the semianalytical SEM is essentially a comparison between the FEM and HPI for the longitudinal integration in (21). Fig. 6 shows the reflection coefficients under different working frequencies by the two numerical methods, as well as

6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 4. Uniform plane wave is impinging onto a PEC-backed stratified dielectric slab, which consists of ten layers with different values of permittivity given by Fig. 5. Fig. 7. Relative errors of reflection coefficients under different working frequencies obtained by the conventional FEM and semianalytical SEM.

Fig. 5. Distribution of relative permittivity of the stratified dielectric slab. (a) Real part of  . (b) Imaginary part of  of the layered medium in Fig. 4.

Fig. 8. Microwave filter containing ten identical cells. The unit of length in this figure is centimeters (cm).

Fig. 9. (a) Conventional FEM mesh and (b) semianalytical SEM mesh for one cell of the microwave filter. Meshes of the other nine cells are the same as the shown one for both cases. Fig. 6. Reflection coefficients under different working frequencies obtained by the conventional FEM, semianalytical SEM, and analytical method.

the analytical method [23]. The relative errors under different frequencies by the conventional FEM and semianalytical SEM are shown in Fig. 7. From the two figures, we can observe that the accuracy of the semianalytical SEM is several orders higher than that of the conventional FEM. Furthermore, with the increase of working frequency, i.e., the decrease of discretization density for a fixed mesh, the numerical error by the conventional FEM becomes larger and larger, while the error by semianalytical SEM is always around or below 10 , which is on the same level of computer’s round-off error defined by double precision. In other words, the longitudinal integration by HPI can achieve machine precision. The second example is a microwave filter with ten identical cells. Each cell is a waveguide discontinuity structure with the geometry and dimensions shown in Fig. 8. A conventional FEM,

as well as semianalytical SEM are employed to calculate the reflection and transmission coefficients corresponding to an incident TE wave with various values of working frequencies. As shown in Fig. 9, a relatively dense conventional FEM mesh is used to discretize the microwave filter (61 440 first-order 3-D edge element in total); while for the semianalytical SEM, each cell is divided into three substructures with only six 2-D spectral elements on the cross section. In other words, only 180 semianalytical spectral elements are used to discretize the whole structure of the microwave filter. Fig. 10 shows the numerical results of reflection and transmission coefficients by the conventional FEM and semianalytical SEM (third-order 2-D SEM for discretization of cross section in this case), it also shows reference results, which are obtained by the conventional FEM with a much denser mesh and higher order of basis functions than that shown in Fig. 9. From this figure, we observe that the microwave filter presents bandgaps

CHEN et al.: SEMIANALYTICAL SEM FOR ANALYSIS OF 3-D LAYERED STRUCTURES

7

Fig. 11. Relative errors of reflection and transmission coefficients of the microwave filter by the semianalytical SEM with different interpolation orders for transverse discretization.

TABLE II SEMIANALYTICAL SEM WITH DIFFERENT INTERPOLATION ORDERS FOR TRANSVERSE DISCRETIZATION

Fig. 10. (a) Reflection and (b) transmission coefficients of the microwave filter by conventional FEM and semianalytical SEM.

TABLE I COMPARISON OF EFFICIENCY AND ACCURACY BETWEEN THE CONVENTIONAL FEM AND THE SEMIANALYTICAL SEM

due to the periodicity along the direction of wave propagation. We also can find that even with a much smaller number of element, the semianalytical SEM can give results closer to reference than results by a conventional FEM. More detailed comparison of efficiency, as well as accuracy between two numerical methods are shown in Table I. From this table, we can find that the semianalytical SEM is more efficient and more accurate than the conventional FEM. The accuracy of the semianalytical SEM can be increased by either refining the SEM mesh on the cross section, i.e., -refinement, or increasing the order of the SEM basis function, i.e., -refinement. We use the same discretization scheme as Fig. 9(b), but with different orders of basis functions for spectral elements on the cross section, to re-solve the microwave filter problem. The relative errors of calculated reflection and transmission coefficients by the semianalytical SEM with different interpolation orders are shown in Fig. 11 and Table II,

from which we can conclude that the numerical errors by the proposed method decrease exponentially as the increase of interpolation orders of spectral elements on the cross section, i.e., the semianalytical SEM, can achieve spectral accuracy. IX. CONCLUSION In this paper, we have discussed a semianalytical SEM specially designed for layered structures. A 3-D piecewise homogeneous structure is decomposed into several substructures based on the distributions of material and geometry along the longitudinal direction. Each substructure is then modeled by a combination of 2-D spectral elements for discretization of the cross section and a high precision algorithm for the integration along the longitudinal direction. Compared to other efficient algorithms such as NMM and LAFE, the semianalytical SEM only requires a set of 2-D meshes, and meanwhile does not need the expensive step of solving eigenproblems corresponding to the semidiscretized substructures. Numerical examples demonstrate that the semianalytical SEM is very efficient and accurate, and it can achieve spectral accuracy with the increase of interpolation order of 2-D spectral elements for the discretization of cross sections of substructures. REFERENCES [1] J. R. Wait, Electromagnetic Waves In Stratified Media. New York: Oxford Univ. Press, 1970. [2] S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett., vol. 77, no. 22, pp. 3490–3492, Nov. 2000. [3] P. Meuris, W. Schoenmaker, and W. Magnus, “Strategy for electromagnetic interconnect modeling,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 20, no. 6, pp. 753–762, Jun. 2001.

8

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

[4] Q. H. Liu and W. C. Chew, “Numerical mode-matching method for the multiregion vertically stratified media,” IEEE Trans. Antennas Propag., vol. 38, no. 4, pp. 498–506, Apr. 1990. [5] H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas Propag., vol. 49, no. 2, pp. 185–195, Feb. 2001. [6] Q. H. Liu and W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: An eigenmode propagation method,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 422–430, Mar. 1991. [7] Q. H. Liu, “Electromagnetic field generated by an off axis source in a cylindrically layered medium with an arbitrary number of horizontal discontinuities,” Geophysics, vol. 58, no. 5, pp. 616–625, 1993. [8] G. X. Fan, Q. H. Liu, and S. P. Blanchard, “3-D numerical mode-matching (NMM) method for resistivity well-logging tools,” IEEE Trans. Antennas Propag., vol. 48, no. 10, pp. 1544–1552, Oct. 2000. [9] D. Jiao, S. Chakravarty, and C. H. Dai, “A layered finite element method for electromagnetic analysis of large-scale high-frequency integrated circuits,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 422–432, Feb. 2007. [10] H. Gan and D. Jiao, “A time-domain layered finite element reduction recovery (LAFE-RR) method for high-frequency VLSI design,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3620–3629, Dec. 2007. [11] H. Gan and D. Jiao, “A recovery algorithm of linear complexity in the time-domain layered finite element reduction recovery (LAFE-RR) method for large-scale electromagnetic analysis of high-speed ICs,” IEEE Trans. Adv. Packag., vol. 31, no. 3, pp. 612–618, Aug. 2008. [12] A. Kirsch and P. Monk, “A finite element/spectral method for approximating the time-harmonic Maxwell system in R3,” SIAM J. Appl. Math., vol. 55, no. 5, pp. 1324–1344, Oct. 1995. [13] G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations. Berlin, Germany: Springer, 2002. [14] J. H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 437–444, Jan. 2006. [15] W. X. Zhong, Duality System in Applied Mechanics and Optimal Control. Norwell, MA: Kluwer, 2004. [16] W. X. Zhong, “On precise integration method,” J. Comput. Appl. Math., vol. 163, no. 204, pp. 59–78, Feb. 2004. [17] J. Chen, B. Zhu, and W. X. Zhong, “On the semi-analytical dual edge element and its application to waveguide discontinuities,” Acta Phys. Sin., vol. 58, no. 2, pp. 1091–1099, Feb. 2009. [18] G. Meurant, “A review on the inverse of symmetric tridiagonal and block tridiagonal matrices,” SIAM J. Matrix Anal. Appl., vol. 13, no. 3, pp. 707–728, Jul. 1992. [19] D. Gottlieb and J. S. Hesthaven, “Spectral methods for hyperbolic problems,” J. Comput. Appl. Math., vol. 128, no. 1–2, pp. 83–131, Mar. 2001. [20] C. W. Crowley, P. P. Silvester, and H. Hurwitz, Jr., “Covariant projection elements for 3-D vector field problems,” IEEE Trans. Magn., vol. 24, no. 1, pp. 397–400, Jan. 1988. [21] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [22] E. Angel and R. Bellman, Dynamic Programming and Partial Differential Equations. New York: Academic, 1972. [23] W. C. Chew, Waves And Fields in Inhomogeneous Media. New York: Wiley, 1999. Jiefu Chen received the B.S. degree in engineering mechanics and M.S. degree in dynamics and control from the Dalian University of Technology, Dalian, China, in 2003 and 2006, respectively, and is currently working toward the Ph.D. degree in electrical engineering at Duke University, Durham, NC. Since 2007, he has been a Research Assistant with the Department of Electrical and Computer Engineering at Duke University. His research interest is fast algorithms for computational electromagnetics and their applications in electronic packaging, electromagnetic compatibility, and signal integrity.

Bao Zhu received the B.S. degree in engineering mechanics from the Dalian University of Technology, Dalian, China, in 2005, and is currently working toward the Ph.D. degree in engineering mechanics at the Dalian University of Technology. From September 2008 to September 2009, he was a visiting student with the Department of Electrical and Computer Engineering, Duke University, Durham, NC. His research interest is time-domain algorithms for computational mechanics and computational electromagnetics.

Wanxie Zhong received the B.S. degree in bridge engineering from Tongji University, Shanghai, China, in 1956. From 1956 to 1962, he was with the Institute of Mechanics, Chinese Academy of Science, as a Research Scientist. Since 1962, he has been with the Dalian University of Technology, Dalian, China where he is currently a Professor of engineering mechanics. He has authored or coauthored over 300 papers in refereed journals and 14 books. His research interests include engineering mechanics, computational mechanics, and optimal control. Prof. Zhong is a member of the Chinese Academy of Science.

Qing Huo Liu (S’88–M’89–SM’94–F’05) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1989. From September 1986 to December 1988, he was with the Electromagnetics Laboratory, University of Illinois at Urbana-Champaign, as a Research Assistant, and from January 1989 to February 1990, he was a Postdoctoral Research Associate. From 1990 to 1995, he was a Research Scientist and Program Leader with Schlumberger-Doll Research, Ridgefield, CT. From 1996 to May 1999, he was an Associate Professor with New Mexico State University. Since June 1999, he has been with Duke University, Durham, NC, where he is currently a Professor of electrical and computer engineering. He has authored or coauthored over 450 papers in refereed journals and conference proceedings. He is currently Deputy Editor-in-Chief of Electromagnetic Waves and Applications and Deputy Editor-in-Chief of Progress in Electromagnetics Research. He is an Editor for Journal of Computational Acoustics. His research interests include computational electromagnetics and acoustics, inverse problems, geophysical subsurface sensing, biomedical imaging, electronic packaging, and the simulation of photonic devices and nanodevices. Dr. Liu is a Fellow of the Acoustical Society of America. He is a member of Phi Kappa Phi and Tau Beta Pi. He is a full member of the U.S. National Committee, URSI Commissions B and F. He is an associate editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. He was the recipient of the 1996 Presidential Early Career Award for Scientists and Engineers (PECASE) presented by the White House, the 1996 Early Career Research Award presented by the Environmental Protection Agency, and the 1997 CAREER Award presented by the National Science Foundation (NSF).

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

9

A Common Electromagnetic Framework for Carbon Nanotubes and Solid Nanowires—Spatially Dispersive Conductivity, Generalized Ohm’s Law, Distributed Impedance, and Transmission Line Model George W. Hanson, Fellow, IEEE

Abstract—General equations are presented for the spatially dispersive conductivity, distributed impedance, Ohm’s law relation, and transmission line model of both carbon nanotubes (CNTs) and solid material nanowires. It is shown that spatial dispersion results in an intrinsic (material-dependent) transmission-line capacitance. Spatial dispersion is numerically unimportant in metal nanowires, but leads to a shift in propagation constant of a few percent for CNTs and semiconducting nanowires. Theoretically, spatial dispersion is important for both nanowires and nanotubes, and is necessary to preserve the inductance–capacitance–velocity relation 2 , where is kinetic inductance, dos is intrinsic dos capacitance, and is electron Fermi velocity. It is shown that in order to obtain the correct intrinsic capacitance, it is necessary to use a charge-conserving form of the relaxation-time approximation to Boltzmann’s equation. Numerical results for the propagation constant of various nanowires and CNTs are presented. The general formulation developed here allows one to compute, and directly compare and contrast, properties of CNTs and solid nanowires.

=

Index Terms—Carbon nanotubes (CNTs), multiconductor transmission lines (MTLs), nanotechnology.

I. INTRODUCTION ARBON nanotubes and metal nanowires have been recently considered as interconnects for future electronic systems [1]–[12]. Various configurations are possible, such as single-wall nanotubes, multiwall nanotubes, and bundles of various nanotube types. Although single-wall nanotubes typically have radius values less than 1–2 nm, control over effective radius can be achieved using multiwall tubes or tube bundles. These systems have been predicted to outperform nanowires in some cases [7]–[12], particularly for relatively large bundles of small-radius single-wall tubes. However, lacking sufficient control over geometry, nanowires may perform better, and can also come as single or bundled structures. Thus, at this stage, both carbon nanotubes (CNTs) and solid nanowires are of considerable interest, from both a theoretical and practical standpoint.

C

Manuscript received April 03, 2010; revised August 12, 2010; accepted September 22, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. The author is with the Department of Electrical Engineering, University of Wisconsin–Milwaukee, Milwaukee, WI 53211 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2090693

In this paper, we consider a common electromagnetic model for CNTs and solid nanowires. We will assume that these struc, where is radius tures are formed by electrically thin ( and is wavelength) conducting materials with length much longer than radius, oriented parallel to the axis. Accordingly, we neglect any transverse current and consider only longitudinal current. The main assumption is that the material response is due to intraband conduction electrons such that Boltzmann’s transport equation provides an appropriate description of the electromagnetic field–material interaction. The analysis is applicable below the range of electron interband transitions, which occurs at tens of THz for CNTs and hundreds of THz for metals, and . for CNTs assumes operation near the Fermi point It is shown that the usual relaxation time approximation (RTA) of Boltzmann’s equation is not adequate to account for spatial dispersion, which is intimately connected with diffusion and intrinsic capacitance. A modified RTA, which preserves charge continuity, is shown to lead to the correct result that preserves the inductance–capacitance–velocity relation . A general expression for the spatially dispersive conductivity is derived, which leads to a common form for the distributed impedance of solid nanowires and CNTs. A general transmission line model is developed, where spatial dispersion is shown to be associated with intrinsic capacitance (sometimes called a finite density-of-states capacitance or, more often, a quantum capacitance). The formulation arises purely from Boltzmann’s equation and Maxwell’s equations, rather than from a transmission line approach that starts with a given configuration of circuit elements. It will be shown that spatial dispersion in metal nanowires is numerically negligible, but leads to effects on the order of a few percent for CNTs and semiconducting nanowires. The same methods also carry over to the analysis of graphene, but this is not included in this paper due to limited space. The Boltzmann model provides a semiclassical treatment of electron dynamics, and in a fully numerical procedure the Boltzmann/Maxwell system can be solved simultaneously. Here it is solved in the limit of small perturbations from equilibrium, leading to the closed-form result (1). As an alternative to the Boltzmann/Maxwell model, a Schrödinger/Maxwell model is developed in [13]–[15]. Schrödinger’s equation is a fully quantum model for electron dynamics. However, it should be noted that the exact many-electron Schrödinger equation (i.e., using the exact potential and the -particle

0018-9480/$26.00 © 2010 IEEE

10

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

wavefunction including exchange correlations) is not actually solved since this is practically impossible, but rather, an envelope function Schrödinger equation (the typical effective mass model) is solved. This approximation has a similar spirt to the fact that Boltzmann’s equation does not consider the movement of individual electrons, but uses statistical mechanics to formulate an equation for the electron distribution function. Therefore, both Boltzmann’s and Schrödinger’s equation (as used) approximate electron dynamics, even if used in a fully numerical simultaneous solution with Maxwell’s equations. A simultaneous numerical solution should provide more accurate results for higher electron energies, beyond the validity of the perturbation approach. Moreover, the method of [13]–[15] allows for boundary conditions on both electrons and photons, whereas the closed-form Boltzmann model used here assumes the structure is infinite for electrons, allowing the definition of material conductivity (in a numerical solution of Boltzmann’s equations boundary conditions on the electron distribution function could be applied). Both the Schrödinger/Maxwell and Boltzmann/Maxwell models should provide similar results for frequencies of interest in this paper. The main new results of the paper are presented in the following sections with most of the derivations appearing in the Appendix. Throughout this paper, SI units are used, and the (suppressed) time dependence is . II. GENERAL RESULTS FOR SPATIALLY DISPERSIVE CONDUCTIVITY, CURRENT-FIELD RELATIONSHIP, AND GENERALIZED IMPEDANCE For the considered class of cylindrical structures the spatially has the general form dispersive conductivity

is a material-specific parameter of order unity defined later for CNTs and for solid nanowires). In [3], the ( concepts of kinetic inductance and quantum capacitance have been introduced for a general nanowire. Both CNTs and solid nanowires satisfy a generalized Ohm’s law (4) where is the usual local Drude conductivity, and is a spa. For CNTs, this relation tial dispersion parameter was first derived in [16], and here we show that it is also applicable to solid nanowires, and that the spatial dispersion parameter requires special treatment. The generalized Ohm’s law (4) is equivalent to a 1-D drift-diffusion model [17] (see also [27] where a fluid model accommodating diffusion is presented), (5) where the diffusion constant is . Thus, one can solve either (4) in the transform domain, or (5) in the space domain. is actually For solid nanowires, the term , but the higher order terms are negligible, and for CNTs, the higher order terms are identically zero. Thus, (6) is equivalent to (1) for CNTs, and provides an excellent approximation to (1) for solid nanowires. The generalized Ohm’s law (4) leads to the general integral equation

(1) (7)

where (2)

In the above expressions, is the spatial wavenumber, is radian frequency, is a phenomenological electron relaxation is the longitudinal electron velocity, is the equilibtime, is the electron charge, is rium Fermi–Dirac distribution, electron energy, is electron quasi-momentum, is a geometrical constant ( and 1 for CNTs and solid nanowires, respectively), and is a material-dependent quantity defined later. The second term in (1), which is absent in the usual RTA solution of Boltzmann’s equation, provides an important contribution, as described later. The distributed impedance ( /m) of these structures can be for CNTs and defined as for solid nanowires. These lead to the general form (3) where per-unit-length quantities and are the resistance, kinetic inductance, and intrinsic (density of states or quantum) capacitance, respectively, of the material, and where

where

is the usual kernel and for CNTs and

for solid cylinders. The role of spatial dispersion can be appreciated by considering an infinitely long structure. then leads to the space-domain current in the usual nonlocal form (8) where is the inverse Fourier transform of . The conductivity can be evaluated using complex-plane analysis, leading to (9) where . The range of nonlocality is the range of for which is appreciably different distance from zero. Since we can represent the delta function as , we have (10)

HANSON: COMMON ELECTROMAGNETIC FRAMEWORK FOR CNTs AND SOLID NANOWIRES

11

Thus, we recover the local limit as , which, . Therefore, a local conas shown later, corresponds to ductivity is obtained in the limit of vanishing electron mean-free path.

(S) is the local Drude where and have the same form, exconductivity. Note that cept for the near-unity factor . The distributed impedance ( m) is

III. SPECIFIC EXPRESSIONS FOR CNTs AND SOLID NANOWIRES, AND NUMERICAL COMPARISONS

(21) where

In this section, we present some specific results for CNTs and solid nanowires (these are derived in the Appendix). For nanowires, the conductivity (1) and the spatial dispersion parameter are

(22) (23) (24) where

(11) (12) (13) where ductivity ( electron density, m

(S/m) is the usual Drude conis the dc conductivity where is the , [18]) and (14)

is the resistance quantum. We have , where is the electron Fermi velocity and in the nanotube, as expected. The values of agree with independently derived expressions [1] (see also [19] agrees with and [20]). The expression for the resistance that obtained in [3], and provides values that are in reasonable agreement with experiment [2]. Note that these values are independent of nanotube radius. This holds as long as the nanotube radius is sufficiently small so that only two electron channels exist (which typically holds for single-wall nanotubes). It is interesting to numerically compare these parameters for and are freCNTs and nanowires. The parameters quency independent, and for the frequency-dependent parameter , we assume GHz. For CNTs, we use the typical m/s and s, such that values k

is a parameter of order unity The distributed impedance ( /m) is

.

m

(25)

nH/ m

(26)

aF

(27)

m m

(15) m where (16) (17) We have included the factor in the definition of to be consistent with and (so that , where is the Fermi velocity), and to obtain a capacitance with the units of F/m, in order to facilitate comparison with CNTs. For small-radius single-wall CNTs, (18)

(29)

For gold nanowires (other typical metals of interest have similar electrical properties) with m s, S/m, and m/s (here we are using the bulk material properties—electron surface and grain-boundary scattering may change these values for nanowires, usually by less than an order of magnitude [21], [22]), we obtain nm

k

m

(30)

nm nm

pH/ m

(31)

k

m

(32)

nm

pH

m

(33)

pF

m

nm nm

pF

(34)

m

(35)

(19) (20)

(28)

m m

(36) (37)

12

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

It can be seen that the spatial dispersion parameters and are somewhat similar, and only differ by one order of magnitude. Accordingly, from (9), it can be seen that the range of , and is similar to . nonlocality is governed by This can be understood since, aside from the factor , which is order 1, and have the same functional form. Since is similar for CNTs and metals, then the main numerical difference comes from the value of in the expression for . These differ by not quite an order of magnitude for the values chosen here (at most there is likely to be only one to two orders of magnitude difference). Thus, single-wall CNTs are, intrinsically, only slightly more nonlocal than metals, and the difference arises from the electron relaxation time. However, as shown below, there is a strong geometric influence that renders spatial dispersion less important in nanowires made from good conductors (which cannot be fabricated at the 1–2-nm dimensions typical of single-wall nanotubes). As shown in the Appendix, in the development of the distributed impedance, the circuit manifestation of spatial disper, the sion is the intrinsic density-of-states capacitance (if capacitance term is absent). For a solid nanowire, other contains a factor than the factor that is of order 1, and is radius dependent. Assuming nm as the smallest aF/ m, whereas for reasonable metal radius, a CNT aF/ m. Thus, the intrinsic capacitance is at least three orders of magnitude larger for a metal nanowire, and grows as . Therefore, the effective intrinsic capacitive reactance is much smaller for a metal nanowire than for a nanotube, and can typically be neglected in all but the smallest nanowires. , and at Similarly, the kinetic inductance involves a factor nm is at least three orders of magnitude smaller for the metal nanowire than for a CNT. The resistance is approximately two orders of magnitude smaller for a metal nanowire than for . a CNT, and quickly decreases as radius increases Since the intrinsic capacitive reactance in metal nanowires can typically be neglected, spatial dispersion in these structures can typically be neglected in numerical computations. However, from (17), it is clear that this is not be the case for a material having lower electron density (typically, lower conductivity). For example, consider a degenerately doped semim . At this degree of doping, conductor having we can treat the material as a metal-like plasma, although accounting for effective mass. For GaAs with this doping level at S/m, s [23], and room temperature, m/s, leading to ( nm) aF/ m, which is similar to that for the CNT. Thus, the capacitive reactance and spatial dispersion in a degenerately doped semiconductor are numerically important. The same would be true of a simple lower conductivity metal. nm at Hz, For a gold nanowire having

(38) (39) (40)

as a reasonwhere the last line comes from assuming able value. It can be seen that the capacitive reactance is clearly negligible, and that the kinetic inductive reactance is three orders of magnitude smaller than the resistance. For a CNT, (41) (42) (43) in the last line. Clearly, in both where we again assume cases, the impedance is dominated by the resistance, although for the CNT, the kinetic inductive reactance is quite important. Although quantitatively incorrect, it is worthwhile to consider nm. Although this a fictitious metal conductor having radius is not realistic, it allows one to reduce geometrical effects and more directly compare nanotubes and metal nanowires (if nanowires could have similar radius values). We obtain nm nm nm

k

m nH

(44)

m aF

(45) m

(46)

and (47) (48) (49) . The resistance is similar to the nanotube value, for whereas the reactance is one order of magnitude smaller. It can be seen that the performance of these two structures would be were larger, which is possible, than the similar. Note that if resistance of the nanotube would decrease. The value s is consistent with a mean-free path of around 500 s leads to a mean-free path of almost nm, whereas 1 m, which is also a reasonable value and which would halve the resistance. IV. TRANSMISSION LINE ANALYSIS In this section, we obtain the modal properties and transmission line parameters for electromagnetically coupled CNTs or nanowires in a homogeneous dielectric . We assume that the nanotubes or wires are infinity long, oriented parallel to the -axis, and that the radius of the th tube is (and, hence, has conductivity ). The formulation proceeds from Maxwell equations and the Boltzmann equation conductivity, and we do not assume a certain transmission line configuration. We first consider two electromagnetically coupled conductors (we assume that the conductors are not electronically coupled; i.e., there is no wavefunction overlap or quantum tunneling). Assume that one conductor carries current , has radius , and , and the other conductor carries current , is centered at , as shown in Fig. 1. has radius , and is centered at The total field for the first conductor is the field due to , plus the field due to current current on the first conductor

HANSON: COMMON ELECTROMAGNETIC FRAMEWORK FOR CNTs AND SOLID NANOWIRES

where

13

and (57) (58)

Fig. 1. Two electromagnetically coupled CNTs or solid nanowires.

on the second conductor , plus any incident field. For the second conductor, we have a similar situation, and thus, from (7), we obtain a set of two coupled integral equations

where for a CNT and is given by (14) for a solid and are the usual modified cylinnanowire, and where drical Bessel functions [24]. Note that contains all information about the conductor’s material properties. The generalization to conductors is clear,

(59) (50) (51)

, where is the center-to-center distance between conductors and . In matrix form, (59) can be written as (60)

(52) for all field on conductor

, where

is the incident

and (53)

Note that we can have a metal nanowire coupled to a CNT. and observation points on the For both source points and first conductor, or both on the second conductor, , where . The approximation is made for convenience, which is equivalent to the assumption of a filamentary current on conductor , and that the field due to the current is the same on the surface of conductor as at the center of conductor , and (for which is a good assumption for all nanoradius values, there is no skin effect). In this way, . Invoking the translational invariance of the structure, and , using the relations

where and are matrices ( is a diagonal matrix with entries ), is a column of unknown excitation column. current amplitudes, and is a To determine the modal propagation constants, we solve ). The propagation the unforced problem (setting that constants are determined as the complex values . Obviously, the modal propagation constant force is dependent on frequency, the system geometry, and on the material properties of each conductor. for all (i.e., idenSince is a diagonal matrix, if tical conductors), then we can write (61) where is a matrix that is only geometry and frequency dependent, is the scalar function (57) that is geometry independent, but material property dependent, and is the identity depends matrix. Although the root that forces on (and hence, on the resistance, kinetic inductance, and inare trinsic capacitance of the conductors), the eigenvalues of simply the eigenvalues of shifted by . Letting be the eigenvalues of , then the eigenvalues of , denoted as , are simply . Therefore, the propagation constants, which are the zero eigenvalues of , can be simply obtained from the material-independent matrix . A. Small Argument Approximation

(54) (55) and enforcing the equations at

Assuming that argument forms [24] leading to

, we can use the small ,

leads to (62)

(56) for

.

14

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 2. Two rows of conductors with odd current symmetry in the vertical direction (e.g., conductors over a ground plane, or two rows of conductors carrying a transmission line mode current) (i.e., having odd symmetry in the vertical direction).

At this point it is useful to consider an array with a symmetry plane, such as a row of conductors above a ground plane or two rows of conductors carrying currents with odd symmetry in the vertical direction, as depicted in Fig. 2. If we number the conductors as indicated in this figure and enforce , etc., we obtain

(63) for

. In matrix form, this can be written as (64)

Importantly, the matrix is now independent of quency, and only depends on geometry. For identical nanotubes, we have

and fre-

(65) The propagation constants are determined from , leading to the propagation values (66) Thus, the dispersive fundamental material-dependent propagation constants can be obtained without a root search in the complex plane, and are given analytically in terms of the static eigenvalues of the matrix , which only depends on geometry. The is associated with the dominant mode smallest eigenvalue of propagation. , such For the special case of two conductors, that

(67) and are the elecwhere trostatic inductance and capacitance, respectively, of a two-conductor configuration. If spatial dispersion is ignored, the term

Fig. 3. (a) Re( )=k versus frequency for a two-conductor array. Several conductor types are shown—solid gold nanowires having a = 50 nm and a = 10 S/m,  2 10 nm, a degenerately doped GaAs nanowire ( 10 s) having a = 10 nm, and a (13,13) CNT. (b). Same as Fig. 3(a), except for Im( )=k .



 2

is absent. As discussed later, the factors of 2 are beand are intrinsic to each conductor, and there are cause two conductors. In the following, we consider two- and four-conductor array for the case of conducgeometries having tors in free space having currents with odd symmetry in the vertical direction (i.e., a transmission line mode of the structure). For the (13,13) nanotubes considered here, which have radius nm, this corresponds to vertical and horizontal separation nm. For nm nanowires, this correnm, and for nm nanowires, sponds to nm. In the following, we rename the propagation as . constant Fig. 3(a) and (b) shows and versus frequency for the fundamental mode of a two-conductor array. nm and nm au nanowires, We consider a (13,13) CNT, and a degenerately doped GaAs nanowire S/m, s) having nm (we ( ignore radial quantization). All results include spatial dispersion, but since ignoring spatial dispersion only changes the results for the CNT and semiconducting nanowire by a few and, at nm, percent (e.g., ), and much less than 1% for the nm ), metal nanowire (at would not noticeably affect the plot given the setting

HANSON: COMMON ELECTROMAGNETIC FRAMEWORK FOR CNTs AND SOLID NANOWIRES

15

V. MULTICONDUCTOR TRANSMISSION LINE (MTL) THEORY In this section, the per-unit-length transmission line parameters are obtained from the coupled integral equations (63) for a system of two rows of conductors with odd current symmetry in (this definition the vertical direction. If we define is consistent with [25], although the used here and in that reference differ by a factor of ), then (64) can be written as (68) where

is a diagonal matrix with entries (69)

Fig. 4. Propagation constants =k for a four-nanotube array of (13,13) CNTs. The center curve is the two-conductor result (two tubes vertically aligned).

and (6), (68) can be written

Using as

(70) With

for homogeneous media and (3), we have

(71) where

Fig. 5. Propagation constants =k for a four-nanowire array of gold nanowires. The center curve is the two-conductor result (two nanowires vertically aligned).

scale of the vertical axis. It can be seen that propagation along the au nanowires is much faster than on the other structures, with much less attenuation. In Fig. 4, fundamental propagation constants for a four-conductor array (two rows of conductors having currents with odd symmetry in the vertical direction) is shown for (13,13) CNTs. The center curve is the two-conductor result, and the two surrounding curves are the four-conductor case (forming even and odd modes of propagation). It can be seen that, at several tens of GHz, the four-nanotube modes coalesces into the two-nanis not shown for otube case. The local approximation the sake of clarity, but it would result in a noticeable shift of the % . Similarly, although not shown, for a GaAs curves nanowire the local and nonlocal results differ by approximately 5%, and for lower doping levels nonlocality becomes even more important for semiconducting nanowires (e.g., if S/m there is a 10% shift between local and nonlocal results, ). Fig. 5 shows propagation constants for a four-conductor array nm and nm. Clearly, of au nanowires for both the thicker nanowires exhibit less separation between even and odd modes (recall that the conductor spacing is larger for the nm conductors).

, and

are diagonal matrices with , and the ohmic resistance diagonal matrix has entries (all matrices are per-unit-length quantities). Note that this result is for two rows of nanotubes having currents with odd symmetry in the vertical direction. reflects the fact that the current flows The factor of 2 in through a conductor in one row and also through its return in the other row. Since is the resistance of each conductor, (note that this is consistent with the the total resistance is usual transmission line formulation, where, e.g., is the total resistance of both conductors). Since and are similarly and intrinsic to the conductors, the factors of 2 and 1/2 in , respectively, are due to the same reason. For conductors above a perfect ground plane, we would remove the factors of 2 since the return row of conductors (i.e., the ground plane) does and . not exhibit , where Equation (71) can be written as the admittance and impedance matrices are , which is the usual MTL formulation for the unforced problem [25]. Note that we have obtained the total capacitance as the series combination of the electrostatic and intrinsic (quantum) capacitance matrices, which is a generalization to the usual scalar transmission line result [1]. Being a series combination, the intrinsic capacitance acts to reduce the total capacitance from the electrostatic value. This reduction of capacitance due to the finite density of states makes sense if one considers the energy charge relationship . For a material with an infinite density of states (i.e., a perfect conductor), we can consider all of the charge to reside at the bottom of the conduction band. If the material has a finite density of states, we need to apply more energy to get the same charge since we need to fill in higher energy states. The requirement of needing more

16

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

energy to produce the same charge is equivalent to reducing the . Here we ignore electron excapacitance since change-correlation energies [26]. , it would appear that and we Also, note that if are left with only an inductance matrix. However, in that case we such that remains finite have a pure TEM mode, and and we recover the correct transmission line result. Transmission line analysis of single CNTs and quantum wires above a ground plane have been given in [1], [3], [4], and [27], among others. It is worth noting that using the homogeneous system of integral equations cast into the homogeneous form of (71) and comparing with the standard MTL equations, one can only identify and up to a constant. For example, the relative values of for two wires without a ground plane, (71) becomes

highlighted in obtaining the correct spatial dispersion relation and associated intrinsic capacitance. To obtain (1), (3), (4), and (7), we start with Boltzmann’s transport equation [18]

(73) where is the phase–space electron distribution function (i.e., the probability of finding an electron at position at time , and having quasi-momentum ), (74) is the electron velocity,

is the electron energy, and (75)

(72) The first term is the electrostatic inductance for the two-wire geometry, and the third term is the electrostatic capacitance for the two-wire geometry. For the case of a ground plane, it is convenient to divide the above equation by 2 such that the first term is the electrostatic inductance for a wire above a ground , the third term is plane, the second term has one factor of the electrostatic capacitance for a wire above the ground plane, the fourth term is the quantum capacitance of one wire above a ground plane, and the last term is the resistance associated with one wire. Obviously, one can multiply (72) by any constant. Thus, one cannot use the integral equation method in the , and since homogeneous case to identify the values of one only obtains the relative values for a given geometry. The formulation presented here serves as an extension of the classical MTL formulation developed for macroradius conductors to nanoradius conductors. VI. CONCLUSION A general formulation has been presented for the spatially dispersive conductivity, distributed impedance, Ohm’s law relation, and transmission line model of both single-wall CNTs and solid material nanowires. Spatial dispersion, which results in an intrinsic (material-dependent) distributed capacitance, is shown to be theoretically important in all structures, and numerically important in CNTs and solid lower conductivity nanowires (including doped semiconducting nanowires), but unimportant in highly conducting metal nanowires.

APPENDIX DERIVATION OF CONDUCTIVITY, GENERALIZED OHM’S LAW, DISTRIBUTED IMPEDANCE, AND INTEGRAL EQUATION In this appendix, the new results presented in the body of the paper are derived, excepting the transmission-line model, which was presented in the previous section. The role of diffusion is

is the force on the electron. In the usual RTA, the collision integral is [18] (76) where

is the equilibrium Fermi distribution (77)

evaluated at energy and chemical potential . This collision term implies that collisions return the system to global equilibrium (exponentially) in time . However, this approximation does not conserve particle number [29, p. 313], [18, problem 13.4], [30], and leads to a quantum capacitance that diverges at low frequency. Here we use the collision ansatz (sometimes called the Bhatnager–Gross–Krook (BGK) model) [29], [31] (78) where is a local equilibrium distribution function after scattering. In this case, collisions do not return the system to equilibrium, but to a state described by , slightly perturbed from equilibrium. It was found that this method leads to the same result as the well-known Mermin correction that conserves particle number, which is typically applied to the Lindhard permittivity [32]) (see [33] for the hydrodynamic case for an electron gas). For a metal, the spatially dispersive conductivity is given in [34]; however, there the authors use (76), and thus, charge conservation is not enforced. with and that belongs We assume to the same local particle density as does . We write the local equilibrium distribution as a small perturbation of the Fermi dis, where so that the collision term tribution, becomes (79)

HANSON: COMMON ELECTROMAGNETIC FRAMEWORK FOR CNTs AND SOLID NANOWIRES

We assume that a small electric field perturbs the distribution from its equilibrium value. The change in the electron distribution is via the electron density

and

17

such that

(80) where is the equilibrium density and is the induced density that varies as the applied field. This, in turn, produces a perturbation in the chemical potential , where for CNTs. Because the local equilibrium distribution will have the same general form as the Fermi distribution, we expand the distribution in terms of the equilibrium value

(87) , Landau damping is not an issue. Since We need to enforce continuity by the proper choice of have

(88) (89)

(81)

(90)

where becomes

. In the RTA (76), the collision term , whereas in (78), we have . It is the presence of that allows an extra degree of freedom to enforce continuity, as shown below (in this case, by the proper choice of ), which is not possible in the approximation (76). The equilibrium density satisfies (82)

. We

where

is A/m , and for a CNT (

is A/m) (91)

Writing where and using

, suppressing

, common to all terms,

, ignoring the small magnetic field and, replacing contribution, and keeping only first-order terms (92)

where we used netic field contribution, we obtain

. Ignoring the small mag-

which converges for have

, where

, we

(83) The current density in the Fourier transform domain is (the 2 in the numerator accounts for spin)

(93)

(94)

(84)

for solid nanowires and for CNTs. where for materials with parabolic (metals) Since or linear (CNT) energy bands, then

since no current flows from the equilibrium distribution, where

(95) (96)

solid nanowire

(85) (97) (86)

For the electrically thin structures of interest here, we can approximate the current as flowing in the direction. Let

(98)

18

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

For the right-side term, we note

where

(108)

(99) and for the left-side term,

In order to obtain (4), write (100)

and use (109)

Therefore, (110) (111) (101)

where

. Therefore,

where (102)

(112) (113)

so that leading to (114) (103) which is (1). Although it is not necessary, it is convenient to assume K since then . For metals and metallic CNTs, we can use this approximation for all temperatures. We (e.g., in three dimensions can therefore write and for a quasi-1-D structure like a CNT, and ). Using , we have

(115) which has the form (4) with (116) and (117)

(104) Note that if all values are equal (which occurs for CNTs, can but not for solid nanowires), than the series be summed in closed form and the approximate equal signs in (112) become equalities. In this case, there is only quadratic spatial dispersion. As an example, for a solid nanowire

and so

(105)

(118)

(106) (119) such that Equation (116) then leads to (13). Since

(107)

(120)

HANSON: COMMON ELECTROMAGNETIC FRAMEWORK FOR CNTs AND SOLID NANOWIRES

(121)

Using

19

, we obtain (132)

we find that where

, leading to , is the usual dc conductivity. The integral

Redefining

leads to (15). Noting (133)

(122) used in (107) leads to (11). If we do not enforce charge contiis missing in (11), nuity [i.e., if we use (76)], the term and instead of (13), we obtain (123) such that . which does not lead to , and . The For a CNT, double integral over the tube length and circumference is evaluated as [5], [16] (124) where is a discrete index that accounts for azimuthal quantization of the electron wave function. Then (125) (126) where the first 2 is for the contribution of holes and the second 2 is for having two inequivalent Fermi points (valley degeneracy of two). Using [35],

. In a similar manner, leads to using (18) for a CNT, the distributed impedance ( /m) is easily found to be (21). In all structures, if spatial dispersion is ignored, the intrinsic capacwould be missing. itance To derive integral equation (7), note that, in a homogeneous space, the electric field-electric current density relationship is [36] (134) where is a 3-D current density (A/m ) and is the support of the current. For an electrically thin solid , where has units A/m , and for a metal CNT , where is a surface current . We can write this generally (A/m) so that , where for a CNT and for as a solid cylinder. We have assumed that the current density is independent of angle around the tube, which is a reasonable through optical frequencies. In this assumption since case, (135) where

(127)

(136)

we then have (18) and (19) and . The conductivity (107) leads to (18), and (116) provides (20). If we do not enforce charge continuity (i.e., without diffusion), we obtain

emerges for both tubes and solid cylinders since for (the and for tubes we have ) solid cylinders we have and where . Considering a single conductor and using (4) with the total electric field being the electric field due to current on the con, we obtain the ductor (135), plus a possible incident field integral equation (7)

(128) which does not lead to the quantum capacitance , nor does it provide . The distributed impedance ( m) is easily obtained. For a solid nanowire, using (11) we have

(137)

(129) (130)

(138) for all

, where

.

ACKNOWLEDGMENT

where (131)

The author would like to thank the anonymous reviewers for helpful comments concerning this problem.

20

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

REFERENCES [1] P. J. Burke, “An RF circuit model for carbon nanotubes,” IEEE Trans. Nanotechnol., vol. 2, no. 1, pp. 55–58, Mar. 2003. [2] S. Li, Z. Yu, C. Rutherglen, and P. J. Burke, “Electrical properties of 0.4 cm long single walled carbon nanotubes,” Nano Lett., vol. 4, no. 10, pp. 2003–2007, 2004. [3] S. Salahuddin, M. Lundstrom, and S. Datta, “Transport effects on signal propagation in quantum wires,” IEEE Trans. Electron Devices, vol. 52, no. 8, pp. 1734–1742, Aug. 2005. [4] J.-O. J. Wesström, “Signal propagation in electron waveguide: Transmission-line analogies,” Phys. Rev. B, Condens. Matter, vol. 54, pp. 11484–11491, 1996. [5] G. Miano, C. Forestiere, A. Maffucci, S. A. Maksimenko, and G. Ya. Slepyan, “Signal propagation in carbon nanotubes of arbitrary chirality,” IEEE Trans. Nanotechnol., 2010, to be published. [6] Z. Yu and P. J. Burke, “Microwave transport in metallic single-walled carbon nanotubes,” Nano Lett., vol. 5, pp. 1403–1406, 2005. [7] A. Naeemi and J. D. Meindl, “Impact of electron-phonon scattering on the performance of carbon nanotube interconnects for GSI,” IEEE Electron Device Lett., vol. 26, no. 7, pp. 476–478, Jul. 2005. [8] A. Naeemi, R. Sarvari, and J. D. Meindl, “Performance comparison between carbon nanotube and copper interconnects for gigascale integration (GSI),” IEEE Electron Device Lett., vol. 26, no. 2, pp. 84–86, Feb. 2005. [9] A. Nieuwoudt and Y. Massoud, “Evaluating the impact of resistance in carbon nanotube bundles for VLSI interconnect using diameter-dependent modeling techniques,” IEEE Trans. Electron Devices, vol. 53, no. 10, pp. 2460–2466, Oct. 2006. [10] Y. Massoud and A. Nieuwoudt, “Modeling and design challenges and solutions for carbon nanotube-based interconnect in future high performance integrated circuits,” ACM J. Emerg. Technol. Comput. Syst., vol. 2, pp. 155–196, Jul. 2006. [11] H. J. Li, W. G. Lu, J. J. Li, X. D. Bai, and C. Z. Gu, “Multichannel ballistic transport in multiwall carbon nanotubes,” Phys. Rev. Lett., vol. 95, pp. 086601: 1–086601: 4, 2005. [12] A. Naeemi and J. D. Meindl, “Compact physical models for multiwall carbon-nanotube interconnects,” IEEE Electron Device Lett., vol. 27, no. 5, pp. 338–340, May 2006. [13] D. Mencarelli, T. Rozzi, C. Camilloni, L. Maccari, A. Di Donato, and L. Pierantoni, “Modeling of multi-wall CNT devices by self-consistent analysis of multi-channel transport,” IOP Nanotechnol., vol. 19, Apr. 2008, Art. ID 165202, DOI: 10.1088/0957-4484/19/16/165202. [14] D. Mencarelli, L. Pierantoni, and T. Rozzi, “Optical absorption of carbon nanotube diodes: Strength of the electronic transitions and sensitivity to the electric field polarization,” J. Appl. Phys., vol. 103, 2008, Art. ID 063103. [15] L. Pierantoni, D. Mencarelli, and T. Rozzi, “Boundary immittance operator for the combined Schrödinger–Maxwell problem of carrier dynamics in nanodevices,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 5, pp. 1147–1155, May 2009. [16] G. Ya. Slepyan, S. A. Maksimenko, A. Lakhtakia, O. Yevtushenko, and A. V. Gusakov, “Electrodynamics of carbon nanotubes: Dynamic conductivity, impedance boundary conditions, and surface wave propagation,” Phys. Rev. B, Condens. Matter, vol. 60, pp. 17136–17149, Dec. 1999. [17] G. W. Hanson, “Drift diffusion: A model for teaching spatial dispersion concepts and the importance of screening in nanoscale structures,” IEEE Antennas Propag. Mag., Jan. 2010, submitted for publication. [18] N. W. Ashcroft and N. D. Mermin, Solid State Physics. Philadelphia, PA: Holt, Rinehart, Winston, 1976. [19] A. G. Chiariello, A. Maffucci, G. Miano, F. Villone, and W. Zamboni, “Metallic carbon nanotube interconnects, part II: A transmission line model,” in IEEE Signal Propag. Interconnects Workshop, 2006, pp. 185–188. [20] A. Maffucci, G. Miano, and F. Villone, “A transmission line model for metallic carbon nanotube interconnects,” Int. J. Circuit Theory Appl., vol. 36, pp. 31–51, 2006.

[21] W. Steinhögl, G. Schindler, G. Steinlesberger, M. Traving, and M. Engelhardt, “Comprehensive study of the resistivity of copper wires with lateral dimensions of 100 nm and smaller,” J. Appl. Phys., vol. 97, pp. 023706: 1–023706: 7, 2005. [22] G. W. Hanson, “Radiation efficiency of nanoradius dipole antennas in the microwave and far-infrared regime,” IEEE Antennas Propag. Mag., vol. 50, pp. 66–77, Jun. 2008. [23] S. M. Sze, Semiconductor Devices. New York: Wiley, 1985. [24] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Washington, DC: Nat. Bureau Standards, 1964. [25] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York: Wiley, 2007. [26] S. Ilani, L. A. K. Donev, M. Kindermann, and P. L. McEuen, “Measurement of the quantum capacitance of interacting electrons in carbon nanotubes,” Nature Phys., vol. 2, pp. 687–691, 2006. [27] G. Miano and F. Villone, “An integral formulation for the electrodynamics of metallic carbon nanotubes based on a fluid model,” IEEE Trans. Antennas Propag., vol. 54, pp. 2713–2724, Oct. 2006. [28] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005. [29] P. C. Clemmow and J. P. Dougherty, Electrodynamics of Partcles and Plasmas. Reading, MA: Addison-Wesley, 1969. [30] R. Kragler and H. Thomas, “Dielectric function in the relaxation-time approximation generalized to electronic multiple-band systems,” Z. Phys. B, Condens. Matter, vol. 39, pp. 99–107, 1980. [31] A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B, Condens. Matter, vol. 2, pp. 835–850, Aug. 1970. [32] N. D. Mermin, “Lindhard dielectric funstion in the relaxation-time approximation,” Phys. Rev. B, Condens. Matter, vol. 1, pp. 2362–2363, Mar. 1970. [33] P. Halevi, “Hydrodynamic model for the degenerate free-electron gas: Generalization to arbitrary frequencies,” Phys. Rev. B, Condens. Matter, vol. 51, pp. 7497–7499, Mar. 1995. [34] M. Dressel and G. Grüner, Electrodynamics of Solids. Cambridge, U.K.: Cambridge Univ. Press, 2002. [35] L. Latessa, A. Pecchia, A. Di Carlo, and P. Lugli, “Negative quantum capacitance of gated carbon nanotubes,” Phys. Rev. B, Condens. Matter, vol. 72, pp. 035455(1)–035455(5), 2005. [36] A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering. Englewood Cliffs, NJ: Prentice-Hall, 1991.

George W. Hanson (S’85–M’91–SM’98–F’09) was born in Glen Ridge, NJ, in 1963. He received the B.S.E.E. degree from Lehigh University, Bethlehem, PA, in 1986, the M.S.E.E. degree from Southern Methodist University, Dallas, TX, in 1988, and the Ph.D. degree from Michigan State University, East Lansing, in 1991. From 1986 to 1988, he was a Development Engineer with General Dynamics, Fort Worth, TX, where he was involved with radar simulators. From 1988 to 1991 he was a Research and Teaching Assistant with the Department of Electrical Engineering, Michigan State University. He is currently Professor of electrical engineering and computer science at the University of Wisconsin–Milwaukee. He coauthored Operator Theory for Electromagnetics: An Introduction (Springer, 2002) and authored Fundamentals of Nanoelectronics (Prentice-Hall, 2007). His research interests include nanoelectromagnetics, mathematical methods in electromagnetics, electromagnetic wave phenomena in layered media, integrated transmission lines, waveguides, and antennas, and leaky wave phenomena. Dr. Hanson is a member of URSI Commission B, Sigma Xi, and Eta Kappa Nu. He was an associate editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 2002 to 2007. He was the recipient of the 2006 S. A. Schelkunoff Best Paper Award of the IEEE Antennas and Propagation Society.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

21

Neural-Network Modeling for 3-D Substructures Based on Spatial EM-Field Coupling in Finite-Element Method Shaowei Liao, Humayun Kabir, Yi Cao, Member, IEEE, Jianhua Xu, Qi-Jun Zhang, Fellow, IEEE, and Jian-Guo Ma, Senior Member, IEEE

Abstract—This paper presents a new neural-network method to describe the electromagnetic (EM) behavior at the interface between the substructures from an internally decomposed EM structure. A set of neural networks is used to represent the EM behavior of the substructure as seen from the interface. This allows EM coupling between substructures to be effectively represented. The method is developed in a finite-element environment. An EM transfer function matrix is formulated to produce training data, allowing neural networks to learn the spatial coupling between EM-field variables at various locations over the interface of the substructure. A new formulation is proposed where trained neural networks are integrated into the finite-element equation for efficient simulation of an overall EM structure. A technique is developed to allow the proposed model to be used with the mesh different from that in neural-network training. Examples show that the proposed method provides better accuracy than conventional neural-network approaches for modeling substructures from an internally decomposed EM problem. Using the proposed model also speeds up finite-element simulation. Index Terms—Computer-aided design (CAD), electromagnetic (EM) modeling, finite-element method, neural networks.

I. INTRODUCTION

R

ECENTLY, neural-network techniques have been recognized as a powerful tool for microwave design and modeling problems. Neural networks can be trained to learn the elecManuscript received September 15, 2009; accepted September 01, 2010. Date of publication November 18, 2010; date of current version January 12, 2011. This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada, by the National Science Foundation of China under Grant 60871057 and Grant 60688101, by the National 111 Project of China, and by the China Scholarship Council under Grant 2007103813. S. Liao is with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China, and also with the Department of Electronics, Carleton University, Ottawa, ON, Canada K1S 5B6 (e-mail: [email protected]). H. Kabir was with the Department of Electronics, Carleton University, Ottawa, ON, Canada K1S 5B6. He is now with the Research and Development Department, COMDEV Ltd., Cambridge, ON, Canada N1R 7H6 (e-mail: [email protected]). Y. Cao is with Research In Motion Limited, Waterloo, ON, Canada N2L 5R9 (e-mail: [email protected]). J. Xu is with the School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: xujh@uestc. edu.cn). Q.-J. Zhang is with the Department of Electronics, Carleton University, Ottawa, ON, Canada K1S 5B6 (e-mail: [email protected]). J.-G. Ma is with the School of Electronic Information Engineering, Tianjin University, Tianjin 300072, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2090405

tromagnetic (EM) behavior of microwave passive components, and trained neural-network models can be used in circuit simulation providing fast and accurate solutions of EM behavior it learnt. These capabilities make neural networks a very useful alternative for microwave passive component modeling and optimization where repetitive model evaluation is required. Once a model is developed, it can be used over and over again. This avoids time-consuming repetitive EM simulation, thus increases efficiency of microwave design [1]–[3]. Conventional neural-network methods for modeling passive components are usually formulated to represent the -parameters (or other microwave network parameters) at the external input–output ports of EM structures, such as vias and interconnects [4], embedded passives [5], [6], microstrip components [7], [8], coplanar waveguide components [9], etc. Neural networks have also been used in EM optimization procedures to speed up the convergence while maintaining the overall accuracy, e.g., [10]. Recently, neural-network methods have been combined with EM simulation for fast EM-based design [11], [12]. In [11], the elements of the coupling matrix of the method of moments are modeled by neural network for efficient filling of the coupling matrix. In [12], a neural network is used to model the Green’s functions to reduce the time needed for the computation of the spatial-domain Green’s functions during the method of moments solution of the integral-equation technique. A review on this topic can be found in [13]. Thist paper develops a fundamentally new method using neural networks to represent the EM behavior at the interface between the substructures from an internally decomposed EM structure. Internal decomposition of structures has become an attractive approach for efficient EM simulation and design [14]–[18]. Unlike conventional decompositions, which decompose a circuit into components or a system into subsystems, the internal decomposition divides an EM structure into substructures even though these substructures are not necessarily conventional EM components. In this paper, internal decomposition is used to produce models of substructures that can be reused for various EM-based simulations and designs. Neural networks are developed to represent the spatial EM behavior of a substructure as seen from the interface. The proposed model is formulated in a finite-element environment [19], [20]. This paper significantly extends the preliminary work in [21]. For a given substructure, a so-called EM transfer function matrix is formulated to produce training data, allowing neural networks to learn the spatial EM-field coupling between EM-field variables at various locations over the interface of the substruc-

0018-9480/$26.00 © 2010 IEEE

22

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 1. Illustration of the overall waveguide structure and its decomposition. Substructure will be considered for neural-network model development. Inand the rest terfaces and are the two interfaces between substructure of the structure. Planes and are external input–output ports of the overall structure.

P

X

G

R

T

X

ture. To simplify the neural-network training, the overall modeling problem is divided into multiple subproblems according to the directions of the mesh edges between which the EM coupling is defined. Each subproblem is represented by a neural network. The trained neural networks, taken as a whole, become a model of the substructure representing the full EM behavior including the coupling between the substructure and the rest part of the overall EM problem. To use the proposed model of a substructure in the finite-element simulation of an overall EM structure, we first formulate the neural-network model to produce an EM transfer function matrix representing the spatial EM-field coupling between every pair of the mesh edges across the entire interface. Second, this transfer function matrix is converted into part of the stiffness matrix of the overall finite-element equation. This formulation eliminates the mesh variables inside the substructure under question from the overall finite-element equation, thus reducing the size of finite-element equation and computational cost. To make the proposed model more flexible, we also develop a technique which makes the model compatible even if the mesh used in simulation is different from that in neural-network training. Examples show that the proposed modeling method is more accurate than the conventional neural-network approach for modeling substructures from an internally decomposed EM problem. Using the proposed model also speeds up finite-element simulation. II. FORMULATION OF NEURAL-NETWORK TRAINING DATA: EM TRANSFER FUNCTION MATRIX OF 3-D SUBSTRUCTURE A waveguide structure is taken as an example to illustrate the proposed modeling method. The waveguide structure could be an empty waveguide, a waveguide filter, a waveguide phase shifter, or any other type of waveguide device. In Fig. 1, the external input–output ports are denoted as reference planes and . The overall structure is decomposed so that a general substructure, denoted as substructure , will be considered for neural-network model development. The internal interfaces between this substructure and the rest of the structure are denoted as interfaces and . In order to find a way for neural network to learn the spatial EM behavior of substructure , we derive a so-called EM transfer function matrix. Following the equivalence principle in [22], we convert the original overall waveguide problem into two equivalent problems, known as the modeling and simulation problems, as shown in Fig. 2.

Fig. 2. Two equivalent problems, i.e., the modeling and simulation problems, for the original waveguide problem. (a) Modeling problem for modeling substructure . Perfect magnetic conductors (PMCs) are placed over the rest of is placed on interfaces and . the structure. Equivalent current density (b) Simulation problem for simulating the overall structure using the model of substructure . A PMC is placed over substructure . Equivalent current denis placed on interfaces and . In these two equivalent problems, sity is the tangential component of electric field intensity on interfaces and .

X

0J~

X

J~

P

G

P

X

P

G

E~ G

In the modeling problem, we will formulate the relationship between EM variables on the interfaces of substructure . This relationship needs to fully represent the EM behavior of substructure such that it can be used in the simulation with the rest of the structure, which can be different from the original one. To derive the relationship for substructure independently from the specifics of the rest of the structure, it is necessary to find a way to represent the EM effect on substructure from the rest of the structure, which can be arbitrary. Following equivalence principle in [22], we first replace the rest of the structure with perfect magnetic conductors. We then place equivalent surface electric current density , representing the EM effect of the rest of the structure, on the surface between substructure and the remaining structure (i.e., interfaces and ), as shown in Fig. 2(a). Here, the EM field in substructure is kept the same as that in the original problem. Later, in the simulation problem, the model of substructure can be incorporated with the rest of the structure for the simulation of the overall waveguide structure. In this case, it is also needed to find a way to represent the EM effect on the rest of the structure from substructure . The equivalence principle in [22] is applied here again. We first use a perfect magnetic conductor to replace substructure , and then place equivalent suron the surface between subface electric current density and the remaining structure (i.e., interfaces and structure ), as shown in Fig. 2(b). In this way, the EM field in the rest of the structure is the same as that in the original problem. Let represent the tangential component of electric field and the rest intensity on the surface between substructure of the structure (i.e., interfaces and ). can be uniquely determined by in the modeling problem and vice versa [22]. Thus, in the finite-element method using edge elements [19], we can derive the following relation: (1)

LIAO et al.: NEURAL-NETWORK MODELING FOR 3-D SUBSTRUCTURES

23

where the vectors and represent the discrete spatial equivalent electric current density and electric field intensity along the edges of the meshes on the interfaces of substructure , respectively. The matrix is called the EM transfer function matrix of substructure . From the simulation problem, we can derive the finite-element equation for it as (2) where the matrix is the finite-element stiffness matrix (i.e., coefficient matrix) of the rest of the structure [19], [20]. The represents the discrete spatial electric field intensity vector along the edges of the meshes in the rest of the structure and also on the interfaces of substructure . The vector contains the information of the incident wave electric field at the external and input–output ports of the overall EM structure [19]. are the free-space wavenumber and wave impedance, respectively. The matrix is assembled from the integral of the inner product of vector basis functions in the surface elements on interfaces and . On the other hand, inspired by the domain decomposition method in finite-element method [15], the stiffness matrix of is reduced to a so-called compressed stiffness substructure . We accomplish this reduction by elimmatrix denoted as inating the unknowns along the edges of the mesh inside substructure and retaining the unknowns along the edges of the meshes only on the interfaces [23]. From (2) and the finite-element equation for the original problem, the EM transfer function of substrucmatrix and the compressed stiffness matrix ture can be related by (3) In this way, we can obtain the EM transfer function matrix , which captures the EM behavior of substructure . This matrix relates the discrete spatial electric field intensity to equivalent electric current density along the edges of the meshes across the interfaces of substructure . It will be used as the training information for neural-network model to learn. III. FORMULATION OF NEURAL-NETWORK MODEL BASED ON EM BEHAVIOR ACROSS THE INTERFACE OF THE SUBSTRUCTURE A. Formulation of the Neural-Network Model Substructure is considered independently, with its own mesh in finite-element formulation. Each edge of the meshes on the interfaces corresponds to a row and also a column in the EM transfer function matrix . Each element in matrix represents the spatial EM-field coupling term between the electric field intensity (i.e., the source) along an edge of the meshes on the interfaces of substructure and the equivalent electric current density (i.e., the corresponding response) along another edge of the meshes on the interfaces of the same substructure. Considering the th edge as a response edge while considering the th edge as an excitation edge, the coupling relation is expressed as (4)

and represent the discrete spatial equivalent elecwhere tric current density and electric field intensity along the response and excitation edges, respectively. The spatial coupling term is equal to the element in the th row and the th column of matrix . Interfaces and of substructure are assumed to be parallel with the – plane. The direction of an edge of the meshes on the interfaces is defined as the angle between the edge and the – plane. The coupling term can be expressed as a function of frequency, geometrical parameter of substructure , coordinates, and directions of response and excitation edges as (5) where represents frequency and the vector represents the and repgeometrical parameter of substructure . resent the coordinates and direction of an edge, respectively. The subscripts and represent the edge indices of response and excitation edges, respectively. Since the function in (5) is very difficult to be derived analytically and numerical solutions are computationally expensive, neural networks become a good choice to represent it. Since there are too many variables in (5), it is difficult to train a single neural network to directly represent (5). For this reason, we split function (5) into four independent subfunctions according to the directions of response and excitation edges. Each subfunction will be represented by a separate neural network. For simplicity, the surface elements on the interfaces of are assumed to be all regular in shape. The disubstructure rection of an edge is defined as either 0 or 90 . In other words, an arbitrary edge is either horizontal or vertical. Now, we can divide the indices of the edges of the meshes on the interfaces of substructure into two sets as index of all horizontal edges index of all vertical edges

(6)

Set contains all the indices of the edges with 0 direction and set contains all the indices of the edges with 90 direction. According to four different combinations of excitation and response edges, the spatial EM-field coupling given by (5) can be divided into four types. These combinations are: 1) horizontal response and vertical excitation edges; 2) vertical response and horizontal excitation edges; 3) horizontal response and horizontal excitation edges; and 4) vertical response and vertical excitation edges. The corresponding four types of coupling terms and , respectively. The suare represented as perscripts and represent horizontal and vertical edges, respectively. We propose to use four separate neural networks to represent the four types of coupling as

(7a) (7b) (7c) (7d)

24

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

where

the functions and represent the input–output relationships of the and four neural networks, respectively. are the vectors containing the synaptic weight parameters of the four neural networks, respectively. The inputs of each neural network include frequency, geometrical parameter, and coordinates of response and excitation edges. The output is the corresponding spatial EM-field coupling term. Different neural networks use different sets of response and excitation edges for training. For example, the input neurons of neural network are (8a) and output neurons are (8b) The proposed neural-network formulations are further illustrated in Fig. 3. In this paper, rectangular mesh is used on the interfaces of the modeled substructure for simplicity. However, the proposed method can be expanded to include triangular mesh, which is able to approximate arbitrary surfaces. In this case, function (5) would not be split into separate functions for or (hor(vertical edges). Instead, the izontal edges) and or function (5) would be directly represented by a neural network and are both continuous variables and are desigwhere nated as inputs to the neural network. Since and represent the inclination angles of the response and excitation edges in 25 , etc.), a triangular mesh is thereany direction (e.g., 49 fore accommodated. This step would be a useful future research direction. B. Training Data Generation and Neural-Network Training In order to make the four neural networks capable of representing the spatial EM-field coupling anywhere and everywhere across the interfaces of substructure , we need to train the four neural networks with the training data from the entire EM transfer function matrix . The row and column indices of each element in matrix are converted into coordinates of the excitation and response edges of the meshes on the interfaces, which become the data for the input neurons. The value of the corresponding element of matrix becomes the target data for the output neuron during neural-network training. With these training data, we can train the four neural networks to learn the spatial EM behavior of substructure . The objective of training is to optimize weights and in the four neural networks to minimize the difference between the outputs of the neural networks and the training data. The specific process of training neural network is formulated

Fig. 3. Four types of spatial EM-field coupling are represented by four neural ;f ;f ; and f represent the spanetworks. Neural networks f tial EM-field coupling between: (a) horizontal response and vertical excitation edges, (b) vertical response and horizontal excitation edges, (c) horizontal response and horizontal excitation edges, and (d) vertical response and vertical excitation edges, respectively.

as (9), shown at the bottom of this page. In a way similar to , the training of other three neural training neural network and ) can networks (i.e., neural networks also be formulated accordingly. The four trained neural networks, taken as a whole, are referred to as the neural-network model of substructure . The proposed model describes the spatial EM-field coupling between the discrete spatial electric field intensity and equivalent electric current density along the edges of the meshes on the interfaces of substructure . Therefore, it can accurately represent the spatial EM behavior of substructure . Fig. 4 shows the flowchart illustrating the procedure of developing the neural-network model for substructure based on spatial EM-field coupling in finite-element method. The overall procedure is summarized in the following steps. Step 1) Initialize frequency and geometrical parameter of substructure . Step 2) Generate mesh for substructure . From the mesh, build the stiffness matrix of the substructure following the standard procedure of finite-element method. Step 3) Calculate the compressed stiffness matrix from the stiffness matrix obtained in Step 2). Use (3) to compute the EM transfer function matrix of substructure from matrix .

(9)

LIAO et al.: NEURAL-NETWORK MODELING FOR 3-D SUBSTRUCTURES

25

Step 6) Export the neural-network model for substructure based on spatial EM-field coupling in finite-element method. The final model contains all four trained neural networks. IV. FINITE-ELEMENT METHOD USING THE PROPOSED MODEL

Fig. 4. Flowchart illustrating the procedure of developing the neural-network model for substructure based on spatial EM-field coupling in finite-element method.

X

Step 4) Convert each element of matrix into a training of each element is consample. Row index verted into coordinates and direction of response edge. Column index of each element is converted into coordinates and direction of excitation edge. The value of becomes the spatial each element of matrix corresponding to EM-field coupling terms the response and excitation edges. Assemble each . training sample of response and exStep 5) According to the directions citation edges, write each training sample into corresponding training data files for four neural networks. Train the four neural networks separately with their corresponding training data files.

A. Incorporation of the Proposed Model Into Finite-Element Equation We consider the finite-element simulation of the waveguide structure shown in Fig. 1 using the proposed model of substructure . Firstly, mesh is generated for the rest of the structure, from which the stiffness matrix of the rest of the structure is built following the standard procedure of finite-element method. Next, the EM transfer function matrix of subat the given frequency and geometrical paramstructure eter is reproduced from the proposed model. To do this, spatial EM-field coupling terms are generated from the neural networks. For a given pair of response and excitation edges of the meshes on interfaces of substructure , we first identify the directions of the response and excitation edges, i.e., horizontal–vertical, vertical–horizontal, horizontal–horizontal, or vertical–vertical. The directions determine which of the four or ) will be neural networks (i.e., used. Coordinates of the response and excitation edges along with the frequency and the geometrical parameter are fed to the chosen neural network to calculate . By sweeping the indices the spatial EM-field coupling term of response and excitation edges, all the spatial EM-field coupling terms can be obtained. These coupling terms are used to assemble the EM transfer function matrix of substructure at the given frequency and geometrical parameter . Finally, we can add the contribution of substructure into the finite-element equation. To do this, matrix of the rest of the structure and matrix of substructure are used to formulate the finite-element equation following the idea in (2). More specifically, the detailed process of incorporating the proposed models of substructures into the finite-element equation is expressed in (10), shown at the bottom of the following page. Generally, the overall EM structure may include multiple substructures represented by multiple proposed models. The mesh is generated by discretizing the rest of the structure, the interfaces between the substructures, and the interfaces between the substructures and the rest of the structure. From the mesh, the stiffness matrix of the rest of the structure can be built. and in (10) are the same as those defined in (2). represents the number of the substructures modeled by the proposed models. and represent the four neural networks for the th substructure . Let represent the number of the edges of the mesh on the interface of the th substructure. Vector in (10), can be formed as .. . th row .. .

(11)

26

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Here, vector is used to assemble the EM transfer function matrix of the th substructure from the spatial coupling terms generated by the neural networks in the proposed model. The content inside the curly bracket of (10) is the compressed of the th substructure. It will be incorpostiffness matrix rated into the stiffness matrix of the overall EM structure by using matrix , which is expressed as (12) where represents the global edge number of the first edge of the mesh on the interface of the th substructure and represents the total number of the edges in the simulation of in (10) can be either the overall EM structure. The matrix an identity matrix or a mesh-change matrix to be described in is assembled from the next part of this section. The matrix the integral of the inner product of vector basis functions in the surface elements on the interfaces of the th substructure, as defined in (2). along the By solving (10), the electric field intensity edges of the meshes on the interfaces of all modeled substructures and in the rest of the structure can be obtained. From the incident wave electric field intensity and the resulting electric field intensity at the external input–output ports, the -parameters of the overall structure can be calculated. B. Use of the Proposed Model in the Simulation With the Mesh Different from That in Training In the process of developing the proposed model for a substructure, mesh must be generated for the substructure. The mesh on the interface of the substructure used during the model development is called the “training mesh” of the model. On the other hand, the mesh on the interface of the substructure, with which the proposed model is used during simulation of the overall EM structure, is called the “simulation mesh” of the model. Generally, during finite-element simulation, the meshes of two connected substructures must match (i.e., the same mesh or identical mesh) on their connecting interfaces. Otherwise, the stiffness matrix of the overall structure cannot be built and the

simulation cannot be carried out. Suppose the substructure represented by a neural-network model previously trained is to be connected to a user-substructure, i.e., the rest of the structure, in an overall EM simulation. It is possible that the mesh on the interface of the user-substructure is different from the training mesh of the model. To continue simulation, it is necessary to use a unified mesh on both sides of the interface. This means that we need to transform the spatial EM-field coupling represented by the proposed model under the training mesh into that of a different mesh. and represent the EM transfer function Let matrices matrices of the th substructure under the training and simulation meshes, respectively. Using the concept of cement technique in the domain decomposition method in finite-element method [15], we can derive the relation between matrices and as (13) where the matrix is an interpolation matrix, which is referred to as “mesh-change matrix” of the th substructure. Each element of matrix is an interpolation coefficient relating the discrete physical quantity along one edge of the training mesh to that along one edge of the simulation mesh. The interpolation coefficient depends only on coordinates and directions of the two edges of the two meshes. Using (13), we can obtain the EM transfer function matrix of the th substructure under the simulation mesh from the proposed model. With the proposed mesh change formula of (13), even if the simulation mesh of a proposed model is different from its training mesh, the proposed model can still be used in finite-element simulation. To incorporate the mesh change forin (10) to be a mula (13) into (10), we allow the matrix mesh-change matrix of the th substructure. In a special case, where the training and simulation meshes are the same, matrix becomes an identity matrix. C. Discussion The conventional neural-network model, which is based on -parameters, is usually suitable for representing the external

(10)

LIAO et al.: NEURAL-NETWORK MODELING FOR 3-D SUBSTRUCTURES

27

input–output port behavior of an EM structure, but if the conventional neural network is used to represent a substructure from an internally decomposed EM structure, accuracy problems may arise. Notice that because of the discontinuities near the interfaces, -parameter representation of the substructure is usually not accurate. Thus, the accuracy of the conventional neural-network approach may be affected. Instead of -parameters, the proposed neural-network model represents the spatial EM-field coupling over the interface of a substructure from an internally decomposed EM structure. The proposed model can accurately represent the EM behavior of the substructure, even if the discontinuities in the substructure are close to the interface. This enables the finite-element method using the proposed model to produce accurate simulation results. Notice that the number of the unknowns in (10) does not include the unknowns along the edges of the meshes inside all modeled substructures. Therefore, there is a net reduction of the number of the unknowns in the finite-element equation (10) over the original finite-element equation of an un-decomposed finite-element simulation. Since finite-element simulation time heavily depends on the time spent on solving the finite-element equation, the proposed formulation can contribute to speeding up finite-element simulation. V. NUMERICAL EXAMPLES A. Neural-Network Modeling for the Internally Decomposed -Plane Filter Substructures This example demonstrates the application of the proposed method in modeling the substructures from an internally decomposed three-pole -plane filter [24], as shown in Fig. 5(a). For illustrative purpose, the filter is internally decomposed into substructures to , as shown in Fig. 5(b). All of these five substructures will be represented by neural-network models. We will develop one neural-network model, called neural-network model 1, for the substructure of Fig. 5(c), and use the model to represent substructure in the overall filter simulation. We also add a geometrical parameter as an input to neural-network model 1. Therefore, the vector of geometrical parameter , as used in (7), is equal to . Since substructures and are symmetrical about -axis, neural-network model 1 could be reused to represent substructure in the simulation. Similarly, and have the same geometry, we because substructures will also develop one neural-network model, called neural-network model 2, for the substructure of Fig. 5(d), and use the and in the simsame model to represent substructures ulation. We will also develop one neural-network model, called neural-network model 3, for the substructure of Fig. 5(e) for future use. In neural-network models 2 and 3, there is no geometrical parameter considered. To develop neural-network model 1, the substructure of Fig. 5(c) is first discretized to generate mesh. In our example, rectangular brick element is used [19]. The mesh of the substructure comprises of 38, 5, and 64 rectangular elements and directions, respectively. There are 12 160 in the (38 5 64) rectangular brick elements with 42 471 edges. From the mesh, the stiffness matrices at different frequency

H

Fig. 5. (a) Three-pole -plane filter (3-D view). (b) Internally decomposed three-pole -plane filter with substructures to (top view). (c) Substructure represented by neural-network model 1. (d) Substructure represented by neural-network model 2. (e) Another substructure represented by neural-network model 3. ( mm, mm, mm, mm, mm, mm, mm, and mm. is geometrical parameter in the neural-network model 1.)

H

w =6

A E

a = 19:05 b = 9:525 l = 13:2 l = 27:6

t=2 w =9 s=1 d

points (ranging from 11.75 to 12.35 GHz) and geometrical parameters (ranging from [13.8 mm] to [14 mm]) are built. is 42 471 42 471, leading to a total of The size of matrix 42 471 unknowns in simulation. The substructure is connected with other substructures through its two interfaces. The meshes on the two interfaces are the same, each comprising of 190 (38 5) rectangular elements with 423 edges. By eliminating the unknowns along the edges of the mesh inside the substructure (41 625 unknowns in total) and retaining the unknowns along the edges of the meshes only on the two interfaces (846 is reduced to the compressed unknowns in total), matrix stiffness matrix of size 846 846. According to (3), matrix is transformed into the EM transfer function matrix , as defined in (1). Matrices at different frequency points and geometrical parameters are further converted into 8 710 416 training samples. Here, spatial symmetry has been considered to reduce the amount of training samples by deleting redundant information. According to the directions of response and excitation edges, these training samples are written into four separate training data files. The numbers of training samples are (1 956 240), (2 407 680), (2 287 296), and (2 059 200) in the four training data files corresponding to horizontal–vertical, vertical–horizontal, horizontal–horizontal, and vertical–vertical

28

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE I COMPUTATIONAL STATISTICS OF VARIOUS METHODS FOR THE -PLANE FILTER EXAMPLES AT 11.9 GHz

H

Fig. 6. Neural-network models 1 and 2 are used in the simulation of the threepole -plane filter shown in Fig. 5(a). Neural-network model 1 is used twice to represent substructures and . Neural-network model 2 is used three times to and . Substructures to are used to assemble represent substructures the overall three-pole -plane filter.

H

B D A;C; E H

A E

combinations of response and excitation edges, respectively. Using NeuroModelerPlus software [25], four neural networks are trained with the four corresponding training data files. There are eight input neurons (i.e., frequency, geometrical parameter, and coordinates and direction of two edges) and one output neuron (i.e., coupling term) for each of the four neural networks. The four trained neural networks, taken as a whole, are referred to as neural-network model 1. The meshes on the interfaces of the substructure are the “training mesh.” The total data generation time is about 2 h, including elimination of the unknowns along the edges of the mesh inside the substructure. The model training time of the four neural networks all together is about 50 h. Thus, it takes about 52 h in total to develop the model. Most of the time is spent on training the four neural networks. Once the model is developed, it can be used over and over again in different simulations without any further training. The more times the model is used, the more the simulation time is saved. In a similar way, neural-network models for the substructures of Fig. 5(d) and (e) are also individually developed. After the models are developed, we can use them in finite-element simulation. Fig. 6 shows the use of neural-network models 1 and 2 in the simulation of the overall three-pole -plane filter of Fig. 5(a). In this simulation, neural-network model 1 is used twice to represent substructures and . Geometrical parameter in neural-network model 1 is set to [14 mm]. Neural-network model 2 is used three times to represent substructures and . Substructures to are used to assemble the overall three-pole -plane filter, as shown in Fig. 6. We first illustrate the default case where the mesh used in the simulation happens to be the same as the training mesh. We follow (10) to formulate the finite-element equation for the overall filter. To use neural-network model 1 to represent substructure , 715 716 coupling terms generated from the four neural networks are used to assemble the EM transfer function matrix at the given frequency point and geometrical parameter . The size

of the reproduced matrix is 846 846. Since the training and of substructure simulation meshes are the same, matrix is an 846 846 identity matrix. In a similar way, matrices and of the other substructures are also obtained. Using those of substructures to ), matrices (i.e., matrices and we then build the stiffness matrix of the overall filter. The finite-element equation for the overall filter is formulated using the overall stiffness matrix including contributions from each of the substructures. There are 2538 unknowns in total. By solving the equation, the -parameters of the overall filter structure are computed from the resulting electric field intensity solution. For comparative purposes, we also simulate the filter using three conventional approaches. First, a full finite-element method is used to simulate the filter as a whole without decomposition. The full finite-element method is implemented using rectangular brick edge elements in MATLAB [26]. For this part of the example, the mesh everywhere in the filter is exactly the same as that used in developing the proposed models of the substructures. Another method used for comparison is the simulation using the conventional neural-network models of substructures to of Fig. 5(b). In this case, the connections between the substructures are interpreted by conventional equivalent circuit ports, and neural networks are trained to represent the two-port -parameters of the substructures. By cascading the -parameters generated by conventional neural-network models of all the substructures according to the sequence in Fig. 5(b), the -parameters of the overall filter are obtained. The third method is the 3-D EM-field simulation using commercial finite-element solver Ansoft HFSS [27]. The HFSS simulation result is used as the reference for comparing the accuracy of different methods, i.e., the finite-element method using the proposed model, the full finite-element method, and the simulation using the conventional neural-network model. Table I shows the computational statistics of the finite-element method using the proposed model and the full finite-element method at 11.9 GHz. It shows that the number of unknowns in the proposed method is much less than (only about 1/64) that in the full finite-element method. Consequently, the proposed method is much faster than the full finite-element method. In this example, it is 12.6 times faster. The simulation time of the proposed method includes three parts: generating

LIAO et al.: NEURAL-NETWORK MODELING FOR 3-D SUBSTRUCTURES

29

TABLE II MEAN SQUARE ERRORS OF THE S -PARAMETER MAGNITUDES OF THE H -PLANE FILTER EXAMPLES FROM VARIOUS METHODS

and of solving the equation. Magnitude and phase of the overall filter computed from various methods are shown in Fig. 7. Good agreement is observed between the proposed method, the full finite-element method, and HFSS. In addition, the result from the proposed method is much more accurate than that from the simulation using the conventional neural-network model. For simplicity, we use mean square errors to quantitatively validate the simulation results in this paper. However, in general, other measures such as the feature selective validation (FSV) method used for quantitative validation of computational EM [28], [29] can also be applied. The mean square errors of the -parameter magnitudes from the proposed methods and the simulation using the conventional neural-network model with respect to the HFSS solution are compared in Table II. Result from HFSS is the reference solution with respect to which the mean square errors are defined. The equation used to calculate the mean square error in the table is given by

(14)

Fig. 7. Magnitude and phase of S and S of the three-pole H -plane filter computed from various methods. In the finite-element method (FEM) using the proposed model, the simulation mesh used is the same as the training mesh for all the neural-network models, i.e., neural-network models 1 and 2. The solution from the proposed method matches well with those of the full FEM and HFSS, and is much more accurate than that of the simulation using the conventional neural-network (NN) model. (a) jS j. (b) jS j. (c) S . (d) S .

mesh for the overall simulation structure outside the modeled substructure (i.e., the rest of the structure in Section IV-A), formulating the finite-element equation following (10), and

where is number of samples and is sample index. is the reference solution and is the -parameters from the two simis a pure number, ulation methods. The mean square error i.e., its unit is 1. It can be seen that the result from the proposed method is much closer to the HFSS solution than that from the simulation using the conventional neural-network model. Next, we illustrate that without retraining, the same neuralnetwork models developed in the previous three-pole -plane filter can be reused in a new structure, i.e., a two-pole -plane filter. Fig. 8 shows the use of the previously trained neural-network models 1, 2, and 3 to simulate the new two-pole -plane filter. In this simulation, neural-network models 1 and 2, without retraining, are used again to represent substructures and , respectively. In this case, geometrical parameter in neural model 1 is set to [13.8 mm]. Neural-network model 3 is used to repand are used to resent substructure . Substructures assemble the overall two-pole -plane filter. In this part of the example, the mesh used in the simulation is still the same as the training mesh. We follow (10) to formulate the finite-element equation. The EM transfer function matrix for each of the three substructures is generated from their corresponding

30

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 8. Without retraining, the same neural-network models developed in the previous three-pole -plane filter can be reused in a new structure, i.e., a twopole -plane filter shown here. Neural-network models 1 and 2 are reused to represent substructures and , respectively. Neural-network model 3 is used to represent substructure .

H

H

F

H

G

neural-network models. The mesh used in the simulation is the of same as that used in training the model, thus matrices all the three substructures are identity matrices. From these maof substructures to ), the trices (i.e., matrices and contribution from each substructure to the stiffness matrix of the overall finite-element equation is determined. The finite-element equation for the overall filter is formulated and there are 1692 unknowns in total. The equation is solved and the -parameters of the overall filter are then computed. For comparative purposes, the full finite-element method, the simulation using the conventional neural-network models of substructures, and HFSS are also used to simulate the two-pole -plane filter. In the full finite-element method, the mesh everywhere in the filter is exactly the same as that used in developing the proposed models of the substructures. The computational statistics of the finite-element method using the proposed model versus the full finite-element method at 11.9 GHz are shown in Table I. It shows that the number of unknowns in the proposed method is much less than (only about 1/82) that in the full finite-element method. The proposed method is about 8.5 times faster than the full finite-element method. The result computed from the proposed method is much closer to HFSS solution than that from the simulation using the conventional neural-network model, as shown in Fig. 9 and also in Table II. It confirms that the proposed neural-network model is reusable as a plug-in 3-D EM module in the simulation of different overall problems. As a further verification of our method, we illustrate that the proposed model can also be used in the simulation even if the mesh used in the simulation is different from that used in training the model. We re-simulate the three-pole -plane filter shown in Fig. 5(a) using a new mesh. Substructures and , shown in Fig. 5(b), are still represented by neural-network model 1, but substructures and , which are regarded as

S

S

H

Fig. 9. Magnitude and phase of and of the two-pole -plane filter computed from various methods. In the finite-element method (FEM) using the proposed model, the simulation mesh used is the same as the training mesh for all the neural-network models, i.e., neural-network models 1, 2, and 3. The solution from the proposed method matches well with those of the full FEM and HFSS, and is much more accurate than that of the simulation using the conventional neural-network (NN) model. (a) j j. (b) j j. (c) . (d) .

S

S

S

S

the rest of the structure, are directly represented by the stiffbuilt following the standard procedure of finess matrix nite-element method. In this simulation, we deliberately make

LIAO et al.: NEURAL-NETWORK MODELING FOR 3-D SUBSTRUCTURES

31

Fig. 10. Magnitude of S of the three-pole H -plane filter computed from various methods. In the finite-element method (FEM) using the proposed model with mesh change, the simulation mesh used is different from the training mesh for neural-network model 1. The solution from the proposed method with mesh change matches well with those of the full FEM and HFSS.

the meshes on the interfaces of substructures and to be different from the training mesh of neural-network model 1. and The mesh on each of the interfaces of substructures , referred to as the new mesh, comprises of 400 (50 8) rectangular elements with 858 edges. This is different from the training mesh of neural-network model 1, which comprises of 190 (38 5) rectangular elements with 423 edges on each of the two interfaces of the substructure. In order to connect substructure with substructures and for the finite-element simulation, the spatial coupling terms computed from neural-network model 1 have to be converted into those corresponding to the new mesh. Therefore, mesh change formula as described in Section IV-B becomes necessary. We again follow (10) to formulate the finite-element equation. The EM transfer function matrix of substructure under the training mesh is first reproduced from neural-network model 1 and its size is 846 846. According to the training and simufor substructure lation meshes, the mesh-change matrix is generated. Matrix is a large sparse matrix, whose size is 1716 846. Each element of matrix is an interpolation coefficient from discrete physical quantity along one edge of the training mesh to that along one edge of the simulation mesh. Here, the interpolation coefficient is calculated from 2-D quadratic interpolation polynomial [30]. After the mesh change, becomes the size of the new EM transfer function matrix of substruc1716 1716. In the same way, matrices and is directly ture are obtained as well. The stiffness matrix built following the standard procedure of finite-element method and to represent the rest of the structure, i.e., substructures . From these matrices (i.e., matrices and of substrucof the rest of the structure), the tures and , and matrix stiffness matrix of the overall three-pole -plane filter is obtained, and the finite-element equation is then formulated and solved. After that, the -parameters are calculated. The solution is compared with those from the full finite-element method and HFSS, as shown in Fig. 10, where good agreement has been observed. This result confirms that the proposed model can be used in finite-element simulation, even if the simulation mesh is different from the training mesh.

Fig. 11. (a) E -plane finned waveguide filter with three bilateral fins (3-D view). (b) E -plane finned waveguide filter with three bilateral fins (side view) is decomposed into five substructures. (c) Substructure modeled by neural-network : mm, b : mm, a : mm, b : mm, model 4. (a p mm, t mm, l l : mm, l : mm, l : mm, l l l l : mm, l : mm, l : mm, l : mm. l ; l ; and l are geometrical parameters in neural-network model 4.)

= 22 85 = 10 15 = 10 66 = 4 29 = 15 =1 = = 0 125 = 0 125 = 0 99 = = = = 5 725 = 16 = 1 01 = 0 115

B. Neural-Network Modeling for the Waveguide Filter Substructures

-Plane Finned

This example demonstrates the application of the proposed method in modeling the substructures of an -plane finned waveguide filter with three bilateral fins [31], as shown in Fig. 11(a). The filter is internally decomposed into substruc, as shown in Fig. 11(b). We will develop one tures to neural-network model, called neural-network model 4, for the substructure shown in Fig. 11(c), and use the model to represent and in the overall filter simulation. We substructures and as inputs also add three geometrical parameters to the model, as shown in Fig. 11(c). Therefore, the vector of . geometrical parameter as used in (7) is equal to To develop neural-network model 4, we first discretize the substructure of Fig. 11(c) to generate mesh. The mesh comprises of 22 and 11 rectangular elements in the and directions, respectively. The number of rectangular elements in the

32

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

direction is a variable (from 47 to 86), which depends on the length of the substructure in the direction. There are 11 374 (22 11 47) to 20 812 (22 11 86) rectangular brick elements with 37 788 to 68 715 edges. From the mesh, the stiffness at different frequency points (ranging from 12 matrices to 15 GHz) and geometrical parameters (ranging from [0.125 mm 0.99 mm 0.115 mm] to [5.725 mm 1.6 mm 5.725 mm] ) are built. The size of matrix is 37 788 to 68 715. The meshes on the two interfaces of the substructure of Fig. 11(c) are the same, each comprising of 242 (22 11) rectangular elements with 517 edges. By eliminating the unknowns along the edges of the mesh inside the substructure (36 754 to 67 681 unknowns in total) and retaining the unknowns along the edges of the meshes only on the two interfaces (1034 unknowns in total), is reduced to the compressed stiffness the stiffness matrix matrix of size 1034 1034. Matrix is transformed into the EM transfer function matrix . Matrices at different frequency points and geometrical parameters are further converted into 28 867 212 training samples. According to the directions of response and excitation edges, these training samples are written into four separate training data files. The numbers of training samples are (7 213 536), (7 213 536), (7 527 168), and (6 912 972) in the four training data files corresponding to horizontal–vertical, vertical–horizontal, horizontal–horizontal, and vertical–vertical combinations of response and excitation edges, respectively. Four neural networks are trained with the four corresponding training data files. There are ten input neurons and one output neuron for each of the four neural networks. The four trained neural networks, taken as a whole, are referred to as neural-network model 4. After neural-network model 4 is developed, we can use it in the finite-element simulation of the overall -plane finned waveguide filter with three bilateral fins, as shown in Fig. 12. In this simulation, neural-network model 4 is used three times to represent substructures to . Geometrical parameter is set to [0.125 mm 0.99 mm 5.725 mm] , [5.725 mm 1.6 mm 5.725 mm] , and [5.725 mm 1.01 mm 0.115 mm] for substructures and , respectively. The rest of the structure, i.e., substrucbuilt tures and , is represented by the stiffness matrix following the standard procedure of the finite-element method. We first illustrate the default case where the meshes on the interfaces of the rest of the structure happen to be the same as the training mesh of neural-network model 4. We follow (10) to formulate the finite-element equation for the overall filter. To use neural-network model 4 to represent substructure , 1 069 156 coupling terms generated from the four neural networks are used to assemble the EM transfer function matrix at the given frequency point and geometrical parameter . The size of the reproduced matrix is 1034 1034. Since the training and simof substructure is a ulation meshes are the same, matrix 1034 1034 identity matrix. In a similar way, matrices and of the other two substructures are also obtained. Using of substructures to these matrices (i.e., matrices and , and matrix of the rest of the structure), we then build the stiffness matrix of the overall filter. The finite-element equation for the overall filter is formulated and solved. There are 19 514 unknowns in total. The -parameters of the overall filter are then computed.

Fig. 12. Neural-network model 4 is used in the simulation of the E -plane finned waveguide filter with three bilateral fins shown in Fig. 11(a). Neural-network model 4 is used three times to represent substructures J to L. The rest of the structure, i.e., substructures I and M , is represented by its stiffness matrix. TABLE III COMPUTATIONAL STATISTICS OF VARIOUS METHODS FOR THE E -PLANE FINNED WAVEGUIDE FILTER EXAMPLES AT 12.5 GHz

For comparative purposes, the full finite-element method, the simulation using the conventional neural-network models of of Fig. 11(b) and HFSS are used to simusubstructures to late the -plane finned waveguide filter with three bilateral fins. Table III shows the computational statistics of the finite-element method using the proposed model and the full finite-element method at 12.5 GHz. It shows that the number of unknowns in the proposed method is much less than (only about 1/8) that in the full finite-element method. The proposed method is 5.8 times faster than the full finite-element method. Magnitude of of the filter computed from the reflection coefficients various methods are shown in Fig. 13. Good agreement is observed between the proposed method, the full finite-element method, and HFSS. In addition, the result from the proposed method is much more accurate than that from the simulation using the conventional neural-network model. Furthermore, the mean square errors of the -parameter magnitudes from these two methods with respect to the HFSS solution are compared in Table IV. It can be seen that the result from the proposed method is much closer to the HFSS solution than that from the simulation using the conventional neural-network model.

LIAO et al.: NEURAL-NETWORK MODELING FOR 3-D SUBSTRUCTURES

Fig. 13. Magnitude of S of the E -plane waveguide filter with three bilateral fins computed from various methods. In the finite-element method (FEM) , the simulation mesh used is the same as using the proposed model the training mesh for neural-network model 4. In the FEM using the proposed , the simulation mesh used is different from model with mesh change the training mesh for neural-network model 4. The solutions from the proposed method (both with and without mesh change) match well with those of the full FEM and HFSS, and are much more accurate than that of the simulation using the conventional neural-network (NN) model.

TABLE IV MEAN SQUARE ERRORS OF THE S -PARAMETER MAGNITUDES OF THE E -PLANE FINNED WAVEGUIDE FILTER EXAMPLES FROM VARIOUS METHODS

Next, we illustrate that without retraining, the neural-network model 4 developed in the previous -plane finned waveguide filter with three bilateral fins can be reused in a new structure, i.e., an -plane finned waveguide filter with four bilateral fins. Fig. 14 shows the use of the previously trained neural-network model 4 to simulate the new filter. In this simulation, neural-network model 4, without retraining, is used again to represent substructures to . In this case, geometrical parameter in neural-network model 4 is set to [0.125 mm 0.99 mm 5.725 mm] , [5.725 mm 1.6 mm 5.725 mm] , [5.725 mm 1.6 mm 5.725 mm] , and [5.725 mm 1.01 mm 0.115 mm] for substructures and , respectively. The rest of the and , is represented by the structure, i.e., substructures built following the standard procedure stiffness matrix of finite-element method. Substructures and have the and of Fig. 11(b), same geometry with substructures respectively. The mesh used in the simulation is still the same as the training mesh. The finite-element equation for the overall filter is formulated and there are 20 031 unknowns in total. The equation is solved and the -parameters of the overall filter are then computed. For comparative purposes, the full finite-element method, the simulation using the conventional neural-network models of

33

Fig. 14. Without retraining, the neural-network model 4 developed in the previous E -plane finned waveguide filter with three bilateral fins can be reused in a new structure, i.e., an E -plane finned waveguide filter with four bilateral fins shown here. Neural-network model 4 is reused to represent substructures P to S . The rest of the structure, i.e., substructures N and T , is represented by its stiffness matrix.

substructures, and HFSS are also used to simulate the -plane finned waveguide filter with four bilateral fins. The computational statistics of the finite-element method using the proposed model versus the full finite-element method at 12.5 GHz are shown in Table III. It shows that the number of unknowns in the proposed method is much less than (only about 1/12 of) that in the full finite-element method. The proposed method is about 9.1 times faster than the full finite-element method. The result computed from the proposed method is much closer to HFSS solution than that from the simulation using the conventional neural-network model, as shown in Fig. 15 and also in Table IV. As a further verification of our method, we illustrate that the proposed model can also be used in the simulation even if the mesh used in simulation is different from that used in training the model. We re-simulate the -plane finned waveguide filter with four bilateral fins of Fig. 11(a) using a new mesh. In this simulation, we deliberately make the meshes on the interfaces of the rest of the structure, i.e., substructures and , connected to substructures and , to be different from the training mesh of neural-network model 4. The mesh on each of the two interfaces, referred to as the new mesh, comprises 375 (25 15) rectangular elements with 790 edges. This is different from the training mesh of neural-network model 4, which comprises 242 (22 11) rectangular elements with 517 edges on each of the two interfaces of the modeled substructure. In order to confor the nect substructures with and substructures with overall simulation, the spatial coupling terms computed from neural-network model 4 have to be converted into those correof the rest sponding to the new mesh. The stiffness matrix of the structure, i.e., substructures and , is built following the standard procedure of finite-element method. We again follow (10) to formulate the finite-element equation. EM transfer function matrix of substructure under the training mesh is first reproduced from neural-network model 4 for and its size is 1034 1034. The mesh-change matrix

34

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 15. Magnitude of S of the E -plane waveguide filter with four bilateral fins computed from various methods. In the finite-element method (FEM) using the proposed model, the simulation mesh used is the same as the training mesh for neural-network model 4. The solution from the proposed method matches well with those of the full FEM and HFSS, and is much more accurate than that of the simulation using the conventional neural-network (NN) model.

substructure is generated and its size is 1580 1034. After the mesh change, the size of the new EM transfer function matrix becomes 1580 1580. In a similar way, matrices and of the other two substructures are obtained as well. From these matrices (i.e., matrices and of substructures to , and matrix of the rest of the structure), the finite-element equation is then formulated and solved. The -parameters are calculated. The solution is compared with those from the full finite-element method and HFSS, as shown in Fig. 13, where good agreement has been observed. C. Neural-Network Modeling for the Left-Handed Transmission Line Substructures In this example, the proposed method is used in modeling the substructure of a coaxial waveguide based left-handed transmission line with five periodic units [32], as shown in Fig. 16(a). Here, we consider a rectangular coaxial waveguide based lefthanded transmission line. In this example, the left-handed transmission line is internally decomposed into seven substructures, i.e., two access regions and five periodic units, as shown in Fig. 16(b). All the five periodic units have the same geometry. We will develop one neural-network model, called neural-network model 5, for the periodic unit of Fig. 16(c), and use the model to represent all the five units in the overall left-handed transmission line simulation. We also add two geometrical paas inputs to neural-network model 5, as shown in rameters . Fig. 16(c) and (d). Symbol is used to represent vector To develop neural-network model 5, we first discretize the periodic unit of Fig. 16(c) and (d) to generate mesh. The mesh and comprises of 15, 15, and 29 rectangular elements in the directions, respectively. There are 6525 (15 15 29) rectangular brick elements with 21 824 edges. From the mesh, The at different frequency points (ranging stiffness matrices from 0.5 to 2 GHz) and geometrical parameters (ranging from [0.6 mm 0.1 mm] to [1 mm 0.2 mm] ) are built. The size of matrix is 21 824. The meshes on the two interfaces of the periodic unit are the same, each comprising of 225 (15 15) rect-

Fig. 16. (a) Left-handed transmission line with five periodic units (3-D view). All the periodic units have the same geometry. (b) Left-handed transmission line with five periodic units (side view) is decomposed into seven substructures, i.e., two access regions and five periodic units. (c) Periodic unit of the left-handed transmission line (3-D view). (d) Periodic unit of the left-handed transmission mm, b mm, p mm, l : mm, line (side view). (a s mm, t mm, c : mm. g and g are geometrical variables in neural-network model 5.)

= 14

=2

= 30 = 24 = 0 01

= 30

= 53

angular elements with 480 edges. By eliminating the unknowns along the edges of the mesh inside the periodic unit (20 864 unknowns in total) and retaining the unknowns along the edges of the meshes only on the two interfaces (960 unknowns in total), is reduced to the compressed stiffness the stiffness matrix matrix of size 960 960. Matrix is transformed into the EM transfer function matrix . Matrices at different frequency points and geometrical parameters are further converted into 12 902 400 training samples. Here, spatial symmetry has been considered to reduce the amount of training samples by deleting redundant information. According to the directions of response and excitation edges, these training samples are written into four separate training data files. The number of training samples is (3 225 600) in each of the four training data files. Four neural networks are trained with the four corresponding training data files. There are nine input neurons and one output neuron for each of the four neural networks. The four trained neural networks, taken as a whole, are referred to as neural-network model 5. After neural-network model 5 is developed, we can use it in the simulation of the overall left-handed transmission line with five periodic units, as shown in Fig. 17. In this simula-

LIAO et al.: NEURAL-NETWORK MODELING FOR 3-D SUBSTRUCTURES

35

TABLE V COMPUTATIONAL STATISTICS OF VARIOUS METHODS FOR THE LEFT-HANDED TRANSMISSION LINE EXAMPLES AT 0.86 GHz

Fig. 17. Neural-network model 5 is used in the simulation of the left-handed transmission line with five periodic units shown in Fig. 16(a). Neural-network model 5 is used to represent each of the five periodic units. The rest of the structure, i.e., the two access regions, is represented by its stiffness matrix.

tion, neural-network model 5 is used to represent each of the five periodic units. Geometrical parameter is set to [1 mm 0.2 mm] . The rest of the structure, i.e., the two access rebuilt following gions, is represented by the stiffness matrix the standard procedure of finite-element method. We first illustrate the default case where the meshes on the interfaces of the rest of the structure happen to be the same as the training mesh of neural-network model 5. We follow (10) to formulate the finite-element equation for the overall filter. To use neural-network model 5 to represent the periodic unit, 921 600 coupling terms generated from the four neural networks are used to assemble the EM transfer function matrix at the given frequency point and geometrical parameter . The size of the reproduced matrix is 960 960. Since the training and simulation meshes are the same, matrix of the periodic unit is a 960 960 identity matrix. Using these matrices (i.e., maof the periodic unit, and matrix of the trices and rest of the structure), we then build the stiffness matrix of the overall left-handed transmission line. The finite-element equation for the overall left-handed transmission line is formulated and solved. There are 33 792 unknowns in total. The -parameters of the overall filter are then computed. For comparative purposes, the full finite-element method, the simulation using the conventional neural-network models of all the substructures of Fig. 16(b), and HFSS are used to simulate the left-handed transmission line. Table V shows the computational statistics of the finite-element method using the proposed model and the full finite-element method at 0.86 GHz. It shows that the number of unknowns in the proposed method is much less than (only about 1/4) that in the full finite-element method. The proposed method is 5.5 times faster than the full finite-element method. Magnitude of the reflection coefficients of the left-handed transmission line computed from various methods are shown in Fig. 18. Good agreement is observed between the proposed method, the full finite-element method, and HFSS. In addition, the result from the proposed method is much more accurate than that from the simulation using the conventional neural-network model. Furthermore, the mean square

Fig. 18. Magnitude of S of the left-handed transmission line with five periodic units computed from various methods. In the finite-element method (FEM) using the proposed model , the simulation mesh used is the same as the training mesh for neural-network model 5. In the FEM using the proposed , the simulation mesh used is different from model with mesh change the training mesh for neural-network model 5. The solutions from the proposed method (both with and without mesh change) match well with those of the full FEM and HFSS, and are much more accurate than that of the simulation using the conventional neural-network (NN) model.

errors of the -parameter magnitudes from these two methods with respect to the HFSS solution are compared in Table VI. It can be seen that the result from the proposed method is much closer to the HFSS solution than that from the simulation using the conventional neural-network model. Next, we illustrate that without retraining, the same neuralnetwork model developed in the previous left-handed transmission line with five periodic units can be reused in a new structure, i.e., a left-handed transmission line with three periodic units. Fig. 19 shows the use of the previously trained neural-network model 5 to simulate the new left-handed transmission line. In this simulation, neural-network model 5, without retraining, is used again to represent the periodic units. In this case, geometrical parameter is set to [0.6 mm 0.1 mm] . The rest of the structure, i.e., the two access regions, is represented by the built following the standard procedure of stiffness matrix the finite-element method. Here, the two access regions shown in Fig. 19 have the same geometry with the two access regions shown in Fig. 16(b). The mesh used in the simulation is still the same as the training mesh. The finite-element equation for

36

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE VI MEAN SQUARE ERRORS OF THE S -PARAMETER MAGNITUDES OF THE LEFT-HANDED TRANSMISSION LINE EXAMPLES FROM VARIOUS METHODS

Fig. 20. Magnitude of S of the left-handed transmission line with three periodic units computed from various methods. In the finite-element method (FEM) using the proposed model, the simulation mesh used is the same as the training mesh for neural-network model 5. The solution from the proposed method matches well with those of the full FEM and HFSS, and is much more accurate than that of the simulation using the conventional neural-network (NN) model.

Fig. 19. Without retraining, the same neural-network model developed in the previous left-handed transmission line with five periodic units can be reused in a new structure, i.e., a left-handed transmission line with three periodic units shown here. Neural-network model 5 is reused to represent each of the three periodic units. The rest of the structure, i.e., the two access regions, is represented by its stiffness matrix.

the overall filter is formulated and there are 32 832 unknowns in total. The equation is solved and the -parameters of the overall filter are then computed. For comparative purposes, the full finite-element method, the simulation using the conventional neural-network models of substructures, and HFSS are also used to simulate the left-handed transmission line with three periodic units. The computational statistics of the finite-element method using the proposed model versus the full finite-element method at 0.86 GHz are shown in Table V. It shows that the number of unknowns in the proposed method is much less than (only about 1/3) that in the full finite-element method. The proposed method is about three times faster than the full finite-element method. The result computed from the proposed method is much closer to HFSS solution than that from the simulation using the conventional neural-network model, as shown in

Fig. 20 and also in Table VI. It further confirms that the proposed neural-network model is reusable as a plug-in 3-D EM module in the simulation of different overall problems. As a further verification of our method, we illustrate that the proposed model can also be used in the simulation even if the mesh used in simulation is different from that used in training the model. We re-simulate the left-handed transmission line with five periodic units of Fig. 16(a) using a new mesh. In this simulation, we deliberately make the meshes on the interfaces of the rest of the structure, i.e., the two access regions, to be different from the training mesh of neural-network model 5. The mesh on each of the interfaces, referred to as the new mesh, comprises of 324 (18 18) rectangular elements with 684 edges. This is different from the training mesh of neural-network model 5, which comprises of 225 (15 15) rectangular elements with 480 edges on each of the two interfaces of the modeled substructure. In order to connect the rest of the structure with the periodic units for the overall finite-element simulation, the spatial coupling terms computed from neural-network model 5 have to be converted into those of the corresponding to the new mesh. The stiffness matrix rest of the structure, i.e., the two access regions, is first built following the standard procedure of finite-element method. We again follow (10) to formulate the finite-element equation. EM transfer function matrix of a periodic unit under the training mesh is first reproduced from neural-network model 5 for the and its size is 960 960. The mesh-change matrix periodic unit is generated and its size is 1368 960. After the mesh change, the size of the new EM transfer function matrix becomes 1368 1368. From these matrices (i.e., matrices and of the periodic unit, and matrix of the rest of the structure), the finite-element equation is formulated and solved. The -parameters are then calculated. The solution is compared with those from the full finite-element method and HFSS, as shown in Fig. 18, where good agreement has been observed. This result further confirms that the proposed model can be used in the

LIAO et al.: NEURAL-NETWORK MODELING FOR 3-D SUBSTRUCTURES

finite-element method, even if the simulation mesh is different from the training mesh. VI. CONCLUSION A new neural-network modeling method has been proposed to describe the EM behavior of a 3-D substructure from an internally decomposed structure. The neural network has been used to represent the spatial EM behavior of the substructure as seen from the interface between the substructure and the rest of the structure. To simplify the neural-network training, the overall modeling problem is divided into multiple subproblems according to the directions of the mesh edges. Each of the subproblems is represented by a neural network. A new formulation has been proposed to connect the proposed models with finiteelement equation, allowing us to use the proposed models in various EM-based simulations and designs. A technique has also been developed to make the proposed model valid even if the simulation mesh is different from the training mesh. Compared with the conventional neural-network model based on -parameters of the external input–output ports, the proposed model is more effective in representing the EM behavior of a substructure from EM decomposition. Three examples have shown that the finite-element simulation using the proposed model is much more accurate than the simulation using the conventional neural-network model. The proposed method is also faster than the full finite-element method. As a possible future work, this method can be extended to include more general meshes such as triangular mesh on the interfaces. Another interesting future direction is to incorporate space mapping to transform the input space of the neural network to a new space of excitation-response functions to simply the neural-network input–output relationship and speedup the training process. By combining EM structural decomposition and neural-network representations, this work helps making EM based design more powerful for microwave computer-aided design (CAD). REFERENCES [1] Q. J. Zhang, K. C. Gupta, and V. K. Devabhaktuni, “Artificial neural networks for RF and microwave design—From theory to practice,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1339–1350, Apr. 2003. [2] Q. J. Zhang and K. C. Gupta, Neural Networks for RF and Microwave Design. Boston, MA: Artech House, 2000. [3] P. Burrascano, S. Fiori, and M. Mongiardo, “A review of artificial neural networks applications in microwave computer-aided design,” Int. J. RF Microw. Comput.-Aided Eng., vol. 9, no. 3, pp. 158–174, Apr. 1999. [4] P. M. Watson and K. C. Gupta, “EM-ANN models for microstrip vias and interconnects in dataset circuits,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2495–2503, Dec. 1996. [5] V. K. Devabhaktuni, M. C. E. Yagoub, and Q. J. Zhang, “A robust algorithm for automatic development of neural-network models for microwave applications,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2282–2291, Dec. 2001. [6] X. Ding, V. K. Devabhaktuni, B. Chattaraj, M. C. E. Yagoub, M. Deo, J. Xu, and Q. J. Zhang, “Neural-network approaches to electromagnetic based modeling of passive components and their applications to highfrequency and high-speed nonlinear circuit optimization,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 436–449, Jan. 2004. [7] J. W. Bandler, M. A. Ismail, J. E. Rayas-Sanchez, and Q. J. Zhang, “Neuromodeling of microwave circuits exploiting space-mapping technology,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2417–2427, Dec. 1999.

37

[8] V. Rizzoli, A. Costanzo, D. Masotti, A. Lipparini, and F. Mastri, “Computer-aided optimization of nonlinear microwave circuits with the aid of electromagnetic simulation,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 362–377, Jan. 2004. [9] P. M. Watson and K. C. Gupta, “Design and optimization of CPW circuits using EM-ANN models for CPW components,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2515–2523, Dec. 1997. [10] G. Antonini and A. Orlandi, “Gradient evaluation for neural networks based electromagnetic optimization procedures,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 5, pp. 874–876, May 2000. [11] E. A. Soliman, M. H. Bakr, and N. K. Nikolova, “Neural networks method of moments (NN-MoM) for the efficient filling of the coupling matrix,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1521–1529, Jun. 2004. [12] J. P. Garcia, F. Q. Pereira, D. C. Rebenaque, J. L. G. Tornero, and A. A. Melcon, “A neural network method for the analysis of multilayered shielded microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 309–320, Jan. 2006. [13] S. Liao, L. Zhang, J. Xu, and Q. J. Zhang, “Neural network modeling for electromagnetic structures,” in Electromagn. Compat. Asia–Pacific Symp., Beijing, China, Apr. 2010, pp. 870–873. [14] M. Righi, W. J. R. Hoefer, M. Mongiardo, and R. Sorrentino, “Efficient TLM diakoptics for separable structures,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 4, pp. 854–859, Apr. 1995. [15] M. N. Vouvakis, Z. Cendes, and J. F. Lee, “A FEM domain decomposition method for photonic and electromagnetic bandgap structures,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 721–733, Feb. 2006. [16] V. D. L. Rubia and J. Zapata, “Microwave circuit design by means of direct decomposition in the finite-element method,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 7, pp. 1520–1530, Jul. 2007. [17] J. C. Rautio, “Perfectly calibrated internal ports in EM analysis of planar circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, 2008, pp. 1373–1376. [18] S. Koziel, J. Meng, J. W. Bandler, M. H. Bakr, and Q. S. Cheng, “Accelerated microwave design optimization with tuning space mapping,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 2, pp. 383–394, Feb. 2009. [19] J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. New York: Wiley, 2002. [20] M. N. O. Sadiku, Numerical Techniques in Electromagnetics, 2nd ed. New York: CRC, 2000. [21] S. Liao, J. Xu, H. Kabir, Q. J. Zhang, and J. G. Ma, “Neural network EM-field based modeling for 3-D substructure in finite element method,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, Jun. 2009, pp. 517–520. [22] R. F. Harrington, Time–Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [23] G. Guarnieri, G. Pelosi, L. Rossi, and S. Selleri, “A domain decomposition technique for finite element based parametric sweep and tolerance analyses of microwave passive devices,” Int. J. RF Microw. Comput.-Aided Eng., vol. 19, no. 3, pp. 328–337, May 2009. [24] K. Chang, Handbook of Microwave and Optical Components. New York: Wiley, 1997. [25] Q.-J. Zhang, NeuroModeler Plus. Dept. Electron., Carleton Univ., Ottawa, ON, Canada, 2005. [26] MATLAB. ver. 7.6.0.324 (R2008a), Mathwork Inc., Natick, MA, 2008. [27] HFSS. ver. 11.0.1, Ansoft Corporation, Pittsburgh, PA, 2007. [28] A. P. Duffy, A. J. M. Martin, A. Orlandi, G. Antonini, T. M. Benson, and M. S. Woolfson, “Feature selective validation (FSV) for validation of computational electromagnetics (CEM). Part I—The FSV method,” IEEE Trans. Electromagn. Compat., vol. 48, no. 3, pp. 449–459, Aug. 2006. [29] A. Orlandi, A. P. Duffy, B. Archambeault, G. Antonini, D. E. Coleby, and S. Connor, “Feature selective validation (FSV) for validation of computational electromagnetics (CEM). Part II—Assessment of FSV performance,” IEEE Trans. Electromagn. Compat., vol. 48, no. 3, pp. 460–467, Aug. 2006. [30] R. Kress, Numerical Analysis. New York: Springer, 1998. [31] J. Boremann and F. Arndt, “Transverse resonance, standing wave, and resonator formulations of the ridge waveguide eigenvalue problem and its applications to the design of E -plane finned waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 8, pp. 104–113, Aug. 1990. [32] S. Liao, P. Yan, J. Xu, and Y. Wang, “Coaxial waveguide based lefthanded transmission line,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 8, pp. 568–570, Aug. 2007.

38

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Shaowei Liao received the B.Sc. degree in electronic and information engineering from the Southwest University of Science and Technology, Mianyang, Sichuan, China, in 2004, and is currently working toward the Ph.D. degree at the University of Electronic Science and Technology of China, Chengdu, China. He is also currently a Visiting Student with Carleton University, Ottawa, ON, Canada. His research interests include metamaterials, computational electromagnetics, and neural-network modeling. Humayun Kabir received the B.Sc. degree in electrical and electronic engineering from the Bangladesh University of Engineering and Technology, Dhaka, Bangladesh, in 1999, the Masters degree in electrical engineering from the University of Arkansas, Fayetteville, in 2003, and the Ph.D. degree in electrical and computer engineering from Carleton University, Ottawa, ON, Canada, in 2009. From 2001 to 2003, he was a Research Assistant with HiDEC, Fayetteville, AR, where he was involved in design, fabrication, and testing of active and passive microwave components. He was also a Teaching and Research Assistant with the Department of Electronics, Carleton University. He is currently with the Research and Development Department, COMDEV Ltd., Cambridge, ON, Canada, where he is involved in modeling and design automation of microwave components for satellite and space applications. His research interests include RF/microwave modeling, design and optimization. He has authored or coauthored over 25 conference and journal publications. Dr. Kabir was the recipient of the Senate Award presented by Carleton University for his outstanding research work at the doctoral level. He was also the recipient of a Natural Sciences and Engineering Research Council (NSERC) of Canada Industrial Postdoctoral Fellowship. Yi Cao (S’06–M’09) received the Ph.D. degree in electrical engineering from Carleton University, Ottawa, ON, Canada, in 2009. From 2009 to 2010, he was a Signal Integrity Researcher with Fidus Systems Inc., working under an Industrial Research and Development Postdoctoral Fellowship (2009–2011) presented by the Natural Sciences and Engineering Research Council (NSERC) of Canada. In 2010, he joined Research In Motion Limited, Waterloo, ON, Canada, as a Digital Hardware Designer, where he has been involved with high-speed modeling and simulation of the next-generation smartphone. He has authored or coauthored over 20 papers in international conferences and journals. He is a contributor to Time Domain Methods in Electrodynamics (Springer, 2008). He is a member of the Editorial Board of the International Journal of RF and Microwave Computer-Aided Engineering and the Journal of Circuits, Systems and Computers. His research interests include neural networks, behavioral modeling for passive and active devices, and their applications to signal/power integrity and EM compatibility analyses of high-speed digital systems. Dr. Cao was a two-time recipient of the Ontario Graduate Scholarship presented by the Ministry of Training, Colleges and Universities of Ontario in 2006 and 2007. He was also the recipient of the Chinese Government Award for Outstanding Self-financed Students Abroad presented by the China Scholarship Council in 2008. Jianhua Xu received the B.Sc., M.Sc., and Ph.D. degrees from the University of Electronic Science and Technology of China, Chengdu, China, in 1983, 1986 and 1989, respectively. He is currently the Dean and a Professor with the School of Physical Electronics, University of Electronic Science and Technology of China. His research interests include microwave logging, metamaterials, and computational electromagnetics.

Qi-Jun Zhang (S’84–M’87–SM’95–F’06) received the B.Eng. degree from the Nanjing University of Science and Technology, Nanjing, China, in 1982, and the Ph.D. degree in electrical engineering from McMaster University, Hamilton, ON, Canada, in 1987. From 1982 to 1983, he was with the System Engineering Institute, Tianjin University, Tianjin, China. From 1988 to 1990, he was with Optimization Systems Associates (OSA) Inc., Dundas, ON, Canada, where he developed advanced microwave optimization software. In 1990, he joined the Department of Electronics, Carleton University, Ottawa, ON, Canada, where he is currently a Full Professor. He has authored or coauthored over 200 publications. He authored Neural Networks for RF and Microwave Design (Artech House, 2000), coedited Modeling and Simulation of High-Speed VLSI Interconnects (Kluwer, 1994), and contributed to the Encyclopedia of RF and Microwave Engineering (Wiley, 2005), Fundamentals of Nonlinear Behavioral Modeling for RF and Microwave Design (Artech House, 2005), and Analog Methods for ComputerAided Analysis and Diagnosis (Marcel Dekker, 1988). He was a Guest Co-Editor for the “Special Issue on High-Speed VLSI Interconnects” for the International Journal of Analog Integrated Circuits and Signal Processing (Kluwer, 1994), and twice was a Guest Editor for the “Special Issue on Applications of ANN to RF and Microwave Design” for the International Journal of RF and Microwave Computer-Aided Engineering (Wiley, 1999 and 2002). He is a member of the Editorial Board of the International Journal of RF and Microwave Computer-Aided Engineering and the International Journal of Numerical Modeling. He is an Associate Editor for the Journal of Circuits, Systems and Computers. His research interests are microwave CAD and neural-network and optimization methods for high-speed/high-frequency circuit design. Dr. Zhang is a Fellow of the Electromagnetics Academy. He is a member on the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He is a member of the Technical Committee on CAD (MTT-1) of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). Jian-Guo Ma (M’96–SM’97) received the B.Sc. degree from Lanzhou University, Lanzhou, China, in 1982, and the Doctoral degree in engineering from Duisburg University, Duisburg, Germany. From April 1996 to September 1997, he was with Technical University of Nova Scotia (TUNS), Halifax, NS, Canada, as a Postdoctoral Fellow. From October 1997 to November 2005, he was with Nanyang Technological University (NTU), Singapore, as a faculty member, where he was also the founding Director of the Center for Integrated Circuits and Systems, NTU. From December 2005 to October 2009, he was with the University of Electronic Science and Technology of China (UESTC), Chengdu, China. Since November 2008, he has been the Technical Director for the Tianjin Integrated Circuit (IC) Design Center, and since October 2009, he has concurrently been the Dean for the School of Electronic Information Engineering, Tianjin University. He has authored or coauthored about 245 technical papers (109 are in SCI cited journals) and two books. He holds six U.S. patents and 15 filed/granted China patents. His research interests are RF integrated circuits (RFICs) and RF integrated systems for wireless, RF device characterization modeling, monolithic microwave integrated circuit (MMIC), RF/microwave circuits and systems, electromagnetic interference (EMI) in wireless, RF identification (RFID) and wireless sensing networks. Dr. Ma was the associate editor of the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS from January 2004 to December 2005. He is one of the founding members of the IEEE Chengdu Section and was the external liaison chair and technical activity chair for the IEEE Chengdu Section. He founded the IEEE Electron Device Society (EDS) Chengdu Chapter and serves as the chapter chair. He is a member of the IEEE University Program Ad Hoc Committee. He was the recipient of the Changjiang Professor awarded by the Ministry of Education of China. He was also the recipient of the Distinguished Young Investigator Awarded presented by the National Natural Science Foundation of China.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

39

Adaptive Sampling Algorithm for Macromodeling of Parameterized S -Parameter Responses Dirk Deschrijver, Member, IEEE, Karel Crombecq, Huu Minh Nguyen, and Tom Dhaene, Senior Member, IEEE

Abstract—This paper presents a new adaptive sampling strategy for the parametric macromodeling of -parameter-based frequency responses. It can be linked directly with the simulator to determine up front a sparse set of data samples that characterize the design space. This approach limits the overall simulation and macromodeling time. The resulting sample distribution can be fed into any kind of macromodeling technique, provided that it can deal with scattered data. The effectiveness of the approach is illustrated by a parameterized H-shaped microwave example. Index Terms—Adaptive sampling, frequency response, multivariate model, parametric macromodel, sequential design.

I. INTRODUCTION

P

ARAMETRIC macromodels are important for the design, study, and optimization of microwave structures. Such macromodels approximate the -parameter response of high-speed multiport systems as a function of frequency, and several layout variables that describe physical properties of the structure. They are frequently used for real-time design space exploration, design optimization, and sensitivity analysis. The calculation of parametric macromodels has received a lot of attention over the past years, and many new modeling approaches were introduced. Most of them are either based on artificial neural network modeling [1], the multivariate Cauchy method [2], Thiele-type interpolation [3], combined rationalmultinomial modeling [4], Kriging [5], radial basis functions [6], vector-fitting-based approaches [7]–[11], and others. In order to simulate all data samples needed to build such a parametric macromodel, many full-wave analyses must be performed. Quite often, the data is collected over a predefined dense grid in the design space. However, since the number of data samples grows exponentially with the number of dimensions, an excessive amount of computer resources may be required. For this purpose, adaptive sampling strategies can be used to determine Manuscript received January 26, 2010; revised July 09, 2010; accepted August 24, 2010. Date of publication November 18, 2010; date of current version January 12, 2011. This work was supported by the Research Foundation Flanders. The work of D. Deschrijver and K. Crombecq was supported under an FWO Fellowship. D. Deschrijver, H. M. Nguyen, and T. Dhaene are with the Department of Information Technology, Ghent University–IBBT, 9000 Ghent, Belgium (e-mail: [email protected]; [email protected]; [email protected]). K. Crombecq is with the Department of Mathematics and Computer Science, University of Antwerp, 2020 Antwerp, Belgium (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2090407

a quasi-minimal distribution of data samples that characterizes the overall system response [4]. Adaptive sampling strategies often select data samples in a sequential way by comparing intermediate macromodels. A drawback of this approach is that the selection of new data samples is highly dependent on the quality of these intermediate models. If their accuracy is degraded, e.g., by choosing a wrong model order or by spurious poles occuring in the design space, then the optimality of the sampling algorithm will break down. It is also noted that the selection of samples is affected by the model type (e.g., polynomial, rational, radial basis functions, etc.), which is not a desirable property. Moreover, it is found that many intermediate macromodels must be calculated before a suitable data distribution is obtained, leading to an overall slow and cost-ineffective procedure. This paper presents a new generic sampling strategy that is able to resolve all these issues [12]. It can be linked directly with the simulator to adaptively select a representative set of data samples before any kind of macromodeling procedure is applied. The resulting distribution of the data can be fed into an arbitrary macromodeling technique, provided that it can deal with scattered data. The effectiveness of the approach is illustrated by a scalable microwave H-antenna example. II. PRELIMINARIES AND NOTATION Parametric macromodeling algorithms compute a multivariate model from a set of parameterized -parameter data . These -parameters samples depend on the frequency and several design parameters . These design parameters are the layout and material parameters, which describe, e.g., the metallizations of a component (lengths, widths, etc.) or its substrate parameters (thickness, dielectric constant, losses, etc.). A full-wave electromagnetic (EM) simulator is used to simulate the data samples over a fixed set of discrete frequenat scattered locations in the design space. The decies sign space is defined as a subspace of that is bounded by the parameter ranges of , while the scattered locations are instances of the design parameters. In this paper, these instances are called data points. To limit the overall simulation cost, an adaptive sampling algorithm is introduced to select a limited set in an intelligent way. Note that of data points contains scalar values each data point that correspond to the design parameters . The goal of the adaptive sampling algorithm is to minimize the number of selected data points, while maximizing the model accuracy.

0018-9480/$26.00 © 2010 IEEE

40

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

III. METHODOLOGY OF ADAPTIVE ALGORITHM The adaptive algorithm selects a reduced set of data points that is used to characterize the overall behavior of the response. To obtain a robust procedure, a tradeoff between two important criteria must be made: exploration and exploitation. • Exploration is the act of exploring the design space in order to find key regions that have not yet been identified before. Note that exploration does not involve the response of the system, but only the location of the data points and the way they are distributed over the design space. It ensures that the design space is filled up with data points that are spread as evenly as possible. • Exploitation means that data points are selected in regions of the design space that are identified as potentially interesting. It is clear that regions where the response is highly dynamic require a finer sampling density than regions where the response shows little or no variation. The new sampling algorithm is a generic approach that combines both criteria in a balanced way [13], [14]. It starts from a small number of initial data points so that the majority of data points can be chosen adaptively. For the exploration criterion, the density of data points is quantified by computing a Voronoi tessellation of the data points and by calculating the volume of each Voronoi cell (Section IV). For the exploitation criterion, the dynamic variation of the response is quantified by computing simple local linear approximation models that are compared with the true system response (Section V). Both criteria are translated into a combined metric function that is used to rank the neighborhood of the data points. Based on this ranking, the undersampled regions of the design space are identified and the optimal location for additional data points is derived (Section VI). This procedure of adding data points is repeated sequentially until the algorithm is terminated. IV. EXPLORATION—VORONOI TESSELLATIONS The density of data points is assessed by computing a Voronoi tessellation [15] of the design space and by estimating the volume of each cell. Cells having a large volume correspond to regions in the design space that are sampled sparsely. Let us assume that a discrete and pairwise distinct set of in the design space is given. The domipoints over is then defined as follows: nance of a point (1) It represents a closed half-plane that is bounded by the perpendicular bisector of and , and separates all points that lie . The Voronoi cell of determines the closer to than portion of the design space that lies in all the dominances of over all other data points in the set (2)

contains all points in the design space lying It is clear that closer to than any other point in . The complete set of tessellates the design space, and is called the cells

Voronoi tessellation corresponding to the set . Computing the Voronoi tessellation is usually done by calculating the Delaunay triangulation from which the Voronoi tessellation is obtained. In order to compute the volume of each Voronoi cell, the unbounded Voronoi cells near the border of the parameter ranges are bounded. The volume (Vol) of each cell can then easily be estimated by means of Monte Carlo methods [16]. To assess the density of the data points around , the folis introduced: lowing normalized metric (3) Note that quantifies the portion of the design space that of . is contained within each Voronoi cell V. EXPLOITATION—LOCAL LINEAR APPROXIMATIONS Regions of the design space with a high dynamical behavior in , a suitable set are identified as follows. For each point neighboring points is chosen (Section V-A). For of each frequency , these neighbors are used to estimate that characterizes the best local linear the gradient at (Section V-B). The response of approximation this approximation is compared to the true response at the neighboring points , and quantifies the dynamic variation in the region of . A large deviation indicates the regions where the data is varying more rapidly. A. Selection of Neighboring Samples To accurately estimate the gradient of the response at a certain point , a set of neighboring data points with

(4)

must be selected that provide as much information as possible. This means that neighbors should cover each direction in the design space equally well. Therefore, neighbors should satisfy two important properties: cohesion and adhesion. • Cohesion implies that they must lie as close to the point as possible such that is minimized (5) • Adhesion implies that they must lie as far away from each other as possible such that is maximized (6) These two properties necessarily conflict with each other so a compromise must be made. In the case of , it is proven in [17] that the optimal configuration is the -dimensional cross-polytope configuration because it maximizes the is the upper adhesion (6) in all dimensions. Since bound for the adhesion value of any neighborhood with cohe, a cross-polytope ratio is defined, sion

DESCHRIJVER et al.: ADAPTIVE SAMPLING ALGORITHM FOR MACROMODELING OF PARAMETERIZED

-PARAMETER RESPONSES

41

which indicates how closely a neighborhood resembles a cross.) polytope (It is assumed that

(7) , then the neighborhood forms a perfect crossIf polytope configuration. Also, if , then all neighboring data points must lie in the exact same spot, reducing the should also be taken adhesion to 0. Since the distance from into account, the cross-polytope ratio (7) is divided by the cohesion, leading to the neighborhood score (8) Note that neighboring data points are ideally chosen in such a way that the neighborhood score (8) is maximized.

Fig. 1. 3-D view of the microwave H-antenna.

B. Gradient and Local Linear Approximation

data points according to the local variation of the response. Both are combined into a global metric that is used for ranking

The gradient of the response

is defined as

(15) (9) It is used to characterize the best local linear approximation for around a specified point as follows [18]: (10) Note that plane through point

is estimated by fitting a hyperbased on its neighbors

(11)

(12) Once the gradient is estimated, the dynamical behavior around point is quantified by comparing the response of with the true response at the neighboring points (13) To obtain a normalized metric

, one defines (14)

The metric quantifies the portion of the dynamic variation in the response that is located near point . VI. ADDITIONAL DATA POINT SELECTION The exploration-based metric in (3) quantifies data points according to the size of their corresponding Voronoi cell, in (14) quantifies while the exploitation-based metric

Data points associated with large values of (15) are located in regions that are likely undersampled, whereas the smaller values of (15) correspond to regions that are sampled sufficiently dense. If the data point with the maximum value of (15) , then the algorithm select an additional data is denoted by . Its exact location is chosen point inside the Voronoi cell in such a way that the distance from the neighbors is maximized. Once the new data point is added to , the procedure is repeated until the algorithm is terminated. VII. EXAMPLE: MICROWAVE H-ANTENNA This example deals with the parametric macromodeling of of a scalable H-shaped microwave the reflection coefficient antenna. Fig. 1 shows a 3-D view of the antenna, which consists of three layers: a top layer with the H-shaped antenna, a bottom layer with the feed line, and a middle slot layer with a rectangular aperture that realizes the coupling between the feed and the antenna. Fig. 2 shows a top view of the three metal layers along with their respective dimensions. A cross section of the structure is shown in Fig. 3, depicting the vertical position of the metal layers in the dielectric. The design parameters of the model are the length of the antenna and the width of the aperture. The frequency range of interest varies beGHz . All data samples are simulated with tween the full-wave planar EM simulator ADS Momentum [19], and the data points in the design space are selected by the proposed adaptive sampling algorithm. A. Adaptive Sample Selection As a first example, the parameter ranges of the model are set to mm and mm . The algorithm starts by simulating an initial set of 24 data points, as shown

42

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 4. Adaptive sampling of 24 scattered data points (dots).

Fig. 2. Top view of the microwave H-antenna.

Fig. 5. Adaptive sampling of 500 scattered data points (dots).

Fig. 3. Cross section of the microwave H-antenna.

in Fig. 4. This set consists of four data points that are located at the corners of the design space and 20 additional data points that are scattered in the design space. Based on the combined metric function (15), the neighborhood of each data point is ranked, and the undersampled regions of the design space are identified. In successive iteration steps, additional data points are selected. Figs. 5–7 show the distribution of 500, 1000, and 2000 data

points that are chosen by the algorithm, respectively. It is seen that the overall design space is well resolved, and that the data points are spread in an adaptive nonuniform way. To validate the effectiveness of the sample distribution, the parametrized frequency response is simulated for a constant mm and a varying length . In terms of the value of design space, this corresponds to the horizontal solid line (red in online version) that is shown in Fig. 7. It is seen from this figure that data points are distributed more densely if has a value in between approximately 5 and 8 mm, as marked by the vertical dashed lines (black). The reason becomes clear when Fig. 8 is considered. If is varied in between these values, the frequency response contains a sharp resonance that moves toward the lower frequencies as the length increases. For other values

DESCHRIJVER et al.: ADAPTIVE SAMPLING ALGORITHM FOR MACROMODELING OF PARAMETERIZED

-PARAMETER RESPONSES

Fig. 8. Magnitude parameterized S -parameter response for W

43

= 2:406 mm.

Fig. 6. Adaptive sampling of 1000 scattered data points (dots).

Fig. 9. Magnitude parameterized S -parameter response for L

= 9 mm.

B. Parametric Macromodeling Fig. 7. Adaptive sampling of 2000 scattered data points (dots).

of , this resonance is located outside the frequency range of interest, leading to a smoother frequency response. As an additional test, the frequency response is simulated for mm and a varying width . This cora constant value of responds to the vertical solid line (red in online version) shown in Fig. 7. Here, it is also found that the data points are distributed more densely if has a value in between approximately 0.7 and 1.9 mm, as marked by the horizontal dashed lines (black). In between these values, the frequency response contains a sharp resonance that moves towards the lower frequencies as the width increases, as shown in Fig. 9. For other values of , this resonance is located outside the frequency range. These results confirm that the dynamical regions of the design space are indeed sampled more densely than other regions where the frequency response shows less variation.

As a second example, the parametric modeling of the same H-shaped antenna is considered. In this case, the parameter mm and ranges of the macromodel are set to mm , and the proposed algorithm is applied to compute a representative set of 225 data points that are scattered over the design space. The distribution of the data points is shown in Fig. 10, and it is observed that data points are sampled more densely in the upper right corner. This area corresponds to the region of the design space where the resonance moves into the upper part of the frequency range from outside, increases. A closer inspection of Figs. 7 as the value of and 10 reveals that a similar distribution of the data points is chosen. As a comparison, the same number of data points are simulated over a classical predefined sampling that does not take the dynamical behavior of the response into account, e.g., a uniform 15 15 grid, as shown in Fig. 11. To compute a parametric macromodel from the simulated data samples, any kind of modeling technique can be applied. In this case, the modeling approach in [20] is adopted because

44

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE I COMPARISON OF MAXIMUM ABSOLUTE ERROR

Fig. 10. Adaptive sampling of 225 scattered data points (dots).

Fig. 12. Magnitude adaptive macromodel and reference data for varying L.

accuracy in the uniform case. This follows from the fact that the data points are sampled more densely in the regions where the frequency response is varying more rapidly. As a final illustration, the response of the adaptive macromodel is compared to the simulated frequency response in Fig. 12, and it shows that an excellent agreement is observed. VIII. DISCUSSION Fig. 11. Uniform sampling of 225 grid-based data points (dots).

it can deal with scattered data and offers the possibility to enforce stability and passivity by construction. First, the selected data points of the adaptive algorithm are used to build a triangulation of the design space. Some linear interpolation inside a simplex is then performed to evaluate the model at intermediate data points. It is noted that this interpolation scheme is local, and therefore, highly sensitive to the distribution and the density of the selected data points. In order to assess the importance of the sampling, some parametric macromodels are computed based on the adaptive sampling (shown in Fig. 10) and the uniform sampling (shown in Fig. 11). The response of both parametric macromodels is evaluated for some arbitrary data points that are located in the most dynamic region of the design space, and the response of these models is compared to the simulation data. These data points are marked by five red asterisks (in online version) in both figures. Table I shows a comparison of the maximum absolute error over all frequencies in each of these data points. It is clear that the accuracy in the adaptive case is indeed better than the

In many practical cases, it is possible to characterize the entire frequency response at a limited (or no) additional cost, when compared to the simulation of a single frequency sample. In the frequency domain, standard commercial simulation tools can calculate the entire frequency sweep by simulating the system at a limited number of frequencies (e.g., using AFS algorithms [21]). Also in the time domain, an entire sweep of frequency samples is calculated by applying a fast Fourier transform (FFT) to the impulse response. Therefore, the adaptive sampling algorithm treats frequency as a separate variable. Note, however, that it is possible to include the frequency as a regular design parameter, by making some minor modifications to the algorithm, particularly in (13). IX. CONCLUSIONS In order to limit the overall simulation and macromodeling time, an efficient adaptive sampling algorithm is proposed for parametric macromodeling of -parameter-based system responses. It can easily be linked to any full-wave EM simulator to select a representative set of scattered data points in a sequential way. Note that the sampling algorithm does not

DESCHRIJVER et al.: ADAPTIVE SAMPLING ALGORITHM FOR MACROMODELING OF PARAMETERIZED

depend on the multivariate macromodeling technique used. The benefits of the approach are illustrated by an example. REFERENCES [1] A. H. Zaabab, Q. J. Zhang, and M. Nakhla, “A neural network modeling approach to circuit optimization and statistical design,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 6, pp. 1349–1558, Jun. 1995. [2] A. Lamecki, P. Kozakowski, and M. Mrozowski, “Efficient implementation of the Cauchy method for automated CAD-model construction,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 7, pp. 268–270, Jul. 2003. [3] R. Lehmensiek and P. Meyer, “Creating accurate multivariate rational interpolation models for microwave circuits by using efficient adaptive sampling to minimize the number of computational electromagnetic analyses,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 8, pp. 1419–1419, Aug. 2001. [4] J. De Geest, T. Dhaene, N. Faché, and D. De Zutter, “Adaptive CADmodel building algorithm for general planar microwave structures,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1801–1809, Sep. 1999. [5] J. P. C. Kleijnen and W. C. M. Van Beers, “Application driven sequential designs for simulation experiments: Kriging metamodels,” J. Operat. Res. Soc., vol. 55, pp. 876–883, Dec. 2004. [6] A. Lamecki, P. Kozakowski, and M. Mrozowski, “CAD-model construction based on adaptive radial basis functions interpolation technique,” in 15th Int. Microw., Radar, Wireless Commun. Conf., May 2004, vol. 3, pp. 799–802. [7] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del., vol. 14, no. 3, pp. 1052–1061, Jul. 1999. [8] D. Deschrijver, T. Dhaene, and D. De Zutter, “Robust parametric macromodeling using multivariate orthonormal vector fitting,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 7, pp. 1661–1667, Jul. 2008. [9] D. Deschrijver and T. Dhaene, “Stability and passivity enforcement of parametric macromodels in time and frequency domain,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 11, pp. 2435–2441, Nov. 2008. [10] T. Dhaene and D. Deschrijver, “Stable parametric macromodeling using a recursive implementation of the vector fitting algorithm,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 2, pp. 415–420, Feb. 2009. [11] P. Triverio, S. Grivet-Talocia, and M. S. Nakhla, “A parameterized macromodeling strategy with uniform stability test,” IEEE Trans. Adv. Packag., vol. 32, no. 1, pp. 205–215, Feb. 2009. [12] K. Crombecq, I. Couckuyt, D. Gorissen, and T. Dhaene, “A novel sequential design strategy for global surrogate modeling,” in Proc. 41th Winter Simulation Conf., Dec. 2009, accepted for publication. [13] D. C. Montgomery, Design and Analysis of Experiments, 5th ed. New York: Wiley, 2000. [14] A. Forrester, A. Sobester, and A. Keane, Engineering Design via Surrogate Modelling: A Practical Guide. New York: Wiley, 2008. [15] F. Aurenhammer, “Voronoi diagrams—A survey of a fundamental geometric data structure,” ACM Comput. Surveys, vol. 23, no. 3, pp. 345–405, 1991. [16] N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Amer. Statist. Assoc., vol. 44, no. 247, pp. 335–341, 1949. [17] H. Cohn and A. Kumar, “Universally optimal distribution of points of spheres,” J. Amer. Math. Soc., vol. 21, no. 1, pp. 99–148, Jan. 2007. [18] T. Hachisuka, W. Jarosz, R. P. Weistroffer, K. Dale, G. Humphreys, M. Zwicker, and H. W. Jensen, “Multidimensional adaptive sampling and reconstruction for ray tracing,” ACM Trans. Graphics, vol. 27, no. 3, p. 10, 2008. [19] Agilent EEsof COMMS EDA,ADS Momentum Software. Agilent Technol. Inc., Santa Rosa, CA, 2009. [20] F. Ferranti, T. Dhaene, and L. Knockaert, “Passivity-preserving interpolation-based parameterized macromodeling of scattered data,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 3, pp. 133–135, Mar. 2010.

-PARAMETER RESPONSES

45

[21] T. Dhaene, J. Ureel, N. Faché, and D. De Zutter, “Adaptive frequency sampling algorithm for fast and accurate S -parameter modeling of general planar structures,” in IEEE MTT-S Int. Microw. Symp. Dig., Orlando, FL, May 1995, vol. 3, pp. 1427–1430. Dirk Deschrijver (M’09) was born in Tielt, Belgium, on September 26, 1981. He received the Master degree (licentiaat) and Ph.D. degree in computer science from the University of Antwerp, Antwerp, Belgium, in 2003 and 2007, respectively. From May to October 2005, he was a Marie Curie Fellow with the Scientific Computing Goup, Eindhoven University of Technology, Eindhoven, The Netherlands. He is currently an FWO Post-Doctoral Research Fellow with the Department of Information Technology (INTEC), Ghent University, Ghent, Belgium. His research interests include robust parametric macromodeling, rational least-squares approximation, orthonormal rational functions, system identification and broadband macromodeling techniques.

Karel Crombecq was born in Antwerp, Belgium, on November 6, 1984. He received the Master’s degree in computer science from the University of Antwerp, Antwerp, Belgium in 2006, and is currently working toward the Ph.D. degree at the University of Antwerp. He is currently with the CoMP Research Group, Department of Computer Science and Mathematics, University of Antwerp. His research interests include distributed surrogate modeling, adaptive sampling techniques, and machine learning.

Huu Minh Nguyen was born in Hanoi, Vietnam, on June 13, 1985. He received the Erasmus Mundus M.Sc. degree in photonics from Ghent University, Ghent, Belgium, in 2008, and is currently working toward the Ph.D. degree in information technology at Ghent University. Since April 2009, he has been with the Department of Information Technology (INTEC), Ghent University. His research interests include supervised machine learning, bioinformatics, and multivariate regression or classification problems.

Tom Dhaene (M’94–SM’05) was born in Deinze, Belgium, on June 25, 1966. He received the Ph.D. degree in electrotechnical engineering from the University of Ghent, Ghent, Belgium, in 1993. From 1989 to 1993, he was a Research Assistant with the Department of Information Technology, University of Ghent, where his research was focused on different aspects of full-wave EM circuit modeling, transient simulation, and time-domain characterization of high-frequency and high-speed interconnections. In 1993, he joined the EDA company Alphabit (now part of Agilent). He was one of the key developers of the planar EM simulator ADS Momentum. Since September 2000, he has been a Professor with the Department of Mathematics and Computer Science, University of Antwerp, Antwerp, Belgium. Since October 2007, he has been a Full Professor with the Department of Information Technology (INTEC), Ghent University, Ghent, Belgium. He has authored or coauthored, or has contributed to over 220 peer-reviewed papers and abstracts in international conference proceedings, journals, and books.

46

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

A General Approach for Sensitivity Analysis of Distributed Interconnects in the Time Domain Natalie Nakhla, Member, IEEE, Michel Nakhla, Fellow, IEEE, and Ramachandra Achar, Senior Member, IEEE

Abstract—With the continually increasing operating frequencies, simulation and optimization for signal integrity in high-speed designs is becoming increasingly important. In this paper, a novel and generic method for sensitivity analysis of distributed interconnects is developed. Advantages of the proposed method are that it is independent of the macromodel used to represent the interconnects, and it does not require the computation of the sensitivity of the macromodel components with respect to the interconnect parameter of interest. Various numerical examples are presented to demonstrate the validity, generality, and efficiency of the new method. Index Terms—Circuit simulation, distributed interconnects, high-speed modules, multiconductor transmission lines, passive macromodels, sensitivity analysis, transient analysis.

I. INTRODUCTION

T

HE RAPID increase in operating speeds, density, and complexity of modern integrated circuits and microwave applications has made interconnect analysis and optimization a challenging task for high-frequency circuit designers. Effects such as reflections, crosstalk, and propagation delays associated with the interconnects have become critical factors, and improperly designed interconnects can result in increased signal delay, ringing, inadvertent, and false switching [1]. Interconnections are present at various levels of the design hierarchy such as on-chip, packaging structures, multichip modules (MCMs), printed circuit boards (PCBs), and backplanes. Designers must make the proper tradeoffs, often between conflicting design requirements, to obtain the best possible performance. Therefore, efficient and accurate sensitivity analysis of circuit response with respect to circuit parameters is of significant importance in identifying critical design components, in tolerance assignments, and in optimizing the overall network performance. In addition, when using optimization tools as part of the design cycle, the sensitivity vector in the form of the gradient of the cost function is crucial to obtain fast convergence. Several techniques have been proposed in the literature for sensitivity analysis of microwave circuits and high speed interconnects [2]–[15]. A large number of these previous publications deal with sensitivity analysis for linear circuits in Manuscript received March 06, 2010; revised September 29, 2010; accepted October 17, 2010. Date of publication December 10, 2010; date of current version January 12, 2011. N. Nakhla is with the School of Information Technology and Engineering, University of Ottawa, Ottawa, ON Canada K1S 5B6 (e-mail: nnakhla@ doe.carleton.ca). M. Nakhla and R. Achar are with the Department of Electronics Engineering, Carleton University, Ottawa, ON Canada K1S 5B6. Digital Object Identifier 10.1109/TMTT.2010.2091205

the frequency domain or sensitivity analysis of the steady state response for nonlinear circuits [5], [6], [8]–[10], [13]. In addition, various recent papers have addressed the problem of sensitivity analysis of high-speed interconnects in the time domain [2]–[4], [7], [12], [14], [15]. A commonly used approach is based on two steps. In the first step, the distributed interconnect represented by the Telegrapher’s equations is replaced by a suitable lumped circuit macromodel [16] consisting of RLC components and delay dependent sources. In the second step, the sensitivity of these lumped components due to variations in the interconnect parameters are evaluated and are subsequently related to the sensitivity of the overall circuit response. However, major drawbacks of this approach are as follows. • It may not be straightforward to obtain the relationship between lumped components and interconnect parameters. For example, if the method of characteristics (MoC) based transmission line macromodeling algorithm [17] is used, the computation of the sensitivity of the eigenvalues and eigenvectors of the propagation matrix are usually required, and if the matrix rational approximation (MRA) [18], [19] macromodel is used, the sensitivity of the coefficients of the approximating rational functions are needed, and • This approach for sensitivity analysis is restricted to the specific macromodel used. In other words, for every new macromodeling technique, a new sensitivity algorithm has to be developed. In this paper, a novel and generic method for sensitivity analysis of distributed interconnects is developed [20]. As discussed above, in the previous approaches, the sensitivity analysis algorithms are highly dependent on the details of the specific macromodel used. In contrast, using the proposed approach, the sensitivity algorithm is entirely independent of the details of the specific macromodel used to represent the lines. As a result, the proposed algorithm does not require the computation of the sensitivity of the specific macromodel elements. For example, if the MRA macromodel is used [18], the lengthy derivation of the sensitivity of the rational function coefficients is not needed. This paper is organized as follows. In Section II, details of the new sensitivity analysis algorithm for multiconductor transmission lines are presented. This includes theoretical basis of the new method, computation of the sensitivity sources, summary of computational steps, and sensitivity analysis in the presence of multiple parameters. Section III provides various numerical examples which demonstrate the validity, generality, and efficiency of the new method. In addition, a brief review of the direct sensitivity approach is provided in the Appendix.

0018-9480/$26.00 © 2010 IEEE

NAKHLA et al.: GENERAL APPROACH FOR SENSITIVITY ANALYSIS OF DISTRIBUTED INTERCONNECTS IN TIME DOMAIN

47

II. DEVELOPMENT OF THE PROPOSED ALGORITHM FOR MTLS In this section, details of the proposed sensitivity analysis algorithm for multiconductor transmission lines are presented. In the proposed method, the sensitivity computation process is significantly simplified. In contrast to the current techniques in the literature, we start by directly differentiating the Telegrapher’s equations with respect to the interconnect parameter of interest. For the benefit of the reader, a brief review of the direct sensitivity method is given in the Appendix. Using the direct sensitivity approach, this results in an equivalent “sensitivity circuit” (or “companion model”), which consists of lumped and distributed components, as well as independent sources. The sensitivity information is obtained by simulating the original and sensitivity circuits using any suitable macromodeling technique. Advantages of the new algorithm are that it is independent of the specific macromodeling technique used to represent the lines and it does not require the computation of the sensitivity of the specific macromodel components with respect to the interconnect parameter of interest. Details of the new algorithm are given below.

Fig. 1. Equivalent sensitivity transmission line circuit.

where

(5)

A. Companion Model for Transmission Lines (6)

The Telegrapher’s equations describing the transmission lines can be described in the following form [16]:

(1) , and where are the p.u.l. parameters of the line, and represent the voltage and current vectors as a function of position and time , and is the number of coupled lines. Differentiating (1) with respect to a parameter (where represents any electrical or physical interconnect parameter of interest) yields the following relationship:

and is the length of the line. The sensitivity equations in (4) can be modeled as a transmission line circuit with independent sources at the end of the lines, as shown in Fig. 1. In this case, the variables are the sensitivities of the circuit voltages and currents, with respect to the parameter . The sensitivity sources are obtained from the solution of the original circuit and can be expressed in the time domain as

(7)

(2) where the sensitivity variables in (2) are defined as

where denotes the inverse Fourier transform operator. The equations in (4)–(7) are general in nature, and therefore, are not constrained by the specific transmission line macromodeling algorithm. Particularly, in contrast to previous techniques [12], [15], the new method does not require the sensitivity of the macromodel components with respect to the interconnect parameter of interest, providing a more robust and versatile approach for sensitivity analysis of TL networks. B. Computation of the Sensitivity Sources

(3) Next, the solution of (2) can be written in the frequency domain as

(4)

It is important to note that the direct calculation of sensitivity sources in (7) can, in general, be computationally expensive. This is because we need to convert the time-domain voltages and currents at all points along the line (at every ) to the frequency domain, using the fast Fourier transform (FFT). Moreover, this step is followed by the numerical evaluation of the integral. To overcome the above issues, in this section, a new and more efficient eigenvalue based approach for obtaining the sensitivity

48

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

sources is presented. In this approach, only the frequency spectrum of the voltages and currents of the near-end of the transmission lines are required. In addition, a closed-form solution for the integral in (7) is obtained. From (7), the sources can be written in the frequency domain as

is given by

where the matrix

(17) The term inside the integral in (16) can be expressed in a general form as follows:

(8) (18)

Using the exponential stamp of the line [1], (9)

where (19)

and moving the terms, which are independent of integral, (8) can be rewritten as

outside the Integrating

with respect to , we get

(10) where

(20) where the elements in

are given by

(11) and

is given by

(21) (12)

Using (20) and (21),

can be expressed as (22)

Defining the following:

Finally, the sensitivity voltage and current sources can be obtained using (22) and

and

(13) (23) we get (14) and are the voltage and current vectors at where the beginning of the transmission lines, respectively. can be diagonalized as Next,

Several techniques exist in the literature to guarantee the accuracy and efficiency of the inverse fast Fourier transform (IFFT) process [21]–[24]. The advantage of using the IFFT is that for data points, it requires only operations; as opposed to for the inverse discrete Fourier transform. One of the requirements for accurate IFFT computations is that the signal must be band-limited [22]. Since a transmission line generally acts as a low-pass filter, this guarantees that

(15) and are the eigenvalues where , respectively, and , and eigenvectors of for and . in (13) can be rewritten as Using (15),

(24) which is a necessary condition for accurate computation of (23), where

represents the frequency

response at the output of the line corresponding to an input source

. Other factors to ensure the accuracy of

the IFFT can be found in [21]–[24]. C. Overall Sensitivity Circuit (16)

Following the approaches described above, the overall sensitivity circuit can be obtained by using the derived companion

NAKHLA et al.: GENERAL APPROACH FOR SENSITIVITY ANALYSIS OF DISTRIBUTED INTERCONNECTS IN TIME DOMAIN

49

E. Proposed Sensitivity Analysis With Respect to Physical Parameters

Fig. 2. Example of a transmission line circuit.

When studying the sensitivity of distributed networks, the is usually required with respect to sensitivity of the output physical parameters (such as width, spacing of conductors, etc.). In (6), the variable can represent either a physical or electrical parameter. It is important to note that the results obtained using the proposed algorithm are equivalent to those obtained using the chain rule, which is given by

(25) where are the output nodes. However, in the case where the sensitivity with respect to length is required, obviously (6) cannot be used. Instead, the following steps can be used to derive the companion model. Differentiating (9) with respect to the length, we get Fig. 3. Sensitivity circuit for the transmission line network in Fig. 2.

models for the transmission lines, as well as the lumped components. To illustrate, consider the transmission line network shown in Fig. 2. The corresponding overall sensitivity circuit is shown in Fig. 3, which has the same topology as the original network. It is important to note that the transmission lines in both the original and sensitivity circuits can be macromodeled using any desired algorithm. D. Summary of Computational Steps The computational steps for the proposed sensitivity analysis algorithm can be summarized as follows. • Step 1: Solve the original network using any macromodel of interest to represent the transmission lines. • Step 2: If the sensitivity with respect to a specific component other than transmission line parameters is required, then the transmission line remains unchanged in the sensitivity circuit (no sources). Use the companion model for this specific component, where the corresponding sources for this component are computed using the results of Step 1. • Step 3: If the sensitivity with respect to transmission line parameters is required, replace the transmission line in the original circuit with its corresponding companion model in the sensitivity circuit (as shown in Fig. 1). Compute the and using the procedure sensitivity sources described in (7)-(23) and the results from Step 1. It is important to note, however, that although in our algorithm it is not required to use the same macromodel in the original and sensitivity circuits, for obvious practical reasons, the accuracy of the two macromodels has to be comparable in order to obtain consistent accurate results. For this reason, although the choice of the specific macromodel is up to the user, we recommend that once the specific macromodel has been selected it should be used in both the original and sensitivity circuits.

(26) Again, the sensitivity equation in (26) can be modelled as a transmission line circuit with independent sources, as shown in Fig. 1, where the sources in this case are given by (27) F. Proposed Sensitivity Analysis in the Presence of Multiple Parameters In the case of sensitivity with respect to multiple parameters, Step 1 of the computational steps does not need to be repeated. In addition, the transmission line sensitivity sources do not need to be entirely recomputed for a new parameter. To illustrate, conin (12), which is the only variable in the sider the variable source computation that is a function of the sensitivity parameter . Thus, when changes, this boils down to simply plugging in matrix in (21), where new values for the elements of the all the other variables need to be computed only once. Similarly, for the simulation of the sensitivity circuit, the presence of the sensitivity sources only affects the right-hand side of the circuit modified nodal analysis (MNA) equations [1]. Therefore, when the parameter is changed, the LU factors of the network are reused and only the right-hand side of the circuit equations are modified according to the change in . This results in several additional forward/backward substitutions, which have a minimal affect on the overall computational cost. III. NUMERICAL EXAMPLES In this section, several examples are presented to validate the accuracy and efficiency of the proposed sensitivity analysis algorithm. It is to be noted that one of the major advantages of the proposed method is its generality, i.e., the transmission line can

50

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

be represented using any macromodel of interest. To demonstrate this, the new method was tested under several scenarios using different types of MTL macromodels, such as conventional lumped segmentation [16], the MoC [17], and the delay extraction based passive macromodeling algorithm (DEPACT) [25]. In general, the p.u.l. parameters of the transmission line are frequency dependent. Several macromodeling techniques exist in the literature, which model frequency-dependent interconnects [26], [27]. Although the proposed sensitivity analysis method can be equally applied to lines with frequency-dependent parameters, for simplicity, the comparisons in this section are done with frequency independent lines. In addition, it is to be noted that, the reported CPU times include the computation of sensitivity sources and IFFT operations, and the CPU cost of the IFFT calculation was between 4%–6% of the total simulation time. In addition, for all the examples, the average discrepancy between results was less than 1% (within the limits of computational noise). Moreover, it is important to note that any changes to interconnect physical parameters affects multiple (if not all) of the elements of the p.u.l. matrices. However, based on (6) and (7), sensitivity with respect to a physical parameter is dependent on matrices. To validate the sensitivity of each entry in the the accuracy of the proposed method, the results of this intermediate step are presented in Examples 1–3. Example 4 presents sensitivity with respect to physical parameters and , and the second part of Example 2 shows the sensitivity with respect to the interconnect length .

Fig. 4. Sensitivity of voltage at far end of line with respect to L for all three macromodeling algorithms compared to the perturbation approach (Example 1).

A. Example 1 The purpose of this example is to illustrate the generality of the proposed sensitivity analysis method with respect to the macromodeling algorithm used to represent the lines. Here we /m, consider a single transmission line circuit (with H/m, /m, F/m) with a length of cm, terminated by a 50- resistor and 1-pF capacitor at the near and far ends, respectively. The lines were macromodeled (for both the original and sensitivity circuits) using three methods: DEPACT [25] (with an approxima), the MoC [17], [28], and uniform lumped tion order segmentation [16] (with 100 sections). The advantage here is that no matter what macromodeling technique is used, the sensitivity analysis model remains unchanged, and no extra effort is required when changing transmission line macromodels. For example, referring to the sensitivity circuit shown in Fig. 1, only the box representing the transmission lines changes with the macromodeling algorithm of choice, but the remainder of the circuit and the computation of the sensitivity sources is the same for all cases. Fig. 4 shows a sample of the results with respect to inductance parameter for all methods compared to the perturbation approach. As seen, the results are in excellent agreement. The total CPU time, which includes simulation of the original circuit, computation of sensitivity sources and simulation of the sensitivity model was 5.23 s for lumped segmentation, 3.23 s for MoC, and 2.93 s for DEPACT. Note that this difference in CPU time comes directly from the difference in simulation speeds for the various transmission line macromodels (this is because

Fig. 5. Sensitivity of voltage at far end of line 2 with respect to (Example 2).

C

the time devoted to computation of sensitivity sources does not change when varying the transmission line macromodel). B. Example 2 In this example, we consider a three coupled line circuit with cm, terminated by 25- resistors and line length of 0.15-pF capacitors at the near and far ends, respectively. The simulation of both the original and sensitivity circuits was performed using both the DEPACT and MoC algorithms. Figs. 5 and 6 show a sample of the sensitivity results with respect to the parameter , compared to the perturbation approach. As seen, the results are in very good agreement. The overall CPU time was 9.1 s for MoC and 8.3 s for DEPACT. Next, the sensitivity analysis was repeated in order to obtain the sensitivity with respect to interconnect length . The sensitivity sources were computed using (27). Fig. 7 shows a sample of the time-domain sensitivity results compared to the perturbation approach. As seen, the results are in excellent agreement. In addition, the overall CPU cost for the proposed method was 5.1 s.

NAKHLA et al.: GENERAL APPROACH FOR SENSITIVITY ANALYSIS OF DISTRIBUTED INTERCONNECTS IN TIME DOMAIN

Fig. 6. Sensitivity of voltage at near end of line 3 with respect to (Example 2).

C

Fig. 7. Sensitivity of voltage at far end of line 3 with respect to interconnect length d (Example 2).

51

Fig. 8. Nonlinear multiconductor transmission line network considered in Example 3.

Fig. 9. Equivalent sensitivity circuit for nonlinear transmission line network in Fig. 8 (Example 3).

C. Example 3 In this example, we consider a nonlinear multiconductor transmission line circuit, as shown in Fig. 8. The line parameters for MTL subnetwork #1 are given by [25] cm nH/cm pF/cm cm

(28)

and the parameters for MTL subnetworks #2 and #3 are

D. Example 4: Sensitivity With Respect to Physical Parameters

cm nH/cm pF/cm cm

For the purpose of illustration, the resistor between node 2 and ground is taken to be a function of the sensitivity parameter , given by , where in this case, and of subnetwork #1. The corresponding sensitivity circuit is shown in Fig. 9, which consists of the companion models for both the transmission lines and the resistor. In this case, , and is obtained from the solution of the original circuit. Both the original and sensitivity circuits , and were simulated using the DEPACT algorithm uniform lumped segmentation (100 sections). Figs. 10 and 11 show a sample of the time-domain sensitivity results compared to the perturbation approach. As seen from the plots, the results are in very good agreement. The overall CPU time was 19 s for lumped segmentation and 9.5 s for DEPACT.

(29)

In this example, sensitivity analysis with respect to physical parameters is computed. Fig. 12 shows the physical structure of the coupled microstrip considered, along with its p.u.l. parameter matrices. Terminations of 25- resistors and 0.15-pF capacitors were used at the near and far ends of the lines, respectively. Several techniques exist in the literature, where given the physical parameters, the p.u.l. electrical parameters can be obtained.

52

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 10. Sensitivity of voltage at node 2 of multiconductor TL network with respect to L of subnetwork #1 (Example 3).

Fig. 11. Sensitivity of voltage at node 5 of multiconductor TL network with respect to L of subnetwork #1 (Example 3).

Fig. 13. Sensitivity of voltage at near end of victim line with respect to interconnect width w (Example 4).

Fig. 14. Sensitivity of voltage at far end of active line with respect to interconnect spacing s (Example 4).

respect to interconnect width and interconnect spacing , respectively, compared to the perturbation approach. As seen, the results are in excellent agreement. In addition, the overall CPU cost for the proposed algorithm was 3.05 s. IV. CONCLUSION

Fig. 12. Coupled lossy microstrip lines used in Example 4.

Most of the commercial circuit simulators also have this feature [29]–[31]. For the purpose of this paper, HSPICE was used to obtain the electrical parameters used in this example [31]. The transmission line was modeled using the DEPACT algorithm in both the original and sensitivity circuits. Figs. 13 and 14 show a sample of the results for sensitivity analysis with

This paper has presented a novel and generic algorithm for time-domain sensitivity analysis of high-speed interconnects. The new method significantly simplifies the sensitivity computation process. The main advantages of the proposed method is that it is independent of the transmission line macromodel used, and it does not require the computation of the sensitivity of the macromodel components with respect to the interconnect parameter of interest. This results in a more robust and versatile approach for sensitivity analysis of interconnect networks. Various numerical examples were provided, which demonstrate the generality, accuracy, and efficiency of the new method.

NAKHLA et al.: GENERAL APPROACH FOR SENSITIVITY ANALYSIS OF DISTRIBUTED INTERCONNECTS IN TIME DOMAIN

53

Fig. 16. Companion model for VCCS.

Fig. 15. Companion model for inductors.

APPENDIX DIRECT SENSITIVITY ANALYSIS OF LUMPED CIRCUITS Sensitivity analysis of lumped circuits has been extensively described in the literature. One common approach is the direct sensitivity method [32]. This method involves the construction of a new sensitivity circuit with the same topology as the original circuit with the voltage and current variables being replaced by the sensitivities of the voltages and currents, respectively. The solution of the sensitivity circuit yields the required sensitivity information. For the benefit of the reader, a brief review of the direct sensitivity approach is presented in this section.

Fig. 17. Companion model for nonlinear components.

source, and the current-controlled voltage source can be obtained in the same manner [32]. C. Sensitivity Circuits for Nonlinear Components Consider a nonlinear component [shown in Fig. 17(a)] with the terminal relationship given by

A. Sensitivity Circuits for RLC Components

(32)

Consider the inductor shown in Fig. 15(a) with the terminal relationship given by

Differentiating (32) yields (33)

(30)

The resulting companion model for (32) is shown in Fig. 17(b), , in parallel which consists of a time-variant conductance , where with an independent current source

Differentiating (30) with respect to , we get

(34) (31) From the above equation, each inductor in the original circuit is replaced by an inductor of the same value, in series with an , as shown independent voltage source equal to in Fig. 15(b), where the voltage is obtained from the solution of the original circuit. Similarly, we can prove that the companion models for resistors and capacitors can be derived using the same approach, as described in [32]. B. Sensitivity Circuits for Dependent Sources Consider the voltage-controlled current source (VCCS) [shown in Fig. 16(a)]. The sensitivity circuit [see Fig. 16(b)] consists of the original controlled current source in parallel is with an independent current source, where the voltage known from the solution of the original circuit. Likewise, the companion models for other dependent sources such as the voltage-controlled voltage source, current-controlled current

and where cuit [32].

is acquired from the solution of the original cir-

REFERENCES [1] R. Achar and M. Nakhla, “Simulation of high-speed interconnects,” Proc. IEEE, vol. 89, no. 5, pp. 693–728, May 2001. [2] C. Jiao, A. Cangellaris, A. Yaghmour, and J. Prince, “Sensitivity analysis of multiconductor transmission lines and optimization for highspeed interconnect circuit design,” IEEE Trans. Adv. Packag., vol. 23, no. 2, pp. 132–141, May 2000. [3] S. Lum, M. Nakhla, and Q. Zhang, “Sensitivity analysis of lossy coupled transmission lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 4, pp. 607–615, Apr. 1994. [4] G. Antonini and L. D. Camillis, “Time-domain Green’s function-based sensitivity analysis of multiconductor transmission lines with nonlinear terminations,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 7, pp. 428–430, Jul. 2009. [5] N. Nikolova, Z. Jiang, L. Dongying, M. Bakr, and J. Bandler, “Sensitivity analysis of network parameters with electromagnetic frequencydomain simulators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 670–681, Jan. 2006.

54

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

[6] R. Liu, Q. J. Zhang, and M. Nakhla, “A frequency domain approach to performance optimization of high-speed vlsi interconnects,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 12, pp. 2403–2411, Dec. 1992. [7] A. Beygi and A. Dounavis, “Sensitivity analysis of lossy multiconductor transmission lines based on the passive method of characteristics macromodel,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 29, no. 8, pp. 1290–1294, Aug. 2010. [8] Y. Fei and A. Opal, “Noise and sensitivity analysis of periodically switched linear circuits in frequency domain,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 7, pp. 986–998, Jul. 2000. [9] J. Bandler, Q. J. Zhang, and R. M. Biernacki, “A unified theory for frequency-domain simulation and sensitivity analysis of linear and nonlinear circuits,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 12, pp. 1661–1669, Dec. 1988. [10] G. Antonini, L. D. Camillis, and F. Ruscitti, “A spectral approach to frequency-domain sensitivity analysis of multiconductor transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 1, pp. 65–67, Jan. 2009. [11] M. H. Bakr, Z. Peipei, and N. K. Nikolova, “Adjoint first order sensitivities of transient responses and their applications in the solution of inverse problems,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 2137–2146, Jul. 2009. [12] M. Jun-Fa and E. Kuh, “Fast simulation and sensitivity analysis of lossy transmission lines by the method of characteristics,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 44, no. 5, pp. 391–401, May 1997. [13] J. Bandler, R. Biernacki, and S. Chen, “Harmonic balance simulation and optimization of nonlinear circuits,” in Proc. IEEE Int. Circuits Syst. Symp., San Diego, CA, May 1992, pp. 85–88. [14] T. Arabi, R. Suarez-Gartner, and R. Pomerleau, “Optimization and sensitivity analysis of multiconductor transmission line networks,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 9, pp. 1827–1836, Sep. 1994. [15] N. Nakhla, A. Dounavis, M. Nakhla, and R. Achar, “Delay-extraction based sensitivity analysis of multiconductor transmission lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3520–3530, Nov. 2005. [16] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York: Wiley, 1994. [17] F. Y. Chang, “The generalized method of characteristics for waveform relaxation analysis of lossy coupled transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 2028–2038, Dec. 1989. [18] A. Dounavis, R. Achar, and M. Nakhla, “Addressing transient errors in passive macromodels of distributed transmission line networks,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2759–2768, Dec. 2002. [19] A. Dounavis, R. Achar, and M. Nakhla, “A general class of passive macromodels for lossy multiconductor transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 10, pp. 1686–1696, Oct. 2001. [20] N. Nakhla, M. Nakhla, and R. Achar, “A general approach for time-domain sensitivity analysis of high-speed interconnects,” in Proc. IEEE 15th Elect. Perform. Electron. Packag. Topical Meeting, Scottsdale, AZ, Oct. 2006, pp. 189–192. [21] A. Edelman, P. McCorquodale, and S. Toledo, “The future fast Fourier transform,” SIAM J. Sci. Statist. Comput., vol. 20, pp. 1094–1114, 1999. [22] E. O. Brigham, The Fast Fourier Transform. New York: PrenticeHall, 2002. [23] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. New York: Cambridge Univ. Press, 2007. [24] P. Duhamela and M. Vetterlib, “Fast Fourier transforms: A tutorial review and a state of the art,” Signal Process., vol. 19, pp. 259–299, 1990. [25] N. Nakhla, A. Dounavis, R. Achar, and M. Nakhla, “DEPACT: Delay extraction-based passive compact transmission line macromodelling algorithm,” IEEE Trans. Adv. Packag., vol. 28, no. 1, pp. 13–23, Feb. 2005. [26] N. Nakhla, A. E. Ruehli, M. Nakhla, R. Achar, and C. Chen, “Waveform relaxation techniques for simulation of coupled interconnects with frequency-dependent parameters,” IEEE Trans. Adv. Packag., vol. 30, no. 2, pp. 257–269, May 2007.

[27] N. Nakhla, M. Nakhla, and R. Achar, “Simplified delay extraction-based passive transmission line macromodeling algorithm,” IEEE Trans. Adv. Packag., vol. 33, no. 2, pp. 498–509, May 2010. [28] S. Grivet-Talocia, H. Huang, A. Ruehli, F. Canavero, and I. Elfadel, “Transient analysis of lossy transmission lines: An efficient approach based on the method of characteristics,” IEEE Trans. Adv. Packag., vol. 27, no. 1, pp. 45–56, Feb. 2004. [29] Agilent Advanced Design System (ADS). Agilent Technol., Santa Clara, CA, 2010. [Online]. Available: http://www.home.agilent.com [30] Cadence Virtuoso Spectre Circuit Simulator. Cadence, San Jose, CA, 2010. [Online]. Available: http://www.cadence.com [31] “HSPICE: Star-Hspice Manual,” Synopsis, Mountain View, CA, 2010. [32] T. Pillage, R. Rohrer, and C. Visweswariah, Electronic Circuit and System Simulation Methods. New York: McGraw-Hill, 1995.

Natalie Nakhla (S’05–M’09) received the B. Eng. degree in telecommunications engineering and M.A.Sc. and Ph.D. degrees in electrical engineering from Carleton University, Ottawa, ON, Canada in 2003, 2005, and 2008, respectively. Dr. Nakhla is currently a Post-Doctoral Fellow with the School of Information Technology and Engineering (SITE), University of Ottawa, Ottawa, ON, Canada. Her research interests include computer-aided design of very large scale integration (VLSI) circuits, simulation and modeling of high-speed interconnects, parallel processing, power systems simulation, numerical techniques, and development of modeling algorithms for electrophysiology. Dr. Nakhla is a reviewer for the IEEE TRANSACTIONS ON ADVANCED PACKAGING and the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. She was the recipient of several prestigious awards, including the 2010 Natural Science and Engineering Research Council (NSERC) Doctoral Prize, the NSERC Post-Doctoral Fellowship, the Carleton University Senate Medal for outstanding research achievement at the doctoral level, the Canada Graduate Scholarship at the doctoral level, and the Carleton University Medal for outstanding academic achievement at the master’s level.

Michel Nakhla (S’73–M’75–SM’88–F’98) received the Ph.D. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 1975. He is currently a Chancellor’s Professor of Electrical Engineering with Carleton University, Ottawa, ON, Canada. From 1976 to 1988, he was with Bell-Northern Research, Ottawa, ON, Canada, as the Senior Manager of the Computer-Aided Engineering Group. In 1988, he joined Carleton University, as a Professor and the Holder of the Computer-Aided Engineering Senior Industrial Chair established by Bell-Northern Research and the Natural Sciences and Engineering Research Council of Canada (NSERC). He is the founder of the high-speed Computer-Aided Design (CAD) Research Group, Carleton University. He serves as a technical consultant for several industrial organizations and is the principal investigator for several major sponsored research projects. His research interests include modeling and simulation of high-speed circuits and interconnects, nonlinear circuits, parallel processing, multidisciplinary optimization, and neural networks. Dr. Nakhla serves on various international committees, including the Standing Committee of the IEEE International Signal Propagation on Interconnects Workshop (SPI), the Technical Program Committee of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS), the Technical Program Committee of the IEEE Conference on Electrical Performance of Electronic Packaging (EPEP), and the CAD committee (MTT-1) of the IEEE MTT-S. He is an associate editor of the IEEE TRANSACTIONS ON ADVANCED PACKAGING and was an associate editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—PART I: REGULAR PAPERS. He has also served as a member of many Canadian and international government-sponsored research grants selection panels.

NAKHLA et al.: GENERAL APPROACH FOR SENSITIVITY ANALYSIS OF DISTRIBUTED INTERCONNECTS IN TIME DOMAIN

Ram Achar (S’95–M’00–SM’04) received the B.Eng. degree in electronics engineering from Bangalore University, Bangalore India in 1990, the M.Eng. degree in micro-electronics from the Birla Institute of Technology and Science, Pilani, India, in 1992 and the Ph.D. degree from Carleton University, Ottawa, ON, Canada, in 1998. He is currently a Professor with the Department of Electronics, Carleton University. Prior to joining the faculty of Carleton University in 2000, he served in various capacities with leading research laboratories, including the T. J. Watson Research Center, IBM, Yorktown Heights, NY (1995), Larsen and Toubro Engineers Ltd., Mysore, India, (1992), the Central Electronics Engineering Research Institute, Pilani, India (1992), and the Indian Institute of Science, Bangalore, India (1990). He is a consultant for several leading industries focused on high-frequency circuits, systems and tools. He has authored or coauthored over 150 peer-reviewed papers in international journals/conferences, six multimedia books on signal integrity, and four book chapters. His research interests include signal/power integrity analysis, circuit sim-

55

ulation, numerical and parallel algorithms, microwave/RF networks, opto-electronic devices, microelectromechanical systems (MEMS), and electromagnetic compatibility (EMC)/electromagnetic interference (EMI) analysis. Dr. Achar is a Professional Engineer in the Province of Ontario. He serves in various capacities, including general co-chair, Steering Committee member, Technical Program Committee (TPC) member of several leading IEEE conferences such as EPEPS, SPI, EDAPS, IWMS, and MNRC. He is a member of the Canadian Standards Committee on Nanotechnology, chair of the joint chapters of the Circuits and Systems (CAS)/Electron Device (ED)/Solid-State Circuit (SSC) societies of the IEEE Ottawa Section. He has been the recipient of several prestigious awards including the Carleton university Research Achievement Award (2004), the Natural Science and Engineering Research Council (NSERC) Doctoral Medal (2000), the University Medal for outstanding doctoral work (1998), the Strategic Microelectronics Corporation (SMC) Award (1997), and the Canadian Microelectronics Corporation (CMC) Award (1996). He was also a corecipient of the IEEE Advanced Packaging Best Transactions Paper Award (2007). His students have also been the recipients of numerous Best Student Paper Awards in international forums.

56

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Efficient Implementation for 3-D Laguerre-Based Finite-Difference Time-Domain Method Yan-Tao Duan, Bin Chen, Member, IEEE, Da-Gang Fang, Fellow, IEEE, and Bi-Hua Zhou, Member, IEEE

Abstract—When the Laguerre-based finite-difference time-domain (FDTD) method is used for electromagnetic problems, a huge sparse matrix equation results, which is very expensive to solve. We previously introduced an efficient algorithm for implementing an unconditionally stable 2-D Laguerre-based FDTD method. We numerically verified that the efficient algorithm can save CPU time and memory storage greatly while maintaining comparable computational accuracy. This paper presents new efficient algorithm for implementing unconditionally stable 3-D Laguerre-based FDTD method. To do so, a factorization-splitting scheme using two sub-steps is adopted to solve the produced huge sparse matrix equation. For a full update cycle, the presented scheme solves six tri-diagonal matrices for the electric field components and computes three explicit equations for the magnetic field components. A perfectly matched layer absorbing boundary condition is also extended to this approach. In order to demonstrate the accuracy and efficiency of the proposed method, numerical examples are given. Index Terms—Computational electromagnetics, finite-difference time-domain (FDTD) method, Laguerre polynomials, perfectly matched layer (PML), unconditionally stable method.

I. INTRODUCTION

T

HE finite-difference time-domain (FDTD) method plays an important role in the solution of electromagnetic problems [1]. However, since it is an explicit time-marching technique, its time step is constrained by the Courant–Friedrich–Levy condition [1]. If the time step size is larger than the Courant limitation, the method becomes unstable. Thus, for solving some problems with fine structures, the time step must be very small. This results in a significant increase in the total CPU time. To overcome the Courant limitation on the time step size, an unconditionally stable FDTD method based on the alternating-direction implicit (ADI) technique has been developed [2]–[4]. The ADI-FDTD method consumes less CPU time than that of the FDTD method for solving problems with fine structures. However, it is found that the numerical dispersion error becomes bigger as the time Manuscript received March 15, 2010; revised July 17, 2010; accepted August 24, 2010. Date of publication December 13, 2010; date of current version January 12, 2011. This work was supported in part by the National Natural Science Foundation of China under Grant 60971063. Y.-T. Duan, B. Chen, and B.-H. Zhou are with the Nanjing Engineering Institute, Nanjing 210007, China (e-mail: [email protected]; chenbin1957@ hotmail.com; [email protected]). D.-G. Fang is with the Millimeter Wave Technique Laboratory, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2091206

step increases [5]. The locally 1-D FDTD [6], [7] has better computational efficiency than ADI-FDTD. Nevertheless, both methods provide the comparable accuracy. Recently, a new unconditionally stable scheme with weighted Laguerre polynomials for the FDTD method was introduced [8]. This marching-on-in-order scheme uses the weighted Laguerre polynomials as the temporal basis functions and the Galerkin’s method as the temporal testing procedure to eliminate the time variable. In this way, the stability condition is no longer affected by the time step. At present, the marching-on-in-order scheme has been widely used by many researchers. To improve the boundary absorbing performance, the Mur’s second-order absorbing boundary condition [9] and the perfectly matched layer (PML) absorbing boundary condition for the Laguerre-based FDTD method were proposed [10]–[12]. Alighanbari et al. introduced a time-domain method that combines the scaling function-based multiresolution time-domain technique with the Laguerre polynomial-based time-integration scheme [13]. This approach leads to a reduction in the size of the produced sparse matrix. Shao et al. introduced a hybrid time-domain method based on the compact 2-D FDTD method combined with weighted Laguerre polynomials to analyze the propagation properties of uniform transmission lines [14]. After that, Chen et al. proposed an unconditionally stable Laguerre-based body-of-revolution FDTD method for analyzing the structures with circular symmetry [15]. However, this marching-on-in-order scheme leads to a huge sparse matrix equation. Directly solving this matrix equation is very expensive, especially for the 3-D cases. The scheme is hardly usable for practical problems. To overcome this problem, we previously introduced an efficient algorithm for implementing the 2-D Laguerre-based FDTD method [16]. Numerical results indicated that the efficient algorithm can provides sufficient accuracy while being computationally efficient. This paper furthers our work of [16] by applying a factorization-splitting scheme to the conventional 3-D Laguerre-based FDTD method. It leads to an efficient unconditionally stable Laguerre-based FDTD method for modeling 3-D cases. The work in this paper is the extension and expansion of [16] where only 2-D formulations were derived and no PML absorbing boundary condition was presented. For a full update cycle the proposed scheme solves six tri-diagonal matrices and computes three explicit equations. Compared with the conventional implementation, the CPU time and memory requirement can be saved greatly. Our simulation results have verified this proposed method.

0018-9480/$26.00 © 2010 IEEE

DUAN et al.: EFFICIENT IMPLEMENTATION FOR 3-D LAGUERRE-BASED FDTD METHOD

II. NUMERICAL FORMULATIONS For simplicity, assume a linear, isotropic, nondispersive, and lossless medium, the 3-D differential Maxwell’s equations can be written as

The first derivative of field components, taking example, with respect to is [17]

57

for

(11) (1)

Inserting (10) into (1)–(6), then using a temporal Galerkin’s testing procedure to eliminate the time variable, we get

(2)

(3)

(12)

(4) (5) (13)

(6) where is the electric permittivity, is the magnetic permeability, , and are the difference operators for the first derivatives along the -, -, and -axes. and are the excitation sources along the - - and -directions, respectively. For consistency of notation and completeness, in Section II-A, we will show the derivations of the formulations for the conventional 3-D Laguerre-based FDTD method.

(14)

(15)

A. Derivations of the Conventional 3-D Laguerre-Based FDTD Method The Laguerre polynomials of order

are defined by (16)

for

(7)

Using the orthogonality of Laguerre polynomials with respect , a set of orthogonal basis functions to the weighted function can be constructed, given by

(17)

(8) where is a time-scale factor. These basis functions are where absolutely convergent to zero as , and also orthogonal with respect to as (9) Using these basis functions, we can expand the electric and magnetic fields in (1)–(6) as

(10)

(18) is chosen in such a way that the waveThe time span forms of interest have practically decayed to zero [8]. For convenience, we define a set of auxiliary matrices as (19)–(24), shown at the bottom of the following page. With some manipulations, (12)–(14) and (15)–(17) can be written in the following matrix forms, respectively: (25) (26)

58

where have

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

. Inserting (26) into (25), we

where is a lower triangular matrix, and is an upper trianand are the difference operators gular matrix. for the second derivatives. Equation (27) can then be written as

(27) where is the 3-by-3 identity matrix. Note that (27) is the produced huge sparse matrix equation for the conventional 3-D Lais guerre-based FDTD method. The product a huge sparse irreducible matrix, and it is independent of the order . After solving (27) with order zero, One can solve (27) of higher orders by using the back-substitution routine repeatedly. The expansion coefficients of the magnetic fields can be obtained from (15)–(17). We can then reconstruct the field components in the time domain with (9) and (10). Note that the time step in the Laguerre-based FDTD method is only used for the calculation of the Laguerre coefficients of the current source excitation in (18) at the preprocessing stage and for the field reconstruction at the post-processing stage. It should be small enough to accurately follow the fast temporal variations of the weighted Laguerre polynomials of order . Since the system of (27) is a huge sparse matrix equation, which is very expensive to solve, the conventional 3-D Laguerre-based FDTD method is hardly usable for practical problems. In Section II-B, we present an efficient algorithm for implementing it.

(30) Here, adding a perturbation term we get the factorized form of (30) as

to (30),

(31) Equation (31) can be solved into two sub-steps with the following splitting scheme [18]–[20]: (32) (33) where is a nonphysical intermediate value. Using the splitting scheme (32) and (33) to solve (28)–(30) leads to

B. Derivations of the Efficient 3-D Laguerre-Based FDTD Method In order to solve (27) efficiently, we decompose into two triangular matrices and as (28)

(29)

(34a)

(19) (20) (21)

(22)

(23) (24)

DUAN et al.: EFFICIENT IMPLEMENTATION FOR 3-D LAGUERRE-BASED FDTD METHOD

59

(34b)

(35a)

(34c)

(34d)

(34e)

(35f) Using the central difference scheme in the space domain to (34), we can obtain discrete space equations for the efficient 3-D Laguerre-based FDTD method [see (35a)–(35f)]. For simplicity, uniform cells are considered here. and are the spatial steps in the -, -, and -directions, respectively,

(35b)

60

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

(35f) (35c)

(35d)

Note that exchanging the two matrices and leads to slightly different update equations. The systems of (35) are six tri-diagonal matrix equations, which can be solved efficiently. In an actual simulation, since the right-hand sides of (35b)–(35f) include unknown values, in order to implement the efficient algorithm, the expansion coefficients of the electric field components must be updated as the following , and sequence: . The expansion coefficients of the magnetic field components can be calculated explicitly from (15)–(17) with , and already updated. One can then reconstruct the field components in the time domain with (9) and (10). C. PML Absorbing Boundary Condition for the Efficient 3-D Laguerre-Based FDTD Method In this section, a PML absorbing boundary condition is developed that can maintain the tri-diagonal matrix form of the efficient Laguerre-based FDTD method. In PML regions, the Maxwell’s equations are expressed as [21]

(36a)

(36b)

(35e)

(36c)

(36d)

(36e)

(36f) and are the PML electric conducwhere tivity and magnetic loss, respectively. According to the derivational procedure in Section II-A, (36) can be written as

DUAN et al.: EFFICIENT IMPLEMENTATION FOR 3-D LAGUERRE-BASED FDTD METHOD

61

(37a)

(42)

(37b)

(37c) (43) Inserting (39) into (38), we have (44) (37d)

Equation (44) is the produced huge sparse matrix equation of the PML for the conventional 3-D Laguerre-based FDTD method. According to the procedure described in Section II-B, we decompose into two triangular matrices and , and add a perturbation term to (44). We can then can solve (44) by factorizing it into two sub-steps using (45)

(37e)

(46) where the matrices

and

are constructed as

(47) (37f)

where , and . Writing (37) in a matrix form leads to (38)

(48) From (45) and (46), with some manipulation, we can obtain discrete space equations of PML for the efficient 3-D Laguerrebased FDTD method.

(39) III. NUMERICAL RESULTS where

(40)

(41)

In order to verify the proposed formulations, three numerical examples are given. First, we use the proposed method to simulate a 3-D parallel-plate capacitor and compare the numerical results with the results obtained from the conventional FDTD method and the ADI-FDTD method. The configuration consists of two 10 cm 10 cm parallel conducting plates in free space with a separation of 1 cm, as shown in Fig. 1. The electromagnetic field

62

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

=9

Fig. 2. Normalized frequency-domain E field along the y axis at x cm cm, calculated by the conventional FDTD scheme, ADI-FDTD and z scheme, and proposed scheme.

= 25

Fig. 1. 3-D parallel-plate capacitor excited by an ideal voltage source at the excitation plane. The computational domain is terminated with the Mur’s firstorder absorbing boundary condition (ABC). (a) Vertical view. (b) Horizontal view.

near the plate edges exhibits a sharp spatial distribution [22]. cm. The following The cell size is Gaussian pulse is taken as the time-domain excitation: (49) where ns, . For the conventional FDTD, ps. For clarity, we choose the time step , the Courant number is defined as where is the time step of the ADI-FDTD method. In , and , respectively. For the this test, we choose , and [23], proposed method, we choose [24]. field at Fig. 2 shows the normalized frequency-domain Hz along the axis at cm and cm, calculated by applying the Fourier transformation to the time-domain results. It can be seen that the result calculated by the proposed method agrees well with that calculated by the FDTD method. Similar results are obtained with the ADI-FDTD method only . The accuracy of the when the Courant number is ADI-FDTD method degrades quickly with the increase of the CFLN value. The second example is related to a shielded microstrip line, as shown in Fig. 3. The microstrip line has a lossless isotropic and thickness dielectric substrate with permittivity mm, a perfect electric conductor strip of width mm, and thickness mm, mm, and mm. A graded grid division is adopted along the -direction and the computational domain consists of 40 10 60

Fig. 3. Cross section of a shielded microstrip line. Perfect electric conductor: PEC.

cells in the -, -, and -directions, respectively, with mm, mm, and mm. The Mur’s first-order absorbing boundary condition [9] is set at the -directional terminals of the computational domain. The excitation is performed by the application of an -directed electric field at mm and the component is calculated at the observation point . The Gaussian pulse (49) is also ps. taken as the excitation source, and In the conventional FDTD, the Courant–Friedrich–Levy stafs). bility condition imposes a tiny time step ( In our method, we choose time duration ns, [23], [24], and we set ps (as small ) to calculate the Laguerre coefficient of the excitaas tion pulse. Fig. 4 shows the components of the electric field at the observation point. The agreement between the conventional FDTD and the efficient Laguerre-based FDTD method is very good. However, the ADI-FDTD method shows large errors . when the Courant number is Table I shows the computational resources (the CPU time and memory size) for the numerical simulations. We can see that the simulation takes 81.2 s for the FDTD method, and 3.9 s for the proposed method. The CPU time for the proposed method is reduced to about 4.8% of the FDTD method. The memory size, which is 2.31 times in comparison with that of the FDTD method, is increased because of the necessity for extra array storage. In addition, the proposed method not only achieves

DUAN et al.: EFFICIENT IMPLEMENTATION FOR 3-D LAGUERRE-BASED FDTD METHOD

Fig. 4. Transient electric fields of the x component at the observation point.

TABLE I SIMULATION RESULTS FOR THE SHIELDED MICROSTRIP LINE

63

cm. The -polarized sinusoidally modulated Gaussian pulse is used as the GHz, . The excitation, where computation domain is terminated by the Mur’s first-order absorbing boundary condition and the proposed PML absorbing boundary condition, respectively. For this example, we choose , and [23], [24]. Fig. 5 shows the relative reflection error at the observation point for the two absorbing boundary conditions. It was computed , where as dB is the field computed in the test domain and is the reference field computed using a larger domain. Compared with Mur’s first-order absorbing boundary condition, the proposed PML absorbing boundary condition shows better absorbing performance. All calculations in this paper have been performed on a Core2 2.4-GHz machine. IV. CONCLUSION An efficient algorithm for implementing unconditionally stable 3-D Laguerre-based FDTD method has been proposed in this paper. Different from the conventional Laguerre-based FDTD, the proposed method solves six tri-diagonal matrices and computes three explicit equations for a full update cycle by using a factorization-splitting scheme. The numerical examples indicate that the proposed method shows much improvement in the computation efficiency compared to the FDTD method, the ADI-FDTD method, and the conventional Laguerre-based FDTD method. In addition, the proposed method provides sufficient computational accuracy, i.e., the errors introduced by the perturbation term are not as severe as the second-order truncation error in the ADI-FDTD method. The proposed PML absorbing boundary condition remains the tri-diagonal matrix form of the efficient Laguerre-based FDTD method and shows good absorbing performance. The solutions of six tri-diagonal matrices for the electric field components are dependent, but each tri-diagonal matrix can be efficiently parallel processed on a computing cluster. Further studies for its applications to solve real problems will be performed in the future work. REFERENCES

Fig. 5. Relative reflection error for different absorbing boundary conditions (ABCs) at the observation point.

about 2.3 times the saving in CPU time in comparison with the ADI-FDTD method, but also gives better results than the ps . It is ADI-FDTD method with mentioning that, when using the conventional Laguerre-based FDTD method to simulate the microstrip line, a 72 000 72 000 sparse matrix is set up. The memory size and the computational time (compared with the proposed method) consumed to directly solve such a matrix are significant. In order to evaluate the numerical performance of the proposed PML absorbing boundary condition, we simulate the radiation of a line electric current source in free space as the third example. The source is applied at the center of the computation domain with 30 30 30 cells. The cell size is

[1] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000. [2] T. Namiki, “A new FDTD algorithm based on alternating-direction implicit method,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 10, pp. 2003–2007, Oct. 1999. [3] F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1550–1558, Sep. 2000. [4] T. Namiki, “3-D ADI-FDTD method-unconditionally stable time-domain algorithm for solving full vector Maxwell’s equations,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 10, pp. 1743–1748, Oct. 2000. [5] F. Zheng and Z. Chen, “Numerical dispersion analysis of the unconditionally stable 3-D ADI-FDTD method,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 5, pp. 1006–1009, May 2001. [6] J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett., vol. 41, no. 19, pp. 1046–1047, Sep. 2005. [7] I. Ahmed, E. K. Chua, E. P. Li, and Z. Chen, “Development of the three-dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3596–3600, Nov. 2008.

64

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

[8] Y. S. Chung, T. K. Sarkar, B. H. Jung, and M. Salazar-Palma, “An unconditionally stable scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 697–704, Mar. 2003. [9] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat., vol. EMC-23, no. 4, pp. 377–382, Nov. 1981. [10] W. Shao, B. Z. Wang, and X. F. Liu, “Second-order absorbing boundary conditions for marching-on-in-order scheme,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 5, pp. 308–310, May 2006. [11] P. P. Ding, G. F. Wang, H. Lin, and B. Z. Wang, “Unconditionally stable FDTD formulation with UPML-ABC,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 161–163, Apr. 2006. [12] Y. Yi, B. Chen, H. L. Chen, and D. G. Fang, “TF/SF boundary and PML-ABC for an unconditionally stable FDTD method,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 2, pp. 91–93, Feb. 2007. [13] A. Alighanbari and C. D. Sarris, “An unconditionally stable Laguerre-based S-MRTD time-domain scheme,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 69–72, 2006. [14] W. Shao, B. Z. Wang, X. H. Wang, and X. F. Liu, “Efficient compact 2-D time-domain method with weighted Laguerre polynomials,” IEEE Trans. Electromagn. Compat., vol. 48, no. 3, pp. 442–448, Aug. 2006. [15] H. L. Chen, B. Chen, Y. T. Duan, Y. Yi, and D. G. Fang, “Unconditionally stable Laguerre-based BOR-FDTD scheme for scattering from bodies of revolution,” Microw. Opt. Technol. Lett., vol. 49, no. 8, pp. 1897–1900, Aug. 2007. [16] Y. T. Duan, B. Chen, and Y. Yi, “Efficient implementation for the unconditionally stable 2-D WLP-FDTD method,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 11, pp. 677–679, Nov. 2009. [17] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. New York: Academic, 1980. [18] G. Sun and C. W. Trueman, “Unconditionally stable Crank–Nicolson scheme for solving the two-dimensional Maxwell’s equations,” Electron. Lett., vol. 39, no. 7, pp. 595–597, Apr. 2003. [19] G. Sun and C. W. Trueman, “Approximate Crank–Nicolson schemes for the 2-D finite-difference time-domain method for TE waves,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2963–2972, Nov. 2004. [20] G. Sun and C. W. Trueman, “Efficient implementations of the Crank–Nicolson scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2275–2284, May 2006. [21] J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 127, pp. 363–379, 1996. [22] S. G. Garcia, T. W. Lee, and S. C. Hagness, “On the accuracy of the ADI-FDTD method,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 31–34, 2002. [23] M. Yuan, J. Koh, T. K. Sarkar, W. Lee, and M. Salazar-Palma, “A comparison of performance of three orthogonal polynomials in extraction of wideband response using early time and low frequency data,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 785–792, Feb. 2005. [24] M. Yuan, A. De, T. K. Sarkar, J. Koh, and B. H. Jung, “Conditions for generation of stable and accurate hybrid TD-FD MoM solutions,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2552–2563, Jun. 2006. Yan-Tao Duan was born in Hebei Province, China, in 1980. He received the B.S. and M.S. degrees in electric systems and automation from the Nanjing Engineering Institute, Nanjing, China, in 2002 and 2006, respectively, where is currently working toward the Ph.D. degree at the Nanjing Engineering Institute. His research interests include computational electromagnetics and electromagnetic pulse (EMP).

Bin Chen (M’98) was born in Jiangsu, China, in 1957. He received the B.S. and M.S. degrees in electrical engineering from the Beijing Institute of Technology, Beijing, China, in 1982 and 1987, respectively, and the Ph.D. degree in electrical engineering from the Nanjing University of Science and Technology, Nanjing, China, in 1997. He is currently a Professor with the Nanjing Engineering Institute, Nanjing, China. His research includes computational electromagnetics and electromagnetic pulse (EMP).

Da-Gang Fang (SM’90–F’03) was born in Shanghai, China. He graduated from the Graduate School, Beijing Institute of Posts and Telecommunications, Beijing, China, in 1966. From 1980 to 1982, he was a Visiting Scholar with Laval University, Quebec, QC, Canada, and the University of Waterloo, Waterloo, ON, Canada. Since 1986, he has been a Professor with the Nanjing University of Science and Technology (NJUST), Nanjing, China. Since 1987, he had been a Visiting Professor with six universities in Canada and Hong Kong. He has authored or coauthored three books, two book chapters, and over 380 papers. He is an Associate Editor of two Chinese journals and is on the Editorial or Reviewer Board of several international and Chinese journals. He holds three patents. His name has been listed in the Marquis Who is Who in the World (1995) and the International Biographical Association Directory (1995). His research interests include computational electromagnetics, microwave integrated circuits and antennas, and electromagnetic (EM) scattering. Prof. Fang is a Fellow of the Chinese Institute of Electronics (CIE). He was the recipient of a National Outstanding Teacher Award, the People’s Teacher Medal, and the Provincial Outstanding Teacher Award. He was the Technical Program Committee (TPC) chair of ICMC 1992, vice general chair of PIERS 2004, a member of the International Advisory Committee of six international conferences, TPC co-chair of APMC 2005, and general co-chair of ICMMT 2008.

Bi-Hua Zhou (M’97) was born in Jiangsu, China, in 1940. She received the B.E. degree in radio communication engineering from the Beijing University of Posts and Telecommunications, Beijing, China, in 1965. From 1988 to 1997, she was the Leader of the Electromagnetic Pulse Laboratory, Nanjing Engineering Institute, Nanjing, China, where she is currently a Professor. Her research interests include lightning, electromagnetic compatibility, and electromagnetic pulse protection. Prof. Zhou was the recipient of three National Science and Technique Awards of China in 1993, 1995, and 2000, respectively.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

65

Linear Circuit Models for On-Chip Quantum Electrodynamics Alpár Mátyás, Christian Jirauschek, Member, IEEE, Federico Peretti, Paolo Lugli, Senior Member, IEEE, and György Csaba

Abstract—We present equivalent circuits that model the interaction of microwave resonators and quantum systems. The circuit models are derived from a general interaction Hamiltonian. Quantitative agreement between the simulated resonator transmission frequency, qubit Lamb shift, and experimental data will be shown. We demonstrate that simple circuit models, using only linear passive elements, can be very useful in understanding systems where a small quantum system is coupled to a classical microwave apparatus. Index Terms—Cavity quantum electrodynamics (QED), circuit modeling, coupled quantum-classical systems, quantum nondemolition measurement.

I. INTRODUCTION AVITY quantum electrodynamics (QED) has long been an active field to study the interaction of electromagnetic radiation with matter, which is a fundamentally important topic in physics [1]–[3]. Recently, cavity quantum electrodynamic experiments were performed in the microwave regime [4]–[7], factor miwhere the cavity consisted of a high-quality crowave coplanar resonator. The quantum system was a charge qubit built from two Josephson junctions. Similar experiments were proposed to bring other nanosystems (molecules [8], nanomagnets [9], [10], flux qubits [10]–[12]) into interaction with coplanar waveguides. One of the most promising aspects of these experiments is that they can provide circuits which integrate quantum mechanical behavior and “conventional” (high frequency) components on a chip. It is well established how to construct circuit models from electromagnetic models for commonly used active or passive components, which obey “classical” circuit theory. It has been also demonstrated how one can build equivalent circuit models

C

Manuscript received May 05, 2010; revised September 14, 2010; accepted September 14, 2010. Date of publication November 29, 2010; date of current version January 12, 2011. This work was supported in part under the Emmy Noether Program (German Physical Society (DPG), JI115/1-1), by the DFG under the SFB 631 Priority Program, by the European Union under the Erasmus Programme, and by the Pázmány Péter Catholic University (ITK) of Budapest. A. Mátyás, C. Jirauschek, F. Peretti, and P. Lugli are with the Institute for Nanoelectronics, Technische Universität München, D-80333 Munich, Germany (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). G. Csaba is with the Institute for Nanoelectronics, Technische Universität München, D-80333 Munich, Germany, and also with the Center for Nano Science and Technology, University of Notre Dame, Notre Dame, IN 46656 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2090406

of basic two-state quantum systems [13]. The goal of this paper is to show a systematic approach for building circuit models for an interacting resonator-quantum system, using linear approximations. This can be useful for developing quantum systems coupled via a resonator [14] or modeling the large external circuitry coupled to the quantum system. The possibility of this modeling approach was mentioned in [5]; to our knowledge, though, our paper is the first to analyze and exploit this model in detail. We investigate an experimental setup, similar to the one in [15], composed of a high- superconducting microwave resonator electrically coupled to a quantum circuit (often referred as transmon-qubit [16]). During the past years, the field of circuit QED went through a considerable growth. Many quantum phenomena known from optical and 3-D microwave systems were also observed in circuit QED and simulated, including systems of entangled quantum circuits [17] and resonators coupled via flux qubits [18]. Such quantum electrodynamic systems need further optimization of both the quantum bit and the resonator, such as increasing qubit decay times and optimizing resonator losses [19]. The natural application area of circuit models is the understanding of the interaction between a relatively complex “classical” circuitry and a relatively simple quantum circuit. These models do not show new physics, but rather facilitate the engineering of coupled microwave-quantum systems. II. DEVICE GEOMETRY The setup of a typical circuit quantum electrodynamic experiment is shown in Fig. 1. A main component is a superconducting coplanar waveguide resonator, displayed in Fig. 1(a) (longitudinal section) and (b) (cross section). A superconducting resonator can easily reach an unloaded quality [4], [20]. The loaded factor in the range of quality factor of the resonators is designed by using finger-type capacitors in the central strip. These typically determine the lifetime of the resonator field (which should be high) and also the duration of the measurement of the nanosystem (which should be low in order to avoid energy relaxation and decoherence of the quantum system). The superconducting resonator is practically lossless and close to resonance, its port-to-port behavior can be modeled by the circuit elements in Fig. 1(c), where the LC parameters are

0018-9480/$26.00 © 2010 IEEE

(1)

66

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 2. Circuit models representing the linear quantum mechanical interaction  ) in case of: (a) electric interaction of the resonator and the qubit (i.e.,  (e.g., charge qubit) and (b) magnetic interactions (e.g., flux qubit). The coupling between the resonator and quantum system is capacitive in (a) and inductive in (b). The quantum system behaves as a linear LC oscillator that changes the frequency of the resonator. The outcoupling capacitors, corresponding to the resonator finger-type gaps, have very small values and thus can be added to the resonant circuits that model the linear interaction.



Fig. 1. Geometry-sketch of the: (a) longitudinal section of the resonator, showing the quantum system-loop that is placed near the field maximum and (b) the cross section of the investigated coplanar waveguide resonator. The simplified equivalent circuit is shown in (c), representing the ideal resonator. Here, the capacitors C and C correspond to the finger capacitors at the two ends of the resonator, and the load resistors are also included. The coupling to the quantum system is capacitive.

Here, is the resonator impedance and is the resonator frequency. The 50- external environment is represented by the in Fig. 1(c), which, together with the fingerload resistors in the circuit model and the resonator type capacitors frequency , give the loaded quality factor of the resonator, . The quantum circuit is placed in the gap of the resonator [see Fig. 1(a)] near the electric field maximum (magnetic field maximum) if it is an electrically (magnetically) interacting quantum system, i.e., charge qubit. Its presence slightly changes the values of the capacitance and inductance of the resonator and, thus, its resonance frequency. The total system can be divided into a resonator part, an interaction part and a quantum system part. The coupling in the case of electrically interacting systems is capacitive. Throughout this paper, we will formulate our models assuming electrically coupled systems. We will also, however, mention the parameters for the magnetic case in Section III. Instead of modeling different geometries, the resonator will be considered as a lumped circuit. For more detailed investigation of resonator geometries, the reader is referred to [10], [12], and [16].

III. METHOD For the modeling of the interacting resonator-quantum system, we use a simplified (lossless) version of the circuit model in Fig. 1(c), and the coupling to the environment is neglected. The Hamiltonian is obtained by adding the energy resonator, the energy stored in the stored in the ideal

interaction of the resonator with the quantum system, and the energy of the quantum system

(2) In the above equation, the different parts are illustrated by the boxes in Fig. 1(c). The resonator energy can be defined analogous to the classical circuit in Fig. 1(c); however, the voltages and energy of the currents here are operators and their expectation value gives the voltage and current of the LC resonant circuit. The resonator and voltage [4], [5] root-mean square current is related to the minimum (or vacuum) field in the resonator. They are further discussed in Appendix A. The interaction energy also contains the resonator voltage . Here, is related to the geometric capacitance of the term , and is related to the voltage resonator-quantum system of the quantum system. The , in the QED framework, are referred to as Pauli matrices and are the basic observables of the two state quantum system. The interaction term is a standard term used in QED, and valid for every electrically interacting quantum system and resonator. The last term in (2), containing represents the energy level of the quantum system. This part can splitting only be derived in terms of quantum mechanics. For simulating the dynamics of the coupled system, we use the Liouville-von Neumann equation (3) with the Hamiltonian (2). Here, is an operator that represents or the resonator voltage either one of the Pauli matrices

MÁTYÁS et al.: LINEAR CIRCUIT MODELS FOR ON-CHIP QED

67

TABLE I SUMMARY OF THE PARAMETERS CALCULATED FROM THE LINEAR RBEs IF THE INVERSION IS CONSTANT (

or current operator. Based on the von Neumann equation, we find the following coupled differential equation system: (4) (5) (6) (7) (8) The above equations are a direct result of the von Neumann and equation for the dissipationless case (if ). We can observe that the first three equations (4)–(6) are the widely used Bloch equations [21] written for a two-level quantum system; the last two are equations for the dynamics of normalized resonator voltage and currents, thus we will call (4)–(8) resonator-Bloch equations (RBEs). Dissipation is phenomenologically described by the decay and decoherence conand [21], and the resonator loss, is added simstants variables are the expectation values of ilar as in [22]. The the Pauli matrices and represent the coherence vector characterizing the state of the quantum bit. The derivation of the RBEs is shown in Appendix A. The only approximation used here is to neglect the correlations between the qubit and field [22], thus writing the expectation value of the normalized voltage and coherence vectors as a product of the individual expectation values (9) An equation system similar to the RBEs was introduced by Jaynes and Cummings [22], written for the coupling of electric fields and dipole moments. We now use this model to find passive circuits describing the interaction of the LC resonatorquantum system. A. Modeling the System for Large Detuning In circuit quantum electrodynamic experiments, measurements are often performed in the so-called dispersive regime, where the qubit-resonator detuning is much larger then the . In this case, off-resonant interaccoupling frequency: tion between the quantum system and the resonator does not remains constant. Thus, switch the quantum system and we can neglect the oscillations of the inversion term in the and perform a linear approximation by keeping RBEs

 )

. This is valid only if the measurement time is much smaller than the decay time . For a large decay time or short measurement time window, the system behaves linearly, and the RBEs reduce to the four coupled linear differential equations (we will call them linear RBEs): (4), (5), (7), and (8) with . These can be modeled by the passive circuits shown in Fig. 2 and represent the interaction of (a) electric and (b) magnetic quantum systems, with the resonator in the dispersive regime. Interaction between the quantum system and resonator occurs through their coupling capacitance in Fig. 2(a) or mutual inductance in Fig. 2(b), depending on the type of interaction (electric or magnetic). In the circuit models, the decoherence time from RBEs (4) and (5) is represented by the parallel resistor with ; and the resonator decay the series resistor, with in case of electric and in case rate , is given by of magnetic interactions. The parameters of the circuits were found as a direct result of the linear RBEs and are summarized in Table I; for the derivation, see Appendix B. This table summarizes the central result of our work. The physical parameters of the resonator are changed from the values in (1) to the values in Table I due to the presence of the quantum system. The circuit models presented in this section provide two resonance peaks in every case, which represent the oscillations of the coupled system. The coupling between the quantum system and the resonator changes the electromagnetic response of the resonator in a way that depends on the state of the quantum system. The quantum state can be read out nondestructively from the measurement on the resonator. In physics, this is often referred to as quantum nondemolition measurement. B. Modeling the System in Its Ground State Next, we will investigate the ground state of the coupled resonator-quantum system. We will also compare our simulation results mainly to measurements done in the ground state. In case of no applied field on the input port, the coupled resonator-quantum system will converge to the common ground , due state in the RBEs. In the case of to the decay times , , and a small applied probe field on the resonator port, the system will oscillate around its ground state and the inversion remains ). unchanged (i.e., Rather than implementing the drive-fields in our RBEs, we start our simulations by taking a nonzero mode occupation of the resonator field. For this we take the initial condition: [0.04; 0; 0.999; 0; 0]. This corresponds to fields in the resonatorquantum system with an average photon number of approximately 0.01. Thus, for low fields in the resonator (approximately

68

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

0.01 photons [5], [15]), the nonlinear term in the RBEs will oscillate near 0, as it is a product of two very small will stay approximately constant and values. The inversion not change from its ground state even when the system has no detuning. As the inversion can be taken constant, the linear approximation done in the previous subsection holds, and we can . take In Section IV, we will compare our derived RBE model and circuit models to experimental data, and show the validity of our approximations.

IV. RESULTS We have performed numerical and circuit simulations on a recently measured resonator-transmon qubit device from [15]. We took as input parameters: resonator frequency GHz, impedance , vacuum Rabi frequency MHz, and cavity decay rate MHz. s, which is The dephasing time was approximately known to be in good agreement with experimental findings [16]. Further, the linewidth of the qubit coherence vector elements is in good agreement with the experiment (approximately 3 MHz) in [15]. Simulations were performed in LTSPICE [24] based on the presented circuit models, using frequency domain ac analysis. The transmission of the circuit in Fig. 2(a) was calculated as the system was for the above parameters, setting in ground state. We investigated the dependence of the result with respect to quantum system frequency, while keeping the resonator frequency fixed. This was done by varying , which was directly related to the parameters of our circuit model. The empty circles in Fig. 3 represent data from [15] that has been transformed from flux to detuning coordinates so that our simulated results can be compared to the measured cavity transmission. We can see excellent agreement of the transmission frequency of our circuit (dashed line) with experimental data (small circles), for positive and negative detuning, showing the validity of the linear approximation and the excellent description of the anticrossing of the resonator-quantum system by the , the peaks are derived circuit model. At approximately . The transmisseparated by the vacuum Rabi frequency sion graphs do not contain amplitude data, but we mention here that the reduction of the peak far from the resonator frequency GHz is also observed in our simulation. In the case of large detuning, the peaks are highly separated and the qubit peak gets highly reduced, while the resonator frequency peak approaches the original resonator frequency; this shows that for large detuning, the two systems are decoupled. After successfully applying our circuit model to the description of the cavity transmission, we performed an analysis on the quantum bit frequency shift (Lamb shift). The cavity transmission was calculated as a function of the detuning; the Lamb shift was extracted by subtracting the bare cavity frequency and from the simulated qubit frequency. This was the detuning done by solving the full RBEs (4)–(8) numerically using a basic Runge–Kutta solver and by employing our circuit models. The simulation results were compared to experimental data from

Fig. 3. Transmission frequency of the resonator as a function of detuning. Small circles represent experimental data from [15] and the dashed lines represent our circuit model. The anticrossing behavior is clearly visible and the . separation is approximately given by the vacuum Rabi frequency at The excellent agreement confirms the usability of our circuit model to calculate cavity transmission.

1=0

Fig. 4. Frequency (Lamb) shift of the quantum bit as a function of detuning. Small circles represent experimental data. Continuous lines represent results extracted from the numerical solutions of the RBEs. Dashed lines indicate results of our passive circuit model [see Fig. 2(a)].

[15] for a detuning ranging from 100 to 600 MHz. The comparison was done by changing the frequency of the qubit and keeping the resonator frequency constant. As shown in Fig. 4, we see an excellent agreement between the experimental Lamb shift (small empty circles) and the numerical solutions of the RBEs (solid line), as well as our linear circuit model (dashed line). Thus, we can affirm that also the frequency shift of the quantum system (Lamb shift) is well described by our circuit model. To this point, we only did simulations of the quantum system in its ground state. For the numerical solution of the full RBEs, we used the initial conditions , as discussed in Section III-B. For the circuit model, we set .

MÁTYÁS et al.: LINEAR CIRCUIT MODELS FOR ON-CHIP QED

69

quantum state the simulation window can be reduced. For a too small simulation window, however, the Fourier transform of the signal becomes inaccurate. V. CONCLUSIONS We have derived circuit models for the understanding and modeling of superconducting coplanar waveguide resonators interacting with quantum systems. They can easily be used to model the cavity transmission, Lamb shift, and quantum nondemolition measurement. We find their application straightforward in understanding experimental data and estimating decay times for an optimal quantum state readout. Our models can be extended, e.g., by adding further circuits to represent additional qubits. Also, embedding quantum systems in large-scale classical circuitry is straightforward in our circuit models. Fig. 5. Theoretical demonstration of our circuit model showing the state dependent resonator frequency shift, which opens the possibility of a quantum state measurement. The quantum system decay time was 20 s, which was large enough not to influence the linewidth of the excited state. The shift is approximately g = . The slight disagreement of the peak positions results from the Fourier transform on the finite simulation window of the output voltage signal calculated with the RBEs.

1

We will now investigate the case when the quantum system is away from its ground state and its frequency is far detuned from the resonator frequency. In this case, one can perform a nondemolition readout of the quantum state, i.e., read out the inversion without changing it. This can be done if the detuning tends to infinity (practically is large enough so that the inversion changes only slightly). We then know from Rabi’s solutions that the inversion will not change (i.e., the resonator is unable to switch the quantum circuit during measurement). For an optimal quantum state readout, however, the state of the quantum system should not decay during measurement. Based on our circuit model and RBEs presented earlier, we now estimate the value of the decay for an optimal readout. time The quantum state readout measurement is performed in the dispersive regime, as discussed in Section III-A. In the case of small decay times , if the quantum system is in its excited state, it will quickly relax to the ground state; this limits the measurement time and, thus, the Fourier window of our numerical simulations. On the other hand, for large decay times, the quantum-state dependent cavity shift can be simulated, as shown in Fig. 5. The simulated cavity-pull linewidths extracted from the Fourier transform of the numerical RBE solutions and from the frequency-domain simulation of our circuit model agree well. This shows that the broadening of the peaks is and cavity decay and mainly due to decoherence not the qubit decay , which was not included in the circuit model. In this limit, the quantum-state dependent cavity pull is s (or more) resolvable, and we find that for this case would be required so that the two linewidths approximately agree. For smaller decay times, the numerical simulations of the RBEs show high asymmetry of the two peaks due to the non-Lorentzian behavior of the spectral line corresponding to ; this is due to the decay of the excited qubit state the state during simulation. In order to avoid the decay of the

APPENDIX A DERIVATION OF THE RBEs For the derivation of the RBEs, we need to review a few concepts widely used and well established in quantum optics. The starting point is to define the resonator voltage, similar as in [4] circuit are defined and [5]. The voltages and currents on the as (10) (11) and represent the emission and absorption Here, the terms operators. The commutator properties of these operators are

(12) The brackets represent a commutator relation [also used in the von Neumann equation (3)]. and represent the root The normalization terms mean square voltage and current [4], [5] on the resonator (13) (14) If we substitute the voltage and current operators [see (10) and (11)] into the general Hamiltonian shown in (2) and use the commutator operations defined in (12), we can get the general Hamiltonian

(15) widely used in cavity QED [5], [21], [25], [26]. For each Pauli matrix , ( ) and for the normalized current and voltage in (10) and (11), we used the von Neumann equation (3) for finding the RBEs. We now show the derivation of the dynamic equation for the resonator normalized current (8). The other RBEs (for coher-

70

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

ence vector and resonator normalized voltage) can be derived analogously. First, the normalized current can be written as

(16) Using the properties of the creation/annihilation operators from (12), the above von Neumann equation simplifies to (17) Using the definition in (10) and taking the expectation values of the time dependent variables in the above equation, we get (8). APPENDIX B DERIVATION OF THE EQUIVALENT LINEAR CIRCUIT and the system does not If the inversion is constant ), we can linearize the RBEs and thus get decay (i.e., an equivalent circuit. In this case, (6) is automatically fulfilled, and the other RBEs (4), (5), (7), and (8) simplify to the linear differential equation system (18) (19) (20) (21) By rewriting the above as two second-order equations, we get

(22) (23) The above equations can be equivalently found by using (24) (25) (26) (27) Introducing the quantum system voltage and current as (28) (29) (24)–(27) change into equations for two capacitively coupled resonant circuits with parameters shown in Table I. ACKNOWLEDGMENT The authors are grateful to Prof. Á. Csurgay, Faculty of Information Technology, Pázmány Péter Catholic University, Budapest, Hungary, and Prof. W. Porod, University of Notre Dame, Notre Dame, IN, for initiating their research on circuit

models of quantum systems. The work on resonators started in collaboration with Prof. R. Gross, Walther Meissner Institute, Garching, Germany.

REFERENCES [1] H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: Coherence in context,” Science, vol. 298, pp. 1372–1377, Nov. 2002. [2] Cavity Quantum Electrodynamics, P. R. Berman, Ed. Boston, MA: Academic, 1994. [3] H. Carmichael, An Open Systems Approach to Quantum Optics Cavity Quantum Electrodynamics. Berlin, Germany: Springer, 1993. [4] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature, vol. 431, pp. 162–167, Sep. 2004. [5] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phy. Rev. A, Gen. Phys., vol. 69, no. 6, pp. 062320-1–062320-11, Jun. 2004. [6] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Resolving photon number states in a superconducting circuit,” Nature, vol. 445, pp. 515–518, Feb. 2007. [7] F. Deppe, M. Mariantoni, E. P. Menzel, A. Marx, S. Saito, K. Kakuyanagi, H. Tanaka, T. Meno, K. Semba, H. Takayanagi, E. Solano, and R. Gross, “Two photon probe of the Jaynes Cummings model and controlled symmetry breaking in circuit QED,” Nat. Phys., vol. 4, pp. 686–691, Sep. 2008. [8] A. André, D. Demille, J. M. Doyle, M. D. Lukin, S. E. Maxwell, P. Rabl, R. J. Schoelkopf, and P. Zoller, “A coherent all-electrical interface between polar molecules and mesoscopic superconducting resonators,” Nat. Phys., vol. 2, pp. 636–642, Sep. 2006. [9] J. Tejada, E. M. Chudnovsky, E. del Barco, J. M. Hernandez, and T. P. Spiller, “Magnetic qubits as hardware for quantum computers,” Nanotechnology, vol. 12, pp. 181–186, Jun. 2001. [10] G. Csaba, A. Matyas, F. Peretti, and P. Lugli, “Circuit modeling of coupling between nanosystems and microwave coplanar waveguides,” Int. J. Circuit Theory Appl., vol. 35, no. 3, pp. 315–324, Apr. 2007. [11] T. Lindström, C. H. Webster, J. E. Healey, M. S. Colclough, C. M. Muirhead, and A. Y. Tzalenchuk, “Circuit QED with a flux qubit strongly coupled to a coplanar transmission line resonator,” Superconduct. Sci. Technol., vol. 20, pp. 814–821, Aug. 2007. [12] G. Csaba, Z. Fahem, F. Peretti, and P. Lugli, “Circuit modeling of flux qubits interacting with superconducting waveguides,” J. Comput. Electron., vol. 6, pp. 105–108, Jan. 2007. [13] A. I. Csurgay and W. Porod, “Equivalent circuit representation of arrays composed of coulomb-coupled nano-scale devices: Modeling, simulation and realizability,” Int. J. Circuit Theory Appl., vol. 29, no. 1, pp. 3–35, Jan. 2001. [14] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature, vol. 449, pp. 443–447, Sep. 2007. [15] A. Fragner, M. Göppl, J. M. Fink, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Resolving vacuum fluctuations in an electrical circuit by measuring the lamb shift,” Science, vol. 322, pp. 1357–1360, Nov. 2008. [16] A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Controlling the spontaneous emission of a superconducting transmon qubit,” Phys. Rev. Lett., vol. 101, no. 8, pp. 080502-1–080502-4, Aug. 2008. [17] A. Blais, J. Gambetta, A. Wallraff, D. I. Schuster, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Quantum-information processing with circuit quantum electrodynamics,” Phys. Rev. A, Gen. Phys., vol. 75, no. 3, pp. 032329-1–032329-21, Mar. 2007. [18] G. M. Reuther, D. Zueco, F. Deppe, E. Hoffmann, E. P. Menzel, T. Weißl, M. Mariantoni, S. Kohler, A. Marx, E. Solano, R. Gross, and P. Hänggi, “Two-resonator circuit quantum electrodynamics: Dissipative theory,” Phys. Rev. B, Condens. Matter, vol. 81, no. 14, pp. 1445101–144510-16, Apr. 2010.

MÁTYÁS et al.: LINEAR CIRCUIT MODELS FOR ON-CHIP QED

[19] D. Williams and S. Schwarz, “Design and performance of coplanar waveguide bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 7, pp. 558–566, Jul. 1983. [20] L. Frunzio, A. Wallraff, D. Schuster, J. Majer, and R. Schoelkopf, “Fabrication and characterization of superconducting circuit QED devices for quantum computation,” Trans. Appl. Superconduct., vol. 15, no. 2, pp. 860–863, Nov. 2004. [21] L. Allen and J. H. Eberly, Optical Resonance and Two Level Atoms. New York: Wiley, 1975. [22] E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE, vol. 51, no. 1, pp. 89–109, Jan. 1963. [23] F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Ph.D. dissertation, Dept. Phys., Stanford University, Stanford, CA, 1962. [24] LTSPICE. Linear Technol., Milipitas, CA, 2008. [Online]. Available: http://www.linear.com/designtools/software/ [25] J. Shen and S. Fan, “Coherent single photon transport in a one-dimensional waveguide coupled with superconducting quantum bits,” Phys. Rev.Lett., vol. 95, no. 21, pp. 213001-1–213001-4, Nov. 2005. [26] D. F. Walls and G. J. Milburn, Quantum Optics. Berlin, Germany: Springer, 1995. Alpár Mátyás was born in Aiud, Romania, in 1983. He received the M.Sc. degrees in electrical engineering from the Pázmány Petér Catholic University of Budapest, Budapest, Hungary, in 2007, and is currently working toward the Ph.D. degree at the Technische Universität München, Munich, Germany. His main research interests are modeling and optimizing quantum cascade lasers in the terahertz and infrared regimes and simulations including the interaction between resonators and quantum systems.

Christian Jirauschek (S’03–M’04) was born in Karlsruhe, Germany, in 1974. He received the Dipl-Ing. and Doctoral degrees in electrical engineering from the Universität Karlsruhe (TH), Karlsruhe, Germany, in 2000 and 2004, respectively. From 2002 to 2005, he was a Visiting Scientist with the Massachusetts Institute of Technology (MIT), Cambridge. Since 2005, he has been with the Technische Universität München, Munich, Germany, initially as a Postdoctoral Fellow, and since 2007, as the Head of an Independent Junior Research Group [Emmy-Noether Program of the German Physical Society (DPG)]. His research interests include modeling in the areas of optics and device physics, especially the simulation of quantum devices and mode-locked laser theory. Dr. Jirauschek is a member of the German Physical Society (DPG) and the Optical Society of America. He was the recipient of a scholarship from the German National Merit Foundation (Studienstiftung des Deutschen Volkes) from 1997 to 2000.

71

Federico Peretti was born in Fermo, Italy, in 1980. He received the Electronic Engineering degree and Doctoral degree in electromagnetism and bioengineering from the Università Politecnica Delle Marche, Rome, Italy, in 2005 and 2009, respectively. From 2006 to 2009, he was a Scientist with the Institute for Nanoelectronics, Technische Universität München, where his interests included modeling in the areas of microwaves and quantum devices, especially the simulation of coplanar resonators for interactions with two-level systems.

Paolo Lugli (M’01–SM’07) received the Laurea degree in physics from the University of Modena, Modena, Italy, in 1979, and the Master of Science in 1982 and Ph.D. degrees in 1985, in electrical engineering from Colorado State University, Fort Collins, in 1982 and 1985, respectively. In 1985, he joined the Physics Department, University of Modena, as Research Associate. From 1988 to 1993, he was an Associate Professor of solid-state physics of the Engineering Faculty, 2nd University of Rome “Tor Vergata.” In 1993, he became a Full Professor of optoelectronics with the 2nd University of Rome “Tor Vergata.” In 2003, he joined the Technische Universität München, Munich, Germany, where he became Head of the newly created Institute for Nanoelectronics. He has authored or coauthored over 250 scientific papers. He coauthored The Monte Carlo Modelling for Semiconductor Device Simulations (Springer, 1989) and High Speed Optical Communications (Kluver, 1999). His current research interests involve the modeling, fabrication, and characterization of organic devices for electronics and opto-electronics applications, the design of organic circuits, the numerical simulation of microwave semiconductor devices, and the theoretical study of transport processes in nanostructures. Dr. Lugli was general chairman of the 2004 IEEE International Conference on Nanotechnology, Munich, Germany.

György Csaba was born in Budapest, Hungary, in 1974. He received the M.Sc. degree from the Technical University of Budapest, Budapest, Hungary, in 1998, and the Ph.D. degree from the University of Notre Dame, Notre Dame, IN, in 2003. From 2004 to 2009, he was a Research Assistant with the Technische Universität München, Munich, Germany. In 2010, he joined the faculty of the University of Notre Dame, as a Research Assistant Professor. His research interests are in circuit-level modeling of nanoscale systems (especially magnetic devices) and exploring their applications for nonconventional architectures such as magnetic computing and physical cryptography.

72

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

A Planar Magic-T Structure Using Substrate Integrated Circuits Concept and Its Mixer Applications Fan Fan He, Ke Wu, Fellow, IEEE, Wei Hong, Senior Member, IEEE, Liang Han, Student Member, IEEE, and Xiaoping Chen

Abstract—In this paper, a planar 180 phase-reversal T-junction and a modified magic-T using substrate integrated waveguide (SIW) and slotline are proposed and developed for RF/microwave applications on the basis of the substrate integrated circuits concept. In this case, slotline is used to generate the odd-symmetric field pattern of the SIW in the phase-reverse T-junction. Measured results indicate that 0.3-dB amplitude imbalance and 3 phase imbalance can be achieved for the proposed 180 phase-reversal T-junction over the entire -band. The modified narrowband and optimized wideband magic-T are developed and fabricated, respectively. Measured results of all those circuits agree well with their simulated ones. Finally, as an application demonstration of our proposed magic-T, a singly balanced mixer based on this structure is designed and measured with good performances. Index Terms—Magic-T, 180 phase-reverse T-junction, slotline, substrate integrated circuits (SICs), substrate integrated waveguide (SIW).

I. INTRODUCTION

A

SLOTINE presents advantages in the design of microwave and millimeter-wave integrated circuits, especially when solid-state active devices are involved. Recently, the substrate integrated circuits (SICs) concept, involving the substrate integrated waveguide (SIW) technique and other synthesized nonplanar structures in planar form with planar circuits, has been demonstrated as a very promising scheme for low-cost, small size, relatively high power, low radiation loss, and high-density integrated microwave and millimeter-wave

Manuscript received December 22, 2009; revised May 21, 2010; accepted September 08, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), in part by the National 973 Project of China under Grant 2010CB327400 and in part by the National Nature Science Foundation of China (NSFC) under Grant 60921063. F. F. He is with the Poly-Grames Research Center, Department of Electrical Engineering, École Polytechnique de Montreal, Montreal, QC, Canada H3C 3A7, and also with the State Key Laboraotry of Millimeter Waves, College of Information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). K. Wu, L. Han, and X. Chen are with the Poly-Grames Research Center, Department of Electrical Engineering, École Polytechnique de Montreal, Montreal, QC, Canada H3C 3A7 (e-mail: [email protected]). W. Hong is with the State Key Laboratory of Millimeter Waves, College of Information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2091195

components and systems [1]–[6]. Alternatively named integrated waveguide structures of the SIW, such as laminated waveguide or post-wall waveguide, can be found in [7] and [8]. The transitions from the SIW to slotline [9] have already been investigated theoretically and experimentally, which provide a design base to integrate SIW circuits with slotline circuits. As a fundamental and important component, the magic-T has widely been used in microwave and millimeter-wave circuits such as balanced mixers, power combiners or dividers, balance amplifiers, frequency discriminators, and feeding networks of antenna array [10], [11]. Following intensive investigations of SIW components and systems in the past ten years, more and more attention is being paid to integrate the conventional magic-T based on SIW technology. Some SIW-based magic-T structures have been proposed and studied [9], [12], [13]. In [12] and [13], magic-T techniques were developed using multilayer low-temperature co-fired ceramic (LTCC) or printed circuit board (PCB) processes. An SIW planar magic-T was successfully designed with relatively narrowband characteristics in [9]. This magic-T consisting of an SIW T-junction, a slotline T-junction, and two phase-reverse slotline-to-SIW T-junctions, and it has an 8% bandwidth centered at 9 GHz with 0.2-dB amplitude and 1.5 phase imbalances. In this paper, a modified version of a planar SIW magic-T, which only consists of a phase-reverse slotline-to-SIW T-junction and an -plane SIW T-junction, is proposed and presented, as shown in Fig. 1, which has smaller size and wider bandwidth than its previous version [9]. Described in Section II are the analysis and discussions of the proposed 180 phase-reversal slotline-to-SIW T-junction with its simulated and measured results. In Section III, the modified planar magic-T structures with direct design and with further optimization are discussed with their transmission line models. Measured results agree with simulated results very well. Additional wideband slotline-to-microstrip and SIW-to-microstrip transitions are designed for port-to-port measurements of microstrip line in support of experimental characterization of the proposed structures. In the end, a singly balanced mixer based on our modified wideband magic-T is developed. All the structures in this paper are simulated with means of the full-wave simulation tool Ansoft HFSS, designed and fabricated on an RT/Duroid 6010 substrate with a dielectric constant of 10.2 and a thickness of 0.635 mm.

0018-9480/$26.00 © 2010 IEEE

HE et al.: PLANAR MAGIC-T STRUCTURE USING SICs CONCEPT AND ITS MIXER APPLICATIONS

73

Fig. 1. Physical 3-D configurations of the modified magic-T.

II. PHASE-REVERSAL SLOTLINE-TO-SIW T-JUNCTION Here, the slotline-to-SIW T-junction acts as a mode converter between the slotline and SIW. Fig. 2(a) depicts the physical 3-D is configuration of the slotline-to-SIW T-junction, where is the SIW width, and the width of metallic slot, is the slotline width. The yellow (in online version) and dark layers are the top metal cover and bottom metal cover. The light gray area means substrate. The slotline and SIW structures intersect with each other in which the slotline extends length into the metallic cover of the SIW with a short-circuited termination. Two via-posts with the diameter of are used to optimize the return loss of the T-junction. Fig. 2(b) shows the cross section at the A–A plane, where the orientation of electric fields is sketched. When the signal is coupled from the slotline into the SIW at the A–A plane, the electric fields of the slotline mode are converted to those of the half-mode SIW (or HMSIW) mode [14] because of overlapped metallic covers on the top and bottom of the SIW. As such, two phase-reverse waves come out of ports P2 and P3. Fig. 2(c) shows the equivalent circuit model of the T-junction. The model is similar to that of an -plane waveguide T-juncand are tion due to their similar electric field conversion. the characteristic admittances of the slotline and HMSIW, reis used instead of the spectively. In the equivalent circuit, SIW characteristic admittance because both of them have almost the same value. Based on the above principle, parameters , , and are mainly dependent on slotline’s length , width , and at the slotline port (port 1), and mainly depends on the SIW width . Therefore, the relationship between parameters of the equivalent circuit and return loss at port 1 is replaced by that between parameters of physical configuration and return loss at port 1. In order to minimize any potential radiation loss while transmitting signal from the slotline to the SIW, a possible minimum width of the slot line is mm. chosen as

W L =4

= 0:2

W W =8

= 7:3

D = 0:6

L = 4 :6

Fig. 2. (a) Physical description and parameters of the slotline-to-SIW T-juncmm, mm, mm, mm, tion. mm, and mm. (b) Electric field distribution at cross section A–A plane. (c) Equivalent circuit for the slotline-to-SIW T-junction.

Fig. 3 shows simulated and measured frequency responses of power dividing and return loss of the 180 phase-reversal slotline-to-SIW T-junction. The imbalance in amplitude and phase are, respectively, 0.3 dB and 3 , as shown in Fig. 4. These results suggest that the junction has broadband characteristics. Fig. 5 presents a photograph of the T-junction. III. MODIFIED PLANAR SLOTLINE-TO-SIW MAGIC-T A. Magic-T Circuit Configuration and Operating Principle Fig. 1 describes the physical 3-D configuration of the proposed magic-T. The yellow (in online version) and dark layers are the top metal cover and bottom metal cover. The light gray area means substrate. The orange areas (in online version) are metallic slots for the SIW. This magic-T consists of an SIW -plane T-junction and a slotline-to-SIW T-junction. Two such T-junctions share the two common arms with 45 rotation. Metallic vias V1 and V2 with diameter are used to construct the SIW -plane T-junction. Ports 1 ( port) and 4 ( port) are sum and difference ports, respectively, while ports 2 and 3 are the power dividing arms. Without the microstrip line-to-SIW and slotline-to-microstrip line transitions, the size of the magic-T is about 20 mm 20 mm. A signal applied to

74

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 6. Corresponding equivalent circuit of the magic-T. Fig. 3. Simulated and measured frequency responses of power dividing and return loss for the 180 slotline-to-SIW T-junction.

Fig. 4. Measured amplitude and phase imbalances of the slotline-to-SIW T-junction. Fig. 7. Simplified equivalent circuits of the magic-T. (a) In-phase. (b) Out-ofphase.

Fig. 5 Photograph of the slotline-to-SIW T-junction. Left and right figures are the top view and bottom view, respectively.

port 1 is split into two in-phase components by metallic via V1. The two components cancel each other at the slotline, while port 4 is isolated. In this case, the four-port junction works as an SIW -plane T-junction and the symmetrical plane A–B becomes a virtual open plane. Otherwise, the four-port junction works as a slotline-to-SIW T-junction and the plane A–B becomes a virtual ground plane when a signal is applied to port 4. The input signal is naturally split into two equal and out-of-phase signals at ports 2 and 3, and port 1 is isolated in this case.

The operating principle of the modified magic-T can also be well explained by its corresponding equivalent circuit at the working frequency shown in Fig. 6, where the slot-to-SIW T-junction can be seen as an ideal transformer and the SIW -plane T-junction as a divider. Parameters , , , and stand for the characteristic impedances, slotline, ground slotline, HMSIW, and SIW, respectively. In the in-phase case, the equivalent circuit model will further be simplified as depicted in Fig. 7(a), when at the working frequency. In the out-of-phase case, the simplified equivalent is shown in . On the basis of the above disFig. 7(b), where and should depend on the positions cussion, distances of the three metallic vias in the magic-T circuit. B. Implementation and Results Based on the above-stated principle, two magic-T structures are designed and fabricated on an RT/Duroid 6010LM

HE et al.: PLANAR MAGIC-T STRUCTURE USING SICs CONCEPT AND ITS MIXER APPLICATIONS

75

Fig. 8. Photograph of the modified magic-T. Left and right figures are the top view and bottom view, respectively.

TABLE I DIMENSIONS OF THE MODIFIED NARROWBAND MAGIC-T

substrate, respectively, with narrowband and wideband characteristics. Thus, the narrowband and wideband cases of the magic-T will be discussed separately. Fig. 8 shows the top view and bottom views of the modified magic-T’s photograph. From this photograph, we can estimate that the size of the magic-T is reduced by near 50% with reference to [9]. 1) Narrowband Case: The two out-of-phase signals cancel each other at port 1 as described in Section II-A, and simultaneously the distance is equal to a quarter of the guide wavelength of the SIW at the working frequency. Thus, the working bandwidth of the return loss at port 4 should be narrow in a similar manner to the previous design [9]. However, the working bandwidth judging from the return loss at port 1 should be wider because the two in-phase signals cancel each other in the slotline at port 4. In this demonstration, the magic-T was designed at 9 GHz. All design parameters of the magic-T are listed in Table I. Fig. 9 shows the return loss and insertion loss of the fabriis lower than 15 dB from 8.7 cated narrowband magic-T. to 9.4 GHz with a 7.8% bandwidth, which has validated the above discussion. Within the frequency range of interest, the minimum insertion loss is 0.7 dB and it is less than 0.8 dB in both in-phase and out-of-phase cases. Simulated and measured isolation characteristics are described in Fig. 10. The isolation is better than 30 dB between ports 1 and 4, and better than 20 dB between ports 2 and 3 over the entire frequency range. As shown in Fig. 11(a) and (b), the maximum phase and amplitude imbalances for both in-phase and out-of-phase cases are less than 1.5 and 0.5 dB, respectively. 2) Wideband Case: The narrowband characteristics of this magic-T have well been confirmed in the above discussion. However, an interesting outcome can be observed in that the return-loss defined bandwidth can be broadened by optimizing the , , , and . When the signal parameter values of flows into the SIW from the slotline in this slotline-to-SIW structure, it would be split into two components and each of at the working frequency, as them will propagate along line

Fig. 9. Simulated and measured frequency responses of the magic-T. (a) Return loss. (b) Insertion loss.

Fig. 10. Simulated and measured isolation characteristics of the magic-T.

shown in Fig. 1. Nevertheless, the propagating directions being different slightly at different frequencies provide a possibility of broadening the bandwidth of the magic-T. In other words, , it is possible for the magic-T to simultaneously realize and , at two different frequencies. In our proposed broadband design, these two frequencies are set at 8.7 and 9.8 GHz. Through optimization, some geometrical parameters of magic-T are changed

76

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 11. Measured results of amplitude and phase imbalance characteristics of the magic-T. (a) Amplitude. (b) Phase.

such that mm, mm, mm, and mm. Fig. 12(a) shows the newly designed magic-T’s simulated and measured return losses at each port. Among the results, simuindicates that the above two geometrical conditions lated for achieving broadband performances are readily satisfied at 8.7 and 9.8 GHz. Measured return loss is better than 15 dB from 8.4 to 10.6 GHz with 23.2% bandwidth. In this broadband frequency range, the insertion loss is less than 0.9 dB and the minimum insertion loss is 0.7 dB in both in-phase and out-of-phase cases, as shown in Fig. 12(b). Measured and simulated isolation curves between port 1 and port 4 or port 2 and port 3 are plotted in Fig. 13. In addition, the amplitude and phase imbalances of the magic-T are 2 and 0.5 dB, respectively, as shown in Fig. 14(a) and (b). Measured results of all circuits agree well with their simulated counterparts.

IV. MODIFIED MAGIC-T’s APPLICATION IN MIXER DESIGN As a practical and straightforward demonstration of our modified magic-T applications, a singly balanced mixer is designed, as shown in Fig. 15. Fig. 16 shows the photograph of the practical mixer. An antiparallel pair of series connected

Fig. 12 Simulated and measured frequency responses of the magic-T. (a) Return loss. (b) Insertion loss.

Fig. 13. Simulated and measured isolation characteristics of the magic-T.

diodes (SMS7630-006LF from Skyworks Inc., Woburn, MA) is adopted. Generally, a quarter-wavelength short stub in the matching circuit is need for providing a dc return and good IF-to-RF and IF-to-local oscillator (LO) isolations. However, a matching circuit is designed between the diode and SIW without using a quarter-wavelength short stub because the SIW open-circuited stubs on is grounded inherently. Two

HE et al.: PLANAR MAGIC-T STRUCTURE USING SICs CONCEPT AND ITS MIXER APPLICATIONS

77

Fig. 16. Photograph of the mixer.

Fig. 17. Measured conversion loss versus LO input power.

Fig. 14. Measured results of amplitude and phase imbalance characteristics of the magic-T. (a) Amplitude. (b) Phase.

Fig. 15. Circuit topology of the proposed mixer.

Fig. 18. Measured conversion loss versus IF frequency.

the right side of the diodes pair are used to provide a terminal virtual grounding point for LO frequency and RF frequency simultaneously. In addition, a low-pass filter is designed to suppress LO and RF signals at IF port. The mixer designed and simulated by the harmonic balance (HB) method in Agilent ADS software combined with measured -parameters of the wideband magic-T structure. Fig. 17 depicts the measured conversion loss versus LO input power level when the IF signal is fixed at 1 GHz with an input power level of 30 dBm and LO frequency is fixed at 10.2 GHz. When the LO input power level is larger than 13 dBm, the conversion loss almost is about 7.4 dB. Fig. 18 shows the measured conversion loss versus IF frequency when the IF signal is swept

from 0.1 to 4 GHz (RF is from 10.1 to 6.2 GHz) with a constant input power level of 30 dBm, and the LO signal is fixed at the frequency of 10.2 GHz with a 13-dBm power level. The measured conversion loss is about 8 0.6 dB over the IF frequency range of 0.1–3 GHz (RF is from 7.2 to 10.1 GHz). Fig. 19 plots the measured conversion loss versus input RF power level, where RF frequency is set at 9.2 GHz and LO frequency is at 10.2 GHz with a power level of 13 dBm, input RF power level is swept from 30 to 5 dBm. The output IF power almost increases with the RF power linearly when the RF power level is less than 3 dBm. On the other hand, when the RF power level is larger than 0 dBm, the mixer is driven into the nonlinearity region. From this figure, it can also be seen that the input 1-dB

78

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 19. Measured IF output power versus RF input power

compression point is around 3 dBm. Moreover, the LO-to-IF isolation is about 40 dB. All the measurement results dictate that this mixer is suitable for wideband applications. V. CONCLUSION The slotline-to-SIW 180 reversal T-junction with its simple equivalent circuit model has been presented. The modified SIW magic-Ts were then developed with narrowband and wideband cases, respectively. The operating principles and transmission line models for both cases have also been presented. Good performances related to the insertion loss, isolation, and balance were observed for our fabricated prototypes designed over the entire -band. Finally, a singly balanced mixer based on the modified magic-T was designed to validate the magic-T. Those novel structures are key components for designing integrated microwave and millimeter-wave circuits and systems such as the antenna feed network and mono-pulse radar. ACKNOWLEDGMENT The authors would like to thank the Rogers Corporation, Rogers, CT, for providing the free samples of the RT/Duroid 6010LM substrate and to S. Dubé and A. Traian, both with the Poly-Grames Research Center, Montreal, QC, Canada, for the fabrication of the experimental prototypes. The authors also thank the anonymous reviewers for their comments and suggestions on this paper. REFERENCES [1] K. Wu, D. Deslandes, and Y. Cassivi, “The substrate integrated circuits—A new concept for high-frequency electronics and optoeletronics,” in Telecommun. Modern Satellite, Cable, Broadcast. Service/TELSIKS 6th Int. Conf., Oct. 2003, vol. 1, pp. P-III–P-X. [2] F. Xu, Y. L. Zhang, W. Hong, K. Wu, and T. J. Cui, “Finite-difference frequency-domain algorithm for modeling guided-wave properties of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 11, pp. 2221–2227, Nov. 2003. [3] F. F. He, K. Wu, W. Hong, J. T. Hong, H. B. Zhu, and J. X. Chen, “Suppression of second and third harmonics using =4 low-impedance substrate integrated waveguide bias line in power amplifier,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 7, pp. 479–481, Jul. 2008.

[4] P. Chen, W. Hong, Z. Q. Kuai, J. F. Xu, H. M. Wang, J. X. Chen, H. J. Tang, J. Y. Zhou, and K. Wu, “A multibeam antenna based on substrate integrated waveguide technology for MIMO Wireless Communications,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1813–1821, Jun. 2009. [5] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 2, pp. 68–70, Feb. 2001. [6] J. X. Chen, W. Hong, Z. C. Hao, H. Li, and K. Wu, “Development of a low cost microwave mixer using a broadband substrate integrated waveguide (SIW) coupler,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 2, pp. 84–86, Feb. 2006. [7] A. Piloto, K. Leahy, B. Flanick, and K. A. Zaki, “Waveguide filters having a layered dielectric structures,” U.S. Patent 5 382 931, Jan. 17, 1995. [8] J. Hirokawa and M. Ando, “45 linearly polarized post-wall waveguide-fed parallel-plate slot arrays,” Proc. Inst. Elect. Eng.–Microw. Antennas, Propag., vol. 147, no. 6, pp. 515–519, Dec. 2000. [9] F. F. He, K. Wu, W. Hong, H. J. Hong, H. B. Zhu, and J. X. Chen, “A planar magic-T using substrate integrated circuits concept,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 6, pp. 386–388, Jun. 2008. [10] T. Tokumitsu, S. Hara, and M. Aikawa, “Very small ultra-wide-band MMIC magic T and applications to combiners and dividers,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1985–1990, Dec. 1989. [11] C. P. Tresselt, “Broad-band high IF mixers based on magic T’s,” IEEE Trans. Microw. Theory Tech., vol. MTT-18, no. 1, pp. 58–60, Jan. 1970. [12] T. M. Shen, T. Y. Huang, and R. B. Wu, “A laminated waveguide magic-T in multilayer LTCC,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 713–716. [13] L. Han, K. Wu, and S. Winkler, “Singly balanced mixer using substrate integrated waveguide magic-T structure,” in Eur. Wireless Technol. Conf., 2008, pp. 9–12. [14] W. Hong, B. Liu, Y. Q. Wang, Q. H. Lai, H. J. Tang, X. X. Yin, Y. D. Dong, Y. Zhang, and K. Wu, “Half mode substrate integrated waveguide: A new guided wave structure for microwave and millimeter wave application,” in Joint 31st Int. Infrared Millim. Waves Conf./14th Int. Terahertz Electron. Conf., Shanghai, China, Sep. 18–22, 2006, p. 219.

Fan Fan He was born in Nanjing, China. He received the M.S. degree in mechanical–electrical engineering from Xidian University, Xi’an, China, in 2005, and is currently working toward the Ph.D. degree in electrical engineering both at Southeast University, Nanjing, China, and the École Polytechnique de Montréal, Montréal, QC, Canada, as an exchange student. His current research interests include advanced microwave and millimeter-wave components and systems.

Ke Wu (M’87–SM’92–F’01) is currently a Professor of electrical engineering and Tier-I Canada Research Chair in RF and millimeter-wave engineering with the École Polytechnique de Montreal, Montreal, QC, Canada. He also holds the first Cheung Kong endowed chair professorship (visiting) with Southeast University, the first Sir Yue-Kong Pao chair professorship (visiting) with Ningbo University, and an honorary professorship with the Nanjing University of Science and Technology and the City University of Hong Kong. He has been the Director of the Poly-Grames Research Center and the Director of the Center for Radiofrequency Electronics Research of Quebec (Regroupement stratégique of FRQNT). He has authored or coauthored over 630 referred papers and a number of books/book chapters. He holds numerous patents. He has served on the Editorial/Review Boards of many technical journals, transactions and letters, as well as scientific encyclopedia as both an editor and guest editor. His current research interests involve SICs, antenna arrays, advanced computer-aided design (CAD) and modeling techniques, and development of low-cost RF and millimeter-wave transceivers and sensors for wireless systems and biomedical applications. He is also interested in the modeling and design of microwave photonic circuits and systems.

HE et al.: PLANAR MAGIC-T STRUCTURE USING SICs CONCEPT AND ITS MIXER APPLICATIONS

Dr. Wu is a member of the Electromagnetics Academy, Sigma Xi, and the URSI. He is a Fellow of the Canadian Academy of Engineering (CAE) and the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities). He has held key positions in and has served on various panels and international committees including the chair of Technical Program Committees, International Steering Committees, and international conferences/symposia. He will be the general chair of the 2012 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS). He is currently the chair of the joint IEEE chapters of the IEEE MTT-S/Antennas and Propagation Society (AP-S)/Lasers and Electro-Optics Society (LEOS), Montreal, QC, Canada. He was an elected IEEE MTT-S Administrative Committee (AdCom) member for 2006–2009. He is the chair of the IEEE MTT-S Transnational Committee. He is an IEEE MTT-S Distinguished Microwave Lecturer (2009–2011). He was the recipient of many awards and prizes including the first IEEE MTT-S Outstanding Young Engineer Award and the 2004 Fessenden Medal of the IEEE Canada.

Wei Hong (M’92–SM’07) was born in Hebei Province, China, on October 24, 1962. He received the B.S. degree from the Zhenzhou Institute of Technology, Zhenzhou, China, in 1982, and the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves, Southeast University, where he is currently a Professor and the Associate Dean of the Department of Radio Engineering. In 1993 and from 1995 to 1998, he was a short-term Visiting Scholar with the University of California at Berkeley and the University of Santa Cruz, respectively. He has authored or coauthored over 200 technical publications. He authored Principle and Application of the Method of Lines (Southeast Univ. Press, 1993, in Chinese) and Domain Decomposition Method for EM Boundary Value Problems (Sci. Press, 2005, in Chinese). He has been engaged in numerical methods for electromagnetic problems, millimeter-wave theory and technology, antennas, electromagnetic scattering and RF technology for mobile communications, etc. Prof. Hong is a Senior Member of the China Institute of Electronics (CIE). He is vice-president of the Microwave Society and Antenna Society of CIE. He has been a reviewer for many technical journals including the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and is currently an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was a two-time recipient of the First-Class Science and Technology Progress Prize issued by the State Education Commission in 1992 and 1994. He was a recipient of the Fourth-Class National Natural Science Prize in 1991, and the Firstand Third-Class Science and Technology Progress Prize of Jiangsu Province. He was also the recipient of the Foundations for China Distinguished Young Investigators and the “Innovation Group” issued by the National Science Foundation of China.

79

Liang Han (S’07) was born in Nanjing, China. He received the B.E. (with distinction) and M.S. degrees from Southeast University, Nanjing, China, in 2004 and 2007, respectively, both in electrical engineering. He is currently working toward the Ph.D. degree in electrical engineering at the École Polytechnique de Montréal, Montréal, QC, Canada. His current research interests include advanced CAD and modeling techniques, and development of multifunctional RF transceivers.

Xiaoping Chen was born in Hubei province, China. He received the Ph.D. degree in electrical engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2003. From 2003 to 2006, he was a Post-Doctoral Researcher with the State Key Laboratory of Millimeter-waves, Radio Engineering Department, Southeast University, Nanjing, China, where he was involved with the design of advanced microwave and millimeter-wave components and circuits for communication systems. In May 2006, he worked as a Post-Doctoral Research Fellow with the Poly-Grames Research Center, Department of Electrical Engineering, Ecole Polytechnique (University of Montréal), Montréal, QC, Canada, where he is currently a Researcher Associate. He has authored or coauthored over 30 referred journals and conference papers and some proprietary research reports. He has been a member of the Editorial Board of the IET Journal. He holds several patents. His current research interests are focused on millimeter-wave components, antennas, and subsystems for radar sensors. Dr. Chen has been a reviewer for several IEEE publications. He was the recipient of a 2004 China Postdoctoral Fellowship. He was also the recipient of the 2005 Open Foundation of the State Key Laboratory of Millimeter-waves, Southeast University.

80

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Exact Synthesis and Implementation of New High-Order Wideband Marchand Baluns Jhe-Ching Lu, Chung-Chieh Lin, and Chi-Yang Chang, Member, IEEE

Abstract—New ultra-wideband high-order Marchand baluns with one microstrip unbalanced port and two microstrip balanced ports are proposed in this paper. The proposed Marchand baluns are synthesized based on an -plane high-pass prototype using the Richards’ transformation. The responses of the synthesized high-order Marchand baluns are exactly predicted at all real frequencies. All circuit elements are commensurate, which means the electrical lengths of all transmission line elements are a quarter-wavelength long at the center frequency. Two fifth-order Marchand baluns with reflection coefficients of 20.53 and 21.71 dB corresponding to 131% and 152% bandwidth, respectively, are synthesized and realized using the combinations of microstrip lines, slotlines, and coplanar striplines. Simulated and measured results are presented. Index Terms—Circuit transformations, high-pass prototype, Marchand balun, planar structures, synthesis, wideband.

I. INTRODUCTION ALUNS [1] are widely used in RF and microwave communication systems. The main feature of baluns is to transform an unbalanced signal to a balanced signal, and vice versa. Thus, baluns can be used in antenna excitations or balanced circuit topologies such as balanced mixers, push–pull amplifiers, and phase shifters. There are many types of baluns as proposed in [1]–[5]. Among the various kinds of baluns, the Marchand balun [6]–[10] is very popular because of its relatively wider bandwidth of amplitude and phase balance. Several methods to realize Marchand baluns have been proposed [6]–[16] In design of a coupled-line Marchand balun, various analysis methods were presented. In [10] and [11], use of the relationships of the power wave in a balun to derive the scattering parameters can analyze a symmetrical Marchand balun, but the exact prediction is only valid at the center frequency. In [12], inclusion of the parameter of the electrical length of the transmission line can predict broadband performances, but the approach lacks generality. Furthermore, to achieve wider bandwidth, multiconductor coupled lines to realize tight couplings

B

Manuscript received May 21, 2010; revised August 30, 2010; accepted August 31, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. This work was supported in part by the National Science Council under Grant NSC98-2221-E-009-034-MY3. J.-C. Lu is with the RF Modeling Program, Taiwan Semiconductor Manufacture Company Ltd. (TSMC), Hsinchu 300, Taiwan (e-mail: Zill_gerching@ hotmail.com). C.-C. Lin and C.-Y. Chang are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2090702

were presented. Another method is the even- and odd-mode analysis method. However, it is limited to the case of a symmetrical coupled-line Marchand balun with the maximally flat responses. Actually, the exact synthesis of conventional Marchand baluns has been presented in [6]–[9]. A Chebyshev response can be synthesized using the synthesis method. While focusing on realizing Marchand baluns with planar coupled-line technologies, useful design values of even- and odd-mode parameters in each coupled line section are available in [9]. Nevertheless, when bandwidth of a balun is a major consideration, one should concern the limited range of practical evenand odd-mode impedance values of coupled lines. In [17], coupled lines with suitable lumped capacitors are utilized to constitute the new miniaturized planar Marchand baluns. The circuits were synthesized based on the bandpass prototype in the Richards’ domain. The key issue to use the Richard’s domain bandpass prototype is to shrink the circuit size. However, the circuit in [17] may not be easy to design a wider band performance due to physical limitations of the coupled lines as described above. Moreover, the use of the capacitors to approximate the series and shunt open-circuit stubs deteriorates the wideband responses. An interesting research on a higher order Marchand balun has been presented in [18]. The analysis of the balun is based on the chain-scattering matrices of the open-circuit series stub, short-circuited shunt stub, and uniform transmission line. With these matrices, the coefficients of the desired polynomial form describing the balun response can be numerically solved due to equations, where is equal to the elethe formulated ment number of the balun. The stepped-impedance transformers embedded in the circuit contribute the number of order to the Marchand balun. A coplanar waveguide with a ground plane and coupled microstrip lines is used to realize the balun. The analysis method and design procedures may be complicated and the implementation of the balun occupies a relatively large circuit size. The purpose of this paper is to propose new higher order wideband Marchand baluns and its novel implementation. The new Marchand balun structure shown in Fig. 1 is with the order higher than the conventional fourth-order Marchand balun. Compared with the conventional fourth-order Marchand baluns and their realizations, the proposed baluns have the advantages of realizable design values and very wideband performances in planar technology. During the synthesizing procedures of the Marchand Balun, one may first perform the circuit analysis on Richards’ circuit elements and then obtain the prescribed characteristic functions [19], [20] to extract element values of the Marchand balun. Thus, the design of the

0018-9480/$26.00 © 2010 IEEE

LU et al.: EXACT SYNTHESIS AND IMPLEMENTATION OF NEW HIGH-ORDER WIDEBAND MARCHAND BALUNS

81

TABLE I RELATIONSHIPS BETWEEN DISTRIBUTED CIRCUITS IN THE f -PLANE AND HIGH-PASS CIRCUITS IN THE S -PLANE, AND THE CORRESPONDING ABCD PARAMETERS

Fig. 1. Distributed circuit of the proposed fifth-order Marchand balun.

II. DERIVATION OF A FIFTH-ORDER MARCHAND BALUN The distributed circuit of the proposed Marchand balun, as shown in Fig. 1, comprises one open-circuit series stub, two short-circuited shunt stubs, one uniform transmission line connected to input port (unbalanced port), one uniform transmission line connected to two short-circuit stubs, and two identical uniform transmission lines connected to each of the bal, anced output ports with impedance values corresponding to , and , and , , and , respectively. The electrical lengths of all the stubs and uniform transmission lines are 90 at the center frequency. Due to differential outputs, the two output ports can be combined into one port. Thus, the two-port distributed circuit can be simplified as shown in Fig. 2(a). Its equivalent -plane high-pass prototype is shown in Fig. 2(b). The Richards’ frequency-domain variable is defined as Fig. 2. Proposed fifth-order Marchand balun. (a) Two-port distributed circuit simplified in Fig. 1. (b) Its S -plane high-pass circuit.

proposed baluns is based on the -plane high-pass prototype using Richards’ transformation , where is the center frequency of the passband, and and are the real frequency domain and Richards’ frequency-domain variables, respectively. By applying proper circuit transformations, the proposed original distributed circuit of the Marchand balun, as shown in Fig. 1, can be converted into a fifth-order -plane high-pass prototype, as shown in Fig. 2. With the aid of the synthesis method, all element values of the synthesized prototype can be obtained. Thus, the design parameters of the original circuit in Fig. 1 can be controlled. For realizing the balun, new combinations of the planar structures are presented for the first time. It will be shown that the microstrip lines, slot lines, and coplanar striplines are utilized to artfully implement the transmission line elements of the balun. The microstrip lines are also arranged on the other side of the slot lines, which saves the circuit area.

(1) where is the center frequency of the passband, and is the real frequency domain variable. The open- and short-circuit stubs in the -plane become a capacitor and inductors, respectively, in the -plane. The interconnecting uniform transmission lines in the -plane are turned into the unit elements (UEs) in the -plane. The description of the parameters of the three important components in high-pass prototype is shown in Table I. To derive the final fifth-order Marchand balun, circuit transformations will be used. Firstly, the circuit transformation to be used is the Kuroda’s identity, as shown in Fig. 3(a) [21]. The Kuroda transformation is now applied to the shunt inductor in Fig. 2(b), thus changing the position of the shunt inductor to another side. The transformed from one side of the UE circuit is shown in Fig. 3(b) with the following transformation equation: (2)

82

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 5. Final fifth-order S -plane high-pass prototype of Marchand balun.

Fig. 3. First transformation. (a) Kuroda identity transformation from [21]. (b) Applying the Kuroda identity in (a) to the circuit in Fig. 2(b).

Fig. 6.

N -order S -plane high-pass prototype of the Marchand balun.

Thirdly, by applying the newly derived circuit transformation, the further transformed circuit can be obtained, which is shown in Fig. 4(b). Finally, the two shunt inductors can be combined, transformer can be absorbed into the load resisand the tance. Consequently, the final fifth-order Marchand balun prototype is shown in Fig. 5 with the following relations: (6) (7) Fig. 4. Second transformation. (a) Newly presented exact circuit transformation. (b) Applying the exact circuit transformation in (a) to the circuit in Fig. 3(b).

(8) (9)

Secondly, observation of the two shunt inductors connected transformer shown in Fig. 3(b) shows that by the one redundant shunt inductor exists. Hence, to combine the redundant element, a new circuit transformation should be derived. Consider the new circuit transformation in Fig. 4(a). -parameters of the left circuit in Fig. 4(a) are The

The design parameters in (2) and (6)–(9) complete the transformation of the balun prototype of Fig. 2(b) into that of Fig. 5. An effect way to increase the order of the Marchand balun is to add nonredundant UEs. Fig. 6 shows the -plane high-pass Marchand balun. prototype circuit of the -order III. SYNTHESIS AND DESIGN OF TWO BALUN EXAMPLES A. Synthesis Procedures

(3)

The

-parameters of the right circuit in Fig. 4(a) are

Before designing the proposed balun, two important points should be addressed in the following. The first point is to determine a characteristic transfer function of the fifth-order Marchand balun. The second point is to apply the exact synthesis to the proposed balun such that the wideband responses can be predicted. By observing the prototype circuit in Fig. 5, the suitable characteristic function exhibiting the Chebyshev responses, which is comprehensively discussed in [20], is given by

(4)

Assume that (3) is equal to (4), the transformed parameters can be obtained as follows:

(5)

(10) , is the filter cutoff frequency where that is used to determine the bandwidth of the balun, specifies equal-ripple value, and and are the unnormalized Chebyshev polynomials of the first and second kinds of degree , respectively. In (10), subscript and denote the number

LU et al.: EXACT SYNTHESIS AND IMPLEMENTATION OF NEW HIGH-ORDER WIDEBAND MARCHAND BALUNS

of high-pass ladder elements (series capacitors and shunt inductors) and UEs, respectively. , the square of the magnitude of the input reGiven flection coefficient is obtained using

The input impedance is then obtained by (13) as

(18)

(11) can then be found with the knowledge that (12) The relationship between input impedance with a normalized source resistance of 1 is

and

83

The following step is to synthesize the element values in Fig. 5. To extract the encountered UE value and to obtain the input impedance of the remaining network, (14) and (15) are used. The standard synthesis procedure of lumped-element ladder networks is then applied to extract the series capacitor and shunt inductor. Thus, the circuit element values of the first balun with a normalized source resistance of 1 are

(13) The circuit prototype to be synthesized is shown in Fig. 5. The first element to be extracted is the left-most UE of the circuit. By applying Richards’ theorem, a UE can be obtained using (14) (19)

where

denotes the impedance value of the th UE and is the input impedance looking from the th UE. The input impedance of the remaining network after removal of the extracted UE is

Substituting (19) into (2) and (6)–(9) and de-normalizing to the 50- system then gives the design parameters of Fig. 1 as follows:

(15) where the common factor of can be cancelled out. The second and third element types to be extracted are the series capacitor and shunt inductor. The method used to synthesize lumped-element ladder networks can be applied to this balun prototype and obtain the element values of the series capacitor and shunt inductor. B. Two Design Examples The two examples of fifth-order equal-ripple wideband Marchand baluns corresponding to the -plane high-pass prototype, as shown in Fig. 5, are considered in this paper. The first balun is designed with the center frequency of 2 GHz, a normalized cutoff frequency of corresponding to a bandwidth of 131%, and a ripple level of corresponding to a return loss of 20.53 dB. Applying these parameters into (10), the characteristic polynomial can be constructed as

(20) of 2 GHz, The second balun is with the center frequency corresponding a normalized cutoff frequency of to a bandwidth of 152%, and ripple level corresponding to a return loss of 21.71 dB. Similarly, follow the synthesized procedures, as described in the first designed balun. The polynomial of the input impedance, the circuit parameters corresponding to Fig. 5 in a normalized source resistance of 1 , and the design parameters of Fig. 1 in the 50- system are shown in (21)–(23), respectively,

(21) (16)

and

The square of the magnitude of the input reflection coefficient is then established by (11), and with the knowledge of (12) it can lead to

(17)

(22)

84

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 9. Photograph of the first designed fifth-order balun. (a) Top view. (b) Bottom view. Fig. 7. Ideal responses of the first and second designed baluns with bandwidth of 131% and 152%, respectively.

Fig. 8. Physical layout of the first designed fifth-order balun with S (unit: millimeters).

= j0:6

(23) The ideal responses of the two designed baluns corresponding to the circuit of Fig. 5 are shown in Fig. 7.

Fig. 10. Measured and simulated performances of the first fifth-order balun. (a) jS j, jS j, and jS j. (b) Amplitude imbalance and phase difference.

IV. PHYSICAL IMPLEMENTATION AND EXPERIMENTAL RESULTS The practical implementations of the two designed wideband baluns comprise the microstrip lines, slotlines, and coplanar striplines. A 0.635-mm-thick RT/Duroid6010 substrate with and a loss tangent of 0.0023 is used to implement the two wideband baluns. The distributed circuits to be directly realized are the circuit shown in Fig. 1. Here, the two short-cirand , the uniform transmission line , the cuit stubs , and other uniform transmission lines open-circuit stub and are implemented by coplanar striplines, a slotline, and microstrip lines, respectively. The microstrip lines ( and ) are constructed on the front side, whereas the slotline ( ) and coplanar striplines are on the back side. Moreover, the slotline also serves as the ground plane of the microstrip lines. This arrangement not only saves the circuit area, but also provides the straightforward connection between coplanar striplines and slotlines. The coplanar striplines and slotline are both balanced transmission lines, which ensure the differential-mode signal

transmission. All the stubs and uniform transmission line sections are with electrical lengths of 90 at the center frequency GHz. of Fig. 8 depicts the detailed physical dimensions of the first balun. The design was accomplished with a commercial electromagnetic (EM) simulator, i.e., Ansoft’s High Frequency Structure Simulator (HFSS). Fine tuning in HFSS was performed to take all the EM effects into consideration. The topand bottom-view photographs of the fabricated first designed balun are shown in Fig. 9(a) and (b), respectively. The top-view photograph shows the realizations of the microstrip lines cor, and in Fig. 1. responding to the circuit elements , While the bottom-view photograph shows the realizations of the two coplanar striplines and the slotline corresponding to , , and , respectively, in Fig. 1. the circuit elements Fig. 10 shows the simulated and measured performances. The measured return losses are better than 10 dB from 0.7 to

LU et al.: EXACT SYNTHESIS AND IMPLEMENTATION OF NEW HIGH-ORDER WIDEBAND MARCHAND BALUNS

Fig. 11. Physical layout of the second designed fifth-order balun with S j : (unit: millimeters).

04

=

85

3.5 GHz. The measured amplitude imbalance is within 1 dB from 0.72 to 3.62 GHz and the measured phase difference is 10 from 0.7 to 3.53 GHz. within 180 The physical layout of the second balun is shown in Fig. 11. It should be pointed out that the synthesized impedance value of is much larger than the intrinsic impedance in free space, section can be removed. which is equal to 377 . Thus, the The top- and bottom-view photographs of the fabricated second designed balun are shown in Fig. 12(a) and (b), respectively. The simulated and measured performances are shown in Fig. 13. The measured return losses are better than 10 dB from 0.52 to 3.68 GHz. The measured amplitude imbalance is within 1 dB from 0.46 to 3.75 GHz and the measured phase imbalance is 10 from 0.46 to 3.62 GHz. within 180 V. CONCLUSION This paper has proposed a planar circuit Marchand balun structure. The novel circuit structure is suitable for the circuit synthesis of the Richard’s domain high-pass prototype. Comparing to the conventional fourth-order Marchand balun, the proposed Marchand balun can easily implement a circuit with the order higher than five. The responses of the synthesized circuits are exact. Two fifth-order examples have been given to show the feasibility of the planar circuit implementation and their wideband performances.

Fig. 12. Photograph of the second designed fifth-order balun. (a) Top view. (b) Bottom view.

Fig. 13. Measured and simulated performances of the second fifth-order balun. (a) jS j, jS j, and jS j. (b) Amplitude imbalance and phase difference.

REFERENCES [1] R. Mongia, I. Bahl, and P. Bhartia, RF and Microwave Coupled-Line Circuits. Norwood, MA: Artech House, 1999, pp. 391–442. [2] A. M. Pavio and A. Kikel, “A monolithic or hybrid broadband compensated balun,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1990, pp. 483–486. [3] B. J. Minnis and M. Healy, “New broadband balun structures for monolithic microwave integrated circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1991, pp. 425–428. [4] K. S. Ang, Y. C. Leong, and C. H. Lee, “Multisection impedance-transforming coupled-line baluns,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 536–541, Feb. 2003. [5] D. Kuylenstierna and P. Linner, “Design of broadband lumped-element baluns with inherent impedance transformation,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 12, pp. 2739–2745, Dec. 2004. [6] N. Marchand, “Transmission line conversion transformers,” Electron, vol. 17, no. 12, pp. 142–145, Dec. 1944. [7] J. Cloete, “Exact design of the Marchand balun,” Microw. J., vol. 23, no. 5, pp. 99–102, May 1980. [8] J. Cloete, “Graphs of circuit elements for the Marchand balun,” Microw. J., vol. 24, no. 5, pp. 125–128, May 1981. [9] C. L. Goldsmith, A. Kikel, and N. L. Wilkens, “Synthesis of Marchand baluns using multilayer microstrip structures,” Int. J. Microw. Millimeter-Wave Comput.-Aided Eng., vol. 2, no. 3, pp. 179–188, 1992. [10] K. S. Ang and I. D. Robertson, “Analysis and design of impedancetransforming planar Marchand baluns,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 402–406, Feb. 2001. [11] S. C. Tseng, C. C. Meng, C. H. Chang, C. K. Wu, and G. W. Huang, “Monolithic broadband Gilbert micromixer with an integrated Marchand balun using standard silicon IC process,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4362–4371, Dec. 2006. [12] C. S. Lin, P. S. Wu, M. C. Yeh, J. S. Fu, H. Y. Chang, K. Y. Lin, and H. Wang, “Analysis of multiconductor coupled-line Marchand baluns for miniature MMIC design,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 6, pp. 1190–1199, Jun. 2007. [13] C. W. Tang and C. Y. Chang, “A semi-lumped balun fabricated by low temperature co-fired ceramic,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, pp. 2201–2204. [14] M. J. Chiang, H. S. Wu, and C. K. Tzuang, “A compact CMOS Marchand balun incorporating meandered multilayer edge-coupled transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 125–128.

86

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

[15] S. A. Maas, The RF and Microwave Circuit Design Cookbook. Norwood, MA: Artech House, 1998, pp. 109–114. [16] Y. S. Lin and C. H. Chen, “Lumped-element Marcand-balun type coplanar waveguide-to-coplanar stripline transitions,” in Proc. Asia–Pacific Microw. Conf., Dec. 2001, pp. 539–542. [17] W. M. Fathelbab and M. B. Steer, “New class of miniaturized planar Marchand baluns,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1211–1220, Apr. 2005. [18] Z. Xu and L. MacEachern, “Optimum design of wideband compensated and uncompensated Marchand baluns with step transformers,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 8, pp. 2064–2071, Aug. 2009. [19] H. J. Orchard and G. C. Temes, “Filter design using transformed variable,” IEEE Trans. Circuit Theory, vol. CT-15, no. 12, pp. 385–408, Dec. 1968. [20] 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. [21] D. M. Pozer, Microwave Engineering, 3rd ed. New York: Wiley, 2005, p. 407. Jhe-Ching Lu was born in Kaohsiung, Taiwan, on May 18, 1982. He received the B.S. degree in electrical engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 2004, and M.S. and Ph.D. degrees in communication engineering from National Chiao-Tung University, Hsinchu, Taiwan, in 2006 and 2009, respectively. In 2009, he joined the Taiwan Semiconductor Manufacture Company Ltd. (TSMC), Hsinchu, Taiwan. He is currently with the RF Modeling Program, TSMC. His research interests include the analysis and design of microwave and millimeter-wave circuits and RF device characterization and modeling.

Chung-Chieh Lin was born in Taipei, Taiwan. He received the B.S. degree in electrical engineering and M.S. degree in communication engineering from National Chiao-Tung University, Hsinchu, Taiwan, in 2007 and 2009, respectively. His research interest is microwave passive component design.

Chi-Yang Chang (S’88–M’95) was born in Taipei, Taiwan, on December 20, 1954. He received the B.S. degree in physics and M.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1977 and 1982, respectively, and the Ph.D. degree in electrical engineering from The University of Texas at Austin, in 1990. From 1979 to 1980, he was with the Department of Physics, National Taiwan University, as a Teaching Assistant. From 1982 to 1988, he was with the Chung-Shan Institute of Science and Technology (CSIST), as an Associate Researcher, where he was in charge of the development of microwave integrated circuits (MICs), microwave subsystems, and millimeter-wave waveguide E -plane circuits. From 1990 to 1995, he returned to CSIST, as an Associate Researcher in charge of the development of uniplanar circuits, ultra-broadband circuits, and millimeter-wave planar circuits. In 1995, he joined the faculty of the Department of Communication, National Chiao-Tung University, Hsinchu, Taiwan, as an Associate Professor and became a Professor in 2002. His research interests include microwave and millimeter-wave passive and active circuit design, planar miniaturized filter design, and monolithic-microwave integrated-circuit (MMIC) design.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

87

Microstrip Branch-Line Couplers for Crossover Application Jijun Yao, Cedric Lee, and Swee Ping Yeo

Abstract—The branch-line coupler may be redesigned for crossover application. The bandwidth of such a coupler can be extended by suitably incorporating additional sections into the composite design. Laboratory tests on microstrip prototypes have shown the return loss and isolation of the three- and four-section couplers to be better than 20 dB over bandwidths of 22% and 33%, respectively. The insertion losses and group delays vary by less than 0.05 dB and 1 ns, respectively, for both prototypes. Index Terms—Microstrip components.

I. INTRODUCTION

W

HEN designers of microwave integrated circuits find it difficult to avoid transmission lines cutting across each other’s path, they usually opt for air-bridges or under-passes [1], but such structures are not the only means available to them. Designers of Butler matrices [2]–[8], for example, have recently been exploring the possibility of employing other structures that are capable of yielding the following scattering matrix for crossover application:

(1)

Recognizing that the cross junction [9] formed by a pair of intersecting lines does not in general yield the crossover characteristics specified in (1), researchers have experimented with modifications of various planar coupler structures. The annular-ring coupler reported in [10] has a simple structure, but tests have shown its crossover bandwidth to be limited. Studies of the composite ring-based structures proposed in [11]–[13] have indicated that optimized designs are able to meet 20-dB targets for the return-loss and isolation parameters in (1) over bandwidths of 20%, but these novel couplers may not gain ready acceptance because of their unconventional shapes. Lange couplers have Manuscript received June 07, 2010; accepted September 03, 2010. Date of publication November 22, 2010; date of current version January 12, 2011. This work was supported by the Singapore Academic Research Fund under AcRF Grant R-263-000-175-112. J. Yao is with the Institute for Infocomm Research, A*STAR, Singapore (e-mail: [email protected]). C. Lee was with the Electrical and Computer Engineering Department, National University of Singapore, Kent Ridge, Singapore. He is now with the DSO National Laboratory, Singapore (email: [email protected]). S. P. Yeo is with the Electrical and Computer Engineering Department, National University of Singapore, Kent Ridge, Singapore (e-mail: eleyeosp@nus. edu.sg). Digital Object Identifier 10.1109/TMTT.2010.2090695

Fig. 1. Multisection branch-line structures (with nominal electrical angles of 90 for all line lengths): (a) basic unit, (b) two-section, (c) three-section, and (d) four-section.

also been re-configured to function as crossovers [6], but these inter-digitated structures require wire bonding and fabrication precision. We revert instead to the multisection branch-line structure, as outlined by Wight et al. [14], in our effort to design microstrip couplers for crossover application. The two-section prototype fabricated by Wight et al. [14] yields a measured bandwidth of 10% with its isolation meeting the 20-dB specification and its return loss better than 18 dB at the center frequency. Kolodniak et al. [15] have experimented with other multisection structures and their window-shaped prototypes yield measured bandwidths of 10% with the isolation meeting the 20-dB target, but the return loss varying between 10–15 dB. For our design optimization in the present paper, we have chosen to impose the 20-dB targets on both isolation and return loss over a bandwidth exceeding the 20% limit faced by the ring-based prototypes tested in [12] and [13]. II. PRELIMINARY CONSIDERATIONS Depicted in Fig. 1 are the structures that we considered in detail. Following the formulation furnished by Kolodniak et al. [15], we have employed the analytical procedure based on the matrices to repodd and even eigenmode approach with resent the various sections, as explained in [16]. We have additionally found the symbolic functional capabilities of MATLAB helpful in facilitating our analysis. Our analytical findings have indicated that it is not possible for the basic branch-line structure sketched in Fig. 1(a) to yield the ideal crossover characteristics listed in (1). If, however, we in (1) and allow for indo not insist on stead, we are then able to derive the following crossover design

0018-9480/$26.00 © 2010 IEEE

88

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 2. Simulation plots of scattering-coefficient magnitudes for one-section and Z Z . coupler depicted in Fig. 1(a) where Z

= 50

=

= 10

condition for the single-section structure based on a return loss of 20 dB: (2) The isolation of the single-section coupler will naturally be adversely affected when is not strictly imposed. Since it is evident from the simulation results plotted in Fig. 2 that the bandwidth of the single-section design is limited, our next step is to consider the two-section coupler sketched in Fig. 1(b). There are no design equations provided by Wight et al. [14] for the two-section structure, and we thus have to derive them based on the ideal-case specifications listed in (1) and

for two-section structure (3)

Fig. 3 presents the simulation plots we generated for a twosection coupler, which has been designed in accordance with (3) for crossover application. Although the return loss is better than 20 dB over a bandwidth of nearly 20%, the plot in Fig. 3(a) shows the isolation between Port 1 and Port 2 deteriorating rapidly away from the center frequency. As a result, the insertion-loss plot is not sufficiently flat as can be seen from the variation of with frequency in Fig. 3(b).

Fig. 3. Simulation plots of scattering-coefficient magnitudes for two-section Z Z Z . coupler depicted in Fig. 1(b) where Z

=

=

=

= 50

In order to meet our design targets for both isolation and return loss, we have found it necessary to opt for the threeand four-section structures sketched in Fig. 1(c) and (d), respectively. By extending our analysis to include additional sections, we have been able to derive the following design equations based on the crossover specifications listed in (1): and

for three-section structure

(4)

and and for four-section structure

(5)

We infer from the simulation results plotted in Fig. 4 for the three-section structure that the crossover bandwidth may be increased by incorporating a third section into the composite design in accordance with (4); in addition, we observe from Fig. 4(b) that the insertion-loss plot has become flatter in the process. As demonstrated in Section III, further improvement can be achieved when we progress to the four-section structure based on (5). To facilitate the connection of our coupler to the external cirfor the cuitry, we have chosen the default value of 50

YAO et al.: MICROSTRIP BRANCH-LINE COUPLERS FOR CROSSOVER APPLICATION

89

TABLE I DIMENSIONS FOR THREE-SECTION PROTOTYPE DEPICTED IN FIG. 5(a)

TABLE II DIMENSIONS FOR FOUR-SECTION PROTOTYPE DEPICTED IN FIG. 5(b)

for lowest frequency of requisite bandwidth highest frequency of requisite bandwidth Fig. 4. Simulation plots of scattering-coefficient magnitudes for three-section coupler depicted in Fig. 1(c), where Z Z ,Z , and Z Z .

=

=

= 64

= 50

= 40

lines at the four ports when generating the numerical results presented in Figs. 2–4. For the design optimization performed in Section III, we have found that removing this default constraint will next be allows for further bandwidth improvement and permitted to vary over the range of characteristic impedance values typically expected for microstrip lines. III. EXPERIMENTAL RESULTS The number of equations contained in either (4) or (5) is less than the number of coupler dimensions. Hence, each set of equations permits a range of possible designs and we have resorted to optimization in order to search for the maximum-bandwidth design that meets the 20-dB targets imposed on the return loss and isolation parameters in (1). Another optimization target is for the insertion-loss plot to be maximally flat over the operating bandwidth. There is no need for elaborate search algorithms and the commonly used Newton technique is sufficiently efficient in finding the minimum of the following error function: where

(6)

Spurious effects due to junction parasitics and other hardware imperfections have been taken into consideration. A survey of the numerical results we amassed during the course of our simulation trials indicates that a tolerance limit of 0.1 mm should be sufficient for the fabrication of our prototypes. Although we have incorporated into (6) the possibility of choosing different ), the optimization results weights for (where are found to be satisfactory even when we retain the default set. ting of Listed in Tables I and II are the coupler dimensions returned by the optimization process for the three- and four-section designs, respectively (when implemented in microstrip form on Rogers 4003C substrate with thickness of 0.78 mm, relative permittivity of 3.38, and loss tangent of 0.0027). Depicted in Fig. 5 are our two prototypes (drawn to scale). Since we have allowed during the iterative process to optimize the coufor pler designs, we need to insert step junctions between the lines and the 50- lines connecting the coupler to the external circuitry. For our three-section prototype, the plots presented in Fig. 6 show generally good agreement between the numerical results and the experimental data (taken by the HP8510C network analyzer). The measured isolation and return loss meet their 20-dB targets over a 22% bandwidth (between 2.3–2.9 GHz), which is wider than that observed from the previous plots in Fig. 4 for the preliminary design based on (4) with the default setting of . Varying between 0.3–0.4 dB over the entire bandwidth, the insertion-loss plot in Fig. 6(c) is also flatter when compared with its counterpart in Fig. 4(b).

90

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 5. Printed circuit board (PCB) layout of: (a) three- and (b) four-section prototypes with dimensions listed in Tables I and II, respectively.

It is possible to increase the crossover bandwidth by opting for the four-section design instead, as can be seen from the plots presented in Fig. 7, which similarly show close agreement between the numerical and measured results. Adopting the common 20-dB benchmark for both isolation and return loss, we infer from the measured plots that our four-section prototype is suitable for crossover application over a 33% bandwidth (between 2.2–3.1 GHz) where the insertion loss remains constant at 0.4 0.05 dB and the group delay varies between 8.5–10 ns. In comparison, we note that the various microstrip couplers reported in [14] and [15] have been designed to function as crossovers over bandwidths of only 10%. Actually, there are no results presented in [15] for the two-section coupler, as the focus of Kolodniak et al.is on the window-shaped prototypes, which, however, do not meet the 20-dB target for the return loss over the entire bandwidth. It should be pointed out that other implementations are possible for the cascaded branch-line couplers as well, e.g., the rectangular-coaxial [3] and slot-line [4] versions have recently been designed for crossover application, but their return-loss plots also do not satisfy the 20-dB specification over the entire bandwidth.

Fig. 6. Predicted and measured results for three-section prototype depicted in Fig. 5(a) with dimensions listed in Table I.

IV. ADDITIONAL EXTENSIONS The progressive improvements we observed for the results obtained thus far for the one-, two-, three-, and four-section designs indicate the possibility of further enhancement by incorporating additional sections into the branch-line structure. The design data we subsequently derived for the five-, six-, and

seven-section designs are listed in Table III (where, for the purpose of completeness, we also reproduce the corresponding data for the structures considered in Sections II and III). We infer from the bandwidth data presented in the bottom row of Table III that a steady improvement can be expected,

YAO et al.: MICROSTRIP BRANCH-LINE COUPLERS FOR CROSSOVER APPLICATION

91

TABLE III KEY COUPLER PARAMETERS OF MULTISECTION BRANCH-LINE DESIGNS FOR CROSSOVER APPLICATION

size has become unacceptably large for use in microwave integrated circuits with space limitations. If there is a need to em, it is suggested that some ploy the -section design with appropriate size-reduction technique be attempted (such as the approaches proposed in [5] or [17]). V. CONCLUSION Although commonly employed as a quadrature coupler in microwave integrated circuits, the microstrip branch-line coupler may also be redesigned for crossover application, but the bandwidth of the two-section prototype reported in [14] is only 10%. We have shown that the crossover bandwidth of such a coupler can be increased by opting for the three- or four-section design. The results we obtained from simulations and measurements have confirmed that the return-loss and isolation parameters of our three- and four-section prototypes meet their 20-dB targets over bandwidths of 22% and 33%, respectively, with the insertion loss and group delay varying by less than 0.05 dB and 1 ns, respectively. Our simulation results have also indicated that additional bandwidth improvements can be gained by extending to five-, six-, and seven-section designs. However, such couplers may not be readily adopted by circuit designers because of size constraints (unless size-reduction techniques such as those proposed in [5] or [17] are attempted as well). REFERENCES [1] T. S. Horng, “A rigorous study of microstrip crossovers and their possible improvements,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 9, pp. 1802–1806, Sep. 1994. [2] S. Gruszczynski and K. Wincza, “Broadband 4 4 Butler matrices as a connection of symmetrical multi-section coupled-line 3-dB directional couplers and phase correction networks,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 1, pp. 1–9, Jan. 2009. [3] Y. Wang, M. Ke, and M. J. Lancaster, “A -band Butler matrix with antenna array based on micro-machined rectangular coaxial structures,” in Eur. Microw. Conf. Dig., 2009, pp. 739–742. [4] T. A. Denidni and N. Nedil, “Experimental investigation of a new Butler matrix using slot-line technology for beamforming antenna arrays,” in Proc. IET Microw. Antennas Propag., Jul. 2008, vol. 2, no. 7, pp. 641–649.

2

Fig. 7. Predicted and measured results for four-section prototype depicted in Fig. 5(b) with dimensions listed in Table II.

e.g., the simulation results show that the seven-section design is capable of yielding a 55% bandwidth (based on 20-dB targets for both isolation and return loss), but the overall coupler

Ka

92

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

[5] W. L. Chen, G. M. Wang, and C. X. Zhang, “Fractal-shaped switchedbeam antenna with reduced size and broadside beam,” Electron. Lett., vol. 44, no. 19, pp. 1110–1111, Sep. 2008. [6] T. N. Kaifas and J. N. Sahalos, “On the design of a single-layer wideband Butler matrix for switched-beam UMTS system applications,” IEEE. Antennas Propag. Mag., vol. 48, no. 6, pp. 193–204, Dec. 2006. -band 32 [7] A. S. Liu, H. S. Wu, C. K. C. Tzuang, and R. B. Wu, “ GHz planar integrated switched-beam smart antenna,” in IEEE MTT-S Int. Microw. Symp. Dig., 2005, pp. 565–568. [8] M. Bona, L. Manholm, J. P. Starski, and B. Svensson, “Low-loss compact Butler matrix for a microstrip antenna,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 9, pp. 2069–2075, Sep. 2002. [9] S. C. Wu, H. Y. Yang, N. G. Alexopoulos, and I. Wolff, “A rigorous dispersive characterization of microstrip cross and T junctions,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 12, pp. 1837–1844, Dec. 1990. [10] F. Tefiku and E. Yamashita, “Improved analysis method for multi-port microstrip annular-ring power dividers,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 3, pp. 376–382, Mar. 1994. [11] F. C. de Ronde, “Octave-wide matched symmetrical reciprocal fourand five-ports,” in IEEE MTT-S Int. Microw. Symp. Dig., 1982, pp. 521–523. [12] Y. Chen and S. P. Yeo, “A symmetrical four-port microstrip coupler for crossover application,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 11, pp. 2434–2438, Nov. 2007. [13] Y. C. Chiou, J. Y. Kuo, and H. R. Lee, “Design of compact symmetric four-port crossover junction,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 9, pp. 545–547, Sep. 2009. [14] J. S. Wight, W. J. Chudobiak, and V. Makios, “A microstrip and stripline crossover structure,” IEEE Trans. Microw. Theory Tech., vol. MTT-24, no. 5, p. 270, May 1976. [15] D. Kholodniak, G. Kalinin, E. Vernoslova, and I. Vendik, “Wide-band 0-dB branch-line directional couplers,” in IEEE MTT-S Int. Microw. Symp. Dig., 2000, pp. 1307–1310. [16] J. Reed and G. J. Wheeler, “A method of analysis of symmetrical four port networks,” IRE Trans. Microw. Theory Tech., vol. MTT-4, no. 4, pp. 246–252, Oct. 1956. [17] K. W. Eccleston and S. H. M. Ong, “Compact planar microstripline branch-line and rat-race couplers,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 10, pp. 2119–2125, Oct. 2003.

Ka

Jijun Yao received the B.S. and M.S. degrees from the Huazhong University of Science and Technology, Wuhan, China, in 1996 and 1999, respectively, and the Ph.D. degree from the National University of Singapore, Singapore, in 2009. He was involved with RF circuit design for four years with Huawei Technologies, Shanghai, China. For three years he was involved with power amplifier (PA) designs with ST Electronics. He is currently a Research Fellow with the Institute for Infocomm Research, A*Star, Singapore. His research interests focus on terahertz, RF passive circuit, and PA design.

Cedric Lee received the Bachelor’s degree in electrical and computer engineering (with a specialization in microwave engineering) from the National University of Singapore, Singapore, in 2009. He then joined the DSO National Laboratory, Singapore. His research interests include planar circuits and antennas.

Swee Ping Yeo received the M.A. degree from University of Cambridge, Cambridge, U.K., in 1981, and the Ph.D. degree from the University of London, London, U.K., in 1985. Following three years with the Singapore Ministry of Defense, he joined the National University of Singapore, Singapore, where he is currently a Professor with the Electrical and Computer Engineering Department. His research interests include electromagnetic modeling, passive components, and six-port reflectometers. Dr. Yeo was the recipient of three Best Paper Awards of the Institution of Electrical Engineers (IEE), U.K.—two for the Electronics Letters Premium in 1985 and one for the Ambrose Fleming Premium in 1988. He was also the recipient of the Outstanding University Researcher Award of the National University of Singapore in 1998.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

93

Modeling the Effects of Interference Suppression Filters on Ultra-Wideband Pulses Adrian Eng-Choon Tan and Karumudi Rambabu

Abstract—In this paper, we study the effects of interference suppression filters (multiple stopbands) on ultra-wideband (UWB) short pulses. Multiple stopbands of the filter, which is placed in the receiver, affect the UWB pulse amplitude, pulsewidth, and ringing; resulting in reduced signal-to-noise ratio and degradation in receiver performance. An analytical method is proposed in this paper to model the effect of stopbands on the pulse shape. In this paper, the effects of an interference suppression filter on three different Gaussian pulses are studied. Theoretical and measured pulses are compared and found to be in good agreement. The proposed method enables the UWB radio designer to predict the distortion of the received signals after passing through interference suppression filters. Index Terms—Multiple stopband filter and ultra-wideband (UWB) short pulses, narrowband interference cancellation, UWB communications.

I. INTRODUCTION LTRA-WIDEBAND (UWB) communication and radar systems operate in the frequency range as low as 1 GHz to as high as 10 GHz with specified power levels. The specified frequency spectrum also accommodates numerous narrowband applications, such as Global System for Mobile Communications (GSM) at 1.8/1.9 GHz; wireless local area network (WLAN) at the industrial, scientific, and medical (ISM) bands, at 2.4/5 GHz; worldwide interoperability for microwave access (WiMAX) at 2 to 6 GHz; and other systems in the ISM bands; along with different types of UWB applications. The operating power levels of the narrowband applications are much higher than the allowed UWB emission, which is limited to average effective isotropic radiated power (EIRP) of 41.3 dBm/MHz between 3.1–10.6 GHz [1]. One of the key issues of the UWB technology is that UWB systems should not interfere with the existing narrowband applications. However, narrowband applications can interfere with the UWB systems [2]. The short pulses used in the UWB systems have power spectral densities that occupy a wide frequency range. The exact power spectral density of a particular UWB system depends on the pulse duration, pulse shape, antenna effect [3], [4], and pulse repetition frequency.

U

Manuscript received February 08, 2010; revised August 24, 2010; accepted September 16, 2010. Date of publication November 29, 2010; date of current version January 12, 2011. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G2R4 (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2090418

Multiple stopband filters [5]–[10] and antennas with notched frequency characteristics [11] are proposed to suppress the interference of the narrowband transmitters onto UWB systems. The effect of stopbands of the filter in the receiver, which removes energy from the spectrum of the UWB pulse, needs to be studied for optimal receiver design. The distortion on the pulse depends on the number of stopbands, attenuation in the stopband, overall insertion loss, and group delay of the filter. The parameters of the pulse that are expected to get affected are pulse amplitude, pulsewidth, and ringing. In the case of multiple stopband filters [12], it is difficult to predict the filter’s effect on the UWB pulse. To minimize pulse distortion, the interference suppression filter should have narrow stopband bandwidth and high rejection in its stopband. Besides that, the group delay of the filter in the passband should be approximately constant. If the group delay is approximately constant in the passband, pulse distortion will be strongly dependent on the characteristics of the stopband. In this paper, we propose an analytical model that can estimate the effect of multiple stopband filter on UWB pulses. This model is developed using two parameters—stopband frequency and bandwidth. Using this model, pulse distortion and ringing caused by the multiple stopbands on received UWB pulse can be estimated. This analysis helps UWB system designers to predict the characteristics of the received pulse. If the estimated received pulse does not meet the required specifications of the system design, the designer can optimize the filter parameters or the transmitted pulse to improve the received pulse characteristics. Section II presents the theoretical analysis that establishes a relationship between the individual notch and its effect on the UWB pulse. Section III studies the effect of different notches on a UWB pulse. In Section IV, the time-domain response of the multiple-stopband filter is measured and compared with the theoretical response to validate the proposed analysis. II. THEORY A. Modeling the Multiple Stopband Filter in Laplace Domain Let the multiple-stopband filter used for narrowband interference suppression in UWB receivers have stopbands. The transfer function of the filter can be modeled in the Laplace domain as (1) where

is the insertion loss of the filter in the passband and is the profile of the stopbands. is the coordinate of the . Laplace plane, where

0018-9480/$26.00 © 2010 IEEE

94

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE I NOTCH PARAMETERS OF THE MULTIPLE STOPBAND FILTER

where [13]

can be modeled as a Gaussian function of variance , is the frequency at which the insertion loss is 4.34 dB (2)

is modeled as a product of the transfer functions of individual notches. Each notch is modeled as a second-order bandstop filter, which is a function of two zeros and two poles. axis, while the poles The two zeros are placed on the [14]. The center are placed near the zeros, at frequency of the th notch, , is fixed by the position of the two zeros. The bandwidth of the th notch, , is determined by the and . proximity of the poles to the zeros, i.e., Thus, a multiple-stopband filter with number of notches can be modeled, in the Laplace domain, as the multiplication of second-order bandstop filters

Fig. 1. Comparing the measured filter magnitude response [see Fig. 1(a)] and group delay [see Fig. 1(b)], Transmission—S (solid line), and the derived transfer function of the notches (dotted line).

(3) The bandwidth of the th notch,

, is related to

as (4)

Frequency response of a multiple stopband filter can be completely constructed using (3) and (4). To demonstrate the proposed analysis, we consider the filter design that has been proposed in [12]. Table I shows the center frequencies and bandwidths of the six notches of the multiple stopband filter of the filter presented in [12]. Fig. 1 compares the measured . (solid line) with the transfer function of the notches, is modeled based on the data presented in Table I. Here The measured of the filter (solid line, Fig. 1) shows the insertion loss, which increases with frequency. The insertion with (12.7 GHz). In loss can be modeled by , of the multiple Fig. 2, the theoretical frequency response, stopband filter is fully modeled (dotted line), and it is compared (solid line). with measured

Fig. 2. Comparing the measured filter response, Transmission—S (solid line), the derived insertion loss due to conductor and dielectric losses (dashed line), and the composite transfer function of the notches with the insertion loss (dotted line).

B. Modeling the Multiple Stopband Filter in Time Domain

, the function that The inverse Laplace transform of models the insertion loss of the multiple stopband filter, is

The time-domain impulse response of a stopband filter can be derived by inverse Laplace transform of its frequency response. Reference [14] shows that the inverse Laplace transform of a bandstop filter can be written as

(5)

where and . As mentioned in Section I, the notches of a narrowband interference rejection filter should be narrow and have high rejection in its stopband, i.e., percentage bandwidth of the notch is small . For this case, the following approximations can be made: and . Equation (5) can be simplified as (6)

(7) Hence, the impulse response of the multiple stopband filter can be approximated as (8)

TAN AND RAMBABU: MODELING THE EFFECTS OF INTERFERENCE SUPPRESSION FILTERS ON UWB PULSES

95

Fig. 4. Comparison between the measured and theoretical impulse response of the multiple notches of the bandstop filter.

III. EFFECT OF BANDSTOP FILTER ON UWB PULSES

Fig. 3. Time- and frequency-domain impulse responses of the notches: N1 : GHz, b : GHz, MHz, in Fig. 3(a) and (b)]; N3 [f [f : GHz, b b MHz, in Fig. 3(c) and (d)]; N4 [f MHz, in Fig. 3(e) and (f)], and N6 [f : GHz, b MHz, in Fig. 3(g) and (h)].

= 1 84 = 191

= 96

= 9 17

= 5 51 = 795

= 3 32 = 317

In this section, we will evaluate the effect of bandstop filter on the UWB pulses. The response of the bandstop filter on UWB pulses can be computed by convolving the modeled filter impulse response, i.e., (8), with the input UWB pulses. To represent a range of UWB pulses, we consider three different pulses: —the Gaussian pulse; —the first derivative Gaussian —the second derivative Gaussian pulse pulse; and (9)

Validity of (6) can be confirmed with the following check. First, specify the bandwidth and center frequency of a notch. Then compute its impulse response using (6). The Fourier transform of the derived impulse response is computed numerically. Successful comparison of frequency characteristics of the impulse response with specified characteristics confirms the validity of (6). Fig. 3 shows the validation of (6) at four different notches with specifications obtained from Table I: GHz, MHz, in Fig. 3(a) and (b); GHz, MHz, in Fig. 3(c) and GHz, MHz, in Fig. 3(e) and (f); (d); GHz, MHz, in Fig. 3(g) and (h). The center frequency and bandwidth of the numerically computed frequency responses are accurate to well within 1 MHz of the specifications. This verifies the approximation involved in (6). The impulse response of the multiple stopband filter, specified in Table I, is computed using (8). In Fig. 4, the modeled impulse response is compared with the measured impulse response. The measured impulse response is obtained by inverse Fourier transform of the measured of the multiple stopband filter [12]. Comparison of the theoretical and measured impulse responses in Fig. 4 shows that there is slight difference in the amplitude and pulsewidth of the main pulse; while the amplitude and duration of ringing for both the responses are similar. This comparison validates the accuracy of (8) as an impulse response for the multiple stopband filter that can predict the pulse distortion.

(10) (11) where and scale the pulses to the specified amplitudes, and and control the width of the pulses. Gaussian pulses can be generated with the techniques suggested in [15]–[18]. Due to the circuit imperfections, the generated Gaussian pulses are always associated with ringing. The solid lines in Fig. 5 show the generated Gaussian pulses using the circuit presented in [15]. These pulses are mathematically modeled using (9)–(11) with the parameters ps, ps, and ps. The measured (solid line) and modeled (dotted line) Guassian pulses are compared in Fig. 5. For the case of the Gaussian pulse, there is slight difference between the modeled and measured pulse due to the presence of ringing. The spectral content of the UWB pulses and be has been shown in Fig. 6. Let , , and the spectral representation of the UWB pulses , respectively. Theoretical and measured , , are compared in Fig. 6. It is observed from Fig. 6 and that the 10-dB bandwidth of the Gaussian pulse is 0–10 GHz; first derivative Gaussian pulse: 1–14 GHz; and second derivative Gaussian pulse 3.5–15.5 GHz. The bandwidths of the UWB pulses encompass all the notches of the multiple-stopband filter in [12]. Let the response of the multiple stopband filter for input and be and , respectively. The output pulses and can be derived as a convolution of the pulses

96

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 5. Measured (solid line) and the theoretical (dotted line) UWB pulses (Gaussian pulse, first derivative Gaussian pulse and second derivative Gaussian pulse) in time domain.

Fig. 7. Gaussian first derivative of Fig. 5 is fed into notch filters with a single notch with the same center frequency ( = 6 4 GHz), but increasingly large bandwidth ( ). The output signals from the filters are as follows: = 320 MHz (dotted line); = 640 MHz (dashed line); = 1 28 GHz (line).

b

f

b

:

b

:

b

A. Effect of Notch Bandwidth on UWB Pulse In this section, we study the effect of increasing notch bandwidth on the UWB pulse. For illustration, the notch frequency of the bandstop filter is chosen at 6.4 GHz, and of bandwidths 320 MHz, 640 MHz, and 1.28 GHz. The source signal considered here is the first derivative Gaussian pulse (10), which has a center frequency of 6.4 GHz and 10-dB bandwidth of 1–14 GHz. The filter output is computed by convolving (6) and (10) with the above specified parameters, and are plotted in Fig. 7. Fig. 7 shows that for increased notch bandwidth, there is a corresponding decrease in the amplitude of the pulse and increased ringing at the tail end of the pulse. However, the length of the ringing gets shorter as the notch bandwidth is increased. B. Effect of Notch Frequency on UWB Pulse

G ! ;G ! G ! G !

Fig. 6. Theoretical plots of ( ) ( ) and ( ) (solid lines) versus gen( )—“x”; ( )—“o”; And ( )—“1.” erated

G !

G !

input pulses with the impulse response of the multiple stopband in (8) can be combined filter (8). As a further simplification, and , resulting in with the input pulses

(12)

(13)

(14) where

has been defined in (6).

In this section, we study the effect of notch frequency on the UWB pulse. Again, the same source signal is considered, which is the first derivative Gaussian pulse with center frequency of 6.4 GHz and 10-dB bandwidth of 1–14 GHz. The notch’s bandwidth is chosen to be 10%, and the notch frequency is chosen to be 2.04, 6.4, and 12.37 GHz. The center frequency of the notch at 6.4 GHz matches to the center frequency of chosen input pulse, while 2.04 and 12.37 GHz corresponds to the frequencies at which the normalized pulse spectral amplitude is at 6 dB. The filter output is shown in Fig. 8, which has been computed by convolving (6) and (10) with the above specified parameters. Here, the ringing of the output pulse is significant for the notch frequency of 6.4 GHz. The pulse amplitude was affected more for the notch frequency at 12.37 GHz. The length of ringing is shortest when the notch frequency is 12.37 GHz, while it is longest at 2.04 GHz. These studies show the effect of notch center frequency and bandwidth on the input UWB pulse. Thus, these results can be used in optimizing the interference suppression filter. IV. MEASUREMENTS AND RESULTS Time-domain experiments are conducted to measure the source pulse and its filtered output. Fig. 9 shows the exper-

TAN AND RAMBABU: MODELING THE EFFECTS OF INTERFERENCE SUPPRESSION FILTERS ON UWB PULSES

Fig. 8. First derivative Gaussian pulse of Fig. 5 is fed into notch filters with a single notch with the same bandwidth of 10%, but decreasing center frequencies (f ). The output signals from the filters are as follows: f = 2:04 GHz (dotted line); f = 6:4 GHz (dashed line); f = 12:37 GHz (solid line).

Fig. 9. Schematics of measurement setup to measure the time-domain response of the multiple stopband filter.

97

Fig. 11. Comparison between the measured (solid line) and derived (dotted line) filter response for Gaussian first derivative input signal.

Fig. 12. Comparison between the measured (solid line) and derived (dotted line) filter response for Gaussian second derivative input signal.

UWB source signals that have been modeled in (9)–(11). The measurement procedure has been shown in Fig. 9. The input pulse is measured with a synchronously triggered 50-GS/s digital sampling oscilloscope. Different pulse forming network configurations are used to achieve the different Gaussian pulses, and have been shown in Fig. 5. The output pulses are measured at the output port of the multiple stopband filter. In Figs. 10–12, the measured output pulses (solid line) are compared with the derived output pulses (dotted line) using (12)–(14). It is clearly evident that the proposed mathematical model accurately predicts the effect of multiple stopband filter on UWB pulses. However, there are some slight differences in pulse shape for the case of the first and second derivative Gaussian pulse. The amplitude, duration, and trend of the ringing at the tail-end of the pulse are also accurately predicted by the model. Fig. 10. Comparison between the measured (solid line) and derived (dotted line) filter response for Gaussian pulse input signal.

V. CONCLUSION

imental setup: a 45-ps step function generator is used as an input step to the pulse forming networks [15] to form the

In the design of UWB radio, the knowledge of received pulse shape is essential for optimal receiver design. Interference suppression filter modifies the shape of the received pulse. The mathematical model proposed in this paper is able to predict

98

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

the pulse distortion due to the filter. The model can also be used as an instructive tool to study the effect of stopbands on UWB pulses. REFERENCES [1] “First report and order regarding the revision of part 15 of the Commission’s rules regarding ultra-wideband transmission systems,” FCC, Washington, DC, FCC ET Docket 98-153, Apr. 2002. [2] L. Zhao and A. M. Haimovich, “Performance of ultra-wideband communications in the presence of interference,” IEEE J. Sel. Areas Commun., vol. 20, no. 9, pp. 1684–1691, Dec. 2002. [3] K. Rambabu, A. E.-C. Tan, K. K.-M. Chan, and M. Y.-W. Chia, “Estimation of antenna effect on ultra-wideband pulse shape in transmission and reception,” IEEE Trans. Electromagn. Compat., vol. 51, no. 3, pp. 604–610, Aug. 2009. [4] K. Rambabu, A. E.-C. Tan, K. K.-M. Chan, and M. Y.-W. Chia, “Experimental verification of link loss analysis for ultra wideband systems,” IEEE Trans. Antennas Propag., Jul. 2010, accepted for publication. [5] W. Menzel and P. Feil, “Ultra-wideband (UWB) filter with WLAN notch,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 595–598. [6] M. Mokhtaari and J. Bornemann, “Ultra-wideband and notched wideband filters with grounded vias in microstrip technology,” in Asia–Pacific Microw. Conf., Dec. 16–20, 2008, pp. 1–4. [7] J. Wang, H.-J. Tang, Y. Zhang, Y.-D. Dong, and K. Wu, “UWB bandpass filter with multiple frequency notched band,” presented at the IEEE MTT-S Int. Microw. Workshop Series on Art of Miniaturizing RF Microw. Passive Compon., Chengdu, China, Dec. 2008. [8] Z.-C. Hao and J.-S. Hong, “Compact UWB filter with double notchbands using multilayer LCP technology,” IEEE Microw. Compon. Lett., vol. 19, no. 8, pp. 500–502, Aug. 2009. [9] A. Vallese, A. Bevilacqua, C. Sandner, M. Tiebout, A. Gerosa, and A. Neviani, “Analysis and design of an integrated notch filter for the rejection of interference in UWB systems,” IEEE J. Solid-State Circuits, vol. 44, no. 2, pp. 331–343, Feb. 2009. [10] Z.-C. Hao, J.-S. Hong, J. P. Parry, and D. P. Hand, “Ultra-wideband bandpass filter with multiple notch bands using nonuniform periodical slotted ground structure,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 12, pp. 3080–3088, Dec. 2009. [11] Y.-Y. Yang, Q.-X. Chu, and Z.-A. Zheng, “Time-domain characteristics of band-notched ultrawideband antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3426–3430, Oct. 2009. [12] K. Rambabu, M. Y. W. Chia, K. M. Chan, and J. Bornemann, “Design of multiple-stopband filters for interference suppression in UWB applications,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, pp. 3333–3338, Aug. 2006. [13] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Washington, DC: Nat. Bureau Standards, 1972, ch. 26.

[14] R. W. Newcomb, “Analysis in the time domain,” in The Circuits and Filters Handbook, R. C. Dorf, Ed., 2nd ed. Boca Raton, FL: CRC, 2003, ch. 25. [15] J. R. Andrews, “Picosecond pulse generators for UWB radars,” Picosecond Pulse Labs., Boulder, CO, Appl. Notes AN-9, May 2000. [16] E. K. Miller, Time-Domain Measurements in Electromagnetics. New York: Van Nostrand, 1986. [17] J. Han and C. Nguyen, “A new ultra-wideband, ultra-short monocycle pulse generator with reduced ringing,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 6, pp. 206–208, Jun. 2002. [18] A. E.-C. Tan, M. Y.-W. Chia, and S.-W. Leong, “Sub-nanosecond pulse-forming networks on SiGe BiCMOS for UWB communications,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1019–1024, Mar. 2006. Adrian Eng-Choon Tan received the B.Eng. and Ph.D. degrees from the National University of Singapore (NUS), Singapore, in 2002 and 2008, respectively. From 2008 to 2009, he was with the Institute for Infocomm Research (I R), as a Research Engineer. He is currently a Postdoctoral Fellow with the University of Alberta, Edmonton, AB, Canada. His research interests include microwave circuits, time-domain analysis and UWB transceiver systems. Mr. Tan was a recipient of the Agency for Science, Technology and Research (A-STAR) Graduate Scholarship.

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

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

99

A Distributed Dual-Band LC Oscillator Based on Mode Switching Guansheng Li, Student Member, IEEE, and Ehsan Afshari, Member, IEEE

Abstract—In this paper, we present a distributed dual-band LC oscillator suitable for low-phase-noise applications. It switches between the odd and even resonant modes of a fourth-order LC resonator. In contrast to other switched-resonator designs, the switches used for mode selection do not carry current, and therefore, do not affect the quality factor of the resonator, which leads to low phase noise. Analysis shows it achieves the same phase-noise figure-of-merit (FoM) as a single-band LC oscillator that uses the same inductor and active core. This was verified by a prototype in a 0.13- m CMOS process. It draws a current of 4 mA from a 0.5-V power supply and achieves a FoM of 194.5 dB at the 4.9-GHz band and 193.0 dB at the 6.6-GHz band, which is the same as the reference standalone LC oscillator. There is good agreement among theory, simulation, and measurement results.

Fig. 1. Schematic of the proposed dual-band oscillator.

Index Terms—Dual-band oscillator, high-order LC resonator, low phase noise, low supply voltage.

I. INTRODUCTION

W

IRELESS design is developing from single-mode to multimode systems that support multiple standards at several frequency bands. A major challenge for these systems is to design local oscillators (LOs) that cover the wide spectrum and meet the stringent phase-noise requirement. This is usually beyond the capability of a single-tank LC voltage-controlled oscillator (VCO) using varactors for continuous frequency tuning. As a result, there has been an increasing interest in LC oscillators that switch between multiple frequency bands [1]–[12]. One multiband scheme is to implement multiple LC oscillators at different frequencies and enable one of them at a time [1]. However, this scheme can be improved, in the sense that there are always one or more inductors in idle. As will be explained later, these idle inductors could have been utilized to enhance the working LO’s phase noise. Another technique is to use a switched resonator, in which the inductance and capacitance of the LC resonator are controlled by MOS switches [2]–[4]. However, these switches usually insert resistance to critical current paths, which degrades the resonator’s quality factor and deteriorates phase noise significantly [5], [6]. Several

Manuscript received May 28, 2010; revised October 15, 2010; accepted October 22, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. This work was supported by the National Science Foundation (NSF) under NSF Grant DMS-0713732. The work of E. Afshari was supported by the NSF under Early Career Award ECCS-0954537. The authors are with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14850 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2091203

other structures were also proposed [7]–[11], and [12] summarized existing wideband/multiband VCO techniques. However, the phase-noise performance of these state-of-the-art multiband oscillators is generally inferior to single-band LC oscillators [13], [14], which demonstrates the tradeoff between phase noise and frequency tuning range in oscillator design. In this paper, we propose a dual-band LC oscillator, which achieves band switching without impairing phase-noise performance. As shown in Fig. 1, two center-taped inductors and four capacitors comprise a differential resonator. It has an even and an odd resonate mode. Their frequencies are not harmonically related. A switching network is used to select one from the two resonant modes, and two p-type field-effect transistor (PFET) pairs are used to compensate the resonator’s energy loss and sustain oscillation. Thus, one can get dual-band LO output from either PFET pair. In contrast to other switched-resonator designs, there is no current going through the switches during steady oscillation because the switches that are turned on only damp the undesired mode. As a result, the working-mode quality factor of the resonator is not affected by the switches, and the oscillator can achieve low-phase noise. Analysis shows, this dual-band oscillator achieves the same phase-noise figure-of-merit (FoM) as a single-band LC oscillator that uses the same inductor and active core (Fig. 2), where the FoM is defined as [15]

(1) in which is the center frequency, is the offset frequency, is the phase noise in dBc/Hz, and is the power consumption in mW. This analysis was verified by a prototype in a 0.13- m CMOS process, in which the dual-band oscillator achieves 3-dBc/Hz lower phase noise while consuming 3-dB

0018-9480/$26.00 © 2010 IEEE

100

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 2. Schematic of a single-band LC oscillator.

more power, compared to a reference single-band LC oscillator using the same inductor and active core. Compared to the two-oscillator scheme, the proposed dual-band oscillator can be considered as two oscillators coupled through capacitors, which readily achieves 3-dBc/Hz lower phase noise than the two independent single-band oscillators. Phase-noise reduction by coupling multiple oscillators has been recognized in the RF and microwave circuit design community [16]–[18]. The phase noise was proved in theory if oscillators are ideally coupled [16]. to scale down as However, coupling is far from trivial in real circuit design, and various structures were proposed [16]–[18]. Along this line, the proposed dual-band oscillator demonstrates the first ideally coupled structure with band switching capability. In a sense, the two inductors, which never idle, are better utilized to enhance phase noise than in the two-oscillator scheme. Moreover, the proposed dual-band structure provides an alternative way of trading power for phase noise, and can achieve low phase noise that is impossible for a single-tank LC oscillator (Fig. 2). For a single-tank LC oscillator, this power versus phasenoise tradeoff is often done by halving the tank impedance and doubling the power consumption, which lowers the phase noise by half [18], but in low-supply-voltage applications, this conventional way may lead to impractically small inductance to meet the stringent phase-noise specifications [18]. In this case, the proposed structure can be used to reduce the phase noise by another 3 dBc/Hz without further scaling the inductance to impractical values. Besides, its band-switching capacity eases the tradeoff between phase noise and frequency tuning. In the remainder of this paper, we will start with the operation of the proposed oscillator in Section II. In Section III, we will discuss its phase-noise performance. In Section IV, we will verify our analysis by a prototype and present measurement results. In closing, we will present a conclusion in Section V. II. OPERATION OF THE DUAL-MODE LC OSCILLATOR As illustrated in Fig. 3, the proposed oscillator consists of two parts: an LC resonator and a trans-conductance network. and , respectively, these two parts form a Denoted by feedback loop. We will study the oscillation condition of the oscillator using this feedback network model. A. LC Resonator Since we use center-taped symmetric inductors and crossconnected differential pairs as an active core, as shown in Fig. 1, the oscillator only works differentially. That is, the voltages at the two terminals of each inductor are opposite in sign. Thus, we are only interested in the resonator’s differential modes. Analysis shows it has two resonant modes: an even mode and an odd mode. Instead of an abstract mathematical derivation, we introduce the two modes in an intuitive way.

Fig. 3. Dual-mode LC oscillator. (a) Physical model. (b) Redrawn physical model as two-port networks, i.e., an LC resonator and a trans-conductance network. (c) Mathematical model.

1) In the even mode, the two LC tanks resonate in phase, i.e., . As illustrated in Fig. 4, the capacitors see a zero voltage drop and do not carry current. Thereby, the resonator can be reduced to two LC tanks. The resonant frequency is easily found to be (2) in which the subscript stands for “even mode.” 2) In the odd mode, the two LC tanks are 180 out of phase, i.e., . As illustrated in Fig. 5, the capacitor sees differential voltage at its two terminals. Thereby, up in it is virtual ground at the center of . Breaking the middle, we get the equivalent circuit on the right side of Fig. 5, which consists of two LC tanks. Thus, the resonant frequency is easily found to be (3) in which the subscript stands for “odd mode.” A complete description of the resonator is to model it as a two-port network, as shown in Fig. 3(b), and use its impedance matrix ( matrix) [19], i.e., (4)

LI AND AFSHARI: DISTRIBUTED DUAL-BAND LC OSCILLATOR BASED ON MODE SWITCHING

101

Fig. 4. Illustration of the even mode of the LC resonator.

Fig. 6. Input impedance of the dual-band LC resonator, with C being tuned. Fig. 5. Illustration of the odd mode of the LC resonator.

and (5), shown at the bottom of this page. The two resonant frequencies are clearly shown by the two terms of each matrix element. For instance, the impedance looking into Port1 , and has two peaks, at when Port2 is open, i.e., and , respectively. As an illustration, we implemented the resonator in a CMOS process. The impedance looking into Port1 is plotted in Figs. 6 and 7. As expected, there are two peaks in each curve, corresponding to the odd and even mode, respectively. The frequencies of the peaks are as predicted by (2) and increases, the odd-mode frequency (3). In Fig. 6, when decreases, whereas the even-mode frequency is not affected; in Fig. 7, when increases, both and decrease. It is worth mentioning that the two frequencies are not harmonically is determined by related, and their ratio .

Fig. 7. Input impedance of the dual-band LC resonator, with C being tuned.

or . Putting everything together, we obtain the two-port network shown in Fig. 3(b), which is described by

B. Trans-Conductance Network As illustrated in Fig. 3, we use a trans-conductance network to model all energy-loss and energy-compensation components in the oscillator. In particular, as in Fig. 3(a): is the parallel conductance of the resonator, and models • the energy loss of the passive components; is the negative conductance of the differential pairs; • is the conductance of switches and in Fig. 1; • • is the conductance of switches and in Fig. 1. or Note that, when a switch turns off, its conductance is zero; when it turns on, or is a positive value

(6) and (7), shown at bottom of the following page. Interestingly, this network also shows even/odd operations. That is, if ap, we plying even voltage to the two ports, i.e., . Thus, each port sees get ; if applying odd an effective conductance of voltage , each port sees an effective conductance . As a result, even and odd modes of the of LC resonator see different energy loss and compensation, which

(5)

102

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

we can use to realize mode/frequency switching. Rigorous analysis is given in Section II-C.

has its poles on the right-half plane. Therefore, only the odd mode can start up. To sum up, the transistors should be sized such that

C. Feedback Network Based on above discussion and the mathematical model in Fig. 3(c), we can derive the response of the oscillator to noise to be current

(8) in which (9)

(10)

From (8), we find even-mode noise can stimulate oscillation at with transfer function and odd-mode noise can with transfer function . Since stimulate oscillation at random noise has both even- and odd-mode components, whether a mode can start up is determined by its transfer or . function and are on, and • In even mode oscillation, switches and are off. Thus, and . Thereby, by sizing transistors such that (11) we can make and . In this case, the even-mode transfer function has on the right-half plane, while the odd-mode its poles has its poles on the left-half transfer function plane. Therefore, only the even mode can start up. and are off, and • In odd-mode oscillation, switches and are on. Thus, and . Thereby, if the transistors are sized such that (12) we can make In this case,

and . has its poles on the left-half plane, while

(13) in order to enable frequency switching, i.e., to guarantee startup of either oscillation mode and damp the other mode. and only damp the unIt is worth noting that wanted mode and do not degrade the working mode quality factor. Take the odd-mode oscillation as an example, in which and . We simulated the input and impedance of the resonator, with changing from 0 to 0.004 . As illustrated in Fig. 8, lowers the even-mode peak, but does not affect the height or width of the odd-mode peak. Intuitively, the even-mode and thus its component imposes a voltage drop across energy is dissipated, but the odd-mode component does not and is not affected. Similarly, in even-mode oscillation, see only damps the odd mode and does not affect the even mode, as illustrated in Fig. 9. Thereby, we can expect good phase-noise performance in the dual-band oscillator. A rigorous analysis of phase noise will be given in Section III. D. Continuous Frequency Tuning In a single-band LC VCO, people use a switched capacitor bank for coarse tune and a varactor for fine tune. However, due to the tradeoff between switch loss and parasitic capacitance, cannot be very large. For instance, [5] demonstrates a state-of-the-art design with a continuous tuning range , which of 3.1–5.2 GHz. This corresponds to includes parasitics from active core and loading, etc. The proposed scheme in Fig. 1 can be used to extend above is implemented as tuning range. For instance, assume switched capacitor and varactor with a tuning range between and ,1 and is the fixed capacitor. Thus, the to odd mode covers a low band from , and the even mode covers a high band to . Since the switches in this from design do not affect the working mode quality factor, they do not need to be extremely wide transistors and thus do not introduce significant parasitic capacitance. Thus, can achieve roughly the same value as the single-band VCO , , and , the above. If even and odd bands have a considerable overlap and cover a . In contrast, in continuous tuning range with conventional single-band VCO, such a tuning range requires 1Parasitic

capacitances of active core and loading are included.

(7)

LI AND AFSHARI: DISTRIBUTED DUAL-BAND LC OSCILLATOR BASED ON MODE SWITCHING

103

and thus varies at the same frequency as LC tank’s state at is actually the ISF function, describing the the oscillator. time-variant phase response to noise [15]. For oscillators with high- LC resonators, the ISF can be well approximated by a sinusoidal function at the same frequency as the oscillator [20]. Assuming the oscillation swing is , one can find the amplitude of the ISF is (14)

Fig. 8. Input impedance of dual-band resonator when

G

changes.

Next, we will consider the dual-band oscillator in Fig. 1. Take odd-mode oscillation as an example. Based on (8) and (10), if a (or ), the induced noise current pulse is injected from . Although the odd-mode voltage jump is current pulse also induces an even-mode component, it decays rapidly and does not change the phase of odd-mode oscillation in a first-order estimate. Thus, assuming the oscillation swing is and by the same reasoning as the single-band oscillator, one can find the amplitude of ISF is (15) If the two oscillators use the same and are at the same frequency, we must have . If also using the same active core and supply voltage, they should have the same voltage swing . Therefore, from (14) and (15), we find (16)

Fig. 9. Input impedance of dual-band resonator when

G

changes.

, which is hardily possible. In an implementation with 8-bit coarse tune bank and varactors in a 65-nm CMOS technology, simulation shows the odd mode covers 2.4–3.2 GHz and the even band covers 3–5 GHz. III. DISCUSSION ON PHASE NOISE Phase noise is a primary concern in oscillator design. Since the proposed dual-band structure can be considered as two coupled single-band oscillators, many analyses and phase-noise reduction techniques proposed for single-band oscillators (e.g., [13], [14], [22]) can be readily adopted here. Thereby, we will focus on the comparison between single- and dual-band oscillator, and demonstrate a 3-dBc/Hz phase-noise improvement. Existing phase-noise theories mainly fall into two categories [23]: frequency- [24] and time-domain techniques [15], [25]. Our analysis is based on the impulse sensitivity function (ISF) theory in [15]. A. ISF We begin with the single-band LC oscillator shown in Fig. 2. If a noise current pulse is injected into the LC tank , there would be a voltage jump of . This at voltage jump induces a phase shift , which depends on the

Intuitively, this result can be explained by the fact that, in the dual-mode oscillator, only half of the injected current pulse induces odd-mode voltage and thus shifts oscillation phase, while the other half induces even-mode component, which is damped by the circuit and does not perturb oscillation phase. To verify this result, we implemented the single- and dualband LC oscillator in a 0.13- m CMOS process and did simulation with SpectreRF. The two oscillators use the same inductor and active core, and the capacitors are set such that they are at the same frequency 5.39 GHz. Fig. 10 shows the simulated voltage waveforms at a terminal of the inductors, along with the ISFs of a noise source taped to the same net. The voltage swings are V, the same for the two oscillators, and considering both work just in the voltage-limited region [21]. The ISFs are at the same frequency as the voltages, and the phase is shown to be most sensitive to noise at voltage zero-crossings [15]. We calculate the root-mean-square (rms) values as the amplitudes of the ISF curves. The ISF of the dual-band oscillator is exactly half of the amplitude of the single-band oscillator. The ISFs of noise sources at other nets also show the same ratio. In the same way, we found the same comparison result for the dual-band oscillator’s even mode, provided the single-band oscillator is set to the same frequency. For the sake of space, we will not repeat the discussion for the even mode here. B. Noise Sources In addition to the ISFs, noise sources are the other factor to determine phase-noise performance. Although the two oscil-

104

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE I SIMULATED PHASE-NOISE CONTRIBUTION AT

1f = 1 MHz

Fig. 10. Simulated ISFs and voltage waveforms of the single- and dual-band oscillator in the odd mode. The two subplots are aligned in time.

Fig. 12. Simulated phase noise of the single-band oscillator and the dual-band oscillator in the odd mode.

Fig. 11. Illustration of switch noise in even mode oscillation.

lators have common noise sources in active cores and inductors, the switches in the dual-band oscillator introduce extra noise sources and need careful checking. Take the even-mode and are on in the oscillation as an example. Switches even-mode oscillation (Fig. 1), and insert two nonzero conduc, as shown in Fig. 11. The noise of the two ’s can tors be modeled as two current sources, which can always be decomposed into its common- and differential-mode components: The common-mode noise is damped by the active cores and the center-tapped symmetric inductors in that they only support differential mode oscillation and the common mode component cannot sustain; the differential-mode noise leads to an odd-mode component in the resonator, which is damped by ’s. In other words, in even-mode oscillation, although the two ’s generate noise, they do not add any even-mode component to the resonator, and thus does not perturb the phase of oscillation. Along the same line, one can find the same result for the odd-mode oscillation. SpectreRF simulation verified this result. Table I lists itemized phase-noise contribution from different parts of the circuits. The switching network accounts for only 0.9% of the total noise and is thus negligible in phase-noise analysis. C. Phase Noise According to the ISF theory of phase noise [15], the phase noise induced by a noise source is proportional to the square of its ISF’s amplitude. Based on the ISF comparison in (16), the

phase noise of the dual-band oscillator, due to one active core of that of the single-band and/or one inductor, is oscillator. On the other hand, the dual-band oscillator has two inductors and two active cores, which introduces twice the noise sources of the single-band oscillator. Therefore, the total phase of the noise of the dual-band oscillator is single-band one, which is a 3-dBc/Hz improvement. Moreover, because the dual-band oscillator has two active cores, it consumes twice the power of the single-band oscillator. Therefore, we conclude the dual- and single-band oscillators have the same FOM, as defined in (1). This was verified by SpectreRF simulation. The single- and dual-band oscillators draw a current of 2 and 4 mA, respectively, from a 0.5-V power supply. We set the dual-band oscillator to be in odd mode, and set the capacitors of the single-band oscillator such that it is at the same frequency. The simulated phase noises are plotted in Fig. 12. As expected, the dual-band oscillator shows 3-dBc/Hz lower phase noise than the single-band oscillator. The same result also holds for a dual-band oscillator working in the even mode and a single-band oscillator at the same frequency. D. Comparison With Other LO Schemes At this point, we can compare the proposed oscillator with other single- and dual-band LO schemes. Compared to a single-band LC oscillator in Fig. 2, the dual-band oscillator can be considered as a coupling structure and provides an alternative way of trading power for phase noise. For a single-band LC oscillator in Fig. 2, the conventional way of reducing phase noise by half is to halve the LC tank’s impedance and double power consumption [18]. However, this method may lead to inductance values too small to be practical,

LI AND AFSHARI: DISTRIBUTED DUAL-BAND LC OSCILLATOR BASED ON MODE SWITCHING

especially when the phase-noise requirement is stringent [18]. In this case, coupling multiple LC oscillators provides an alternative way of trading power for phase noise [16]–[18]. It is proven in theory, for ideally coupled oscillators, the phase while the power increases to times of noise reduces to a single oscillator [16]. Along this line, the proposed oscillator can be recognized as an ideally coupled structure, which can lower phase noise by another 3 dBc/Hz without scaling down inductances to impractical values. Although this 3-dBc/Hz benefit can also be achieved by directly connecting two LC tanks in parallel, the proposed dual-band structure provides an additional band-switching capability, which eases the tradeoff between frequency tuning range and phase noise. Compared to a dual-band scheme with two independent oscillators [1], the dual-band oscillator readily get a 3-dBc/Hz improvement in phase noise by coupling them together. If two independent LC oscillators work alternatively, there is always one inductor in idle, which could have been used to improve phase noise of the working oscillator. In this sense, the proposed dual-band structure makes better use of the inductors, which never idle, to get 3-dBc/Hz lower phase noise, while keeping the capability of synthesizing two frequencies. With respect to other dual-band structures, it is not easy to make a fair quantitative comparison directly. A single-band LC oscillator should be a good reference here. As discussed above, the proposed dual-band oscillator can achieves the same phasenoise FoM as a single-band one, which is seldom possible for other designs [2]–[12]. Moreover, since the proposed dual-band structure can be considered as two coupled single-band LC oscillators, many phase-noise reduction techniques proposed for single-band LC oscillators [13], [14] can be readily adopted in the proposed dual-band design. IV. VERIFICATION BY PROTOTYPES We implemented a dual-band (Fig. 1) and a single-band oscillator (Fig. 2) in a 0.13- m CMOS technology with 0.5-V power supply. The two use the same PFET pair m m and the same center-taped symmetric inductor (1.138 nH) built on the two thick metals. We use metal–insulator–metal (MIM) capacitors, the values of which are set such that the oscillators work at the desired frequencies. A. Switch Network In the prototype, the switch network was modified from Fig. 1 to 13, mainly because the large voltage swing may turn switches on while they should remain off. Take as an example. In an odd-mode oscillation, it is turned off by setting its gate voltage to zero. However, the oscillation voltage at the inductor termion shortly. nals can drop below 420 mV, which can turn This leads to a leaking current, which degrades the quality factor of the resonator. To solve this problem, we used a capacitive voltage divider in Fig. 13. In this way, switch transistors only see half of the voltage swing and the leaking current is eliminated. m m Meanwhile, the switch transistors are twice as wide as before to provide the same damping effect. It is worth mentioning that the modified switch network is not an essential part of the structure in that there are other ways to

105

Fig. 13. Modified switching network to overcome current leakage.

Fig. 14. Simulated input impedance of implemented resonator, loaded by active core. Two peaks appear at 5.4 and 7.7 GHz, corresponding to odd- and even-mode resonance, respectively.

avoid leakage current, e.g., CMOS instead of pMOS active core. We will not go to details for the sake of space. B. Fabrication and Measurement Result A single- and dual-band oscillator were fabricated in a 0.13- m CMOS process with the frequency of the single-band oscillator close to the odd mode of the dual-band oscillator. Since our purpose here is to verify the functionality of switching and analysis on phase noise, we did not include varactors for continuous tuning in this prototype design. Fig. 14 shows simulated input impedance of the implemented dual-mode LC resonator, loaded with active core and switch network. Fig. 15 shows a die photograph of the dual-band oscillator. Both oscillators were measured with an Agilent 8564EC spectrum analyzer. The measured phase noises are in Figs. 16–18. Tables II and III summarize and compare results from simulation and measurement. We find that, although the measured frequencies of both oscillators drop due to parasitics, the measured phase noise agrees well with simulation. Besides, the frequency of the dual-band oscillator drops more than that of the single band, which can be explained by the parasitics of the long metal traces between the two inductors. In terms of phase noise, the single-band oscillator is at about the same frequency as the odd mode of the dual-band oscillator. As expected, both simulation and measurement show the

106

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 15. Die photograph of the proposed dual-band oscillator.

Fig. 18. Measured phase noise of the even mode of a dual-band oscil114:8 dBc@600 kHz with lator: L(1f ) = 97:7 dBc@100 kHz and f = 6:594 GHz, I = 3:99 mA, and V = 0:510 V.

0

0

TABLE II SUMMARY OF SIMULATION RESULTS

Fig. 16. Measured phase noise of a single-band oscillator: L(1f ) = 98:3 dBc@100 kHz and 115:8 dBc@600 kHz with f = 5:341 GHz, I = 2:01 mA, and V = 0:504 V.

0

0

TABLE III SUMMARY OF MEASUREMENT RESULTS

The even-mode oscillates at higher frequency and its FoM is about 2 dB lower than the odd mode, largely because of the drop in the resonator’s . Fig. 17. Measured phase noise of the odd mode of a dual-band oscillator: L(1f ) = 102:3 dBc@100 kHz and 119:3 dBc@600 kHz with f = 4:936 GHz, I = 4:00 mA, and V = 0:506 V.

0

0

3-dBc/Hz phase-noise improvement of the dual-band oscillator, compared to the single-band oscillator. Besides, the dual-band oscillator consumes twice the power of the single-band one. Thus, the two oscillators have the same phase-noise FoM, which verifies our analysis in Section III.

V. CONCLUSION In this paper, we presented a distributed dual-band oscillator suitable for low-phase-noise applications. In contrast to other switched-resonator designs, there is actually no current going through the switches, which leads to low phase noise. The design and analysis were verified by a prototype implemented in a 0.13- m CMOS process. There is good agreement among theory, simulation, and measurement results.

LI AND AFSHARI: DISTRIBUTED DUAL-BAND LC OSCILLATOR BASED ON MODE SWITCHING

ACKNOWLEDGMENT The authors would like to thank the Metal Oxide Semiconductor Implementation Service (MOSIS) and Taiwan Semiconductor Manufacturing Company (TSMC) University Shuttle Program for chip fabrication, O. Momeni, R. K. Dokania, Y. Tousi, and W. Lee, all with Cornell University, Ithaca, NY, for helpful technical discussions, and their wives M. Azarmnia and S. Lang for their support. REFERENCES [1] A. Kral, A. Behbahani, and A. A. Abidi, “RF-CMOS oscillators with switched tuning,” in Proc. IEEE Custom Integr. Circuits Conf., 1998, pp. 555–558. [2] S.-M. Yim and K. K. O, “Demonstration of a switched resonator concept in a dual-band monolithic CMOS LC-tuned VCO,” in Proc. IEEE Custom Integr. Circuits Conf., 2001, pp. 205–208. [3] N. D. Dalt, E. Thaller, P. Gregorius, and L. Gazsi, “A compact tripleband low-jitter digital LC PLL with programmable coil in 130 nm CMOS,” IEEE J. Solid-State Circuits, vol. 40, no. 7, pp. 1482–1490, Jul. 2005. [4] Z. Li and K. K. O, “A low-phase-noise and low-power multiband CMOS voltage-controlled oscillator,” IEEE J. Solid-State Circuits, vol. 40, no. 6, pp. 1296–1302, Jun. 2005. [5] D. Hauspie, E. Park, and J. Craninckx, “Wideband VCO with simultaneous switching of frequency band, active core, and varactor size,” IEEE J. Solid-State Circuits, vol. 42, no. 7, pp. 1472–1480, Jul. 2007. [6] N. T. Tchamov, S. S. Broussev, I. S. Uzunov, and K. K. Rantala, “Dualband LC VCO architecture with a fourth-order resonator,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 3, pp. 277–281, Mar. 2007. [7] B. Catli and M. M. Hella, “A 1.94 to 2.55 GHz, 3.6 to 4.77 GHz tunable CMOS VCO based on double-tuned double-driven coupled resonators,” IEEE J. Solid-State Circuits, vol. 44, no. 9, pp. 2463–2477, Sep. 2009. [8] G. Cusmai, M. Repossi, G. Albasini, A. Mazzanti, and F. Svelto, “A magnetically tuned quadrature oscillator,” IEEE J. Solid-State Circuits, vol. 42, no. 12, pp. 2870–2877, Dec. 2007. [9] A. Goel and H. Hashemi, “Frequency switching in dual-resonance oscillators,” IEEE J. Solid-State Circuits, vol. 42, no. 3, pp. 571–582, Mar. 2007. [10] J. Borremans, A. Bevilacqua, S. Bronckers, M. Dehan, M. Kuijk, P. Wambacq, and J. Craninckx, “A compact wideband front-end using a single-inductor dual-band VCO in 90 nm digital CMOS,” IEEE J. Solid-State Circuits, vol. 43, no. 12, pp. 2693–2705, Dec. 2008. [11] S.-W. Tam, H.-T. Yu, Y. Kim, E. Socher, M. C. F. Chang, and T. Itoh, “A dual band mm-wave CMOS oscillator with left-handed resonator,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., 2009, pp. 477–480. [12] Z. Safarian and H. Hashemi, “Wideband multi-mode CMOS VCO design using coupled inductors,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 8, pp. 1830–1843, Aug. 2009. [13] P. Andreani and A. Fard, “More on the 1=f phase noise performance of CMOS differential-pair LC tank oscillators,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2703–2712, Dec. 2006. [14] A. Mazzanti and P. Andreani, “Class-C harminic CMOS VCO with a general result on phase noise,” IEEE J. Solid-State Circuits, vol. 43, no. 12, pp. 2716–2729, Dec. 2008. [15] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. [16] H.-C. Chang, X. Cao, U. K. Mishra, and R. A. York, “Phase noise in coupled oscillators: Theory and experiment,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 604–615, May 1997. [17] H. Jacobsson, B. Hansson, H. Berg, and S. Georgovian, “Very low phase-noise fully-integrated coupled VCOs,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, vol. 1, pp. 577–580. [18] L. Romanò, A. Bonfanti, S. Levantino, C. Samori, and A. L. Lacaita, “5-GHz oscillator array with reduced flicker up-conversion in 0.13- m CMOS,” IEEE J. Solid-State Circuits, vol. 41, no. 11, pp. 2457–2467, Nov. 2006.

107

[19] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005. [20] P. Andreani and X. Wang, “On the phase-noise and phase error performances of multiphase Lc CMOS VCOs,” IEEE J. Solid-State Circuits, vol. 39, no. 11, pp. 1883–1893, Nov. 2004. [21] A. Hajimiri and T. H. Lee, “Design issue in CMOS differential LC oscillators,” IEEE J. Solid-State Circuits, vol. 34, no. 5, pp. 717–724, May 1999. [22] J. J. Rael and A. A. Abidi, “Physical processes of phase noise in differential LC oscillators,” in Proc. IEEE Custom Integr. Circuits Conf., May 2000, pp. 569–572. [23] A. Suarez, S. Sancho, S. Ver Hoeye, and J. Portilla, “Analytical comparison between time- and frequency-domain techniques for phase noise analysis,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2353–2361, Oct. 2002. [24] K. Kurokawa, “Noise in synchronized oscillators,” IEEE Trans. Microw. Theory Tech., vol. 16, no. 4, pp. 234–240, Apr. 1968. [25] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: A unifying theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I, Fund. Theory Appl., vol. 47, no. 5, pp. 655–674, May 2000.

Guansheng Li (S’10) received the B.Sc. and M.Sc. degrees in electronic engineering from Tsinghua University, Beijing, China, in 2005 and 2007, respectively, and is currently working toward the Ph.D. degree in electrical and computer engineering at Cornell University, Ithaca, NY. In 2010, he was an Intern with Qualcomm Inc., San Diego, CA, where he designed wide tuning-range voltage-control oscillators (VCOs) and frequency synthesizers. His current research interest is RF integrated circuit design with a focus on multiphase and multiband oscillator design and frequency synthesis. He has also conducted research on wireless communications and networking, studying cross-layer optimization and wireless network coding. He is a reviewer for Wireless Communications and Mobile Computing. Mr. Li is a reviewer for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and the IEEE SENSORS JOURNAL. He was named a Jacobs Scholar at Cornell University in 2007.

Ehsan Afshari (S’98–M’07) was born in 1979. He received the B.Sc. degree in electronics engineering from the Sharif University of Technology, Tehran, Iran, in 2001, and the M.S. and Ph.D. degrees in electrical engineering from the California Institute of Technology, Pasadena, in 2003, and 2006, respectively. In August 2006, he joined the Faculty of Electrical and Computer Engineering, Cornell University, Ithaca, NY. His research interests are high-speed and low-noise integrated circuits for applications in communication systems, sensing, and biomedical devices. Prof. Afshari was the chair of the IEEE Ithaca Section and chair of Cornell Highly Integrated Physical Systems (CHIPS). He is a member of the Analog Signal Processing Technical Committee, IEEE Circuits and Systems Society. He was the recipient of the National Science Foundation CAREER Award in 2010, Cornell College of Engineering Michael Tien Excellence in Teaching Award in 2010, Defense Advanced Research Projects Agency Young Faculty Award in 2008, and Iran’s Best Engineering Student Award presented by the President of Iran in 2001. He was also the recipient of the Best Paper Award of the Custom Integrated Circuits Conference (CICC) in 2003, First Place of the Stanford-Berkeley-Caltech Inventors Challenge in 2005, the Best Undergraduate Paper Award of the Iranian Conference on Electrical Engineering in 1999, the Silver Medal of the Physics Olympiad in 1997, and the Award of Excellence in Engineering Education of the Association of Professors and Scholars of Iranian Heritage (APSIH) in 2004.

108

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Broadband Differential Low-Noise Amplifier for Active Differential Arrays Óscar García-Pérez, Daniel Segovia-Vargas, Member, IEEE, Luis Enrique García-Muñoz, José Luis Jiménez-Martín, and Vicente González-Posadas

Abstract—In this paper, differential low-noise amplifiers are presented as a very powerful solution for radio astronomy applications. A fully differential amplifier topology has been analyzed and implemented in microstrip technology with discrete surface mount components. The amplifier design is made for an active receiving dense antenna array. Thus, the differential amplifier source impedance is no longer 50 , but 150 from the proposed bunny-ear antennas. A full characterization in terms of gain and noise has been undertaken. Source–pull measurements have been included in order to evaluate the performance of the amplifiers operating with variable source impedances. Noise temperatures below 55 K have been obtained for the differential design in the 300–1000-MHz band for the 150- impedance. In addition, the results for different scanning angles are also presented.







Index Terms—Active array, active impedance, differential lownoise amplifier (LNA), radio astronomy.

I. INTRODUCTION

T

HE NEXT-GENERATION radio telescope is projected to be formed by millions of antennas and receivers placed in a huge area (i.e., see square kilometre array (SKA) [1]). It is intended to work in a very large bandwidth (70 MHz–10 GHz) divided in smaller sub-bands. The lower sub-bands are intended to be covered with both sparse and dense aperture arrays, whereas small dish reflectors are intended for higher frequency bands [1]. More concretely, the midfrequency band, which ranges from 300 MHz to 1 GHz, is intended to be covered by a dense array of millions of cheap tapered slot antenna (TSA) elements, with scanning capabilities up to 45 . For this purpose, many research groups are currently working in the development of small array demonstrators in order to validate all the technologies that will be necessary for the final implementation of the SKA telescope (e.g., [2]–[5]).

Manuscript received March 15, 2010; revised August 31, 2010; accepted September 28, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. This work was supported in part by the Instituto Geográfico Nacional (IGN) under the Square Kilometre Array Design Studies (SKADS) Project and under Contract TEC2009-14525-C02-01. O. García-Pérez, D. Segovia-Vargas, and L. E. García-Muñoz are with the Department of Signal Theory and Communications, Carlos III University, 29811 Leganés, Madrid, Spain (e-mail: [email protected]; [email protected]; [email protected]). J. L. Jiménez-Martín and V. González-Posadas are with the Department of Ingeniería Audiovisual y Comunicaciones, Polytechnic University of Madrid (UPM), 28031 Madrid, Spain (e-mail: [email protected]; vgonzalz@diac. upm.es). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2091199

The TSA elements (i.e., Vivaldi or bunny-ear antennas [6]) are balanced antennas that must be fed in a differential way. In order to amplify the differential signals coming from each antenna element, either a passive balun with a single-ended lownoise amplifier (LNA) or a differential LNA is required. Since including a passive balun as the first element of the receiver chain can introduce an unnecessary increment in the noise temperature (in the order of 10 K), the option of a differential LNA directly connected to the antenna ports can be a more proper solution in terms of system noise temperature [7]. Furthermore, differential amplifiers have other inherent advantages in comparison with single-ended ones, which include interference rejection (e.g., noise introduced by the power supply, external interferences, etc.), higher dynamic range, and reduced even-order harmonic distortion [8]. As a counterpart, the characterization and measurement of differential devices is not as immediate as with conventional single-ended devices. Thus, it makes necessary the use of some ad hoc gain and noise measurement techniques [9]–[13]. During the last years, several broadband differential LNAs for radio astronomy applications have been proposed. Three possible solutions for the differential LNA design could be considered, which are: 1) commercial amplifiers; 2) monolithic microwave integrated circuits (MMICs); and 3) hybrid solutions. For the commercial amplifier case there are not differential devices working in this frequency range with the bandwidth and low noise level required for this application. Secondly, many of the proposed ad hoc designs are MMIC devices based on different types of transistor technologies (i.e., CMOS [13], SiGe BiCMOS [14]–[16], or GaAs HEMT [17]). However, they present some constraints due to the low- factor of the printed inductors and the large size needed for the planar coupling capacitors, especially when working at frequencies of some hundreds of megahertz. Thus, they usually require the use of external chip inductors and capacitors, increasing the total size of the circuit, so the advantage of the compactness provided by the MMIC is partially lost. Thirdly, other differential LNA designs based on discrete transistors have been presented as well. In [18], a differential LNA based on discrete transistors was presented, but working at higher frequencies and in a narrower relative bandwidth than the present work. In addition, all the previous amplifiers have to be integrated with the antenna, but none of them have been characterized considering a non-50- and varying antenna impedance. This paper addresses the characterization of differential LNAs taking into account the varying impedance of the antenna. The design of a broadband differential LNA based on discrete elements, covering the bandwidth from 300 MHz

0018-9480/$26.00 © 2010 IEEE

GARCÍA-PÉREZ et al.: BROADBAND DIFFERENTIAL LNA FOR ACTIVE DIFFERENTIAL ARRAYS

109

TABLE I PRELIMINARY LNA DESIGN REQUIREMENTS

to 1 GHz is proposed. In this way, chip components like high- -factor inductors and high value capacitors can be easily utilized. Although, two different topologies, balanced and fully differential [19], were initially proposed by the authors, this paper focuses on the differential one and improves its performance to bandwidth ratios of 3.3:1 and noise temperature around 50 K. In addition, this is achieved for an active antenna design where the antenna impedance (provided by each element of a bunny-ear antenna array) is different from 50 . In this case, the antenna has been designed to exhibit mean 150active impedance in differential mode. Thus, the noise and gain circles of the LNA are indispensable in order to know the system performance with this nonconstant antenna impedance. This measurement is undertaken by means of tunable source impedances (source–pull). Although this method has been widely used for single-ended devices [20], only some theoretical descriptions have been presented for differential amplifiers [21]. Finally, the design requirements are detailed in Table I [22]. II. DESIGN CONSIDERATIONS Simultaneously obtaining low noise, high gain, and unconditional stability is a difficult task, especially in broadband LNA designs, and different tradeoffs should be addressed. The minimum noise figure of a field-effect transistor (FET) is proportional to the drain current from a threshold level [23]. This implies using low current values to reduce the noise contribution from the FET at a price of degrading the gain response. A batch of measurements with the manufactured prototypes using different transistor currents is a good way to find the most adequate voltage supply levels. On the other hand, stabilization usually requires reducing the gain by the use of resistors, which degrades the noise figure of the amplifier [24]. A parametric study is very helpful in order to obtain the padding resistor values, which ensure the stability with the lowest cost in terms of noise figure. One important additional remark is that the proposed amplifier cannot be considered as isolated from the radiating element and has to work as a unique entity with the antenna forming a differential active antenna. In this way, no matching network would be needed since the differential broadband amplifier would directly match the antenna impedance. This implies that the antenna must provide the optimal impedance for the corresponding circuit function to constitute, in this case, a low-noise broadband differential amplifying active antenna. Very few broadband active antennas have been proposed until now (e.g., [25]) and most of them have been based on single-ended amplifiers. One exception to the previous com-

Fig. 1. Scheme of the: (a) antenna element and (b) array configuration.

ment are the transmitting active antennas based on push–pull power amplifiers (balanced configuration) [26]. However, for the current case, we are concerned with differential broadband receiving active antennas, in which differential LNAs are required. In addition, the proposed solution is a differential pair that provides improved common-mode suppression, which is quite advantageous due the inherent protection against external interferences. For the present design, the antenna used is an array of balanced TSAs placed together, where the mutual coupling between elements is very strong and leads to the array wide band performance (Fig. 1). The full design and optimization process of the antenna array can be found in [6]. For the array design, the mutual coupling is critical and the antenna impedance should be calculated as the so-called active impedance of each antenna element in the presence of all the other elements. This active impedance will determine the differential amplifier source impedance condition. It should be noticed that, because of the mutual coupling, the active impedance depends on the scanning angle of the array. In order to reduce the computational cost, and considering the large array dimensions of the final application, an infinite array simulation has been undertaken using Ansoft HFSS software. This analysis assumes that every element supports identical currents and voltages so all the elements provide the same antenna impedance. The final design has been optimized to provide well-matched antennas with respect to 150 in the band of interest (300–1000 MHz), and for scanning angles from broadside to 45 . The impedance curves of the antenna array are depicted in Fig. 2. III. PROPOSED FULLY DIFFERENTIAL TOPOLOGY This prototype is a cascade of two differential amplifiers interconnected with an RC matching network, in which the and in Fig. 3) are chosen for stability padding resistors ( along the whole bandwidth without compromising the noise figure. The stability parameter used in this analysis is the geometric stability factor. This factor can be obtained as a function of the (differential excitations in the mixed-mode -parameters input and output planes) of the amplifier as

(1)

110

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 2. Simulated active antenna impedance for different scanning angles based on an infinite array analysis (50- Smith chart).

Fig. 4. (a) Simulated noise temperature and (b) geometric stability factor of the differential amplifier for several R2 values and R3 = 100 .

Fig. 3. Electric schematic of the fully differential amplifier prototype.

The necessary and sufficient condition for unconditional stability of this equivalent two port device is that . The parametric study of the amplifier response as a function of and is shown in Figs. 4 and 5, respectively. It can be seen that this topology seems somewhat sensitive to instabilities. This is due to the fact that the FETs of both branches are interconnected so power transfers between the branches may appear. A value of 10 has been chosen. However, for the series resistor , is not enough to stabilize the circuit so a shunt this resistor, resistor, , has been also added. This type of padding reduces the gain as well as improves the output matching, contributing to the stabilization of the circuit [24]. Additionally, the response is equalized by means of the capacitor . A value for of of 100 is used in the prototype. The transistors used in this design are the ATF34-143 low noise pseudomorphic HEMT (pHEMT) from Avago Technologies, San Jose, CA. The design of the current sources has been and done by making use of an FET. Two series inductors ( in Fig. 3) have been included to increase the impedance of the current source. Additionally, several grounded shunt capacitors are used to protect the RF circuit from different dc transient

in Fig. 3). All the components for the prototype are peaks ( shown in Table II. One of the advantages of the differential pair in comparison with the balanced amplifier in [19] is that the first one presents a differential-mode gain (DMG) much higher than the common-mode gain (CMG). The relationship between both gains is known as common-mode rejection ratio (CMRR), and is a standard differential amplifier parameter, which indicates how much of the common-mode signal (e.g., interferences) will appear at the output of the amplifier (2) It must be noted that the CMRR of the balanced design is equal to one, since it exhibits equal DMG and CMG. Thus, obtaining a high CMRR is very important in order to mitigate as much as possible all the undesirable common-mode signals. With regard to the physical implementation of the circuit, all the components are surface mount devices (SMDs). The circuit board is shown in Fig. 6. It has been built in microstrip technology with an Arlon 25N substrate (height mm, dielectric constant ). A thicker substrate has been used in order to reduce the influence of the parasitic feedback capacitances, which may cause instabilities in the amplifier. The dimensions of the manufactured circuit are 60 mm 50 mm.

GARCÍA-PÉREZ et al.: BROADBAND DIFFERENTIAL LNA FOR ACTIVE DIFFERENTIAL ARRAYS

111

Fig. 7. Measurement scheme in a 50- environment.

IV. 50-

Fig. 5. (a) Simulated noise temperature and (b) geometric stability factor of the differential amplifier for several R3 values and R2 = 10 .

TABLE II FULLY DIFFERENTIAL AMPLIFIER COMPONENTS

Fig. 6. Top view of the implemented fully differential amplifier prototype.

MEASUREMENTS

The measurement of the differential devices is not as obvious as with the single-ended ones, and they require some specific methods that are described in the literature [12]. The easiest way to measure this type of devices is by using a pure-mode vector network analyzer (PMVNA) [10]. The problem is that this type of equipments is not always available since it is a network analyzer for a very specific use. Furthermore, the differential noise measurements are not solved yet with this type of devices. Another option, by making use of conventional noise or network analyzers, is to measure the four-port differential device by pairs of ports, and processing these data in order to obtain the desired parameters. That is, making transformations from the conventional -parameters matrix to a mixed-mode -parameters matrix with which one can obtain the parameters relative to the different combinations between input/output in differential/common mode [9]. A similar procedure can also be used to obtain the noise figure of a differential device, as described in [13]. The problem with these methods is that one needs to take several measurements to fully characterize the device-under-test (DUT). This is not problematic when doing a 50- characterization, but can be intractable in the source–pull measurements since this characterization needs, by itself, a lot of measurements with several impedance conditions. In any case, the method described in [9] has been used in this work in order to obtain the differential mode and common mode gains for the CMRR. Furthermore, it is also useful to check that no significant power is transmitted between different modes (i.e., common-todifferential or differential-to-common). The third option, and actually the main method used in this paper, is by making use of input and output baluns, as represented in Fig. 7. This technique is used to convert a four-port device into a two-port circuit, easily measurable by conventional noise or network analyzers. The limitation is that one can only do a characterization in differential-to-differential mode. However, this is actually the main characterization we are interested in because it is the main mode of operation of the amplifier. Another issue that may be taken into account is that the desired and actual measurement reference planes are not the same. Thus, it is necessary to properly eliminate the influence introduced by the balun in the gain and noise measurements [11]. The measurements have been taken by using broadband baluns, model ETC1-1-13 of M/A-COM, Lowell, MA, which can work from 4.5 to 3000 MHz. A previous characterization of these baluns has been done, obtaining insertion losses lower

112

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 8. Gain and noise measurements in a 50- environment of the fully differential amplifier.

than 1 dB, and amplitude and phase unbalances of 0 0.7 dB and 180 8 , respectively, along all the frequency range. It must be noted that the method described in [11] only corrects the insertion losses, but not the unbalances introduced by the balun, thus it is very important to obtain unbalances close to their ideal values (i.e., 0 dB in amplitude and 180 in phase). The error provided by the unbalances of these baluns has been estimated by simulation and is lower than 0.1 dB in terms of noise figure. The gain and noise curves obtained for the fully differential amplifier are shown in Fig. 8. A gain value over 25 dB and a noise temperature under 125 K dB can be seen in the operating bandwidth. Good agreement between simulation and measurement is obtained with the gain curve, and not so good with the noise curve. In this case, it has been also obtained a CMRR over 26.5 dB in all the frequency bandwidth by means of the measured mixed-mode -parameters. These 50- measurements have been taken in order to make a comparison between simulations and measurements and see the goodness of the simulations in the entire working band. It must be emphasized that the 50- conditions will not be the ones when the antenna is connected to the differential amplifier. V. SOURCE–PULL MEASUREMENTS A. Source–Pull Measurement Scheme The source–pull method is a technique that consists of obtaining some parameters of a DUT (e.g., gain, noise) for several source impedance conditions. It is very useful in this case, because the differential LNAs are going to be connected to a differentially fed antenna matched to 150 in differential mode [6]. The desired source impedance condition is provided by the differential antenna [see Fig. 9(a)]. An impedance tuner is used to synthesize different source impedance conditions during the LNA measurement. In our case, the 1819A from Maury Microwave Corporation, Ontario, CA (a manual coaxial tuner with three tunable shunt stubs) has been used. The impedances are synthesized by modifying the length of such stubs [see Fig. 9(b)]. Actually, the impedance seen from the input ports of the DUT will not be directly the impedance synthesized by the tuner, but the impedance seen through the input balun used

Fig. 9. Source–pull procedure: (a) antenna impedance, (b) impedance synthesized by the tuner, (c) impedance transformed when including the input balun, and (d) measurement scheme.

in the measurement [see Fig. 9(c)]. If this balun is completely characterized (i.e., -parameters matrix), obtaining the differential input impedance is easy. It must be noticed that, in the case of automatic source–pull equipments, the losses of the tuner are automatically de-embedded. However, in the case of a manual tuner, the losses should be de-embedded by the designer. This loss factor can be calculated as (3) where super-index indicates the tuner, whose port 1 is the one connected to the measurement equipment and port 2 is the one connected to the input balun. Let us assume that the port of the real balun connected to the tuner [see Fig. 9(c)] is port number 1, and the other two ports are numbered as 2 and 3. This balun can be seen as an equivalent two port device if ports 2 and 3 are excited in differential mode. Its , obtained from equivalent two-port mixed-mode matrix, the three-port standard matrix is then given by

(4) where sub-index indicates the single-ended port 1, sub-index indicates the differential mode excitation between ports 2 and indicates the input balun. 3 of the balun, and super-index

GARCÍA-PÉREZ et al.: BROADBAND DIFFERENTIAL LNA FOR ACTIVE DIFFERENTIAL ARRAYS

113

Once the parameters have been obtained, the equivalent impedance parameters can be calculated as

(5)

where is the reference impedance. Finally, the last step is seen from to calculate the differential input impedance ports 2 and 3 when connecting the impedance tuner at port 1. This impedance is given by (6) is the impedance synthesized by the tuner (seen where from the input balun). The measurement scheme is shown in Fig. 9(d), including the impedance tuner, both input and output baluns and the DUT. The procedure to eliminate the influence of the baluns is exactly the same as the one described for the 50- cases [11]. The final result is a level map drawn on the Smith chart for each frequency point at which the characterization is done (Fig. 10). Thus, the estimation of the characteristics of the DUT is immediate for a given antenna impedance. It must be noted that the characterization is done in a discrete number of impedance points so it is necessary to make an interpolation of the data in all the Smith charts. In our case, we use the load–pull tool included in the AWR Microwave Office software. Thus, the source–pull measurement procedure can be summarized as follows: 1) characterization of the impedance tuner, synthesizing at different frequencies several desired impedances [see Fig. 9(b)]; 2) characterization of the input and output baluns: scattering parameters, losses, and noise in the two branches; 3) measurement of the gain and the noise of the cascaded system formed by the DUT and the two baluns, using the test-bench shown in Fig. 9(d), for several impedance con; ditions 4) de-embedding of the impedance tuner; 5) de-embedding of the gain and the noise of the DUT using the procedure described in [11]; 6) tansformation of the impedances given by the tuner to the corresponding actual differential source impedances seen by the DUT, by using the measured -parameters matrix of the input balun and operating with (4)–(6);

Fig. 10. Measured gain and noise source–pull curves at 350, 650, and 1000 MHz.

7) representation of and for several source impedances, , in the Smith chart, and interpolation of the data to obtain the corresponding gain and noise level curves; 8) with this level map, it is possible to estimate the gain and the noise of the DUT for any source impedance condition at each frequency.

114

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE III SOURCE–PULL MEASUREMENTS OF THE DIFFERENTIAL LNA

impedance of 150 due to the limitations to synthesize arbitrary complex load impedances. The gain has also been obtained from the mixed-mode -parameters with a 150- impedance [9]. Fig. 11 shows the obtained noise temperature with the hot/cold procedure and the gain obtained with the mixed mode method. In addition, all the values obtained for the source–pull method have also been drawn. A good agreement between the different methods is observed. VI. CONCLUSION This paper has focused on the design of broadband differential LNAs for differential active TSA arrays. It has been demonstrated that designs with discrete elements can be a valid solution for this type of systems. On the other hand, the obtained results have shown that the noise performance of the LNAs is strongly dependent with the source impedance. Thus, a source–pull analysis is indispensable to fully characterize this type of receivers. This is especially critical in this case, where the antenna impedance is not 50 matched, and moreover the antenna depends on other factors such as the frequency and the scanning angle of the array. Thus, a novel source–pull procedure for differential devices has been defined. Good results are achieved for variable impedance and scanning angles up to 45 . ACKNOWLEDGMENT

Fig. 11. Noise temperature from the hot/cold method and gain for the mixed S -parameter method. The discrete values for the source–pull method are also shown.

B. Source–Pull Measurement Results The source–pull measurements have been undertaken by evaluating the gain and the noise at three frequencies (300, 650, and 1000 MHz), with 35 different impedance points at each frequency. Considering the previous results at 50 , no abrupt variations with frequency are expected so three frequency points are considered to be enough. By means of the graphs shown in Fig. 10, the features of the amplifier (working with a perfectly matched 150- differential source and with the bunny-ear active impedance of Fig. 2) have been estimated. The gain and noise results are shown in Table III. The results presented in this table correspond to two different situations: a fixed source impedance of 150 ; and the simulated active antenna impedance at different array scanning angles [6]. For that last case, scanning angles up to 45 have been considered: 0 , 22.5 , and 45 . For the case of fixed impedance, it can be seen that the variation of both the gain and noise values is not very large. A gain higher 26 dB and a noise temperature lower dB is obtained. For the scanning case, than 55 K the gain varies between 35–25 dB for the worst case while the noise temperature varies between 45–65 K for the worst case. These results are considerably better than the ones obtained for the balanced topology in [19]. Finally, in order to have a comparison for the entire frequency bandwidth, a hot/cold noise measurement has been done using the method described in [17]. This method is used only for an

The authors wish to thank the Instituto Geográfico Nacional (IGN), Yebes, Spain, and, especially, J. A. López-Fernández, T. Finn, and J. M. Serna, all with IGN, for support during this work. The authors also wish to thank J. G. bij de Vaate, Netherlands Institute for Radio Astronomy (ASTRON), Dwingeloo, The Netherlands, for his help during the hot/cold measurement procedure. REFERENCES [1] P. E. Dewdney, P. J. Hall, R. T. Schilizzi, and T. J. W. Lazio, “The square kilometre array,” Proc. IEEE, vol. 97, no. 8, pp. 482–1496, Aug. 2009. [2] J. G. Bij de Vaate, L. Bakker, E. E. M. Woestenburg, R. H. Witvers, G. W. Kant, and W. van Cappellen, “Low cost low noise phased-array feeding systems for SKA pathfinders,” in Int. Antenna Technol. Appl. Electromagn. Symp., Feb. 2009, pp. 1–4. [3] O. Garcia-Perez, L. E. Garcia-Muñoz, J. M. Serna-Puente, V. Gonzalez-Posadas, J. L. Vazquez-Roy, and D. Segovia-Vargas, “Differential active antennas for the SKA project,” in Eur. Antennas Propag. Conf., Mar. 2009, pp. 1316–1319. [4] M. Arts, R. Maaskant, E. de Lera Acedo, and J. G. bij de Vaate, “Broadband differentially fed tapered slot antenna array for radio astronomy applications,” in Eur. Antennas Propag. Conf., Mar. 2009, pp. 566–570. [5] D. DeBoer et al., “Australian SKA pathfinder: A high-dynamic range wide-field of view survey telescope,” Proc. IEEE, vol. 97, no. 8, pp. 1507–1521, Aug. 2009. [6] E. Lera-Acedo, L. E. Garcia-Muñoz, V. Gonzalez-Posadas, J. L. Vazquez-Roy, R. Maaskant, and D. Segovia-Vargas, “Study and design of a differentially fed tapered slot antenna array,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp. 68–78, Jan. 2010. [7] A. van Ardenne, H. Butcher, J. G. bij de Vaate, A. J. Boonstra, J. D. Bregman, B. Woestenburg, K. van der Schaaf, P. N. Wilkinson, and M. A. Garrett, “The aperture array approach for the square kilometre array,” White Paper, May 2003. [Online]. Available: http://www.skatelescope.org/ [8] W. R. Eisenstadt, B. Stengel, and B. M. Thomson, Microwave Differential Circuit Design Using Mixed Mode S -Parameters. Norwood, MA: Artech House, 2006.

GARCÍA-PÉREZ et al.: BROADBAND DIFFERENTIAL LNA FOR ACTIVE DIFFERENTIAL ARRAYS

[9] D. E. Bockelman and W. R. Eisenstadt, “Combined differential and common-mode scattering parameters: Theory and simulation,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1530–1539, Jul. 1995. [10] D. E. Bockelman and W. R. Eisenstadt, “Pure-mode network analyzer for on-wafer measurements of mixed-mode S -parameters of differential circuits,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1071–1077, Jul. 1997. [11] A. A. Abidi and J. C. Leete, “De-embedding the noise figure of differential amplifiers,” IEEE Trans. Solid-State Circuits, vol. 34, no. 6, pp. 882–885, Jun. 1999. [12] “Balanced device characterization,” Agilent Technol., Santa Clara, CA, App. Note, Feb. 2004. [Online]. Available: http://www.home.agilent. com/upload/cmc_upload/All/EPSG084733.pdf [13] L. Belostotski and J. W. Haslett, “A technique for differential noise figure measurement of differential LNAs,” IEEE Trans. Instrum. Meas., vol. 57, no. 7, pp. 1298–1303, Jul. 2008. [14] J. Lintignat, M. L. Grima, S. Darfeuille, B. Barelaud, B. Jarry, S. Barth, S. Bosse, P. Meunier, and P. Gamand, “BiCMOS differential low noise amplifier for radioastronomy application,” SKADS, ASTRON, The Netherlands, SKADS Memo T12, 2008. [Online]. Available: http://www.skads-eu.org/ [15] M. L. Grima, S. Barth, S. Bosse, B. Jarry, P. Gamand, P. Meunier, and B. Barelaud, “A differential SiP (LNA-filter-mixer) in silicon technology for the SKA project,” in Eur. Microw. Integr. Circuits Conf., Oct. 2007, pp. 331–334. [16] J. Lintignat, S. Darfeuille, B. Barelaud, L. Billonnet, B. Jarry, P. Meunier, and P. Gamand, “A 0.1–1.7 GHz, 1.1 dB NF low noise amplifier for radioastronomy application,” in Eur. Microw. Integr. Circuits Conf., Oct. 2007, pp. 231–234. [17] J. Morawietz, R. H. Witvers, J. G. B. de Vaate, and E. E. M. Woestenburg, “Noise characterization of ultra low noise differential amplifiers for next generation radiotelescopes,” in Eur. Microw. Conf., Oct. 2007, pp. 1570–1573. [18] N. Roddis, “Noise measurements for the SKA,” presented at the RadioNet Eng. Workshop: Low Noise Figure Meas. Cryogen. Room Temperatures, Jun. 2009. [19] O. Garcia-Perez, V. Gonzalez-Posadas, J. L. Jimenez-Martin, J. M. Serna-Puente, E. Garcia-Muñoz, and D. Segovia-Vargas, “Design of differential low noise amplifiers for UWB antennas in the low band of the SKA project,” presented at the URSI Gen. Assembly, Aug. 2008. [20] L. Belostotski and J. W. Haslett, “Evaluation of tuner-based noise parameter extraction methods for very low noise amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 1, pp. 236–250, Jan. 2010. [21] O. Garcia-Perez, L. E. Garcia-Muñoz, V. Gonzalez-Posadas, and D. Segovia-Vargas, “Source–pull characterization of differential active antennas for radio-astronomy applications,” in Eur. Wireless Technol. Conf., Sep. 2009, pp. 84–87. [22] R. T. Schilizzi, P. Alexander, J. M. Cordes, P. E. Dewdney, R. D. Ekers, A. J. Faulkner, B. M. Gaensler, P. J. Hall, J. L. Jonas, and K. I. Kellermann, “Preliminary specifications for the square kilometre array,” SKA, Manchester, U.K., SKA Memo 100, Dec. 2007. [Online]. Available: http://www.skatelescope.org/ [23] C. A. Liechti, “Microwave field-effect transistors,” IEEE Trans. Microw. Theory Tech., vol. MTT-24, no. 6, pp. 279–300, Jun. 1976. [24] U. Delpy, “Stabilize transistors in low-noise amplifiers,” Microw. RF, vol. 45, no. 5, pp. 81–88, May 2006. [25] D. Segovia-Vargas, D. Castro-Galan, L. E. Garcia-Muñoz, and V. Gonzalez-Posadas, “Broadband active receiving patch with resistive equalization,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 1, pp. 56–64, Jan. 2008. [26] W. R. Deal, V. Radisic, Y. Qian, and T. Itoh, “Integrated-antenna pushpull power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1418–1425, Aug. 1999. Oscar García-Pérez was born in Madrid, Spain, on December 23, 1984. He received the Engineer degree in telecommunications from Carlos III University, Madrid, Spain, in 2007, and is currently working toward the Ph.D. in communications at Carlos III University. His main research interests are the study of microwave active circuits with metamaterial structures and broadband differential LNAs for radio astronomy applications.

115

Daniel Segovia-Vargas (M’98) was born in Madrid, Spain, in 1968. He received the Telecommunication Engineering degree and Ph.D. degree from the Polytechnic University of Madrid, Madrid, Spain, in 1993 and in 1998, respectively. From 1993 to 1998, he was an Assistant Professor with Valladolid University. Since 1999, he has been an Associate Professor with Carlos III University, Madrid, Spain, where he is in charge of the microwaves and antennas courses. Since 2004, he has been the leader of the Radiofrequency Group, Department of Signal Theory and Communications, Carlos III University. He has authored or coauthored over 110 technical conference, letters, and journal papers. His research areas are printed antennas and active radiators and arrays, broadband antennas, left-handed (LH) metamaterials, terahertz antennas, and passive circuits. He has also been an expert of European Projects Cost260, Cost284, and COST IC0603. Luis Enrique García-Muñoz is currently an Associate Professor with the Universidad Carlos III de Madrid, Madrid, Spain. He has managed or participated in several national and European research projects on areas such as antennas and array design. He has coauthored over 50 papers in international journals and conferences. He holds three patents. His current research interests include terahertz antennas, array design, truncation in antenna arrays, and radio astronomy instrumentation. José Luis Jiménez-Martín was born in Madrid, Spain, in 1967. He received the Radio-Communication Technical Engineering (B.S. in electrical engineering), Telecommunications Engineering (M.S.), and Ph.D. degrees from the Universidad Politécnica de Madrid, Madrid, Spain, in 1991, 2000, and 2005, respectively, and the Master’s degree in high strategic studies from CESEDEN (organization pertaining to the Spanish Ministry of Defense), Madrid, Spain, in 2007. He is currently an Associate Professor with the Technical University of Madrid, Madrid, Spain. He has authored or coauthored over 40 technical conference, letters, and journal papers. His interests are related to oscillators, amplifiers, microstrip antennas, composite right/left-handed (CRLH) lines and metamaterials, and microwave technology. Vicente González-Posadas was born in Madrid, Spain, in 1968. He received the Ing. Técnico degree in radio-communication engineering from the Polytechnic University of Madrid (UPM), Madrid, Spain, in 1992, the M.S. degree in physics from the Universidad Nacional de Educación a Distancia (UNED), Madrid, Spain, in 1995, and the Ph.D. degree in telecommunication engineering from Carlos III University , Madrid, Spain, in 2001. He is currently an Associate Professor with the Technical Telecommunication School, Polytechnic University of Madrid. He has authored or coauthored over 80 technical conference, letters, and journal papers. His interests are related to active antennas, microstrip antennas, CRLH lines and metamaterials, and microwave technology.

116

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

A 0.18-m Dual-Gate CMOS Device Modeling and Applications for RF Cascode Circuits Hong-Yeh Chang, Member, IEEE, and Kung-Hao Liang

Abstract—A merged-diffusion dual-gate CMOS device model is presented in this paper. The proposed large-signal model consists of two intrinsic BSIM3v3 nonlinear models and parasitic components. The parasitic elements, including the substrate networks, the distributed resistances, and the inductances, are extracted from the measured -parameters. In order to verify the model accuracy, a cascode configuration with the proposed dual-gate device is employed in a low-noise amplifier. The dual-gate model is also evaluated with power sweep and load–pull measurements. In addition, a doubly balanced dual-gate mixer is successfully demonstrated using the proposed model. The measured results agree with the simulated results using the proposed device model for both linear and nonlinear applications. The advanced large-signal dual-gate CMOS model can be further used as an RF sub-circuit cell for simplifying the design procedure. Index Terms—BSIM, CMOS, dual gate, low-noise amplifier (LNA), mixer, modeling.

I. INTRODUCTION

D

UE TO the expansion of the modern wireless communications, the low-power low-cost transceivers are very attractive. Advanced CMOS technologies are suitable for integrating the RF front-end and the baseband digital circuits in a single chip owing to their superior performance and mass production [1]–[3]. In general, power amplifiers (PAs), low-noise amplifiers (LNAs), voltage-controlled oscillators (VCOs), and mixers are essential building blocks in the RF front-end system. To compare with the analog and digital circuits, a few transistors are usually employed in the RF circuits. The performance of the RF circuits is strongly related to the layout and the parasitic effects of the interconnections. The topologies of the RF circuits can be generally categorized following sub-circuit cells: cascode configuration, differential pair, cross-coupled pair, and inductor–capacitor (LC) passive components. Therefore, an accurate model of the RF sub-circuit cell is necessary for simplifying the design procedure. The cascode configuration is usually employed in the RF circuit designs because the output resistance is increased, while

Manuscript received May 13, 2010; revised August 12, 2010; accepted October 06, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. This work was supported in part by the National Science Council of Taiwan under Grant NSC 96-2221-E-008-117-MY3 and Grant NSC 99-2221-E-008-097-MY3, and by the Chip Implementation Center (CIC), Taiwan. The authors are with the Department of Electrical Engineering, National Central University, Jhongli City, Taoyuan 32001, Taiwan (e-mail: [email protected]. edu.tw). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2091201

the Miller’s capacitor is reduced. Therefore, the circuit performance at high frequency is further enhanced. A wideband LNA was proposed using a Darlington cascode configuration in an SiGe BiCMOS technology to further enhance the circuit performance [4]. A folded-cascode configuration is a choice for the low-voltage applications [5], but the pMOS characteristics are usually not compared to the nMOS in the microwave and millimeter-wave regimes. For the ultra-broadband applications, the CMOS LNA can be realized using a shunt-feedback inverter with splitting load inductive peaking technique [6], nevertheless, the pMOS is also required for the LNA design. On the other hand, the cascode configuration can be employed in the mixer design [7]. A folded-cascode even-harmonic mixer was proposed for the low-voltage applications [8]. The low-voltage mixers can also be realized using a folded-switching topology [9]. The cascode configuration realized using a dual-gate device has been used in many RF circuits, such as a distributed amplifier (DA) [10], a variable gain amplifier (VGA) [11], an LNA [12], and mixers [13], [14]. The dual-gate device has the advantage of an increased capability due to the ability of the two independent gates. Therefore, the dual-gate device can be performed such functions as gain control and mixing, as well as reduced feedback and improvement in signal gain [15]. Furthermore, the parasitic effects of the cascode interconnections can be reduced using the dual-gate device, especially for the RF circuits. The layout routing is also simplified to enhance the circuit performance. A preliminary model-extraction methodology of the dual-gate CMOS was proposed in our previous work [16]. In this paper, we illustrate the layout structure and the corresponding physical model of the dual-gate device. An analytical extraction procedure is proposed for the dual-gate CMOS device. The intrinsic nonlinear model consists of two SPICE-based BSIM3v3 cores. Moreover, additional parasitic network is capable of the device modeling at high frequencies. The BSIM is widely adopted as the standard CMOS model for the analog and digital circuit designs below gigahertz. To extend frequency limitation, the passive network was included to present the substrate effects for modeling the high-frequency parasitic [17] and [18]. Furthermore, modified scaleable RF large-signal models can be seen in [19] and [20]. The small-signal characteristics of the proposed model are verified by approaching in a dual-gate LNA. The power sweep and load–pull evaluations are also presented for the proposed dual-gate model, and the measurement agrees with the model simulation. Moreover, a dual-gate mixer is successfully demonstrated to verify the nonlinear characteristics of the proposed dual-gate model. The proposed RF CMOS dual-gate model contains the merits of

0018-9480/$26.00 © 2010 IEEE

CHANG AND LIANG: 0.18- m DUAL-GATE CMOS DEVICE MODELING AND APPLICATIONS FOR RF CASCODE CIRCUITS

117

Fig. 1. Layout of the proposed dual-gate device.

simplicity, accuracy, and practicality for the RF circuit applications. To the best of the authors’ knowledge, this work is the first attempt to address an RF self-defined dual-gate model and successfully apply to the RF cascode circuits. This paper is organized as follows. Section II presents the proposed dual-gate model and the parameter extraction method. Section III shows the RF dual-gate circuit designs for the LNA and mixer. Section IV describes the dual-gate model verification and circuits experimental results. Conclusions are given in Section V. II. MODEL DESCRIPTION AND PARAMETER EXTRACTION METHOD The dual-gate device, LNA, and mixer are designed using a Taiwan Semiconductor Manufacturing Company (TSMC) single-poly, six-metal (1P6M) 0.18- m CMOS standard bulk process. A nMOS typical has a unity current gain frequency of 50 GHz and a maximum oscillation frequency of 85 GHz. Thin-film resistors, metal–insulator–metal (MIM) capacitors, and spiral inductors are also available in this process. The cascode configuration with merged diffusion active area is performed as a dual-gate device. The layout of the proposed dual-gate nMOS transistor is shown in Fig. 1, and the multifinger gate is adopted with a total gatewidth of 96 m 16 fingers). The parasitic capacitance and the (i.e., 6 m effective silicon substrate resistance can be reduced since the active area of the dual-gate device is smaller than the conventional cascode structure [12]. The cross section of the dual-gate device and the equivalent circuit model are shown in Fig. 2(a) and (b), respectively. Two BSIM3v3 intrinsic models are used to describe the dc behavior and the nonlinear components, , , , , and of the dual-gate such as intrinsic , device. The extrinsic (bias-independent) components are , , , , , , , , and . The parasitic elements of the proposed dual-gate model are essential and are to predict high-frequency characteristics. the poly-gate resistances, which depend on the gatewidth, gate and length, and the effective sheet resistances [21]. represent the contact resistances of the source and the drain, respectively. The resistances are related to the area of the drain and are the series parasitic inductances of and source. the drain and source metal connections, and the inductances

Fig. 2. (a) Cross section of the dual-gate device. (b) Equivalent circuit model.

are affected by the interconnection of the layout structure. and represent the parasitic substrate capacitance and resistance, respectively, and they are formed by the depletion region beneath the n -doped region inside the p-well. The depletion region beneath the drain is significant since the positive bias is applied to the drain of the device. The substrate parasitic effect of the source can be negligible since the source of the device is is connected to the p-type substrate with a dc bias of 0 V. the coupling capacitance between gate1 and gate2 , is the substrate resistance of the n -doped diffusion and region between the field-effect transistor1 (FET1) and FET2. In order to simplify the extraction methodology, the dualgate device is treated as the cascode configuration with parasitic components. Therefore, the extraction strategy can start with a single-gate device and then stack two FETs in common-source and common-gate configuration. The extrinsic elements of the single-gate FET can be extracted by the - and -parameters converted from the measured -parameters. The bias- and fre, , and are obtained quency-independent resistance using the equations in [21] (1) (2) (3) From (1)–(3), the resistances , , and of the single-gate FET can be extracted from the - and -parameters. The extracted results are applied to the SPICE-based BSIM3v3 core model, and two models are stacked in the cascode configura, , , tion, as shown in Fig. 2(b). The resistances in the proposed equivalent model are obtained based and on the extracted equations (1)–(3). For the -parameters measurement, the , , drain (D) are defined as port1, port2, and port3, respectively. The coupling capacitance and the

118

n -doped diffusion resistance lowing equations:

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

are extracted using the fol-

(4) (5) From (1)–(5), the transistor is biased in the linear region, and the analytical assumption in [21] is used to simplify the extraction and are the series parasitic inductances at the method. drain and source of the dual-gate device, respectively. The inductances are used for fitting the -parameters up to microwave and are associated with the metal frequency. In general, connection and the effective layout dimension. and can be expressed as (6) (7) and where is the total gate finger of the device, are the normalized capacitance of the p-n junction doping capacitor and the normalized substrate resistance, respectively and are typically equal to 0.26 pF and [22], [23]. 90.56 k , respectively. The noise parameters of the dual-gate device are based on the two intrinsic BSIM3v3 nonlinear models. Below 10 GHz, the can be approximated by [24] minimum noise figure (8) where is the proportional constant of the drain current noise. is generally dominated by the transconWe can see that and the gate parasitic resistance . Therefore, ductance the accuracy of the noise model can be further improved by fitand . is obtained from ting the two parameters of is obtained from the -paramethe dc I–V measurement. ters measurement, and it can be calculated by the real part of converted from the -parameters. A noise the -parameter deembedding procedure for on-wafer high-frequency measurement of MOSFETs can be adopted for the evaluation of the noise parameters [25]. For the proposed dual-gate device model, the noise evaluation is only characterized via the noise figure measurements of the LNA and mixer due to the limitation of the noise parameter measurement system. The noise parameters of the BSIM core are based on the design kit provided by the foundry. The extraction flowchart of the dual-gate device is shown in Fig. 3. The proposed dual-gate device modeling is established using the following procedure. 1) Measure one-port -parameter of the open and short pads for the deembedding procedure of the input and output ground–signal–ground (GSG) pads. 2) Measure two-port -parameters of the single-gate device, which is the common-source configuration. The drain-tois swept 0.1, 0.9, and 1.8 V, and the source voltage is swept 0.6, 0.9, 1.2, 1.5, and 1.8 V. gate-to-source

Fig. 3. Extraction flowchart of the dual-gate device.

Calculate the measured -parameters with an open- and short-circuit deembedding technique. 3) Measure the dc I–V characteristics of the single-gate device. Base on the BSIM v3.3 model to fit the dc I–V curves. The nonlinear core extraction procedure can be found in [26]. First, perform the dc-parameter extractions. Second, perform capacitance parameter extractions. The procedures are all based on the BSIM3v3 extraction steps [27]. , , and 4) Extract the intrinsic capacitances from the measured -parameters in step 2). 5) Use (1)–(3) to extract the parasitic resistances , , and , and apply the resistances to the BSIM3v3 core model. 6) Stack two single-gate FETs in the cascode configuration. Measure three-port -parameters based on a two-port vector network analyzer (VNA). The measurement procedure of the three-port -parameters is presented in Section IV. 7) Use (4) and (5) to extract the coupling capacitance and the diffusion resistance , respectively. 8) Use (6) and (7) to obtain the substrate capacitance and resistance , respectively. 9) Above 10 GHz, fit the three-port -parameters of the model to the measured results with varying the parasitic and . inductances 10) Check whether the accuracy of the dual-gate model is good enough or not via the dc I–V curves and the three-port -parameters. If not, repeat the step from step 1) to 9). The final extracted results for the parasitic components of the dual-gate device are summarized in Table I.

CHANG AND LIANG: 0.18- m DUAL-GATE CMOS DEVICE MODELING AND APPLICATIONS FOR RF CASCODE CIRCUITS

119

TABLE I EXTRACTION RESULTS OF THE PARASITIC COMPONENTS

Fig. 4. Equivalent schematic of the dual-gate device.

Fig. 6. Schematics of the: (a) LNA and (b) doubly balanced mixer.

Fig. 5. I–V characteristics of the dual-gate device.

. The Miller capacitance can be also reduced owing to the cascode topology [29]. The extraction results of the gate-toare plotted in Fig. 7 for the single- and drain capacitance can be obtain using dual-gate devices, and the capacitance

III. DUAL-GATE CIRCUIT DESIGNS FOR LNA AND MIXER The proposed dual-gate device is the cascode connection of two equal-width single-gate FETs with merged-diffusion area. The equivalent schematic of the proposed dual-gate device is shown in Fig. 4, which consists of FET1 and FET2. The simulated dc I–V characteristics of the dual-gate device are plotted in is a function of the gate-to-source Fig. 5. The drain current (lower FET) and (upper FET). Hence, the voltages transconductance of the dual-gate device is controlled by both and . There are three possible operation modes of the dual-gate device in Fig. 5, such as linear-active (A), active-active (B), and active-linear (C) regions. The optimum dc-biased region of the dual-gate device depends on the circuit operation mechanism. To evaluate the device model, an LNA and a doubly balanced mixer are designed using the proposed dual-gate device, and the schematics of the LNA and mixer are shown in Fig. 6(a) and (b), respectively. A dual-gate device with a source degeneration is utilized for the LNA design. The input impedance inductor can be close to 50 at the passband frequency with the proper and [28]. The noise figure and the input inductances of match are both improved using the source degeneration inductor

(9) With the same gate periphery, we can see that the Miller capacitance of the dual-gate device is obviously smaller then the single-gate device. Therefore, the frequency response of the LNA and the port-to-port isolations of the mixer can be both enhanced using the dual-gate device. In order to further improve the small-signal gain and noise figure of the LNA, the dual-gate device is biased in the active-active region (i.e., ) to maximize the transconductance of the device. The cascode configuration of the doubly balanced mixer is used to verify the nonlinear characteristics of the dual-gate model. The mixer core is composed of four dual-gate devices as the doubly balanced architecture. The RF and local oscillator (LO) signals are applied into the lower gate and upper gate of the device, respectively. The four dual-gate devices are all biased in the active-active region for high conversion gain and low noise figure. However, the dc bias condition is varying with the LO power since the amplitude of the LO signal is larger than the RF signal. The LO power

120

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 7. Extracted gate-to-drain capacitances devices.

C

of the dual- and single-gate

should be properly selected to maintain the dc bias condition. The drain voltage of the lower FET of the dual-gate device will be influenced by the injected LO signals. The dc bias condition of the dual-gate device is based on the LO power level, which can modulate the maximum transconductance of the lower FET. The operation principle of the dual-gate mixer is the transconductance modulation by switching the lower FET between the linear and active regions over the LO cycle and [13]. The high output impedance pMOS transistors are performed as active loads to further enhance the conversion gain. Two common-source transistors and are designed as IF output buffers to achieve the impedance match for driving an output load of 50 [30]. Moreover, the proposed dual-gate model is also suitable for other RF circuit designs, such as VCO, VGA, switches, and DA. IV. DUAL-GATE MODEL VERIFICATION AND CIRCUITS EXPERIMENTAL RESULTS Based on the proposed device modeling and extraction methodology, the dual-gate model has been successfully accomplished. The proposed dual-gate device is measured via on-wafer probing. The dc I–V characteristics are measured using an HP 4142B dc source/monitor with Agilent ICCAP software. The ICCAP software is only used for the instrument control of the dc I–V, -parameters measurements, and the parameter extraction of the single-gate device. The model prediction and the measured dc I–V characteristics are plotted voltage is fixed at 1.2 V, the in Fig. 8. In Fig. 8(a), the voltage is from 0 to 1.5 V with a step of 0.2 V, and is from 0 to 2.5 V. In Fig. 8(b), the the drain voltage voltage is fixed at 1.2 V, and the voltage is from 0 to 1.5 V. As can be observed, good agreement between the model prediction and the measurement is obtained. The -parameters measurement is utilized for high-fre, , and of the dual-gate quency verification, and the device are port1, port2, and port3, respectively. Due to the limitation of the instrument, the three-port -parameters of the dual-gate device is measured using an HP 8510C VNA. The three-port -parameters measurement procedure is shown in Fig. 9, and the detail is summarized in follows.

Fig. 8. DC I–V characteristics of the model prediction and the measured results. : V, V : V, and V : . (b) V : V (a) V : V with a step of 0.2 V. and V

=025 =015

=015

=12

=12

1) Connect and to the VNA port1 (VNA1) and port2 via (VNA2), respectively. The bias is applied to the an Agilent 11612B broadband dc bias-T. The measured -parameters of , , and are obtained. 2) Connect and to the VNA1 and VNA2, respectively. via the bias-T. The measured The bias is applied to the -parameters of , , , and are obtained. and to the VNA1 and VNA2, respectively. 3) Connect The bias is applied to the via the bias-T. The measured -parameters of and are obtained. 4) The -parameters deembedding can be performed using an open dummy pad method [31] or an open- and short-circuit method [32]. The presented results are based on the openand short-circuit method. The model and measured -parameters are plotted in Fig. 10 is from 50 MHz to 15 GHz. In Fig. 10(a), the drain voltage 1.1 V with a dc current of 5.8 mA, and the voltage and voltage are 1.2 and 0.8 V, respectively. In Fig. 10(b), the is 1.8 V with a dc current of 24.7 mA, the drain voltage voltage and the voltage are 1.2 and and the 1.8 V, respectively. Also, the model predication results for the dual-gate device agree with the measured -parameters. The three-port -parameters measurements can be performed using the proposed measured procedure, but the accuracy decreases as the measured frequency is higher than 15 GHz due to the limitation of the deemedding method. For the characterization higher than 15 GHz, the three-port -parameters can be measured using the commercial standard multiport VNA, such

CHANG AND LIANG: 0.18- m DUAL-GATE CMOS DEVICE MODELING AND APPLICATIONS FOR RF CASCODE CIRCUITS

121

Fig. 9. Three-port S -parameters measurement procedure using a two-port VNA for the dual-gate device.

Fig. 10. Model and measured S -parameters results frequency up to 15 GHz. (a) V : ,V : V, and V : I : mA . (b) V : V, and V : V I : mA . : V, V

22

=11 =12 = 0 8 ( = 5 78 =12 = 1 8 ( = 24 7 )

)

=

as the Agilent PNA-X series four-port VNA, with the on-chip calibration kits. Moreover, the calibration methods can be based on the short-open line-thru (SOLT) or thru-reflection-line (TRL) calibration. To verify the large-signal accuracy of the proposed dual-gate device model, the power-sweep and load–pull measurements are adopted. The power sweeps are measured using an Agilent ESG 4438C signal generator and an HP 8596E spectrum analyzer. The measured and simulated power sweeps of the proposed dual-gate device are plotted in Fig. 11(a) at 2.4 GHz, where the source and load impedance of the dual-gate device is 2 V, and the voltage are both 50 . The drain voltage and the voltage are 1 and 1.2 V, respectively. From the measured results, the proposed dual-gate device features a

Fig. 11. Measured and simulated power performance at 2.4 GHz with the dc bias of V V, V V, and V : V: (a) with a load of 50 and (b) with the optimum load.

=2

=1

=12



maximum small signal gain of 10.3 dB and an output saturation power of 7 dBm. The power contours are measured using a Maury MT-986 load–pull measurement system. With an input power of 5 dBm, the simulated and measured power contours at 2.4 GHz are plotted in Fig. 12 for the proposed dual-gate device, and the bias condition is the same as the power sweep. of the model and meaThe optimum load impedances and , respectively. The surement are

122

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 14. Simulated and measured and S -parameters results of the dual-gate CMOS LNA. The small signal gain is 10.4 dB at 2.4 GHz.

0

Fig. 12. Measured ( ) and simulated ( ) output power contours with an input power of 5 dBm at 2.4 GHz (power step of contours: 0.5 dBm).

0

Fig. 15. Photograph of the PCB for the dual-gate mixer measurement.

Fig. 13. Chip photographs of the: (a) dual-gate LNA with a chip size of 905 640 m and (b) mixer with a chip size of 920 700 m .

2

2

measured maximum output power of the dual-gate device is 12.1 dBm. With the optimum load, the measured and simulated power sweeps of the dual-gate device are plotted in Fig. 11(b) at 2.4 GHz. For the power-sweep measurement, the optimum via the impedance is matched to a load impedance of 50 output tuner of the load–pull measurement system. The measured maximum power gain and output power are 15.3 dB and 12.1 dBm, respectively. The load–pull and power-sweep results of the model prediction also agree with the measured results, and the proposed model is successfully evaluated with the dc I–V, linear -parameters, and large-signal microwave power testing.

The chip photographs of the LNA and the mixer are shown in Fig. 13(a) and (b), respectively. The chip sizes of the LNA and the mixer are 905 640 and 920 700 m , respectively. The LNA is also measured via on-wafer probing. The measured -parameters of the LNA are plotted in Fig. 14, and the dc supply voltage is 1.8 V with a dc current consumption of 7.2 mA. At 2.4 GHz, the LNA demonstrates a small-signal gain of 10.4 dB, input/output return losses of better than 10 dB, a noise figure of within 2.3 dB, and an input 1-dB compression of higher than 6 dBm. point For the mixer measurements, the differential inputs at RF and LO ports are connected to two off-chip 2.4-GHz rat-race baluns using bonding wires. The photograph of the printed circuit board (PCB) for the dual-gate mixer measurement is shown in Fig. 15, which is fabricated using FR4 PCB with a thickness of 1 mm. Two off-chip capacitors are adopted as the dc block at the differential IF outputs and the capacitances are both 15 pF. Moreover, the off-chip capacitors and resistors are mounted on the PCB for low-frequency bypass and stability. For the simulations of the dual-gate mixer, the bounding wire is modeled as an inductor of 1 nH. At an RF frequency of 2.4 GHz with an IF output frequency of 100 MHz, the simulated and measured conversion gains versus LO power are plotted in Fig. 16 for the dual-gate mixer, and the dc supply voltage is 1.1 V with a total dc current consumption of within 5 mA. The measured conversion gain is saturated

CHANG AND LIANG: 0.18- m DUAL-GATE CMOS DEVICE MODELING AND APPLICATIONS FOR RF CASCODE CIRCUITS

123

TABLE II CIRCUITS PERFORMANCE SUMMARY

Fig. 16. Simulated and measured conversion gains versus LO input power. TABLE III COMPARISONS OF THE PREVIOUSLY REPORTED MIXERS AND THIS WORK

V. CONCLUSION Fig. 17. Simulated and measured conversion gains versus IF frequency, LO dBm at 2.3 GHz. power

= 06

when the LO power is up to 6 dBm. The measured maximum conversion gain of the mixer is 8.5 dB. The difference between simulation and measurement is increased as the LO power increases up to 2 dBm. The difference is caused by the injection LO power, especially for high LO power, since the drain voltage of the lower FET of the dual-gate device is affected by the LO power, and the issue has been discussed in Section III, the mixer design. The simulated and measured noise figures of the mixer are 15.2 and 16.5 dB, respectively. The simulated and measured conversion gains versus IF frequency of from 50 to 300 MHz are plotted in Fig. 17, where the LO frequency is 2.3 GHz with is higher than 19 dBm. a power of 6 dBm. The input The measured 3-dB IF bandwidth is wider than 300 MHz. The measured LO-to-RF, LO-to-IF, RF-to-IF isolations of the mixer are better than 56, 39, and 42 dB, respectively. The performance summary of the LNA and the mixer are summarized in Table II. The simulated results agree with measured results. The proposed dual-gate model has been successfully evaluated with the linear and nonlinear RF circuits. Therefore, the proposed dual-gate model can be treated as a cascode sub-circuit cell for simplifying the RF design procedure. The comparisons of the previously reported mixers and this work are summarized in Table III. The proposed doubly balanced dual-gate mixer demonstrates good conversion gain with low dc power consumption.

This paper presents a dual-gate CMOS device modeling and extracted procedure to accomplish an RF large-signal dual-gate model. Two BSIM3v3 modes are employed in the core of the dual-gate device. In addition, the parasitic components are applied to the model to further enhance the accuracy of the model up to 15 GHz. The proposed dual-gate model has been successfully evaluated with the dc I–V, -parameters, power sweep, and load–pull measurements. Moreover, a single-stage LNA and a doubly balanced mixer are designed using the proposed dualgate model. The measured results agree with the simulated results using our proposed dual-gate model. The proposed dualgate device can be further used for the RF integrated circuit designs to avoid the uncertain parasitic effects, since the interconnection of the transistors is simplified. ACKNOWLEDGMENT The chip was fabricated by the TSMC, Hsinchu City, Taiwan, through the Chip Implementation Center (CIC), Hsinchu City, Taiwan. The authors would like to thank Prof. G. D. Vendelin, and Prof. J.-S. Fu, both with National Central University, Jungli City, Taiwan, for the discussions. Thanks also goes to the Allstron Corporation, Taoyuan, Taiwan, for providing the on-wafer RF probes. REFERENCES [1] D. K. Shaeffer, A. R. Shahani, S. S. Mohan, H. Samavati, H. Rategh, M. M. Hershenson, M. Xu, C. P. Yue, D. Eddleman, and T. H. Lee, “A 115-mW, 0.5-mm CMOS GPS Receiver with wide Dynamic-Range active filters,” IEEE J. Solid-State Circuits, vol. 33, no. 12, pp. 2219–2231, Dec. 1998.

124

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

[2] M. Zargari, D. Su, C. P. Yue, S. Rabii, D. Weber, B. Kaczynski, S. Mehta, K. Singh, S. Mendis, and B. Wooley, “A 5-GHz CMOS transceiver for IEEE 802.11a wireless LAN,” IEEE J. Solid-State Circuits, vol. 37, no. 12, pp. 1688–1694, Dec. 2002. [3] S. K. Reynolds, B. A. Floyd, T. Beukema, T. Zwick, U. Pfeiffer, and H. Ainspan, “A direct-conversion receiver IC for WCDMA mobile systems,” IEEE J. Solid-State Circuits, vol. 38, no. 9, pp. 1555–1560, Sep. 2003. [4] M.-D. Tsai, C.-S. Lin, K.-Y. Lin, and H. Wang, “A 10.8 mW low-noise amplifier in 0.35-m SiGe BiCMOS for UWB wireless receivers,” in IEEE Radio Wireless Symp. Dig., Feb. 2006, pp. 39–42. [5] M.-D. Tsai, R.-C. Liu, C.-S. Lin, and H. Wang, “A low-voltage fullyintegrated 3.5–6 GHz CMOS variable gain low noise amplifier,” in Eur. Microw. Conf. Dig., Oct. 2003, pp. 13–16. [6] S.-F. Chao, J.-J. Kuo, C.-L. Lin, M.-D. Tsai, and H. Wang, “A DC–11.5 GHz low-power, wideband amplifier using splitting-load inductive peaking technique,” IEEE Microw. Wireless Compon. Lett, vol. 18, no. 7, pp. 482–483, Jul. 2008. [7] K. W. Kobayashi, A. K. Oki, D. K. Umemoto, T. R. Block, and D. C. Streit, “A novel self-oscillating HEMT-HBT cascode VCO-Mixer using an active tunable inductor,” IEEE J. Solid-State Circuits, vol. 33, no. 6, pp. 870–876, Jun. 1998. [8] M.-F. Hung, C.-J. Kuo, and S.-Y. Lee, “A 5.25-GHz CMOS folded-cascode even-harmonic mixer for low-voltage applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 660–669, Feb. 2006. [9] V. Vidojkovic, J. V. D. Tang, A. Leeuwenburgh, and A. H. M. V. Roermund, “A low-voltage folded-switching mixer in 0.18-m CMOS,” IEEE J. Solid-State Circuits, vol. 40, no. 6, pp. 1259–1264, Jun. 2005. [10] R. Santhakumar, P. Yi, U. K. Mishra, and R. A. York, “Monolithic millimeter-wave distributed amplifiers using AlGaN/GaN HEMTs,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2008, pp. 1063–1066. [11] T. Kashiwa, T. Katoh, T. Ishida, Y. Kojima, and Y. Mitsui, “A highperformance Ka-band monolithic variable-gain amplifier using dualgate HEMTs,” IEEE Microw. Guided Wave Lett., vol. 7, no. 8, pp. 251–252, Aug. 1997. [12] R. Fujimoto, K. Kojima, and S. Otaka, “A 7-GHz 1.8-dB NF CMOS low-noise amplifier,” IEEE J. Solid-State Circuits, vol. 37, pp. 852–856, Jul. 2002. [13] P. J. Sullivan, B. A. Xavier, and W. H. Ku, “Doubly balanced dualgate CMOS mixer,” IEEE J. Solid-State Circuits, vol. 34, no. 6, pp. 878–881, Jun. 1999. [14] W. R. Deal, M. Biedenbender, P. H. Lin, J. Uyeda, M. Siddiqui, and R. Lai, “Design and analysis of broadband dual-gate balanced lownoise amplifiers,” IEEE J. Solid-State Circuits, vol. 42, no. 10, pp. 2107–2115, Oct. 2007. [15] R. S. Pengelly, Microwave Field-Effect Transistors-Theory, Design and Applications. New York: Res. Studies Press, 1982, ch. 2. [16] K. H. Liang and Y. J. Chan, “A 18-m dual-gate CMOS model for the design of 2.4 GHz low noise amplifier,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., Jun. 2006, pp. 353–356. [17] W. Liu, R. Gharpurey, M. C. Chang, U. Erdogan, R. Aggarwal, and J. P. Mattia, “RF MOSFET modeling accounting for distributed substrate and channel resistances with emphasis on the BSIM3v3 SPICE model,” in Electron Devices Meeting Tech. Dig., 1997, pp. 309–312. [18] J. J. Ou, X. Jin, I. Ma, C. Hu, and P. R. Gray, “CMOS RF modeling for GHz communication IC’s,” VLSI Technol. Dig., pp. 94–95, 1998. [19] C.-W. Kuo, C.-C. Hsiao, C.-C. Ho, and Y.-J. Chan, “Scalable largesignal model of 0.18 m CMOS process for RF power predictions,” Solid State Electron., vol. 47, pp. 77–81, 2002. [20] C.-C. Ho, C.-W. Kuo, C.-C. Hsiao, and Y.-J. Chan, “A 0.18 m p-MOSFET large-signal RF model and its application on MMIC design,” Solid State Electron., vol. 47, pp. 1117–1122, 2003. [21] S. H. M. Jen, C. C. Enz, D. R. Pehlke, M. Schroter, and B. J. Sheu, “Accurate modeling and parameter extraction for MOS transistors valid up to 10 GHz,” IEEE Trans. Electron Devices, vol. 46, no. 11, pp. 2217–2227, Nov. 1999. [22] C. W. Kuo, C. C. Hsiao, C. C. Ho, and Y. J. Chan, “Scalable largesignal model of 0.18 m CMOS process for RF power predictions,” Solid State Electron., vol. 47, pp. 77–81, 2002. [23] C. C. Ho, C. W. Kuo, C. C. Hsiao, and Y. J. Chan, “A 0.18 m p-MOSFET large-signal RF model and its application on MMIC design,” Solid State Electron., vol. 47, pp. 1117–1122, 2003.

[24] M. C. King, Z. M. Lai, C. H. Huang, C. F. Lee, M. W. Ma, C. M. Huang, Y. Chang, and A. Chin, “Modeling finger number dependence on RF noise to 10 GHz in 0.13 m node MOSFETs with 80 nm gate length,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., Jun. 2004, pp. 171–174. [25] C.-H. Chen and M. J. Deen, “A general noise and S -parameter deembedding procedure for on-wafer high-frequency noise measurements of MOSFETs,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 5, p. 1005, May 2001. [26] A. Korbel, T. Schulz, S. Mecking, and U. Langmann, “Extraction of extended BSIM3v3.2.2 model card of vertical 130 nm Si p-MOSFET for circuit simulation,” Proc. Inst. Elect. Eng.—Circuits Devices Syst., vol. 149, pp. 264–270, Aug. 2002. [27] W. Liu, X. Jin, J. Chen, M.-C. Jeng, Z. Liu, Y. Cheng, K. Chen, M. Chan, K. Hui, J. Huang, R. Tu, P. K. Ko, and C. Hu, BSIM3v3.2.2 MOSFET Model User’s Manual. Berkeley, CA: Univ. California at Berkeley, 1999. [28] D. K. Shaeffer and T. H. Lee, “A 1.5-V, 1.5-GHz CMOS low noise amplifier,” IEEE J. Solid-State Circuits, vol. 32, no. 5, pp. 745–759, May 1997. [29] P. R. Gray and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, Third ed. New York: Wiley, 1993, ch. 7. [30] K.-H. Liang, H.-Y. Chang, and Y.-J. Chan, “A 0.5–7.5 GHz ultra low-voltage low-power mixer using bulk-injection method by 0.18-m CMOS technology,” IEEE Microw. Wireless Compon. Lett, vol. 17, no. 7, pp. 531–533, Jul. 2007. [31] H. Cho and D. E. Burk, “A three-step method for the deembedding of high-frequency S -parameter measurement,” IEEE Trans. Electron Devices, vol. 38, no. 6, pp. 1371–1375, Jun. 1991. [32] M. Myslinski, W. Wiatr, and D. Schreurs, “A three-step procedure utilizing only two test structures for de-embedding transistor from on-wafer S -parameter measurements,” in Proc. Int. 15th Microw., Radar, Wireless Commun. Conf., 2004, vol. 2, pp. 674–677.

Hong-Yeh Chang (S’02–M’05) was born in Kinmen, Taiwan, in 1973. He received the B.S. and M.S. degrees in electric engineering from National Central University, Jhongli City, Taoyuan, Taiwan, in 1996 and 1998, respectively, and Ph.D. degree from the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, in 2004. From 1998 to 1999, he was with Chunghwa Telecom Laboratories, Taoyuan, Taiwan, where he was involved in the research and development of code division multiple access (CDMA) cellular phone system. In 2004, he was a Post-Doctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University, where he was involved with research on advanced millimeter-wave integrated circuits. In February 2006, he joined the faculty of the Department of Electrical Engineering, National Central University, Jhongli City, Taiwan, as an Assistant Professor. His research interests are microwave and millimeter-wave circuit and system designs. Dr. Chang is a member of Phi Tau Phi. Kung-Hao Liang was born in Taipei, Taiwan, in 1980. He received the M.S. and Ph.D. degrees in electrical engineering from National Central University, Jhongli City, Taoyuan, Taiwan, in 2004 and 2008, respectively. His doctoral dissertation concerned the RF-CMOS modeling and the low-voltage and low-power mixers design of wireless applications. Upon completion of the doctoral degree, he joined the High Speed Silicon Laboratory, University of California at Santa Barbara, where his research has been focused on the 60-GHz super-regenerative receiver for short-distance and low-power applications. Since 2009, he has been involved with the RF-modeling Program,Taiwan Semiconductor Manufacturing Company (TSMC), Hsinchu City, Taiwan, where he is responsible for advantage RF-CMOS modeling and RF-based cell development.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

125

Six-Port Gigabit Demodulator Joakim Östh, Adriana Serban, Owais, Magnus Karlsson, Shaofang Gong, Member, IEEE, Jaap Haartsen, and Peter Karlsson

Abstract—This paper presents measurement results for a six-port-based demodulator designed for a center frequency of 7.5 GHz and with a bandwidth of 1 GHz for operation in the ultra-wideband band. The demodulator includes the six-port correlator, diodes, and amplifiers needed to recover the baseband data. Measurement results show that the prototype supports data rates at 1.7 Gbit/s with bit-error rate 5 10 5 if a two-tap linear equalizer is used and bit-error rate 4 10 3 if only threshold detection is used. The measured performance of the used six-port correlator including the amplifiers is presented and their influence on the overall system performance is discussed. Limitations in the present system and possible improvements are also considered. Fig. 1. Schematic of a six-port based receiver. The six-port correlator is the circuit inside the dashed box. The part to the right of the correlator does analog processing to recover the baseband data in I and Q channels.

Index Terms—Demodulator, six-port correlator.

I. INTRODUCTION

T

HERE IS an increasing demand for high speed wireless data transmission and the use of a radio architecture, which is based on the six-port correlator is a promising solution to achieve multigigabit data rates [1]. In the U.S., the Federal Communications Commission (FCC) released the 3.1–10.6-GHz frequency spectrum in 2002 for ultra-wideband (UWB) communications [2]. However, other countries may have additional restrictions to protect other narrowband communication systems. For worldwide operation, the band between 6–9 GHz can be used for multigigabit data rates at short distances. The six-port correlator was first used in reflection coefficient measurements [3], [4], and its use as a communication device has been studied in [5]. Since then there has been much interest in the six-port correlator for communications [6]–[11] and its uses in high-data-rate communication systems [1]. However, the reported data rates from receivers based on the six-port correlator have been well below 1 Gbit/s [6], [7], [9], [12], [13]. In this paper, the performance from a prototype built for gigabit data rates is presented and discussed. II. SYSTEM DESIGN The different parts of a six-port based receiver [8] utilizing differential amplifiers are shown in Fig. 1, and the fabricated prototype is shown in Fig. 2. The analysis starts from the point Manuscript received January 19, 2010; revised August 10, 2010; accepted October 07, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. This work was supported by Sony Ericsson Communication AB and Vinnova. J. Östh, A. Serban, Owais, M. Karlsson, and S. Gong are with the Department of Science and Technology ITN, Linköping University, S-601 74 Norrköping, Sweden (e-mail: [email protected]). J. Haartsen and P. Karlsson are with Sony Ericsson Mobile Communications AB, 221 88 Lund, Sweden. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2091198

Fig. 2. Photograph of the prototype demodulator.

where the RF signal already has passed the antenna, filter, and low-noise amplifier (LNA). Hence, the RF signal enters port 2 on the six-port correlator, denoted with a “2” in Fig. 1. The continuous wave local oscillator (LO) signal enters port 1 on the six-port correlator. The RF and LO signals are assumed to have the same frequency. The building blocks of the six-port correlator are in the dashed-line box and consists of one power divider and three 90 branch-line couplers. The modulated RF signal and the continuous wave LO signal are added in the six-port correlator with different phase shifts. Each of the four outputs, denoted by “3,” “4,” “5,” and “6” are input to a nonlinear device. It may be a field-effect transistor (FET) or a diode or any other nonlinear device with a square law characteristic. In this case, a diode is used due to its low complexity. The nonlinear operation will, among other frequencies, generate the demodulated baseband signal. Amplifying and taking the difference between specified ports results in in-phase (I) and quadrature-phase (Q) data in two different paths, without any dc offset in the ideal case. A. Substrate Parameters The six-port correlator, instrumentation amplifier, and the complete demodulator were manufactured on a printed circuit

0018-9480/$26.00 © 2010 IEEE

126

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE I PCB PARAMETERS

Fig. 3. Schematic of the designed instrumentation amplifier.

board (PCB). The used substrate was Rogers 4350B. The substrate parameters are shown in Table I. B. Detector Diodes

where is the RF frequency. The RF and LO frequencies are assumed to be equal

To implement the squaring functionality required to recover the baseband signals, a matched pair of zero-bias Schottky diodes, BAT15-07LRH from Infineon Technology, Neubiberg, Germany, is used. C. Six-Port Correlator To be able to process high symbol rates, it is important that the six-port correlator has a bandwidth large enough and that gain and phase behave well to avoid too much signal distortion. A separate PCB for the six-port correlator designed for the 7–8-GHz UWB band was thus manufactured to measure its performance. The same layout is later used for integration of the six-port correlator with the other parts necessary to build a complete demodulator.

(1) (2) The complex output on port due to the input on port 1 and 2 is (3) and are the -parameters from LO input (port 1) where and RF input (port 2) to output port , where can be 3, 4, 5, or 6. The corresponding real part of is then squared if the diode is modeled with a square function. The output signal on port , , after the diode and low-pass filtering, can be expressed as

D. Instrumentation Amplifier Referring to Fig. 1, there are two instrumentation amplifiers in the system, one in the I and the other in the Q channel. The function of the instrumentation amplifiers is to amplify and recover the baseband I and Q data. Fig. 3 shows the schematic of the implemented instrumentation amplifier. The instrumentation amplifier is built around three operational amplifiers, LMH6703 from National Semiconductors, Santa Clara, CA. The operational amplifier works with current feedback. It has a bandwidth of 1.2 GHz at a voltage gain of 2 dB and a slew rate of 4200 V/ s. Both the bandwidth and slew rate are important parameters to allow for processing of high-speed data.

(4) and output from Q is The output from I is example, the I channel output is

. For

(5) III. MODELING OF OUTPUT BASEBAND I AND Q DATA The mathematical model describes the output of the six-port correlator at port 3–4 and port 5–6. The output is a function of the RF input on port 2, the LO input on port 1, and the -parameters of the six-port correlator. The modulated RF signal and the continuous wave LO signal are represented by and , respectively, and is the phase difference between them. is the and are the baseband I amplitude of the LO signal and , and Q amplitude, respectively. Angular frequency

The following can be observed for I and Q outputs. 1) The nonlinear parts in (4) and (5) contribute in two difwill introduce ferent ways, which are: a) squaring of and a dc signal and b) squaring of the baseband signals introduces nonlinearity. If , the dc offset , the nonlinearity can be eliminated, and if distortion can be eliminated. 2) Assuming the requirement for no dc offset and no nonlinearity is fulfilled, I or Q can be extracted without any

ÖSTH et al.: SIX-PORT GIGABIT DEMODULATOR

127

Fig. 4. Block diagram of the measurement setup.

crosstalk from the other channel for a suitable choice of , if the following is satisfied: where or . This shows clearly how the performance of the six-port correlator affects the output signals. The detected or estimated I and Q signals can be written in matrix form (6) Neglecting any possible dc offset and nonlinearity distortion in (5), the constants and in (6) can be calculated

signal generator E8267 from Agilent Technologies, to modulate an RF carrier at 7.5 GHz. The output power was set to 15 dBm. The modulated signal is then fed to the RF input (port 2) of the six-port correlator. A continuous wave at 7.5 GHz is generated with a vector network analyzer, VZM from Rohde&Schwartz, Munich, Germany, and the output power is set to 15 dBm and is fed to the LO port (port 1). The I and Q outputs, respectively, from the six-port-based demodulator is connected to an oscilloscope, Infiniium 5483D from Agilent Technologies, to measure the regenerated baseband I and Q data. The oscilloscope samples with 2-Gs/s and 1M data points are saved for each channel. It is important that the phase relation between the RF and LO signal is correct to avoid crosstalk between the I and Q channel, as described in Section III. Before the measurement starts, the following process is done manually to do suboptimal tuning of the phase relation. 1) Transmit M-QAM signal at the given symbol rate on the I (Q) channel. 2) Deactivate modulation on the Q (I) channel. 3) Tune the phase until the variation on the Q (I) channel is as low as possible. 4) Activate modulation on the Q (I) channel. 5) Do the measurement. The process is then repeated with I and Q interchanged in run 2 to have at least two data sets for each modulation order and symbol rate.

B. Data Processing (7) (8) In a similar way, the constants and can be calculated. In and (controlled by the value of ), an ideal case, and hence, the I (Q) channel can be detected without distortion from the Q (I) channel. In reality there is always some crosstalk ) that will distort the constellation. ( IV. MEASUREMENT SETUP AND DATA PROCESSING A. Measurement Setup Measurement on the complete system with a setup, as shown in Fig. 4, is used. To generate the baseband signal, an arbitrary waveform generator, N6031 from Agilent Technologies, Santa Clara, CA, was used. The sampling clock of the arbitrary waveform generator is 1.25 Gs/s, and hence, all the symbol rates used, will be an integer fraction of this rate, i.e., symbol rate Msymbols/s

(9)

. where The output baseband I and Q signals from the arbitrary waveform generator are fed to a vector signal generator, PSG vector

I and Q test data is generated in MATLAB with the help of a maximum length PN9 sequence, and no pulse shaping is used. The generated test data is then transmitted to the arbitrary waveform generator. The generated I and Q waveforms and the measured I and Q data from the six-port demodulator are saved to allow future processing. The measured data is processed in MATLAB to calculate bit-error rate and symbol-error rate and for plotting the data. Two different cases are considered for the bit-error-rate and symbol-error-rate calculation, with and without an equalizer. 1) Without Equalizer: In this case, threshold detection is used to detect the symbols. Before the detection, the measured and reference data are scaled and aligned in time. Measured and reference data are then down sampled to have one sample per symbol. The middle sample in each symbol is used. A complex baseband signal from the measured and reference data is then created. The complex baseband signals are then used with the MATLAB functions “qamdemod” and “demod” to demodulate the signals. Finally, the “biterr” and “symerr” functions are used to calculate the bit-error rate and symbol-error rate. 2) With Equalizer: In this case the MATLAB functions “equalize,” “lineareq,” and “lms” are utilized on the received complex baseband signal to equalize the received signal. The equalizer utilizes two taps and a training length of 512 symbols. The equalized output signal is then demodulated in the same way as when no equalizer was used. The bit-error rate and symbol-error rate are then calculated.

128

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 5. Simulated and measured performance of the six-port correlator. (a) Transfer function between port 3 and RF input. (b) Transfer function between port 4 and RF input.

Fig. 6. Measured magnitude and phase response of the instrumentation amplifier for channel 1 and 2. (a) Magnitude. (b) Phase.

V. RESULTS A. Six-Port Correlator Simulated and measured performance on the six-port correlator from RF input (port 2) to two of the output ports (port 3 and 4) is shown in Fig. 5. Fairly good agreement between simulation and measurement, as well as a good performance in the 7–8-GHz band can be observed. Fig. 7. Measured I baseband signal and its corresponding ideal transmitted I baseband signal.

B. Instrumentation Amplifier Each instrumentation amplifier has two inputs and one output, as shown in Fig. 3. Measured forward transfer function from each of the inputs to the output is shown in Fig. 6 when the other input is terminated. The observed gain difference between the two paths will mainly contribute to the dc offset, but may also be a small contribution to the crosstalk. From measurement of the complete demodulator, the peak-to-peak output amplitude on I or Q channel is about 300 mV and from the same measurement the rise time is estimated. The rise time is estimated from the time required to go from a low to a high value on the I or Q baseband signal. The measured rise time is from 2 to 3 ns. A first estimation on the maximum symbol rate can be based on the estimated rise time [14] (10) The shortest rise time was measured to 2 ns. Therefore, the symbol rate should not be more than Msymbols/s

(11)

Other factors as over/under-shot and ringing and allowed variation of the detected amplitude level may further lower the maximum symbol rate. C. Complete Demodulator Fig. 7 shows the measured I baseband signal after the demodulator. Comparing it with the ideal transmitted waveform, the measured and transmitted signals are shifted and scaled to facilitate the comparison. The symbol rate was 250 Msymbols/s and 16-QAM modulation was used. At this symbol rate, the transmitted and received signals follow each other well, and hence, allow for a low bit-error rate and symbol-error rate. It can, however, be observed that there are some overshoots and ringing when the signal changes level and, depending on the value at the sampling point, some spreading of the detected amplitude levels is observed. This spreading may introduce miss detection if a higher modulation order is used or if the spreading is substantial. Measured bit-error-rate and symbol-error-rate results for 16-QAM signals at 250, 312.5, and 416.67 Msymbols/s are

ÖSTH et al.: SIX-PORT GIGABIT DEMODULATOR

129

TABLE II MEASURED BIT-ERROR RATE AND SYMBOL-ERROR RATE FOR 16-QAM SIGNALS

TABLE III MEASURED EVM FOR 16-QAM SIGNALS

Fig. 9. Measured constellations for 64-QAM signal at 250 Msymbols/s, i.e., data rate 1.25 Gbit/s. (a) Without equalizer. (b) With equalizer.

VI. DISCUSSION

Fig. 8. Measured constellation for 16-QAM signal at 416.67 Msymbols/s, i.e., data rate 1.67 Gbit/s. (a) Without equalizer. (b) With equalizer.

summarized in Table II. The corresponding error vector magnitude (EVM) is presented in Table III. The root mean square (rms) EVM is used and the reference signal is used for normalization. From Table II, it can be seen that at a data rate of 1.67 Gbit/s (416.67 Msymbols/s and 16-QAM) the demodulator works with an acceptable bit-error-rate/symbol-error-rate performance. Fig. 8 shows a typical constellation for a 16-QAM signal at 416.67 Msymbols/s without and with the equalizer. It can be seen that the equalizer improves the constellation. Fig. 9 shows the constellation when the modulation order is increased to 64-QAM. The performance is much worse, despite the lower symbol rate compared to the 16-QAM case. Possible reasons for these limitations can be listed as follows: • bandwidth of the amplifier; • over/under shoot and decay time of the step response in the instrument amplifier (IA); • crosstalk between I and Q channel; • nonlinearity distortion introduced by the diodes; • slew-rate (SR) limitations in the IA; • nonidealities due to real components and measurement setup.

During the measurement of the demodulator, it was observed that the phase between RF and LO signal is drifting with time. This is one contribution to the spreading of the constellation points because the location of points is dependent of the relative phase between RF and LO signal. The phase difference also gives crosstalk that limits the maximum QAM order that can be used if I (Q) channel is estimated without any concern of the value on the Q (I) channel. If both the received I and Q channel are considered, the transmitted I (Q) channel can be recovered for any phase difference by inverting the matrix given in (6), and are known or can this requires that the constants be estimated. No carrier recovery circuit is utilized in this work, but the receiver architecture used in this paper can be utilized to support carrier recovery, as shown in [15]. Conversion gain, phase noise, and signal-to-noise (SNR) all are dependent of the LO power used [16]. Future measurements where both the LO and RF input power is changed may give additional information how the bit-error rate, dynamic range, and other performance metrics changes as a function of input power. The effect of a real channel has not been considered; emphasis was instead on the performance of the RF and analog circuitry in the demodulator to support high data rates. Some improvements can be made on both the measurement setup, such as high quality cables for the RF and LO signal and better synchronization between the instruments. On the PCB side, it is possible to do some miniaturization that may improve the performance. A. Measurement Limitations Due to the use of an integer fraction sample clock of the arbitrary waveform generator, it was impossible to test symbol rates between 416.67–625 Msymbols/s. For example, to get 2 Gbit/s, a 16-QAM signal with a symbol rate of 500 Msymbols/s is required. Another source of error is that the actual output from the arbitrary waveform generator is not an ideal waveform due to bandwidth limitations and as the symbol rate increases the actual waveform becomes less and less ideal. Observations of the correlation between the transmitted and received baseband signals also indicated that the optimum sampling point changes over time when high symbol rates are used, indicating that the sampling clocks on the arbitrary waveform generator and the oscilloscope are not completely synchronized.

130

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 10. Distortion introduced in the constellation due to deviation from the optimum sampling point instances. (a) At optimum sampling point. (b) 5% deviation from optimum sampling point.

B. Effect of Sampling Point Instance If the input symbols change faster or close to the rise time of the system, the output waveform starts to resemble a triangular waveform. If the input symbol rate is slower than the rise time of the system, the output waveforms look more like a square wave. Therefore, in the vicinity of a sample point, the amplitude is almost constant if the symbol rate is low. On the other hand, if the symbol rate is high, the amplitude may change in the vicinity of the sampling point. The effect of sampling a symbol at: 1) the optimum point and 2) at 5% offset was simulated for a signal at 625 Msymbols/s, as shown in Fig. 10. The effect is not negligible if the symbol time is close to, or shorter than the rise time on I and Q outputs. This contributes to the spreading of the amplitude levels. An estimation of the actual error of the amplitude in the real system is estimated based on the oscilloscope sample rate, 2 Gs/s, i.e., a ns and the measured rise time of sample time of the I (Q) baseband output, which was measured to be between 2–3 ns. The maximum error occurs for the fastest rise time and is (12)

The measured peak-to-peak amplitude was about 300 mV, may introduce for an offset of 5%, mV or an an error of error of 1.25%. The worst case error is estimated to occur for and gives an error of 12.5%. C. Performance Above 2 Gbit/s Measurements with a 16-QAM signal at the maximum symbol rate of 625 Msymbols/s, i.e., 2.5 Gbit/s, were also conducted. At this symbol rate, both the bandwidth of the used six-port correlator and the limitations in the instrumentation amplifier makes it hard to detect the symbols without any processing of the data. Utilizing the equalizer function, it is, however, possible to achieve a quite good constellation, as shown in Fig. 11. At this speed, the actual sampling point is sensitive since the instrumentation amplifier does not have enough time to come close to its final value before the next symbol. Sampling at a higher speed may give additional insight

Fig. 11. Measured constellation for 16-QAM signal at 625 Msymbols/s, i.e., data rate 2.5 Gbit/s. (a) Without equalizer. (b) With equalizer.

into the signal waveform. Unfortunately, the used oscilloscope only allows for 3.2 samples/symbol when the symbol rate is 625 Msamples/s. VII. CONCLUSION A prototype demodulator operating at 7.5 GHz with a bandwidth of 1 GHz based on the six-port correlator was manufactured. Measurements were conducted to verify its functionality and performance. Demodulation of a 16-QAM signal with a data rate of 1.67 Gbit/s is possible with a bit-error rate about 4 10 if simple threshold detection is used. Utilizing a two tap linear equalizer, the bit-error rate is about 4.8 10 . The present limitation in the system is due to bandwidth limitation in the used operational amplifiers and not in the six-port correlator. It is demonstrated that a demodulator built with the six-port correlator can process gigabit data rates. It may find use in high data-rate applications, e.g., in the 7–8-GHz UWB band. ACKNOWLEDGMENT The authors acknowledge Agilent Technologies Inc., Santa Clara, CA, for their long-term technical support to Linköping University, Norrköping, Sweden. The authors further acknowledge G. Knutsson, Linköping University, for his help and support with manufacturing of prototypes. REFERENCES [1] S. Gong, M. Karlsson, A. Serban, J. Osth, O. Owais, J. Haartsen, and P. Karlsson, “Radio architecture for parallel processing of extremely high speed data,” in Proc. IEEE Int. Ultra-Wideband Conf., Vancouver, BC, Canada, Sep. 2009, pp. 433–437. [2] “Revision of part 15 of the Commissions rules regarding ultra-wideband transmission systems, first report and order,” FCC, Washington, DC, FCC ET Docket 98-153, Feb. 2002. [3] G. F. Engen and C. A. Hoer, “Application of an arbitrary 6-port junction to power-measurement problems,” IEEE Trans. Instrum. Meas., vol. IM-21, no. 4, pp. 470–474, Nov. 1972. [4] G. Engen, “A (historical) review of the six-port measurement technique,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2414–2417, Dec. 1997. [5] J. Li, R. Bosisio, and K. Wu, “Computer and measurement simulation of a new digital receiver operating directly at millimeter-wave frequencies,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 12, pp. 2766–2772, Dec. 1995. [6] S. Tatu, E. Moldovan, K. Wu, and R. Bosisio, “A new direct millimeter wave six-port receiver,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2517–2522, Dec. 2001.

ÖSTH et al.: SIX-PORT GIGABIT DEMODULATOR

131

Ka

[7] S. Tatu, E. Moldovan, K. Wu, R. Bosisio, and T. Denidni, “ -band analog front-end for software-defined direct conversion receiver,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2768–2776, Sep. 2005. [8] P. Hakansson, D. Wang, and S. Gong, “An ultra wideband I/Q demodulator covering from 3.1 to 4.8 GHz,” Trans. Electron. Signal Process., vol. 2, no. 1, pp. 111–116, 2008. [9] P. Hakansson and S. Gong, “Ultra-wideband six-port transmitter and receiver pair 3.1–4.8 GHz,” in Proc. Asia–Pacific Microw. Conf., 2008, pp. 1–4. [10] N. Seman, M. E. Bialkowski, S. Z. Ibrahim, and A. A. Bakar, “Design of an integrated correlator for application in ultra wideband six-port transceivers,” in IEEE Int. Antennas Propag. Soc. Symp., 2009, pp. 1–4. [11] J. C. Schiel, S. O. Tatu, K. Wu, and G. Bosisio, “Six-port direct digital receiver (SPDR) and standard direct receiver (SDR) results for QPSK modulation at high speeds,” in IEEE MTT-S Int. Microw. Symp. Dig., 2002, pp. 931–934. [12] E. Djoumessi and K. Wu, “Tunable multi-band direct conversion receiver for cognitive radio systems,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 217–220. [13] E. E. Djoumessi, S. O. Tatu, and K. Wu, “Frequency-agile dual-band direct conversion receiver for cognitive radio systems,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 1, pp. 87–94, Jan. 2010. [14] J. N. Downing, Fiber Optic Communications. Clifton Park, NY: Cengage Learning, 2005. [15] S. O. Tatu, E. Moldovan, G. Brehm, K. Wu, and R. G. Bosisio, “ -band direct digital receiver,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 11, pp. 2436–2442, Nov. 2002. [16] S. M. Winter, H. J. Ehm, A. Koelpin, and R. Weigel, “Six-port receiver local oscillator power selection for maximum output SNR,” in IEEE Radio Wireless Symp., 2008, pp. 151–154.

Ka

Joakim Östh was born in Bollnäs, Sweden, in 1980. He received the M.Sc. degree from Linköping University, Norrköping, Sweden, in 2006, and is currently working toward the Ph.D. degree in communication electronics from Linköping University. From 2006 to 2008, he was involved with hardware and software development with Qamcom Technology AB, Gothenburg, Sweden. His main research interest has been communication electronics including RF design, wireless communications, and high-speed data transmissions.

Adriana Serban received the M.Sc. degree in electronic engineering from Politehnica University, Bucharest, Romania, in 1978, and the Ph.D. degree in communication electronics from Linköping University, Norrköping, Sweden, in 2010. From 1981 to 1990, she was with the Microelectronica Institute, Bucharest, Romania, as a Principal Engineer, where she was involved in mixed integrated circuit design. From 1992 to 2002, she was with Siemens AG, Munich, Germany, and Sicon AB, Linköping, Sweden, as an Analog and Mixed Signal Integrated Circuit Senior Design Engineer. Since 2002, she has been a Lecturer with Linköping University, Norrköping, Sweden, where she teaches analog/digital system design and RF circuit design. Her main research interest has been RF circuit design and high-speed integrated circuit design.

Owais received the B.E. and Master degrees from the University of Engineering and Technology, Peshawar, Pakistan, in 1996 and 2000, respectively, and is currently working toward the Ph.D. degree in communication electronics at Linköping University, Norrköping, Sweden. He is currently an Assistant Professor with the Comsats Institute of Information Technology, Abbotabad, Pakistan. His main research interest has been Antenna design and high-speed data transmissions.

Magnus Karlsson was born in Västervik, Sweden, in 1977. He received the M.Sc., Licentiate of Engineering, and Ph.D. degrees from Linköping University, Norrköping, Sweden, in 2002, 2005, and 2008, respectively. In 2003, he joined the Communication Electronics Research Group, Linköping University, where he is currently a Senior Researcher. His main research involves wideband antenna techniques, wideband transceiver front-ends, and wireless communications.

Shaofang Gong (A’03–M’03) was born in Shanghai, China, in 1960. He received the B.Sc. degree from Fudan University, Shanghai, China, in 1982, and the Licentiate of Engineering and Ph.D. degrees from Linköping University, Norrköping, Sweden, in 1988 and 1990, respectively. From 1991 to 1999, he was a Senior Researcher with the Microelectronic Institute–Acreo, Norrköping, Sweden. From 2000 to 2001, he was the Chief Technology Officier (CTO) with a spin-off company of the Microelectronic Institute–Acreo. Since 2002, he has been a Full Professor of communication electronics with Linköping University, Norrköping, Sweden. His main research interest has been communication electronics, including RF design, wireless communications, and high-speed data transmissions.

Jaap Haartsen received the Master and Ph.D. degrees in electrical engineering (both with honors) from the Delft University of Technology, Delft, The Netherlands, in 1986 and 1990, respectively. He has been active in the area of wireless communications for the past 24 years. He began his industrial career in the U.S. in early 1991 when he joined the Research Department on Advanced Mobile Systems, Ericsson-GE JV, Raleigh–Durham, NC. In mid–1993, he became involved with indoor communication systems in Lund, Sweden. In 1994, he laid the foundations for the system that later became known as Bluetooth wireless technology. In 1997, he returned to Holland. He played an active role in the creation of the Bluetooth Special Interest Group (SIG), which was founded in February 1998, and served as chairman for the SIG air protocol specifications group from 1998 to 2000, driving the standardization of the Bluetooth radio interface. During that time, he played an important role in obtaining worldwide regulatory approval for Bluetooth technology. From 2001 to Fall 2004, he served as Chief Scientist with Ericsson Technology Licensing AB, a company fully dedicated to Bluetooth Internet Protocol (IP). In 2007, he joined Sony Ericsson JV, as an Expert on wireless systems. In March 2010, he left Ericsson JV and became Chief Technical Officer (CTO) of Tonalite BV, a privately owned company that addresses wireless wearable products. From 2000 to 2008, in parallel with his industrial work, he was a Part-Time Professor with the University of Twente, where he taught and led research in the area of wireless communications. He has authored or coauthored numerous times in major journals and magazines. He holds over 100 patents in the area of mobile and local radio communications. Dr. Haartsen has been the recpient of several awards related to his title as “Inventor of Bluetooth.”

Peter Karlsson received the Ph.D. degree in applied electronics from the Lund Institute of Technology, Lund, Sweden, in 1995. He then joined Telia Research, where he was involved in the design and analysis of high-capacity broadband radio communications systems. In 2000, he was a Research Fellow with the University of Bristol, in 2001, he was Manager of Mobile System Innovation, Telia Research, and then became an Expert with of radio communications with TeliaSonera in 2002. In 2007, he joined Sony Ericsson as a Technology Strategy Manager with the CTO office, and since 2009, is the Head of the research and technology communications and networking focus area. He has authored or coauthored over 70 conference and journal papers in the mobile and wireless communications area.

132

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Differential Amplifier Characterization Using Mixed-Mode Scattering Parameters Obtained From True and Virtual Differential Measurements Oliver Schmitz, Student Member, IEEE, Sven Karsten Hampel, Student Member, IEEE, Hanno Rabe, Tobias Reinecke, and Ilona Rolfes, Member, IEEE

Abstract—This paper examines the differences in large signal mixed-mode scattering parameter characterization of differential amplifiers arising from virtual and true differential probing. The analysis is carried out by means of differential gain compression curves obtained from exemplary amplifier test assemblies with variable common mode rejection ratio. Based on analytical derivations involving basic nonlinear circuit theory, mathematical closed form transfer functions of these amplifiers are presented that enable an a-priori estimation of the occuring measurement error. Numerical simulations complete the theoretical investigations by giving additional physical insights onto the individual amplifier nodal voltages effecting the compression. Experimental results obtained from true and virtual differential compression curve measurements of the considered amplifier topologies are finally compared to the results gained from the theoretical considerations. Based on these results, the reason for the deviation between virtual and true differential measurements is addressed and upper and lower bounds for these deviations are given that are in accordance with the results reported in literature. Index Terms—Differential amplifiers, gain compression, measurement, mixed-mode, scattering parameters, true differential, virtual differential.

I. INTRODUCTION

D

IFFERENTIAL circuit and device topologies are more and more replacing traditional single-ended architectures in many wireless system applications. This is due to the fact that differential topologies are generally more immune to electromagnetic coupling and feature improved even-order linearity. In addition, the inherently available voltage swing is doubled, which is a very important aspect, especially for highly integrated circuits, which have to cope with very low supply voltages. One way of characterizing such differential devices is to simply measure the individual single-ended scattering parameters either by making subsequent measurements with a conventional two-port vector network analyzer (VNA) or

Manuscript received June 03, 2010; revised September 15, 2010; accepted September 24, 2010. Date of publication October 21, 2010; date of current version January 12, 2011. The paper is based on the content of a paper in the European Microwave Conference, Rome, Italy, September 29–October 1, 2009. O. Schmitz, S. Hampel, H. Rabe, and T. Reinecke are with the Institut für Hochfrequenztechnik und Funksysteme, Leibniz Universität Hannover, 30167 Hannover, Germany (e-mail: [email protected]; [email protected]; [email protected]). I. Rolfes is with the Lehrstuhl für Hochfrequenzsysteme, Ruhr-Universität Bochum, 44780 Bochum, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2084095

by employing a multi-port VNA capable of providing all single-ended scattering parameters simultaneously. However, single-ended scattering parameters do not directly describe the device-under-test (DUT) operating mode, since differential devices are intended to operate in differential mode. The use of supplemental external equipment, such as baluns or hybrids, together with a two-port VNA allows for differential and common mode scattering parameter characterization. By solely involving baluns however, this method does not deliver the entire DUT information, since it is not possible to characterize the common mode behaviour. A full system characterization employing hybrids leads to a multitude of measurements that have to be carried out. Furthermore, limited accuracy and the need for compensation and deembedding procedures has to be taken into account for both of these measurement procedures. A method for full, mixed mode DUT characterization (including differential and common mode as well as conversion from one mode into the other) with simply using a two-port VNA without any additional external equipment has been shown in [1] for the first time. This so-called virtual differential measurement method includes subsequent single-ended scattering parameter measurements and mathematical post processing. Here, only one signal source is needed that is switched between the individual single-ended ports in order to virtually form a differential probing stimulus. The validity of this method has been proven in [2] for active devices operating in their linear region. The underlying theory has been extended to really generalized DUT configurations within the past years (see [3] and [4]). However, limited accuracy compared to scattering parameters obtained from true differential and common mode measurements has been observed. In [2], true mode characterization had been carried out with a so-called pure mode network analyzer [5] capable of providing probing stimuli with appropriate amplitude and phase relations and therefore eliminating the need for additional external equipment such as baluns or hybrids. Apart from possible inaccuracies appearing for small signal characterization [2], heavy deviations between virtual and true differential measurements have been reported for large signal measurements, especially in terms of gain compression curve characterization of active differential devices. These effects have first been investigated in [6], restricted on the forward of an exemplary balanced amplifier. Here, no siggain nificant deviation in and could be observed, no matter if the considered DUT was probed virtually or

0018-9480/$26.00 © 2010 IEEE

SCHMITZ et al.: DIFFERENTIAL AMPLIFIER CHARACTERIZATION USING MIXED-MODE SCATTERING PARAMETERS

truely differential. However, a significant difference in terms and has been discernible. Further investiof gations by means of cascaded differential amplifier building blocks have been carried out in [7]. In case of a nonlinear amplifier with no common mode rejection in front of a linear one with large common mode rejection, all four parameters revealed strong deviations between true and virtual mode probing. Although being the fundamental work on this topic, the results presented therein merely reflect individual cases incidentally dedicated to the underlying circuit topology, while the question of the fundamental mismatch between virtual and true differential measurements for general balanced amplifiers is still not answered yet. Based on the initial results presented in [8], this article addresses the open question whether the deviation in true and virtual differential mixed-mode scattering parameter measurements can systematically be dedicated to particular circuit or DUT characteristics and whether these deviations occur within certain bounds. The answer to these questions will finally clarify whether a differential amplifier should necessarily be characterized truely differential or whether it is sufficient to measure virtually differential. To answer those questions, a general behavioural model is needed that is capable of representing the most common differential amplifier circuit topologies. Starting from an emitter degenerated common emitter amplifier, a general block diagram of a nonlinear multiple-input-multiple-output (MIMO) system is derived in Section II serving as system-theoretical representation for a huge class of amplifier topologies. This block diagram is investigated analytically in Section III. Closed form expressions relating input and output quantities are derived, that enable direct calculation of mixed-mode scattering parameters in cases of true and virtual differential probing. Section IV then presents selected results as a-priori estimation of the expected measurement error. Theoretical bounds for the measurement error are derived that are finally verified by experimental results of test circuit assemblies in Section V. The article will close with a brief conclusion in Section VI, answering where and why a true differential probing of balanced systems is necessary.

133

Fig. 1. Schematic and idealized small signal equivalent circuit for resistive emitter degeneration. (a) Schematic. (b) Equivalent circuit.

wherein denotes the transistor transconductance and and represent the collector and emitter resistance, respectively. This linearized model can further be extended by the inclusion of nonlinear effects. Besides the nonlinear relations between voltage and charge in case of capacitances or between voltage and current in case of resistances, the predominant source of nonlinearity within weakly nonlinear active circuits is the nonlinearity of the input voltage to output current relation of the individual transistor device. This effect can be considered within a power series expansion of the underlying transconductance [10] according to

(2) with being the sum of AC and DC currents, being being the the DC voltage in the given operating point and AC voltage that steers the transconductance. of Fig. 1(b), Besides the small signal transconductance corresponding to higher order transconductance terms (3)

II. MODEL DERIVATION In order to derive a model that is capable of explaining possible differences in the compression curve characterization of differential amplifier stages with true or virtual differential probing stimuli, a closer look shall be taken into one of the key elements within almost every differential amplifier implementation - the differential pair. To keep it as simple as possible, the focus shall first be the basic circuit topology each differential pair is comprised of. Therefore, Fig. 1 shows the schematic and the idealized small signal equivalent circuit of a so-called emitter degenerated common-emitter circuit, which is basically equivalent to a source degenerated common-source stage in MOS technology. As indicated by Fig. 1(b), this circuit can be considered as series-feedback stage with a linear voltage transfer function according to [9] (1)

are involved in generating higher order output currents. Certainly, these currents have to be accounted for when investigating amplifier compression, so that the linearized model of Fig. 1(b) has to be extended by consideration of (2) and (3). In a first approximation it shall further be concentrated on third order distortion effects, since they are first and foremost responsible for compression behaviour, especially at 1 dB compression level [11]. This restriction is frequently made in the relevant literature [12], [13] and finally enables the compact and comprehensive analytical investigation, that is presented in Section III. Consequently, the nonlinear feedback system depicted in Fig. 2 is obtained, which is subsequently defined in voltage, current and again voltage domain. The depicted model involves a third in parallel with the first order order transconductance term , whose first and third order curtransconductance rents are summed and establish a negative feedback via the feed-

134

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 2. Nonlinear feedback system arising from emitter degenerated amplifier.

TABLE I DIMENSIONLESS COEFFICIENTS OF NONLINEAR FEEDBACK SYSTEM

Fig. 3. Nonlinear feedback system arising from a bipolar differential pair.

back resistance . The output voltage evolves from the multiand inversion. plication of these output currents with The block diagram representing the differential counterpart of the emitter degenerated common emitter configuration, the differential pair, results from duplication of the forward path in Fig. 2 with the feedback path remaining unaffected as shared path of both input and output nodes. In order to obtain a model completely defined in one single domain, the dimensionless paand shall further be introduced with being rameters being the third order voltage transfer functhe linear and tions of the individual differential pair parts (the two involved being the dimensionless feedbipolar transistors) as well as back factor. Their physical representations are given in Table I. In this way, resistances and transconductances are represented by dimensionless terms and the obtained model is completely defined in voltage domain. The resulting nonlinear system is depicted in Fig. 3. It is a nonlinear MIMO system, since it involves two input and two output ports, which can also be regarded as one differential or common mode input and one differential or common mode output port. This model can now be investigated by means of time-domain based numerical simulations or can be solved directly after applying further analytical considerations. Here, both procedures share the common objective to obtain signal level dependent transfer functions in order to theoretically analyze compression behaviour for true and virtual differential probing. The essential analytical steps shall be presented within the following section. III. ANALYTICAL INVESTIGATIONS In case of a true differential stimulus with and presenting a proper 180 phase shift, the differential output signal can easily be calculated as a function of the differential input,

Fig. 4. Subsystem for calculating E as function of V

.

since the feedback in Fig. 3 is not existent (the output signals are summed in opposite phase at the summing point). This simply and relating input and leaves the polynomial given by output quantities. In case of a virtual differential or subsequent single-ended stimulus at the input, the feedback path has to be accounted for. In order to derive a closed form analytical input-output relation, it is therefore necessary to gradually divide the nonlinear MIMO system in Fig. 3 into single subsystems, whose analytical transfer functions can successively be solved for by means of a power series expansion approach. Fig. 4 shows the first subsystem, which allows to derive the auxiliary term already introduced in Fig. 3 as a function of . The resulting subsystem is comprised of a linear transfer and a polynomial given by and element given by defining the negative feedback path. In order to derive as a , it is now necessary to map this subsystem function of onto a new target polynomial enabling the deduction of closed form expressions. A seventh order polynomial according to (4) is proposed to be an accurate approximation of the original subsystem defined by and . A second auxiliary term (see Fig. 4) is introduced, so that can be stated as: (5) with

given by (6)

Combining (4) and (6) and applying the obtained result to (5) yields

(7) With the expansion of (7) and subsequent combination with the to can now be determined by target polynomial of (4), means of equating the individual coefficients. Based on these reat port sults, the transfer function relating the input voltage at port 3 shall be calculated. This 1 to the output voltage derivation is based on the block diagram depicted in Fig. 5. Here, to composes the the previously derived polynomial with feedback path, whereas the polynomial given by and con-

SCHMITZ et al.: DIFFERENTIAL AMPLIFIER CHARACTERIZATION USING MIXED-MODE SCATTERING PARAMETERS

Fig. 5. Subsystem for calculating V

as function of V

135

Fig. 6. Subsystem for calculating V

as function of V

Fig. 7. Subsystem for calculating V

as function of V

.

.

stitutes the forward path. The procedure is similar to the derivations presented in (4)–(7). Again, an auxiliary term is defined behind the summing point in Fig. 5. This term can be denoted as

(8) where

can be written as

.

and the target polynomial (14) (9)

The target polynomial combining by

and

shall be given

(10) In order to determine the individual coefficients to , (8) and (10) are combined and the obtained result is applied to (9). The expansion of the resulting equation and subsequent comparison of coefficients yields:

Equations (12) and (14) are combined and the obtained result is applied to (13). Equating the coefficients with those of the target to . polynomial yields to can be investiFinally the transfer function of gated. As indicated by Fig. 7, the resulting system consists of a to and to . cascade of polynomials relating The target polynomial is given by (15) in (14) by the expression given in (10) deSubstituting to livers the final coefficients

(11) with being the linear open loop voltage gain of the individual single-ended parts of the system of Fig. 3. For a complete analytical description of the nonlinear MIMO system in case of a single ended excitation at port 1, the transfer functo is needed as well. For this reason, the tion relating to shall be investigated in a first transfer function of step. The corresponding block diagram is given in Fig. 6. Three equations can be denoted (12) (13)

(16) Thus, the analytical description of the nonlinear MIMO system of Fig. 3 in case of a single-ended excitation at port 1 is completely derived. Due to the presumed symmetry of the given to corresponds system, the transfer function relating to , while the polynomial asto the function relating with is equivalent to the polynomial comsociating and . This has to be considered for the calculabining tion of the output signal when the system is excited single-ended at port 2.

136

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 8. Single stage differential amplifier topologies evolving from different common mode rejection ratios.

IV. MODEL-BASED ANALYSIS The applicability of the model in Fig. 3 to commonly used amplifier topologies shall be investigated in a next step. For reasons of simplicity the following analysis shall be focused on single-stage differential systems without mode-conversion. The latter has already been implied throughout the derivation of the model in Section II, since the two single-ended parts of the differential amplifier have both been described by the same polynomial involving and . The fundamental system behaviour can now be defined by a proper choice of the dimensionless parameters. The voltage gain of the individual single-ended amplifiers (single transistor devices) involved is determined by . Each particular amplifier linearity in terms of 1 dB compression and third order intermodulation distortion can be fixed by choice (see [11]). Besides voltage gain and linearity of the of single amplifiers, the common mode rejection of the resulting differential amplifier is another important behavioural characteristic. Usually it is specified in terms of the common mode rejection ratio (CMRR), which represents the ratio of differential and common mode voltage gain. For the model in Fig. 3 the logarithmic CMRR can be stated as (17) and Thus, fixing the individual dimensionless parameters , decides for a certain amplifier architecture and at the same time for a certain electrical behaviour. Corresponding to the chosen CMRR, these topologies range from a pseudo differendB up to an ideal, current source tial amplifier with . As depicted in Fig. 8, biased, differential pair for the resistively degenerated differential pair with the individual bipolar transistor emitter connections sharing a common resistor to ground can be found between those two limiting cases. , the analytical deFor all feedback factors scription given in the previous section simply represents an adequate approximation, since instead of seventh order polynomials an infinite power series would have been necessary for an exact transfer function derivation. For the two limiting cases, the pseudo and ideal differential amplifier, however, an exact solution can be given with respect to the model in Fig. 3.

In case of a pseudo differential amplifier, the excitation of port 1 solely results in an output signal at port 3, while probing at port 2 leads to an output signal at port 4. Here the input voltand the output voltages are directly ages und . The related by means of the polynomial given by resulting system coefficients arising from (11) and (16) with can then be stated as

(18) For an ideal differential pair with to and to can be derived as

, the coefficients

(19) Before the analysis can start, the definition of the different probing stimuli with respect to a uniform power normalization has to be clarified. Therefore, Fig. 9 shows an exemplary differential DUT that is probed by a single-ended stimulus with in Fig. 9(a) and with a true differential stimulus amplitude in Fig. 9(b). Since represents the corwith amplitude responding single-ended reference impedance, the differential DUT is probed with an input power of (20) in case of virtual differential probing in Fig. 9(a). This power can be thought of as being split into the corresponding differential and common-mode input power

(21)

SCHMITZ et al.: DIFFERENTIAL AMPLIFIER CHARACTERIZATION USING MIXED-MODE SCATTERING PARAMETERS

137

, which is identical to The differential voltage gain assuming ideally matched input and output ports, can now be denoted as

(27) The term in (27) denotes the gain compression curve of a pseudo differential pair (index p) at frequency over input voltage obtained from true differential probing (index t). The input referred 1 dB compression point can now be computed by (28) leading to Fig. 9. Probing voltage waveforms serving as reference for a uniform power level normalization. (a) Virtual differential. (b) True differential.

In order to guarantee a uniform reference, the amplitude of the true differential probing stimuli in Fig. 9(b) is chosen in a way that the differential input power equals the single-ended input power

dB

(29)

In a next step the problem shall be investigated for virtual differential probing. Port 1 is therefore excited single-ended with while port 2 remains unaffected. Considering the amplitude coefficients of (18) yields

(22) (30)

thus leading to (23) This normalization ensures that the DUT is probed with the same power in virtual differential and true differential mode. Alternatively it can be postulated that

for the output voltages at port 1 and port 2. Thus, for the singleended scattering parameters (index v) according to the port notation of Fig. 3 it follows:

(24)

(31)

ensuring that the differential part of the single-ended input power equals the amount of power contained within the true differential stimulus. This kind of normalization has been adapted in [6] and [7] resulting to

Exciting port 2 yields similar results for the remaining single ended scattering parameters

(25)

(32)

However, the postulation introduced in (22) represents the normalization as it has been implemented in the pure mode VNA used to obtain the experimental results in the next section. It shall therefore serve as background for the following analysis. The analysis starts with the pseudo differential pair probed by a true differential stimulus. Therefore, port 1 and port 2 are exwith amplitude cited with a sinusoidal voltage at frequency and respectively, leading to as proposed by (23). The resulting output port voltages at fundacan then be stated as mental frequency

According to the superposition method proposed in [1] the voltage gain compression curve of the pseudo diffrential pair obtained in virtual differential mode can be written as

(33) The input referred 1 dB compression point can now be obtained in the same way as has been done in (28) leading to

(26)

dB

(34)

138

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 10. Nodal voltages in case of true and virtual differential probing of a pseudo differential pair.

Fig. 12. Nodal voltages in case of true and virtual differential probing of an ideal differential pair.

Fig. 11. Normalized compression curves obtained for true and virtual differential probing in case of a pseudo differential pair.

Fig. 13. Normalized compression curves obtained for true and virtual differential probing in case of an ideal differential pair.

The results in (29) and (34) already show that a virtual differential probing leads to an underestimation of the input referred . This be1 dB voltage compression point by a factor of comes obvious when the corresponding nodal voltages of the different probing stimuli are examined in detail. Fig. 10 shows the time-domain waveforms in front of and behind the feedback summing point obtained from simulations. Since the feedback path is not existent in case of a pseudo dif, the virtual and true differential ferential amplifier probing stimuli behind the summing point keep their amplitudes. For the same power signal level, this results in probing exactly the same single-ended physical system (represented by and ) either with amplitude or with amplitude . If the calculated voltage compression curves of (27) and (33) are now transferred into power referred compression curves with respect to a given reference impedance , the normalized characteristics of Fig. 11 are obtained for true and virtual differential probing. Here, the analytically derived results are compared to those obtained from numerical simulation with Matlab. Both approaches yield identical results and indicate an underestimation of the input referred 1 dB compression point by 3 dB with dB in case of virtual differential probing. This dB is in accurate compliance with the factor of separating the voltage referred compression points of (29) and (34). The same analysis shall now be carried out for the ideal differential pair. In case of a true differential stimulus, the compression curve of (27) still holds, since solely changing the common mode rejection does not effect the true differential operating

mode. Resorting to the coefficients of (19) in case of a virtual differential stimulus however, finally yields

(35) as voltage domain gain compression curve for the ideal differential amplifier (index i), leading to dB

(36)

as input referred 1 dB compression point. As can be seen, this time the virtual differential computation method overestimates the compression point by a factor of . Once again, the nodal voltage waveforms offer additional insight. According to Fig. 12 the single-ended amplitude is heavily attenuated through its way passing the feedback summing node, so that for remains for probing a uniform power signal level only the individual single-ended parts of the differential amplifier. The feedback entirely cancels out the common mode part included within the single ended excitation. Fig. 13 compares the corresponding power referred gain compression curves obtained analytically and taken from numerical simulations. Here, a virtual differential characterization results in a shift dB with dB between virtual and true of dB differential probing.

SCHMITZ et al.: DIFFERENTIAL AMPLIFIER CHARACTERIZATION USING MIXED-MODE SCATTERING PARAMETERS

139

Fig. 15. Compression curves of differential bipolar amplifier measured at and R . 500 MHz for R

= 0

Fig. 14. PCB photograph and schematic of realized differential bipolar amplifier test circuit assembly. (a) PCB photograph. (b) Schematic.

In summary the obtained results are in accordance to what has been observed in [6] and [7] in so far, as Dunsmore measured deviations within a 0 dB to 6 dB range. This bias in deviation boundaries can be ascribed to the fact that the alternative power normalization method presented in (24) has been employed for these results. With the theoretical results of this section a framework is created that represents the effects in large signal mixed-mode scattering parameter characterization of a huge class of amplifiers. The validity of this framework shall therefore be verified experimentally within the following section. V. EXPERIMENTAL VERIFICATION In order to verify the results obtained from the modelling approach presented in the previous section, multiple measure-

= 27

ment campaigns have been carried out. This verification has been done by determining the compression points for a differential amplifier circuit at different CMRRs using virtual differential and true differential probing. According to the circuits used for the theoretical derivations in the preceding section, a common-emitter amplifier topology has been chosen as test circuit. The realized bipolar differential amplifier is shown in Fig. 14. The figure depicts the schematic and the associated PCBlayout. The particular functional parts of the circuit are highlighted. It is composed of two single-ended common-emitter circuits involving Avago AT-41533 bipolar transistor devices. , which Both emitters are connected to the feedback-resistor in turn is connected to ground. Capacitors and resistors are inserted at the input and output ports to decouple the ports from DC and to achieve a match to the external reference impedance for the differential mode and for the of , the CMRR of common mode, respectively. By means of , the the circuit can be set individually. By choosing and the common-mode small-signal differential gain gain become identical with about 10 dB at 500 MHz. up to is reduced to approxiBy increasing dB. The CMRR can therefore be adjusted between mately , the limits of 0 dB and 16 dB. For a variety of values of the 1 dB compression curves in terms of have been measured using the power-sweep mode of the Rohde&Schwarz ZVA24 four-port network analyzer with true differential option. A comparison of virtual- and true differential measurements is dB and dB. shown in Fig. 15 for As already predicted by the model in the previous section, the input referred 1 dB compression point is underestimated by 3 dB in case of . A condB with dB trary behaviour is expected for sufficiently large values of the shows small deviations in CMRR. The insertion of the true differential compression, due to the change of biasing conditions at the bipolar transistor devices. The 1 dB compression point of the corresponding virtual measurement however, is now approximately 2.2 dB higher than in the true differential case. This measurement already shows that depending on the individual circuit CMRR, deviations in the 1 dB compression dB range. points occur within the predicted

140

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

1

Fig. 16. Deviation P dB in 1 dB compression point characterization over CMRR (arising from different degeneration resistors).

Besides the highest and lowest possible CMRR of the circuit, have been another five variations of the feedback resistor tested and compared to the analytically derived values. The results are shown in Fig. 16. Herein, the black curve indicates the deviation in 1 dB compression point characterization dB obtained from the analytical investigations presented in the preceding section. Apart from the limiting cases elaborately examined therein, dB has been calculated analytically for finally yielding the black curve over different CMRR. The black boxes mark the measured results. The results derived analytically and those obtained from measurement dedB) bescribe a similar characteristic with a slight offset ( tween theory and measurement. This offset can be ascribed to the fact that the DUT experiences little changes in the operating and therefore slightly point for different emitter resistors changes its underlying linear and nonlinear behaviour. A possibility for obtaining dB over CMRR without affecting the DUT biasing conditions is to make use of the frequency dependent behaviour of the realized amplifier for a fixed . Fig. 17 shows the measured and and the resulting CMRR of the implemented amplifier test circuit in the . According to frequency range up to 2 GHz with this, CMRRs ranging from 8 dB at roughly 2 GHz up to approximately 25 dB at nearly DC can be achieved. Fig. 18 shows the deviations (black boxes) in 1 dB compression point characterization extracted from compression curves measured in true and virtual differential mode at various frequencies according to Fig. 17 and compares them to results, that have been predicted analytically (black line). Once again, measured and calculated results show a proper match especially for dB) and high ( dB) CMRRs. However, again a low ( dB) separates the results obtained theoreticertain shift ( cally from those taken from measurements. As to eliminate that this mismatch might be due to a too simplified modelling approach (solely accounting for third order distortion effects in the individual single-ended parts of the differential DUT) further experimental investigations have been carried out. Fig. 19 shows the differential output spectrum measured for the pseudo differat 3 dB backoff from 1 dB comential amplifier pression. As expected for a differential amplifier, the third order

Fig. 17. Small signal characteristics of differential bipolar amplifier for R

27 .

=

1

Fig. 18. Deviation P dB in 1 dB compression point characterization over CMRR (arising from frequency sweep).

distortion effects provide the most considerable part of the spectrum besides the fundamental frequency. However, even order distortion can be observed, which is presumably due to marginal imbalances occurring in the single-ended amplifier parts. In addition to third and even order distortion, Fig. 19 shows signal parts at five and seven times the fundamental frequency indicating the influence of fifth and seventh order nonlinearity. This measured spectrum has then been taken as basis for the estimation of the underlying nonlinear coefficients by means of a nonlinear least-square fitting algorithm [14]. Besides a linear and third order coefficient this estimated parameter set includes a fifth and seventh order term as well. The simulated spectrum arising from this parameter set is also shown in Fig. 19. As can be seen, both spectra offer a nearly ideal match confirming the proper functionality of the estimation algorithm. For this parameter set, again dB has been simulated over CMRR and no significant difference to the analytically derived characteristics of Figs. 16 and 18 could be observed. A further reason for the mismatch between theory and measurement could be a potential imbalance of the single-ended amplifier parts already indicated by the even order distortion shown in Fig. 19. Therefore, an amplifier with two different parameter

SCHMITZ et al.: DIFFERENTIAL AMPLIFIER CHARACTERIZATION USING MIXED-MODE SCATTERING PARAMETERS

141

REFERENCES

Fig. 19. Output spectrum of the pseudo differential amplifier measured at 3 dB backoff from 1 dB compression and simulated odd order spectral parts following from estimated coefficients.

sets, with one set being a scaled version of the other one, has been simulated. The results are also given in Fig. 18. Herein, describes the scaling factor that relates one single-ended part of the amplifier to the other one. As can be seen, an increasing imbalance leads to a flattened characteristic for high CMRR values and therefore provides a closer approximation of the measured results in this area. This result already gives a first hint on the occurring offset between measurement and simulation. However, further research is necessary to clearly identify further non-idealities affecting the predicted deviation. Altogether, both kinds of experimental verification validate the argumentation of the previous sections. The variety of presented measurement results confirms the analytically derived deviation bounds and the applicability of the model.

[1] D. E. Bockelmann and W. R. Eisenstadt, “Combined differential and common-mode scattering parameters: Theory and simulation,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1530–1539, Jul. 1995. [2] D. E. Bockelmann, W. R. Eisenstadt, and R. Stengel, “Accuracy estimation of mixed-mode scattering parameter measurements,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 1, pp. 102–105, Jan. 1999. [3] A. Ferrero and M. Pirola, “Generalized mixed-mode S-parameters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 458–463, Jan. 2006. [4] H. Erkens and H. Heuermann, “Mixed-mode chain scattering parameters: Theory and verification,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 8, pp. 1704–1708, Aug. 2007. [5] D. E. Bockelmann and W. R. Eisenstadt, “Calibration and verification of a pure-mode network analyzer,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 7, pp. 1009–1012, Jul. 1998. [6] J. Dunsmore, “New methods and non-linear measurements for active differential devices,” in Proc. 2003 IEEE MTT-S In. Microw. Symp. Digest, Jun. 2003, pp. 1655–1658. [7] J. Dunsmore, “New measurement results and models for non-linear differential amplifier characterization,” in Proc. 34th Eur. Microw. Conf., Sep. 2004, pp. 689–692. [8] O. Schmitz, S. K. Hampel, H. Rabe, J. Simon, and I. Rolfes, “On the validity of single-ended mixed-mode S-parameter measurements for differential active devices,” in Proc. 39th Eur. Microw. Conf., Sep. 2009, pp. 691–694. [9] P. R. Gray and R. G. Meyer, Analysis and Design of Analog Integrated Circuits. New York: Wiley, 1993. [10] P. Wambacq and W. Sansen, Distortion Analysis of Analog Integrated Circuits. Norwell, MA: Kluwer, 1998. [11] B. Razavi, RF Microelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1998. [12] W. Sansen, “Distortion in elementary transistor circuits,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 3, pp. 315–325, Mar. 1999. [13] K. L. Fong and R. G. Meyer, “High-frequency nonlinearity analysis of common-emitter and differential-pair transconductance stages,” IEEE J. Solid-State Circuits, vol. 33, no. 4, pp. 548–555, Apr. 1998. [14] L. Ljung, System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice Hall PTR, 1999.

VI. CONCLUSION This paper has systematically derived limits for the deviations in compression curve characterization between true and virtual differential excitation in case of single stage differential amplifier circuits without mode conversion. Based on the derived models, an a priori estimation of these limits has been presented and verified by measurements to be within a predicted dB range. The occurring deviations could systematically be dedicated to solely the common mode rejection of the considered amplifier, so that a prediction of the measurement error depending on the CMRR of the underlying circuit could be given. However, using the derived results for simply measuring virtual differential and employing a subsequent error correction would strictly be limited to single stage amplifiers. Besides the CMRR of the entire differential amplifier the individual single stage compression becomes important in a multistage design. In summary, the obtained results show that an accurate compression curve characterization for differential amplifier topologies with unknown CMRR is only possible by incorporating true differential probing stimuli. ACKNOWLEDGMENT The authors would like to thank Rohde & Schwarz GmbH, Munich, Germany, for providing their four-port VNA ZVA24 with true differential option.

Oliver Schmitz (S’05) was born in Hannover, Germany, in 1979. He received the Dipl.-Ing. degree in electrical engineering from the University of Hannover, Germany, in 2006. Since 2006, he has been a Research Assistant at the Institute of Radiofrequency and Microwave Engineering, Leibniz Universität Hannover, working toward the Ph.D. degree. His current research interests include the design of wideband and multi-standard transceiver components in RF-CMOS and the analysis and modelling of nonlinear systems.

Sven Karsten Hampel (S’05) was born in Hildesheim, Germany, in 1979. He received the Dipl.-Ing. degree in electrical engineering from the University of Hannover, Germany, in 2006. He has been working as a Research Assistant towards the Ph.D. degree at the Institute of Radio Frequency and Microwave Engineering, Leibniz Universität Hannover, since 2006. His main research is focused on the design of reconfigurable and wideband radio front-ends for multistandard applications in RF-CMOS. His work also includes the research on inductorless approaches. Furthermore he is interested in antenna design for wideband and MIMO/Diversity communication systems as well as the investigation of High Impedance Surfaces.

142

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Hanno Rabe received the Dipl.-Ing. degree in electrical engineering from the University of Hannover, Germany, in 2007. Since 2007, he has been a Research Assistant at the Institute of Radiofrequency and Microwave Engineering, Leibniz Universität Hannover and is currently working towards the Ph.D. degree. His areas of work include the research on imaging radar systems and microwave metrology.

Tobias Reinecke was born in Hildesheim, Germany, in 1983. He started studying electrical engineering at the University of Hannover, Germany, in 2004 and is currently working towards the Dipl.-Ing. degree. His research interests include analog circuits and microwave metrology.

Ilona Rolfes (M’06) was born in 1973 in Hagen, Germany. She received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from the Ruhr-University Bochum, Germany, in 1997 and 2002, respectively. From 1997 to 2005 she has been with the Research Group High Frequency Measurements at the Ruhr-University Bochum as a Research Assistant. From 2005 to 2009 she worked as a Junior Professor in the Department of Electrical Engineering at the Leibniz Universität Hannover, Germany, where in 2006 she became Head of the Institute of Radiofrequency and Microwave Engineering. Since 2010 she has led the Department for High Frequency Systems at the Ruhr-University Bochum. Her fields of research are concerned with high frequency measurement methods for vector network analysis, material characterization, noise characterization of microwave devices as well as sensor principles for radar systems and wireless solutions for communication systems.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

143

Efficiency Enhancement of Doherty Amplifier Through Mitigation of the Knee Voltage Effect Junghwan Moon, Student Member, IEEE, Jangheon Kim, Member, IEEE, Jungjoon Kim, Student Member, IEEE, Ildu Kim, and Bumman Kim, Fellow, IEEE

Abstract—This paper presents an approach to maximize the efficiency of a Doherty power amplifier (PA) with the knee voltage effect. Since the carrier PA with 2 opt , which is the usual matching impedance for a carrier PA at a low power region, does not reach to the saturated operation at the 6-dB back-off power level, the maximum efficiency could not be achieved. However, the carrier amplifier can be driven into the saturation using the load impedance larger than 2 opt and can deliver the maximum efficiency even under the knee voltage effect. The optimized design for the maximum efficiency at the back-off level is derived. The optimized amplifier is analyzed and simulated in terms of its load modulation behavior, efficiency, and output power, then compared with the conventional Doherty PA. The enhanced performance is demonstrated by the Doherty PA built using CREE’s GaN HEMT CGH40045 devices at 2.655 GHz. For worldwide interoperability for microwave access applications with a 7.8-dB peak-to-average power ratio, the proposed PA delivers an efficiency of 49.3% at an output power of 42 dBm with an acceptable linearity of 23.1 dBc. The linearity is improved to 43 dBc by employing a digital feedback predistortion technique, satisfying the system linearity specification. Index Terms—Doherty, efficiency enhancement, gallium nitride, knee voltage, load modulation, power amplifier (PA), worldwide interoperability for microwave access (WiMAX).

I. INTRODUCTION S WIRELESS communication systems evolve to handle higher data rates, the signals of the systems vary rapidly and have large peak-to-average power ratios (PAPRs). This is due to the complex modulation schemes for efficient spectrum usage. To satisfy the linearity requirements for the systems, such as wideband code division multiple access, orthogonal frequency division multiple access, and worldwide interoperability for microwave access (WiMAX), power amplifiers (PAs) should have a capability of generating the peak power

A

Manuscript received September 03, 2009; revised February 10, 2010; accepted July 28, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. This work was supported by The Ministry of Knowledge Economy (MKE), Korea, under the Information Technology Research Center (ITRC) Support Program supervised by the National Information Technology Industry Promotion Agency (NIPA) (NIPA-2010-(C1090-1011-0011)), by the World Class University (WCU) Program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (Project R31-2008-000-10100-0), and by the Brain Korea 21 Project in 2010. J. Moon, J. Kim, and B. Kim are with the Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Gyungbuk 790-784, Korea (e-mail: [email protected]; [email protected]; [email protected]). J. Kim and I. Kim are with the Samsung Electronics Company Ltd., Suwon, Gyeunggi, Korea (e-mail: [email protected]; ildu12.kim@ samsung.com). Digital Object Identifier 10.1109/TMTT.2010.2091207

and should be able to linearly generate all power levels following the signal distribution. Therefore, PAs operate mostly at the average power level, back-off from the peak power by PAPR. The efficiency at the operation point is low, leading to an amplification with a low efficiency. There are a few methods to solve the low efficiency problem: envelope elimination and restoration, envelope tracking, and Doherty PA [1]–[23]. The first two systems provide good efficiency by adjusting the drain/collector bias to minimize the dissipated power consumptions. There are some problems in the systems, however, such as difficulty of the delay adjustment of the RF and envelope paths and the complexity of the linkage between the PA and bias modulator. Moreover, they require a highly efficient bias modulator, as well as the PA [4]–[7]. The Doherty PA can accomplish the high efficiency by employing two PAs: one PA, the carrier amplifier, is operational from a low power region and another PA, the peaking amplifier, is turned on at a high power region. The output powers of the two amplifiers are combined by the self-adjusted load modulation technique, and the Doherty amplifier enhances the efficiency at the low power region [8]–[23]. A significant amount of research has been focused on the Doherty PA to improve its efficiency. Asymmetrical Doherty PAs, such as an -way Doherty PA [13], an extended Doherty PA [14], and a multistage Doherty PA [15] have been proposed to extend the peak efficiency region over a wider range of output power. In addition, the Doherty PA based on a saturated PA [16]–[19], such as the class-F and inverse class-F, shows an enhanced maximum efficiency. Although the previous works provide good theoretical analyses, they are carried out using an ideal device without considering the knee voltage effect. Moreover, it is difficult to obtain the maximum efficiencies of the Ideal Doherty at the peak and back-off power levels simultaneously, as shown in Fig. 1. Thus, we have evaluated the efficiency of the Doherty PAs optimized at the each theoretical maximum region. Doherty Type-I shows a degraded efficiency at the peak power level, resulting from the imperfect load modulation of the Doherty PA. On the other hand, Doherty Type-II has a reduced efficiency at the back-off power level because the carrier PA does not reach to the saturated operation when the peaking PA is turned on. Based on these profiles, the estimated overall efficiencies for amplification of the modulated signal with PAPR of 7.8 dB are computed using a routine coded in MATLAB and summarized in Table I [16]. The results show that the Doherty amplifier should be optimized for the high efficiency at the back-off region, but the efficiency at the peak power region is not as important.

0018-9480/$26.00 © 2010 IEEE

144

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 1. Ideal efficiencies of the class-B and two-way Doherty PAs. The probability density and power generation distribution of a WiMAX signal with a 7.8-dB PAPR are also depicted.

TABLE I ESTIMATED EFFICIENCY OF TWO-WAY DOHERTY AMPLIFIERS FOR A SIGNAL WITH A PAPR OF 7.8 dB

The knee voltage of the carrier PA plays an improtant role for the maximum efficiency at the back-off region. We have found that the load impedance of the carrier amplifier should to obtain the saturated operation when the be larger than peaking PA is turned on. Though the basic design concept of the PA considering the knee voltage effect has been discussed previously [20], this study presents more detailed analyses and experimental results. We examine the basic operation of the Doherty PA in Section II. Section III describes the detailed behavior of the PA using a real device simulation. To verify our analyses and simulations, we implement the amplifier using CREE GaN HEMT CGH40045 devices at 2.655 GHz and compared with the conventional Doherty PA for operation with continuous wave (CW) and mobile WiMAX 1FA signals, which are explained in Section IV. The experimental results clearly show that the knee voltage effect should be considered to properly design the Doherty PA with high efficiency at the back-off level. Finally, the digital feedback predistortion (DFBPD) technique is applied to achieve highly linear performance [24], satisfying the system specifications.

Fig. 2. Load line of the carrier amplifier. (a) Ideal case with zero knee voltage. (b) Practical case with nonzero knee voltage.

The perfect load modulation of the carrier PA with a zero-knee voltage, depicted in Fig. 2(a), indicates that the output powers and are and , respecfor both tively. The efficiencies ( ’s) are also the same at and , respectively; the carrier PA is in equally saturated states for both cases. In this operation, , , and are given by

(1) where

II. ANALYSIS FOR OPERATIONAL CHARACTERISTICS OF DOHERTY PA A. Doherty PA Operation With Knee Voltage Effect Theoretically, the Doherty PA attains the highest efficiencies at the 6-dB back-off and the peak power levels. The maximum efficiencies at the two points can be achieved through the perfect load modulated operation of the carrier and peaking PAs and the infinite output impedance of the peaking PA at the turn-off state.

and are the fundamental and dc current components of the carrier PA biased at the conduction angle , reis equal to because of the spectively. In the ideal case, with zero knee voltage. zero on-resistance

MOON et al.: EFFICIENCY ENHANCEMENT OF DOHERTY AMPLIFIER

145

In a real device, the effect of the knee voltage should be considered. Fig. 2(b) shows a load line with the knee voltage . and , the load impedances are written by For

(2) (3) where

In with follows:

,

can be derived using (2) and (3) as

(4) represents the maximum current when the carrier PA has the load impedance of . Thus, the output powers for the and cases can be calculated as follows:

(5) and are shown in Fig. 3(a). For the calculation, we assume a of 30 V and of 8 A with uniform trans-conductance. is delivers more fixed at 210 . As shown, the carrier PA with because of the power than the expected value of enlarged voltage and current swings from to and from to , respectively. The efficiency at the 6-dB back-off region deteriorates because the carrier PA does not reach to the saturation state when the peaking PA begins conducting. To maximize the efficiency at the 6-dB back-off power region, the carrier PA with nonzero knee voltage should have a , like case-II in Fig. 2(b) beload impedance larger than cause the carrier PA with cannot be fully saturated at the back-off region, as depicted in Fig. 3(a). In this case, the output power can be written as (6) where

Since should be a half of urated state at the back-off region,

to obtain the fully sat, the maximum current,

Fig. 3. (a) Maximum output powers employing R , R as R varies from 0 to 0.8 . (b) Load impedances for R as R varies from 0 to 0.8 .

R

, and R ,R , and

when the carrier PA has a load impedance larger than can be calculated as follows:

,

(7) Fig. 3(b) shows the required load impedance with varying . As increases, the load impedance of the carrier PA in the , leading to low power region also increases, larger than the fully saturated state of the PA at the 6-dB back-off level. To verify the knee voltage effects on the Doherty operation, we implement the carrier PA using a Cree GaN HEMT CGH40045 device with a 45-W peak envelope power at 2.655 GHz. First, the output of the PA is matched to from 50 . Its load impedance is then modulated by adjusting the offset line and output termination impedance [11]. Fig. 4 shows the load modulation results for the carrier PA employing the output termination impedances of 50, 100, and 130 . As expected, the PA with 100 delivers maximum efficiency at 45.3 dBm, which is a 1.9-dB back-off power from the maximum output power of 47.2 dBm with the output termination

146

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 4. Load modulation results for carrier amplifier employing the load impedances of 50, 100, and 130 .

Fig. 6. Fundamental currents of the conventional and proposed Doherty PA and  ). according to the input voltage (

= 210

= 150

The fundamental currents of the conventional and proposed Doherty PAs based on the input voltage are represented in Fig. 6 and given by (8) (9) Fig. 5. Operational diagram of the proposed Doherty amplifier.

and impedance of 50 . On the other hand, the PA with 130 its maximum efficiency at the 2.9-dB back-off point.

has (10)

B. Load Modulation Behavior of Doherty Amplifier With Optimized Carrier PA The fundamental operating principle and efficiency characteristic of Doherty amplifier have been described in [8]–[23]. To design Doherty amplifier properly using a real device, the knee voltage effect should be considered as described in Section II-A. Fig. 5 shows the operational diagram of the proposed Doherty PA used to analyze the load modulation behavior, efficiency, and output power. Compared to the conventional one, it has a scaled . load impedance of For simplicity, it is assumed that each current source is linearly proportional to the input voltage signal with a maximum . The ideal harmonic short circuits are provided current of so that the output power and efficiency can be determined by using only the fundamental and dc components. The load impedances of the carrier and peaking amplifiers are determined by the current ratios. Since the load impedance of the proposed carrier PA is larger than that of the conventional one, the current of the carrier amplifier cannot reach to the maximum value. However, the peaking PAs for both the conventional and proposed amplifiers can generate the same maximum output current. Thus, the load impedances cannot be modulated properly in this structure, leading to the degradation of output power. In addition, the peaking PA should be driven harder than the carrier PA for the proper current generation [12], [17].

(11) where

, , , and denote the fundamental and dc currents of the carrier and peaking PAs, respectively. As explained in Section II-A, the load impedance of the carrier amplifier in , and the voltage the proposed Doherty PA is larger than swing of the amplifier is increased, as shown in Fig. 2(b). To

MOON et al.: EFFICIENCY ENHANCEMENT OF DOHERTY AMPLIFIER

147

generate the same output power from the conventional and proposed PAs at the 6-dB back-off level, the current of the proposed Doherty PA should be lower than that of the conventional Doherty amplifier by . indicates the uneven power drive ratio [12]. By using the currents of the carrier and peaking PAs, the load impedances of the two PAs are given by

(12)

(13)

where

By using the lossless quarter-wave transmission line in Fig. 5, is defined by

Since the carrier PA of the proposed Doherty amplifier has a larger load impedance than the conventional PA, the carrier amplifier operates in the heavily saturated region after the peaking amplifier is turned on. To prevent clippings of the voltage and current of the carrier amplifier, the load impedance of the carrier amplifier should be lowered by generating more current from the peaking amplifier. Thus, the peaking amplifier should be overdriven by a factor of . For linear operation of the carrier PA without voltage clipping, the voltage across the current source of the carrier cell should satisfy the following condition at the maximum input voltage: (14) From (14),

can be inferred and is given by (15)

and . In this analysis, we use signifies the necessary ratio of to to achieve the fully saturated operation of the carrier PA at the back-off region. For the conventional case, , , and are equal to 1. Thus, the fundamental load impedance of the carrier PA is modto , while that of the peaking PA varies ulated from from to . On the other hand, for the proposed Doherty PA, and are greater than 1, but is less than 1. Thus, the fundamental load impedances of the carrier and peaking PAs to are modulated from

Fig. 7. (a) Load impedances of the conventional and proposed Doherty PA according to the input voltage. (b) Fundamental voltages of the carrier (V ) and peaking (V ) amplifiers for the conventional and proposed Doherty PAs.

and from to , as described in Fig. 7(a). of 4 V is asTo explore the load modulation behavior, sumed and is calculated by (2). At the low power region , the load impedance of the carrier , i.e., . At the high power rePA is larger than , the load gion impedance maintains a value larger than conventional PA. In contrast, the load impedance of the peaking PA sustains a smaller value than that of the conventional PA; this results in a slight power degradation. Fig. 7(b) shows the resulting fundamental voltages of the carrier and peaking amplifiers for the conventional and proposed Doherty PAs. As expected, when compared to the conventional PA at the low power region, the carrier amplifier of the proposed amplifier has a larger fundamental voltage because of the larger load impedance. Therefore, the carrier PA can be fully saturated, delivering the high efficiency. By using the currents and load impedances of the carrier and peaking PAs, the efficiency is estimated through MATLAB. Fig. 8 represents the calculated efficiencies based on the above analand . To calculate the efficienysis for , , , and cies, , , , and are determined by

148

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 8. Efficiencies of the conventional and proposed Doherty PA according to the output power.

Fig. 9. Simulated load modulation results for the carrier PA with the load impedances of 50, 100, 110, and 120 .

, which are 30 V, 4 V, 4.2129 A, and 3.6384 A, respectively. The resulting , , , and are 1.1725 and 0.9235, 2.3158, and 1.2494, respectively. In the low power region, the proposed Doherty PA has higher efficiency than that of the conventional PA because of the larger load impedance. In particular, the proposed Doherty PA delivers its maximum efficiency of 67.9% at the 6 dB back-off output power, which is an increase of about 5% when compared to the conventional one. Although the maximum output power and efficiency at the maximum output power are slightly degraded because of the imperfect load modulation, the proposed Doherty scheme improves efficiency for amplification of modulated signals without any additional circuitry. III. SIMULATION RESULTS In Section II, we proposed a Doherty PA design considering the knee voltage effect and analyzed efficiency and output power of the PA. To verify the analysis and to investigate the real behavior of the PA, we perform ADS simulation using CREE’s large-signal model for CGH40045 GaN HEMT. The unit cell PA is designed to deliver the maximum efficiency while maintaining a high output power. The quiescent current of the carrier cell is set to 200 mA and the gate voltage of the peaking cell is adjusted to turn on at the 6-dB back-off power level. To find the load impedance of the carrier PA delivering the maximum efficiency at the 3-dB back-off from peak output power, we simulate the load–pull test. Fig. 9 illustrates the simulated efficiencies for the carrier PA with output termination impedances of 50, 100, 110, and 120 . As expected, the carrier PA with 100 delivers its maximum efficiency at an output power of 45.4 dBm, which is 2.3-dB back-off power from the maximum output power of 47.7 dBm. For the PA with 120 , its maximum efficiency is achieved at the 2.9-dB back-off power level. As a result, we selecte 120 for the carrier PA at the low power region, i.e., before the peaking PA turns on. Compared to the analytical value of 130 obtained from Fig. 3, the output termination impedance of 120 is reasonable. Fig. 10 shows the calculated the carrier and peaking load impedances of the conventional and the proposed Doherty PAs according to the output power level. For the conventional case, the

Fig. 10. Simulated load impedances of the conventional and proposed Doherty PAs as a function of the output power.

load impedance of the carrier PA modulates 100 to 50 , and that of the peaking PA modulates to 50 . On the other hand, for the proposed case, the load impedance of the carrier PA remains 120 until the peaking PA turns on. At the high power region, the load impedance of the carrier PA remains higher than 50 , while that of the peaking PA is lower than 50 . These results agree with the analysis given in Section II-B. Fig. 11 represents the simulated efficiencies of the conventional and proposed Doherty PAs according to the output power. Due to the larger load impedance at the low power region, the proposed Doherty PA provides higher efficiency. In particular, the proposed PA delivers the maximum efficiency of 68.8% at the low power region, which is an improvement of about 10% compared to that of the conventional PA. At the high power region, however, the output power and efficiency of the proposed Doherty PA are slightly degraded because of the inevitable imperfection of load modulation. However, the proposed Doherty PA delivers the desired efficiency for amplification of the modulated signal. IV. IMPLEMENTATIONS AND MEASUREMENT RESULTS In Section II-A, we have shown that, for the Doherty PA, the carrier amplifier should have a load impedance larger than

MOON et al.: EFFICIENCY ENHANCEMENT OF DOHERTY AMPLIFIER

149

Fig. 11. Simulated efficiencies of the conventional and proposed Doherty PAs according to the output power.

Fig. 12. Schematic diagram of the proposed Doherty PA.

100 to improve the efficiency at the back-off region before the peaking PA turns on. This result is verified using ADS simulation, as explained in Section III. To experimentally validate the proposed Doherty scheme, a 2.655-GHz Doherty PA is fabricated using two CREE CGH40045 GaN HEMTs. For a good efficiency, the two unit cell PAs are optimized for efficiency with reasonable output power [16], [17]. Fig. 12 shows the schematic diagram of the proposed Doherty PA. In contrast with the conventional Doherty PA, its output combiner consists of a 50transmission line with length of and a parallel connected capacitor to easily change the output load impedance. In the exare optiperiment, the offset line lengths of the carrier PAs mized to deliver maximum performances with 100- and 130loads. The offset line lengths of the peaking PAs are also adjusted to block the output power leakage from the carrier to the peaking PA; the transformed output impedance is 1200 . A. CW and WiMAX Tests of the Doherty Amplifiers Fig. 13 shows the measured power characteristics, drain dc currents, and drain efficiencies of the conventional and proposed Doherty PAs for a CW signal. In the experiments, the turn-on power level of the peaking PA is adjusted by the . When the peaking PA is turned off, gate–source voltage the load impedance of the proposed carrier PA is higher than that of the conventional PA. Thus, the dc current level of the proposed PA is lower than that of the conventional PA, leading

Fig. 13. Measured performances of the conventional and proposed Doherty amplifiers for a CW signal. (a) Output power versus input power. (b) Drain dc supply current as a function of the output power. (c) Drain efficiency according to the output power.

to an improved efficiency at the low power region. However, the peak power of the Doherty PA is degraded because the imperfect load modulation causes an impedance mismatch for load at the peak power region. Although the peak power of the proposed Doherty PA is slightly degraded, the efficiency is increased over a broad output power range.

150

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE II MEASURED PERFORMANCE AT THE 8-dB BACK-OFF OUTPUT POWER FOR A CONVENTIONAL DOHERTY PA AND THE 8- AND 7.5-dB BACK-OFF OUTPUT POWERS FOR A PROPOSED DOHERTY PA

Fig. 15. Experimental setup for linearization.

Fig. 14. Measured performances of the conventional and proposed Doherty amplifiers for a WiMAX signal. (a) Drain efficiency. (b) ACLR.

Fig. 14(a) shows the measured efficiencies for a mobile WiMAX signal with a 10-MHz signal bandwidth and a 7.8-dB PAPR. Compared to the conventional Doherty PA, the efficiency of the proposed PA is improved over a broad average output power level. Fig. 14(b) represents the measured adjacent channel power ratio (ACLR) of the implemented PAs. Since the proposed Doherty PA has a larger load impedance than that of the conventional PA at the low power region and cannot achieve perfect load modulation at the high power region, the ACLR is slightly degraded over the power range. The ACLRs of Doherty PAs are worse than that of the comparable class-AB PA because the peaking PAs are biased at a deep class-C to turn off the PAs until the carrier PA is fully saturated and are optimized for efficiency rather than linearity. In Table II, we summarize the measured efficiencies and ACLRs at the 8-dB back-off output power for a conventional Doherty PA and the 8- and 7.5-dB back-off output powers for the proposed Doherty PA. These results allow us to conclude that the proposed Doherty PA has an excellent efficiency with acceptable linearity. B. Linearization Performance of the Amplifier To confirm the suitability of the proposed Doherty amplifier as the main PA of a base-station or repeater linear power amplifier (LPA) system, we apply the DFBPD linearization

Fig. 16. Measured WiMAX spectra before and after the DFBPD linearization of the Doherty PA at an average output power of 42 dBm.

technique to the proposed Doherty amplifier. Two 1024-entry AM/AM and AM/PM lookup tables are accomplished by a MATLAB program using the DFBPD algorithm, as shown in Fig. 15 [24]. The measured spectra before and after compensation for the memoryless nonlinearity by the DFBPD technique are described in Fig. 16. After the linearization, the ACLR at an offset of 6.05 MHz is 43 dBc, which is an improvement of approximately 18 dB at an average output power of 42 dBm. This spectrum satisfies the emission mask of the mobile WiMAX system, type-G. The measured signal constellation diagrams before and after the linearization are described in Fig. 17. By linearizing with DFBPD, the relative constellation error (RCE) is 34.5 dB, which is an improvement of about 16.8 dB. These results clearly indicate that, after employing the linearization technique, the proposed Doherty PA is well suited as a highly efficient main PA of the LPA for amplification of

MOON et al.: EFFICIENCY ENHANCEMENT OF DOHERTY AMPLIFIER

151

can deliver the maximum efficiency at the 6-dB back-off region because the carrier PA is saturated enough at this level while the conventional PA is not. Therefore, the optimized PA delivers better efficiency for amplification of the modulated signals. To experimentally validate the amplifier, we have implemented and tested the Doherty PAs using CREE GaN HEMT CGH40045 devices at 2.655 GHz. The experimental results clearly show that the proposed Doherty PA delivers better efficiency at the back-off output power level than the conventional PA with satisfactory linearity. The linearity performance can be further improved within the system specification by adopting the DFBPD linearization technique. ACKNOWLEDGMENT The authors would like to thank to CREE Inc., Durham, NC, for providing the GaN HEMT transistors used in this work. REFERENCES

Fig. 17. Measured WiMAX signal constellation diagram of the Doherty PA at an average output power of 42 dBm. (a) Before linearization. (b) After linearization.

TABLE III MEASURED LINEARIZATION PERFORMANCE OF THE PROPOSED DOHERTY AMPLIFIER AT AN AVERAGE OUTPUT POWER OF 42 dBm FOR WIMAX 1FA SIGNAL

the modulated signal. The measured linearization performances are summarized in Table III. V. CONCLUSION We have investigated the knee voltage effect on the operation of the Doherty PA, in particular the carrier PA. From this exploration, we have found that the efficiency of a Doherty PA at the back-off region is lower than the expected value due to the effect. To overcome this problem, the carrier PA should employ . The modified Doherty PA the load impedance larger than

[1] F. H. Raab, P. Asbeck, S. Cripps, P. B. Kenington, Z. B. Popovic´, N. Pothecary, J. F. Sevic, and N. O. Sokal, “Power amplifiers and transmitters for RF and microwave,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 814–826, Mar. 2002. [2] J. Choi, D. Kang, D. Kim, J. Park, B. Jin, and B. Kim, “Power amplifiers and transmitters for next generation mobile handset,” J. Semiconduct. Technol. Sci., vol. 9, no. 4, pp. 249–256, Dec. 2009. [3] L. Kahn, “Single-sideband transmission by envelope elimination and restoration,” Proc. IRE, vol. 40, no. 7, pp. 803–806, Jul. 1952. [4] D. F. Kimball, J. Jeong, C. Hsia, P. Draxler, S. Lanfranco, W. Nagy, K. Linthicum, L. E. Larson, and P. M. Asbeck, “High-efficiency envelopetracking W-CDMA base-station amplifier using GaN HFETs,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 11, pp. 3848–3856, Nov. 2006. [5] F. Wang, D. F. Kimball, J. D. Popp, A. H. Yang, D. Y. C. Lie, P. M. Asbeck, and L. E. Larson, “An improved power-added efficiency 19-dBm hybrid envelope elimination and restoration power amplifier for 802.11g WLAN applications,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4086–4099, Dec. 2006. [6] F. Wang, A. H. Yang, D. F. Kimball, L. E. Larson, and P. M. Asbeck, “Design of wide-bandwidth envelope tracking power amplifiers for OFDM applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1244–1255, Apr. 2005. [7] I. Kim, Y. Y. Woo, J. Kim, J. Moon, J. Kim, and B. Kim, “High-efficiency hybrid EER transmitter using optimized power amplifier,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 11, pp. 2582–2593, Nov. 2008. [8] S. C. Cripps, RF Power Amplifiers for Wireless Communications, 2nd ed. Norwood, MA: Artech House, 2006. [9] W. H. Doherty, “A new high efficiency power amplifier for modulated waves,” Proc. IRE, vol. 24, no. 9, pp. 1163–1182, Sep. 1936. [10] F. H. Raab, “Efficiency of Doherty RF power-amplifier systems,” IEEE Trans. Broadcast., vol. BC-33, no. 3, pp. 77–83, Sep. 1987. [11] Y. Yang, J. Yi, Y. Y. Woo, and B. Kim, “Optimum design for linearity and efficiency of microwave Doherty amplifier using a new load matching technique,” Microw. J., vol. 44, no. 12, pp. 20–36, Dec. 2001. [12] J. Kim, J. Cha, I. Kim, and B. Kim, “Optimum operation of asymmetrical-cells-based linear Doherty power amplifiers-uneven power drive and power matching,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1802–1809, May 2005. [13] Y. Yang, J. Cha, B. Shin, and B. Kim, “A fully matched -way Doherty amplifier with optimized linearity,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 986–993, Mar. 2003. [14] M. Iwamoto, A. Williams, P. F. Chen, A. G. Metzger, L. E. Larson, and P. M. Asbeck, “An extended Doherty amplifier with high efficiency over a wide power range,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2472–2479, Dec. 2001. [15] N. Srirattana, A. Raghavan, D. Heo, P. E. Allen, and J. Laskar, “Analysis and design of a high-efficiency multistage Doherty power amplifier for wireless communications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 852–860, Mar. 2003. [16] J. Moon, J. Kim, I. Kim, J. Kim, and B. Kim, “Highly efficient three-way saturated Doherty amplifier with digital feedback predistortion,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 8, pp. 539–541, Aug. 2008.

N

152

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

[17] J. Kim, J. Moon, Y. Y. Woo, S. Hong, I. Kim, J. Kim, and B. Kim, “Analysis of a fully matched saturated Doherty amplifier with excellent efficiency,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 2, pp. 328–338, Feb. 2008. [18] S. Goto, T. Kunii, A. Inoue, K. Izawa, T. Ishikawa, and Y. Matsuda, “Efficiency enhancement of Doherty amplifier with combination of class-F and inverse class-F schemes for S -band base station application,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, pp. 839–842. [19] Y. Suzuki, T. Hirota, and T. Nojima, “Highly efficient feed-forward amplifier using a class-F Doherty amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 77–80. [20] J. Moon, Y. Y. Woo, and B. Kim, “A highly efficient Doherty power amplifier employing optimized carrier cell,” in Proc. 39th Eur. Microw. Conf., Sep. 28–Oct. 2 2009, pp. 1720–1723. [21] K. Horiguchi, S. Ishizaka, T. Okano, M. Nakayama, H. Ryoji, Y. Isota, and T. Takagi, “Efficiency enhancement of 250 W Doherty power amplifier using virtual open stub techniques for UHF-band OFDM applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, pp. 1356–1359. [22] N. Ui, H. Sano, and S. Sano, “A 80 W 2-stage GaN HEMT Doherty amplifier with 50 dBc ACLR, 40% efficienecy, 32 dB gain with DPD for W-CDMA base station,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 1259–1262. [23] R. Sweeney, “Practical magic,” IEEE Microw. Mag., vol. 9, no. 2, pp. 73–82, Apr. 2008. [24] Y. Y. Woo, J. Kim, J. Yi, S. Hong, I. Kim, J. Moon, and B. Kim, “Adaptive digital feedback predistortion technique for linearizing power amplifier,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 5, pp. 932–940, May 2007.

0

Junghwan Moon (S’07) received the B.S. degree in electrical and computer engineering from the University of Seoul, Seoul, Korea, in 2006, and is currently working toward the Ph.D. degree at the Pohang University of Science and Technology (POSTECH), Pohang, Gyungbuk, Korea. His current research interests include highly linear and efficient RF PA design, memory-effect compensation techniques, digital predistortion (DPD) techniques, and wideband RF PA design. Mr. Moon was the recipient of the Highest Efficiency Award of the Student High-Efficiency Power Amplifier Design Competition, IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) in 2008.

Jangheon Kim (S’07–M’09) received the B.S. degree in electronics and information engineering from Chon-buk National University, Chonju, Korea, in 2003, and the Ph.D. degree in electrical engineering from the Pohang University of Science and Technology (POSTECH), Pohang, Gyungbuk, Korea, in 2009. He is currently a Senior Engineer with the Network Division, Digital Media and Communications, Samsung Electronics Company Ltd., Suwon, Korea. From 2009 to 2010, he was a Post-Doctoral Fellow with the University of Waterloo, Waterloo, ON, Canada. His current research interests include highly linear and efficient RF PA design, digital predistortion (DPD) techniques, and highly efficient transmitter for wireless communication systems. Dr. Kim was the recipient of the Highest Efficiency Award of Student HighEfficiency Power Amplifier Design Competition of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) in 2008 and 2010 and the MICROWAVE AND WIRELESS COMPONENTS LETTERS Outstanding Reviewer Award in 2010.

Jungjoon Kim (S’10) received the B.S. degree in electrical engineering from Han-Yang University, Ansan, Korea, in 2007, the Master degree in electrical engineering from the Pohang University of Science and Technology (POSTECH), Pohang, Gyungbuk, Korea, in 2009, and is currently working toward the Ph.D. degree at the POSTECH. His current research interests include RF PA design and supply modulator design for highly efficient transmitter systems.

Ildu Kim received the B.S. degree in electronics and information engineering from Chon-nam National University, Kwangju, Korea, in 2004, and the Ph.D. degree from the Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 2010. In 2010, he joined the Samsung Electronics Company Ltd., Suwon, Gyeunggi, Korea. His current research interests include highly linear and efficient RF PA design, LPA system design, and highly linear and efficient RF transmitter architectures.

Bumman Kim (M’78–SM’97–F’07) received the Ph.D. degree in electrical engineering from Carnegie Mellon University, Pittsburgh, PA, in 1979. From 1978 to 1981, he was engaged in fiber-optic network component research with GTE Laboratories Inc. In 1981, he joined the Central Research Laboratories, Texas Instruments Incorporated, where he was involved in development of GaAs power field-effect transistors (FETs) and monolithic microwave integrated circuits (MMICs). He has developed a largesignal model of a power FET, dual-gate FETs for gain control, high-power distributed amplifiers, and various millimeter-wave monolithic microwave integrated circuits (MMICs). In 1989, he joined the Pohang University of Science and Technology (POSTECH), Pohang, Gyungbuk, Korea, where he is currently a POSTECH Fellow and a Namko Professor with the Department of Electrical Engineering and Director of the Microwave Application Research Center. He is involved in device and circuit technology for RF integrated circuits (RFICs) and PAs. He has authored over 300 technical papers. Prof. Kim is a member of the Korean Academy of Science and Technology and the National Academy of Engineering of Korea. He was an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, a Distinguished Lecturer of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), and an Administrative Committee (AdCom) member.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

153

Modeling and Design Methodology of High-Efficiency Class-F and Class-F-1 Power Amplifiers Joon Hyung Kim, Gweon Do Jo, Jung Hoon Oh, Young Hoon Kim, Kwang Chun Lee, and Jae Ho Jung

Abstract—In this paper, efficiency-limiting physical constraint effects imposed on the knee voltage, along with a variation of the optimum load resistance, are investigated for highly efficient Class-F and Class-F-1 amplifiers. First, for an accurate analysis and comparison, new current waveform models are identified, and a realistic approach incorporated using a nonzero knee voltage and voltage-dependent nonlinear capacitance is employed to derive the voltage waveforms of the amplifiers. An analysis is performed to show the efficiency, output power, power gain, and output power compression points for both modes. Using this knowledge, along with a complete performance comparison, we provide a direction for optimizing the amplifier design. The analytic results are further verified based on the measured results of 3.54-GHz Class-F and Class-F-1 amplifiers using a commercial 60-W peak-to-envelope power gallium–nitride device. The experimental results show that Class-F and Class-F-1 amplifiers operate at drain efficiencies of 69.9% and 69.4% at saturated output powers of 47.4 and 47.2 dBm, respectively. These remarkably similar performances have excellent agreement with the predicted analysis at our operational frequency. Index Terms—Class-F power amplifier (PA), inverse Class-F PA, nonlinear capacitance, on-resistance.

I. INTRODUCTION N MODERN wireless communication systems such as third-generation partnership project long-term evolution (3GPP-LTE) and mobile worldwide interoperability for microwave access (WiMAX), since a modulation scheme utilizes orthogonal frequency division multiplexing (OFDM), which has a high peak-to-average ratio (PAPR) resulting in a degradation of efficiency, the high efficiency of the power amplifier (PA) is an important issue, including an improvement in thermal management, size reduction, and the enabling of a lower cost base station implementation, such as in remote radio head (RRH) systems. It is a challenge, therefore, to achieve highly efficient PA architectures. Recently, advanced transmitters

I

Manuscript received June 10, 2010; revised October 02, 2010; accepted October 05, 2010. Date of publication November 15, 2010; date of current version January 12, 2011. This work was supported in part by the Information Technology (IT) Research and Development (R&D) Program of MKE/KEIT [10035173]. The authors are with the Mobile RF Team, Electronics and Telecommunications Research Institute (ETRI), Yuseong-gu, Daejeon 305-700, Korea (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2090167

for high efficiency, such as Doherty amplifiers and envelope tracking (ET) amplifiers incorporated with a digital pre-distortion (DPD) function, have been vigorously investigated [1]–[3]. To successfully implement these architectures, highly efficient PAs are greatly needed. In previous works, either a Class-F or Class-F- amplifier has been the leading candidate, as each has been known to reduce the dissipated power by minimizing the overlap between the current and voltage waveforms [4]–[12]. Class-F and Class-F- amplifiers can be generally obtained by applying an even or odd harmonic control circuit into the load network. Thus, the voltage waveform of a Class-F amplifier is similar to a square wave, whereas the Class-Famplifier has a half-sinusoidal voltage waveform. Unlike a Class-F amplifier, whose maximum voltage swing is less than double the nominal supply voltage value, the peak voltage of a Class-F- amplifier can be larger than twice the dc supply value at the expense of device stress relative to the device’s breakdown voltage. Thus, peak voltage has often been difficult to realize with some voltage–current controlled devices such as laterally diffused metal–oxide semiconductor (LDMOS) and gallium–arsenide (GaAs) devices. However, due to advances in wide bandgap semiconductor technology, particularly GaN high electron-mobility transistors (HEMTs), a large rail voltage swing has become feasible; hence, allowing a Class-F- amplifier with high efficiency to be achieved. As this breakdown voltage issue has been resolved, several papers in the literature have stated its superiority to a Class-F amplifier for a given transistor [7]–[9]. Result comparisons have been previously carried out as a way to consider either the ideal waveform or different operating conditions, and some previous works have reached different conclusions [6], [9], [11]. Therefore, it is important to formulate an ultimate comparison model in terms of accuracy. In order to accurately compare Class-F and Class-F- modes, the formulation of current waveforms associated with a quiescent current should be emulated. In addition, an investigation of the influence of real constraints, such as knee voltage and nonlinear parasitic capacitance, is necessary. The purpose of this paper is to provide a clear explanation of the RF performance of Class-F and Class-F- amplifiers under the same operational conditions. Furthermore, an optimum design guideline to maximize the efficiency at high frequency is provided. This paper is organized as follows. In Section II, to study the performance of Class-F and Class-F- amplifiers, we introduce the analytical models of current and voltage waveforms in detail. An optimized design methodology is also pro-

0018-9480/$26.00 © 2010 IEEE

154

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 1. Configuration of Class-F and Class-F- amplifiers, and a simplified analytical model of an FET.

vided, and the efficiency, output power, and power gain are compared under optimum conditions. Section III provides a comparison between the simulated and theoretical performance results of the amplifiers. In Section IV, the measured results of 3.54-GHz PAs utilizing a 60-W peak-to-envelope power (PEP) device are given for confirmation of our analysis. Section V draws our conclusions. II. BASIC THEORIES Raab has reported that an attainable efficiency due to wave shaping can be obtained by controlling only the first few harmonic frequencies [13]. Furthermore, at high frequencies, it is difficult to realistically control the harmonic impedance up to . For this reason, the following analysis is derived based on an assumption of a third-harmonic termination. However, the following analysis can simply be extended to higher harmonic frequencies as well.

impedance condition. First, in the Class-F amplifier, assuming that the third-harmonic termination is ideally opened and delivers no third-harmonic current, the truncated current waveform in (1) cannot provide a third-harmonic component when the device is undergoing conduction. This means that the arbitrary third-harmonic current component of an intrinsic drain curin (1) attempts to flow to the ground node based on inrent finite termination, as shown in Fig. 1. When a device is in an off state and the input voltage remains below the transistor’s turn-on is trapped at zero, and thus, a voltage, the intrinsic current shorted third-harmonic current is not generated. By the elimination of the third-harmonic current component, the current waveis constructed from the dc, form for a Class-F amplifier fundamental, and second-harmonic, and higher harmonic comhas to be ponents. Since the third-harmonic component of removed for the formation of , can be easily ex. It is here noted that pressed as occurs at due to the domthe peak level of inant second-harmonic manipulation, and is naturally limited . Therefore, the boundary condition by must be satisfied. For this reason, can be expressed as

Boundary Condition for Class-F (2)

A. Current Waveforms For the sake of simplicity, the drain current characteristic of the device will be assumed as follows, which is also depicted in Fig. 1. Minor modifications can be applied if more additional realistic parameters are added. The field-effect transistor (FET) output is basically a voltage-controlled current source, controlled by an input gate–source voltage. The presence of the current waveform in the device is intrinsically dependent on the conduction angle. If an input voltage is applied to a gate node (i.e., all short-harmonic terminations are imposed at the device input), the drain current waveform is truncated by the conduction angle generating dc, fundamental, and harmonic components, which can be represented as a Fourier-series expansion [14]

where is an arbitrary coefficient related to the third-harmonic is a trans-conductance, and current to be absorbed, is a maximum input voltage. As previously mentioned, the maxwhen imum value of a current waveform is limited by . In this respect, if is a negative value, a Class-F amplifier needs slightly lower input power to achieve the maximum numerator, which is current boundary due to the dependent on both the magnitude of the input voltage and the trans-conductance of the given device. This effect results from both the omission of the third harmonic and the manipulation of the second harmonic. The output current in (2) is not comprised of a third-harmonic component, yielding a boundary condition that allows the coefficient to be obtained, which is given by (3)

(1) is the notation of the upper boundary of the drain where and decurrent, represents the conduction angle, and note Fourier coefficients. Since the current waveform based on (1) contains all harmonic components, except when the conduc(Class-A), in order to tion angle becomes (Class-B) or adopt this current model for Class-F and Class-F- amplifiers, the model must be modified considering the infinite harmonic

Also, the

coefficient can be extracted as

(4) is only dependent on the conduction angle. In It is noted that becomes null when the conduction angle is a particular case, (Class-B) or (Class-A). In this case, the current waveform

KIM et al.: MODELING AND DESIGN METHODOLOGY OF HIGH-EFFICIENCY CLASS-F AND CLASS-F- PAs

in (2) becomes identical to the result of (1). Using (2) and (3), the dc, fundamental, and harmonic components can be expressed as coefficient the conduction angle and

155

coefficient is obtained using the following condition, and is given by

(9)

(5)

(10) The dc, fundamental, and harmonic component coefficients for a Class-F- amplifier are given by

(11)

(6) (12)

(7)

(13)

In the Class-F- configuration, the same method can be applied by simply substituting the harmonic condition. To compare the two modes with the equivalent condition, the maximum input voltage of a Class-F mode is also applied, and the . must corresponding maximum current is therefore satisfy the following specific conditions:

As a consequence, for both of the modes, the new output current waveforms are conveniently expressed as a function of only the conduction angle. According to the results from (3) to (13), two striking phenomena are demonstrated. The first is that the third-harmonic current component of a Class-F- amplifier can be generated even in Class-B mode, which essentially has no third-harmonic content. This result resides in the fact that infinite second-harmonic impedance allows a third-harmonic component to be generated. The second phenomenon is that, according to (7) and (13), the current waveforms are built by not only the second- and third-harmonic components, but also higher harmonic contents even though a third-harmonic termination condition is imposed. This implies that harmonic current are self-generated, and are constrained components above by of the transistor. Fig. 2 shows the normalized fundamental and harmonic currents from the dc component using the previous equations.

(8)

where . Unlike a Class-F amplifier, since the second-harmonic current component is zero for a Class-F- configuration, an unknown

156

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 2. Normalized dc and harmonic current components as a function of the conduction angle.

The nonnegative second-harmonic component of a Class-F amplifier is consistently observed as establishing a half-sinusoidal waveform. By contrast, for a Class-F- amplifier, to properly generate a squared waveform, the third-harmonic of a current is “out-of-phase” with respect to the fundamental component. It can also be seen that the magnitude of the normalized fundamental component of a Class-F amplifier drastically decreases compared to the monotonic decrease in a Class-F- amplifier as the conduction angle increases. This peculiar behavior can be expected from the essential drain current waveform characteristic, in which the magnitude of the dc components increases more rapidly, whereas the fundamental components have a small variation with respect to the increase in conduction angle. For this reason, the efficiency of a Class-F amplifier can be expected to be more sensitive to variations of the conduction angle. The harmonic component magnitudes have opposite dependences on the conduction angle. As the quiescent dc current increases toward Class-A mode, the shaping effects weaken, and the output currents for both modes have a tendency to be half-sinusoidal waveforms, as depicted in Fig. 3. In a particular case, in , since the origin which the Class-A biasing strategy is conducting current has essentially no harmonic components, the current waveforms become sinusoidal. Thus, one is not able to control the current waveforms regardless of the harmonic loading condition. Looking at it in another way, the conduction angle is responsible for the shaping of the current, and must be determined within . Otherwise, there is no way to manipulate the harmonic components. Nevertheless, a particular method in a Class-A bias point has been reported using the clipping phenomenon generating a third-harmonic component, particularly for a Class-F- amplifier. However, due to a high overdrive, the efficiency and output compression point are reduced [11]. B. Comparison of Knee-Voltage Effect A significant power device parameter that limits the achievable maximum efficiency in RF amplifiers is the on-resistance

Fig. 3. Current waveforms for Class-F and Class-F- amplifiers as a function of  !t.

=

at low drain–voltage levels (corresponding to the linear region of an FET). When a current waveform reaches a maximum level, the drain voltage swings of Class-F and Class-F- amplifiers are limited by the knee voltage related to the on-resistance. Assuming that the dc I–V curve in the linear region of an FET ), the knee voltage is is linear (corresponding to constant given by (14) where is the peak level of an output current swing. In is consistently observed the case of a Class-F amplifier, as previously described, and is identical to . at In the case of a Class-F- amplifier, the third-harmonic component shifts the peak level of the current point to , as depicted can be obtained using the first in Fig. 3. A phase shift of derivative of the current waveform. If the current waveform of a Class-F- amplifier is built using three dominant components, dc, fundamental, and third-harmonic components, can be calculated as

otherwise (15) The knee voltages of both modes are expressed by

(16) Fig. 4 shows the normalized knee-voltage ratio for both modes. It can be seen that the knee voltage of a Class-Famplifier is smaller compared with that of a Class-F amplifier because the peak level of the current waveform of a Class-F , which is higher than that of a Class-Famplifier is amplifier. In practice, RF amplifiers are biased for Class-B or

KIM et al.: MODELING AND DESIGN METHODOLOGY OF HIGH-EFFICIENCY CLASS-F AND CLASS-F- PAs

Fig. 4. Normalized knee voltage for Class-F and Class-F- amplifiers.

deep class AB whose conduction angle is assumed as . Fig. 4 reveals that the knee voltage of a Class-F amplifier is approximately 1.35 times larger than that of a Class-Famplifier [9]. In this respect, the available output voltage swing of a Class-F- amplifier can be larger than that of a Class-F amplifier at the appropriate fundamental load condition. As the conduction angle increases further, the third-harmonic component of a Class-F- amplifier reduces so that the current waveform becomes a half sinusoid waveform, which is similar to that of a Class-F amplifier. Consequently, their knee voltages are equal to each other.

157

Fig. 5. Voltage waveforms for Class-F and Class-F- amplifiers as a function of  !t under different conditions: MF (dashed line); MVG (solid line).

=

Using the previous solution with the knee-voltage effect, the optimum voltage waveforms are expressed by means of predetermined coefficients [5], [16]. The optimum load impedance for the fundamental and harmonic frequencies can be obtained using (5), (7), (12), and (13), which are given by

(17)

C. Voltage Waveforms The higher self-generated currents described above can provide output powers at their frequencies by either the high or low impedances seen by the drain node. Since only perfect impedance matching for second and third harmonics is considered, these uncontrollable high-frequency impedance values are assumed to be greater than zero and less than infinity. Although a spurious output power can be expected, the amount is small enough to be ignored compared to the fundamental and second and third components. In this respect, the voltage waveforms for Class-F and Class-F- amplifiers can be formulated considering only harmonic frequencies up to third. Taking into account the three harmonic implementations as well as the knee-voltage effect, the aim of the harmonic shaping of a voltage waveform is to improve the fundamental voltage component with respect to the un-manipulated design approach, such as in a classical Class-AB configuration. General solutions for shaping the appropriate voltage waveforms and maximizing the efficiencies have been introduced [5], [15], [16]. One method is a “maximally flat (MF)” condition in which all the minimum points collapse , zeroing the second derivative of the drain voltage. at Another method, called a “maximum voltage gain (MVG)” condition, is used to maximize the fundamental voltage components by means of controlling the optimum harmonic loading condition. In terms of high-efficiency operation, the second approach is more fruitful for attaining maximum efficiency if the breakdown voltage of a device is sufficiently high.

(18)

(19)

(20) The results expressed by (18) and (20) are quite different from the assumption of our current waveform in which open-terminations are suggested at the second (for Class-F- ) and third (for Class-F) harmonics. The different results obtained from this analysis are justified under the assumption of adopting only harmonics up to third for optimization of the voltage waveform. and are so small as to be of negligible effect, If the harmonic impedance is regarded as an infinite value, which justifies our current expressions as previously described. Fig. 5 shows the estimated drain voltages of Class-F and Class-F- amplifiers at a drain voltage of 30 V. As expected, the voltage waveforms are observed to be quasi-square and half-sinusoid, respectively. The peak level of the voltage waveform of a Class-F-

158

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 7. Simplified circuit model of an RF PA with an output impedance matching network.

D. Nonlinear Capacitance Effect

Fig. 6. Drain efficiency and optimum fundament load resistance of Class-F and Class-F- amplifiers as a function of the conduction angle (for the simulation, a 60-W commercial GaN FET is used).

amplifier is approximately 1.5 times larger than that of a Class-F amplifier when an MVG condition is applied. Using (17)–(20), the drain efficiencies of the two modes can be calculated. To calculate the load resistance, a drain bias voltage of 30 V and maximum current of 10 A were used. Fig. 6 shows a plot of the drain efficiency versus conducting angle. The Class-F- amplifier has the highest peak drain efficiency at 80.48% compared with the Class-F amplifier, 76.37%, under Class-B and deep Class-AB conditions. As expected from Fig. 2, as the conducting angle increases, a wide difference in efficiency is seen. The optimum fundamental load resistance of a Class-F amplifier is kept nearly constant, whereas that of a Class-F- amplifier changes slightly as the conduction angle varies. This behavior is accounted for by the nearly constant fundamental current components of a Class-F amplifier in the range , while the current waveform for a Class-Famplifier changes from squared to sinusoidal, increasing the fundamental component. The required fundamental load resistance of a Class-F amplifier is approximately 4.6 , which is smaller than the 7.3 of the Class-F- amplifier in our simulation. Note here that the small load resistance of a Class-F amplifier will contribute toward minimizing the degradation of efficiency under a variable load condition. This contribution will be discussed in the next paragraph. The output powers and drain efficiencies are readily computed using (17)–(20) yielding the following:

At a high operation frequency, the influence of nonlinear capacitances has to be considered, as they affect the load impedance of the amplifier to be changed, resulting in a degradation of efficiency as well as a reduction of output power. Strictly speaking, except for the shorted harmonic loading condition, it is virtually impossible that the optimum load condition be kept constant owing to the variations of nonlinear capacitances related to the drain voltage waveform during a single period. Fortunately, a GaN HEMT has almost a constant drain–source capacitance as a function of the drain voltage. Nevertheless, since PAs are commonly designed to operate in a common-source topology for an acceptable amount of output power, one has to consider not only the drain–source capacitance, but also the gate–drain capacitance. To verify the effect of nonlinear capacitance, we consider the simplified equivalent circuit shown in Fig. 7. The drain voltage and are mainly considerable, dependent capacitances and change in tandem depending on the state of the voltage waveform. Applying the well-known Miller effect with a high power gain of over 10 dB, the gate–drain capacitance can be lumped, and is regarded as the output capacitance connected from the drain node to ground. Thus, the total nonlinear capacand . The voltage dependence itance is a synthesis of of the capacitances may be extracted from the small signal pa, a simple empirical rameters. For gate–drain capacitance hyperbolic tangent expression has been used [17] (22) is the initial value, controls the magnitude of where is responsible for the center the capacitance nonlinearity, and abruptness of the capacitance nonlinearity, and denotes the time-variant voltage waveform of the Class-F and , Class-F- amplifiers. Considering the output matching, including the initial capacitance value, as shown in Fig. 7, and , the variation of output capacitance assuming a constant can be modeled as

(23)

(21)

denotes the initial capacitance value within the where output matching network (OMN), and represents the and . Fig. 8 shows the extracted nonlinear sum of

KIM et al.: MODELING AND DESIGN METHODOLOGY OF HIGH-EFFICIENCY CLASS-F AND CLASS-F- PAs

159

condition yields voltage degradation and phase shift factors by , which is expressed as the addition of

(24) defines the reactance variation of the nonlinear where , and represents the capacitance, which is equal to optimum fundamental resistance for both modes. Applying the results in (24) to (17) and (19), the new voltage waveforms influenced by the variation of nonlinear capacitances are given by

(25) (26)

Fig. 8. Normalized nonlinear capacitance versus drain voltage for a GaN device.

Since the voltage waveforms in (25) and (26) are time variant, the corresponding average output powers of both modes can be calculated using an integral of one period, and are expressed by (27)

capacitances from a 60-W commercial GaN model. In our simulation, a 2 variation of the nonlinear capacitance is observed over a 6-V (knee voltage) to 80-V (maximum voltage) drain voltage range. For the simulation results based on (22), a of 0.5 pF, of 3.8 pF, and of 10 pF are used for the offset capacitance parameters, and a of 0.04 is used for controlling the nonlinearity. In both the Class-F and Class-F- amplifiers, a large variation of capacitance is undesirable as it changes the optimum resistance for the fundamental to time-variant impedance with the reactance value, resulting in a degradation of efficiency. Unfortunately, no matter what arbitrary value the initial capacitance within an OMN has, a variation in load condition is inevitable. In other words, the optimum fundamental resistance and infinite harmonic condition in (18) and (20) are only satisfied when the voltage waveform passes through a specific point corresponding to a pre-determined initial capacitance. The variation of nonlinear capacitance of a Class-F amplifier is observed to be smaller in comparison than that of a Class-F- amplifier. Obviously, such behavior arises from a small available drain to in Fig. 8. For voltage swing, which is from this reason, it is expected that a Class-F amplifier has a benefit of minimizing the efficiency degradation. According to (23), the initial capacitance value can impose restrictions on the as. For instance, if an undesirable value of initial pect of or , is chosen in a design, a capacitances, such as significant variation of load impedance may occur due to the exthroughout the whole capacitance value range. ploring of Therefore, in order to minimize , the initial capacitance is carefully treated, and is matched by an OMN. For a quantitative investigation, if load impedance , including the initial capacitance in Fig. 7, is matched to the optimum fundamental resistance and infinite harmonic impedance values, this matching

(28) The drain efficiencies are then given by (29) (30) To determine the optimum initial capacitance values of both modes, we can intuitively expect that the appropriate is approximately the midpoint between and for a Class-F amplifier because the squared waveform induces and the capacitance variation to have only two states, . Unlike in a Class-F amplifier, the optimum for a Class-F- amplifier is a little different from the center point, as the drain voltage waveform of a Class-F- amplifier is a half-sinusoid waveform exploring the entire range of nonlinear capacitance trajectory. From further simulation, we find the optimum value given by

(31) and represent the maximum capaciwhere tance values corresponding to different knee voltages. in (31), it can be expected From a proper value of that there will be a reciprocal movement of the fundamental impedance, from positive to negative, in the imaginary region during the operation, as shown in Fig. 9, implying that the fundamental load line varies from the capacitive load to

160

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 9. Fundament load impedance trajectory with different initial capacitance values. Fig. 11. Predicted drain efficiency as a function of on-resistance and operating frequency.

Fig. 10. Computed voltage waveform with (dashed line) and without (solid line) the effect of nonlinear capacitance.

the inductive load with the optimum resistance as its center. The computation shows that a variation of fundamental load impedance of a Class-F amplifier takes place in a small region compared with a Class-F- amplifier. This result arises for two reasons. The first is that a variation of voltage-dependent nonlinear capacitance is smaller due to the squared voltage waveform. The second is based on a small fundamental load , which is relatively insensitive to a variation of resistance reactance of the nonlinear capacitance. In this respect, the predicted voltage waveform of a Class-F amplifier with the effect of nonlinear capacitance is similar to the ideal one, whereas the voltage waveform of a Class-F- amplifier is shifted and distorted by time-variant nonlinear capacitance, as shown in Fig. 10. E. Comparison of RF Characteristics Since voltage degradation factor is strongly dependent on the magnitude of the fundamental load resistance, as well as the frequency, a correct performance estimation must be achieved by considering the on-resistance together with the operational frequency. Fig. 11 shows the estimated drain efficiencies of Class-F and Class-F- amplifiers with a deep Class-AB bias as

Fig. 12. Output power performance of Class-F and Class-F- amplifiers.

a function of on-resistance and operating frequency in our device at three different frequencies (0.8, 2, and 3.54 GHz). At for both amplifiers is high a low operational frequency, enough to allow the on-resistance to contribute somewhat to the efficiency, rather than the factor, leading us to conclude that a Class-F- amplifier delivers a superior efficiency compared with a Class-F amplifier. However, as the operating frequency increases up to 3.54 GHz, the coefficient of a Class-F- amplifier begins to affect the efficiency. In this situation, a Class-F- amplifier has a better performance only if the on-resistance is larger than a specific value (0.5 in our device). If the on-resistance is assumed to be smaller than 0.5 , the superior performance of a Class-F- amplifier corresponding to a lower knee voltage vanishes. The analytic results in Fig. 11 can be widely applicable in both low- and high-power devices. Because a low-power device has a high fundamental resistance value in conjunction with a , and a high-power device has the opposite charachigh teristic, the trend shown from the results in Fig. 11 is generally applied, although a small change can be expected.

KIM et al.: MODELING AND DESIGN METHODOLOGY OF HIGH-EFFICIENCY CLASS-F AND CLASS-F- PAs

Fig. 13. Simulated time-domain voltage and current waveforms for a: (a) Class-F- amplifier at  at  =  , and (d) Class-F amplifier at  = 1:3 .

Fig. 12 shows the output power performances of two modes at deep Class-AB bias points versus an input power related to . is relative to the gate voltage and trans-conductance and it determines the input compression point of RF amplifiers. Assuming that the trans-conductance at induces a 1-dB compression point in this paper, when the gate input voltage is low, comparing (6) and (12), a Class-F amplifier has a better capability to convert from a given input power to the fundamental components. However, as the input power is further increased, the fundamental-to-dc component ratio of a Class-F amplifier is more rapidly reduced. For this reason, the linear power gain of a Class-F amplifier is approximately 1-dB higher than that of a Class-F- amplifier, whereas a Class-F amplifier has a worse output compression performance relative to the linearity. To summarize the above comparisons between Class-F and Class-F- amplifiers, a proper design strategy must be determined based on the following considerations. 1) To maximize the efficiency while maintaining an acceptable power gain of a device, a Class-B or deep Class-AB biasing strategy is highly recommended. 2) At a low frequency, a Class-F- configuration is a more reasonable choice than a Class-F configuration due to the lower knee voltage. 3) At a high frequency (beyond 3 GHz), the amplifier topology should be chosen by considering both the on-resistance and voltage degeneration factor. 4) While a Class-F amplifier has a 1-dB higher power gain to relax the system budget, a Class-F- amplifier delivers

=  , (b) Class-F-

161

amplifier at 

= 1:3 , (c) Class-F amplifier

a higher input compression point associated with the linearity. In the following sections, a deep Class-AB biasing strategy is used for simulation and measurement according to the consideration in 1) above, unless otherwise noted. III. SIMULATION AND NUMERICAL RESULTS In this section, numerical results are given in comparison with the harmonic balance simulation results in Agilent Technologies’ Advanced Design System (ADS). The un-packaged large-signal model of a GaN 60-W PEP CGH35060 device from CREE Inc., is used in the simulation. The given device has a breakdown voltage greater than 120 V. The transistor’s input is optimally matched with the fundamental frequency, and all harmonic impedances are short circuited. Since our device has an of 27–29 GHz (depending on the bias strategy), a ninth-order harmonic-balance simulation is accomplished. The output load impedance is set up to the third-harmonic frequency, which is 10.62 GHz, and the higher order output harmonics beyond 10.62 GHz at the output of the transistor are terminated by 50 . Fig. 13(a) and (b) shows the probed voltage and current signals of a Class-F- amplifier at different conduction angles, and Fig. 13(c) and (d) represents the signals of a Class-F amplifier. As analyzed in Section II, it was observed that the increase of quiescent current is not responsible for the shaping of a voltage waveform, but it seriously shapes the current waveform into a half sinusoidal regardless of the output loading condition. A comparison between the predicted and simulated efficiency at three different conduction angles is carried out in Fig. 14. The

162

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 15. Proposed OMN for: (a) Class-F and (b) Class-F- amplifiers. Fig. 14. Calculated and simulated drain efficiencies as a function of the conduction angle.

difference between the model-base efficiency and predicted efficiency is shown here. We cautiously predict that the reason for the difference is due to an imperfect model library or simulator [18]. Compared to the measurement results in the next section, the predicted results have better agreement than the simulated ones. A predicted drain efficiency of 70.3% for a Class-F- amplifier is observed, while that of a Class-F amplifier is almost the same at 70.19% with a saturated output power of 48.4 dBm at a conduction angle of (deep Class-AB). Despite the difference of absolute efficiency between the simulated and predicted results, Fig. 14 clearly indicates that the performance efficiencies of the two modes are almost identical at a 3.54-GHz frequency as investigated.

Fig. 16. Realized: (a) Class-F and (b) Class-F- amplifiers.

stub with a length of to be infinite when

allows the second-harmonic impedance is given by

(32) IV. IMPLEMENTATION AND MEASUREMENTS A. Implementation of PAs Fortunately, the excellent performance of a Class-F- amplifier has been previously verified below 3 GHz [7]–[9]. For successful comparison of Class-F and Class-F- amplifiers at a high frequency, we have designed and implemented 3.54-GHz-band PAs. In order to simplify the OMN, new OMNs are proposed for each case shown in Fig. 15. In Section II, the output harmonic impedances, including the appropriate initial capacitance in (31) seen to load, are required to be infinite. If the tuning lines in Fig. 15 compensate and tune out both the package parasitic components and the optimum initial capacitance simultaneously, simple OMNs can be realized as depicted. In a Class-F amplifier, an OMN can be implemented using only a quarter-wave transmission bias line. In addition, output matching circuit A has an optimum fundamental load resistance in conjunction with the infinite second-harmonic impedance condition. The harmonic control circuit of a Class-F- amplifier can also be achieved using two additional stubs shown in Fig. 15(b). A stub allows the third-harmonic impedance to be zero, and an additional

The matching network, B, is designed to contribute to an optimum fundamental resistance by considering the additional impedance of the two stubs, and it should also have infinite second-harmonic impedance. Based on this concept, the proposed matching networks not only reduce the matching area, but also provide a good tolerance when the PAs are implemented. Based on the proposed matching network along with our analysis, highly efficient PAs have been implemented using the packaged GaN device shown in Fig. 16. The OMNs in Fig. 15 are quite similar to each other. For confirmation of the appropriate operation for both modes, we have measured the harmonic impedance of each OMN, which is zero at the second and third frequencies. B. Measurement Results Fig. 17 shows the measured results of the implemented RF amplifiers with the proposed matching network under continuous wave (CW) operation at 3.54 GHz. The quiescent current of a GaN HEMT for both amplifiers is 100 mA (1% of ) . The Class-Fcorresponding to a conduction angle of amplifier delivers a maximum drain efficiency of 69.9% and

KIM et al.: MODELING AND DESIGN METHODOLOGY OF HIGH-EFFICIENCY CLASS-F AND CLASS-F- PAs

163

TABLE I PERFORMANCE COMPARISON WITH REPORTED PAs USING GaN HEMT TECHNOLOGY

Fig. 17. Measured and output power, power gain, and drain efficiency for a 3.54-GHz CW signal for Class-F and Class-F- amplifiers.

Fig. 18. Output spectral density of a 10-MHz-3G LTE with 8.3-dB PAPR before and after pre-distortion for Class-F and Class-F- amplifiers.

a saturated output power of 47.4 dBm, while the maximum drain efficiency of the Class-F amplifier is 69.4% at an output power of 47.2 dBm. These similar measurements support the analysis results well, and also confirm that the effect of output load impedance variation cancels out the advantage of the Class-F- amplifier with a lower knee voltage in 3.54-GHz applications. Even if the measured power gain of a Class-F amplifier is 1.2 dB higher than that of a Class-F- amplifier, the saturated output power is almost the same. This indicates that the linearity of a Class-F amplifier is slightly worse than that of a Class-F- amplifier, as we expected. To verify the suitability for base-station applications and the nonlinearity, the spectral characteristics of PAs were measured using a 10-MHz 3G LTE signal with a PAPR of 8.3 dB, as shown in Fig. 18 [19]. Input and output OFDM signals are generated and analyzed using an in-house simulator adopting a 3 oversampling system clock of 92.16 MHz and a 1024 inverse fast Fourier transform (IFFT) engine [20]. The DPD algorithm used in the experiment adopts

an indirect learning architecture [21]. An adaptive filter structure for the pre-distorter is based on the memory polynomial, which has four-delay terms and fifth order, including even-order terms as well [22]. The adaptation algorithm is a recursive least squares (RLS) algorithm with a forgetting factor of 0.999. Fig. 18 shows that the Class-F- amplifier has somewhat better adjacent channel leakage ratio (ACLR) performance compared with that of a Class-F amplifier. It is evident that the output compression point related to the linearity is slightly lower than that of a Class-F- amplifier, as described in Section II. The measured output power spectral densities after pre-distortion using memory effect mitigation are almost the same, and both of the two modes have an ACLR below 50 dBc at an average output power of 40 dBm. In this situation, both PAs have a high power added efficiency (PAE) of above 32%. In comparison to comparable high-power PAs using packaged GaN devices, operating at a frequency higher than 2 GHz and delivering a saturated power of over 20 W, the proposed PAs provide high-efficiency performance at a high operating frequency with a simple output termination. A performance comparison with previous studies is given in Table I. As a consequence, the proposed PA design methodology accommodates the optimum PA guidelines for Doherty and ET transmitter systems. V. CONCLUSIONS To achieve an amplifier with high efficiency, fully realized Class-F and Class-F- amplifiers have been analyzed and demonstrated at beyond 3 GHz. For both modes, the effects of critical parameters used to determine the efficiency, in particular, knee voltage and nonlinear capacitance, have been discussed. We have also provided a design guideline in establishing the choice of amplifier type within the framework of the operating frequency. For comparison, we have designed and implemented 3.54-GHz Class-F and Class-F- amplifiers. Except for the power gain and input dB performance, they have almost the same drain efficiency and output power at the saturated region. The measured results are in very good agreement with the predicted results, and clearly demonstrate that the essential device characteristics and operational frequency must be taken into account when determining the type of amplifier to be used.

164

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

REFERENCES [1] S. C. Jung, O. Hammi, and F. M. Ghannouchi, “Design optimization and DPD linearization of GaN-based unsymmetrical Doherty power amplifiers for 3G multicarrier applications,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 9, pp. 2105–2113, Sep. 2009. [2] J. Kim, J. Moon, Y. Y. Woo, S. Hong, I. Kim, J. Kim, and B. Kim, “Analysis of a fully matched saturated Doherty amplifier with excellent efficiency,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 2, pp. 328–338, Feb. 2008. [3] J. J. Jeong, D. F. Kimball, M. Kwak, P. Draxler, C. Hsia, C. Steinbeiser, T. Landon, O. Krutko, L. E. Larson, and P. M. Asbeck, “High-efficiency WCDMA envelope tracking base-station amplifier implemented with GaAs HVHBTs,” IEEE J. Solid-State Circuits, vol. 44, no. 10, pp. 2629–2639, Oct. 2009. [4] F. H. Raab, “Maximum efficiency and output of class-F power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1162–1166, Jun. 2001. [5] P. Colantonio, F. Giannini, G. Leuzzi, and E. Limiti, “On the class-F power amplifiers,” Int. J. RF Microw. Comput.-Aided Eng., vol. 9, pp. 129–149, Feb. 1999. [6] D. Schmelzer and S. I. Long, “A GaN HEMT class-F amplifier at 2 GHz with 80% PAE,” IEEE J. Solid-State Circuits, vol. 42, no. 10, pp. 2130–2136, Oct. 2007. [7] C. J. Wei, P. DiCarlo, Y. A. Tkachenko, R. McMorrow, and D. Bartle, “Analysis and experimental waveform study on inverse class-F mode of microwave power FETs,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, pp. 525–528. [8] S. Gao, P. Butterworth, A. Sambell, C. Sanabria, H. Xu, S. Heikman, U. Mishra, and R. A. York, “Microwave class-F and inverse class-F power amplifiers designs using GaN technology and GaAs pHEMT,” in Proc. Eur. Microw. Conf., Sep. 2006, pp. 1719–1722. [9] Y. Woo, Y. Yang, and B. Kim, “Analysis and experiments for high efficiency class-F and inverse class-F power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 1969–1974, May 2006. [10] P. Wright, A. Sheikh, C. Roff, P. J. Tasker, and J. Benedikt, “Highly efficient operation mode in GaN power transistors delivering upwards of 81% efficiency and 12 W output power,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2008, pp. 1147–1150. [11] A. Sheikh, C. Roff, J. Benedikt, P. J. Tasker, B. Noori, J. Wood, and P. H. Aaen, “Peak class-F and inverse class-F drain efficiencies using Si LDMOS in a limited bandwidth design,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 7, pp. 473–475, Jul. 2009. [12] H. C. Park, G. H. Ahn, S. C. Jung, C. S. Park, W. S. Nah, B. S. Kim, and Y. G. Yang, “High-efficiency class-F amplifier design in the presence of internal parasitic components of transistors,” in Proc. Eur. Microw. Conf., Sep. 2006, pp. 184–187. [13] F. H. Raab, “Class-E, class-C, and class-F power amplifiers based upon a finite number of harmonics,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 8, pp. 1969–1974, May 2001. [14] S. C. Crips, RF Power Amplifiers for Wireless Communications. Norwood, MA: Artech House, 2006. [15] F. H. Raab, “Class-F power amplifiers with maximally flat waveforms,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 11, pp. 2007–2012, Nov. 1977. [16] P. Colantonio, F. Giannini, G. Leuzzi, and E. Limiti, “High efficiency low-voltage power amplifier design by second-harmonic manipulation,” Int. J. RF Microw. Comput.-Aided Eng., vol. 10, pp. 19–32, Dec. 1999. [17] C. Fager, J. C. Pedro, N. B. Carvalho, and H. zirath, “Prediction of IMD in LDMOS transistor amplifiers using a new large-signal model,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2834–2842, Dec. 2002. [18] P. Wright, J. Lees, J. Benedikt, P. J. Tasker, and S. C. Cripps, “A methodology for realizing high efficiency class-J in a linear and broadband PA,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 12, pp. 3196–3204, Dec. 2009. [19] 3GPP TS 36.104, v9.5.0, “3rd Generation Partnership Project: Technical specification group radio access network; evolved universal terrestrial radio access (E-UTRA); base station (BS) radio transmission and reception (release 9), 2010–09. [Online]. Available: http://www. 3gpp.org

>

[20] J. H. Kim, J. H. Jung, S. M. Kim, C. S. Park, and K. C. Lee, “Prediction of error vector magnitude using AM/AM, AM/PM distortion of RF power amplifier for high order modulation OFDM system,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 2028–2030. [21] C. Eun and E. J. Powers, “A new volterra predistorter based on the indirect learning architecture,” IEEE Trans. Signal Process., vol. 45, no. 1, pp. 223–227, Jan. 1997. [22] L. Ding and G. T. Zhou, “Effects of even-order nonlinear terms on power amplifier modeling and predistortion predistortion lienarization,” IEEE Veh. Technol., vol. 53, no. 1, pp. 156–162, Jan. 2004. [23] T. Kikkawa, M. Nagahara, N. Adachi, S. Yokokawa, S. Kate, M. Yokoyama, M. Kanamura, Y. Yamaguchi, N. Hara, and K. Joshin, “High power and high-efficiency AlGaN/GaN HEMT operated at 50 V drain bias voltage,” in IEEE Radio Freq. Integr. Circuits Symp. Dig., Jun. 2003, pp. 167–170. [24] A. Al Tanany, A. Sayed, and G. Boeck, “A 2.14 GHz 50 watt 60% power added efficiency GaN current mode class D power amplifier,” in Proc. Eur. Microw. Conf., Oct. 2008, pp. 432–435. [25] Y. S. Lee and Y. H. Jeong, “A high-efficiency class-E GaN HEMT power amplifier for WCDMA applications,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 8, pp. 622–624, Aug. 2009. [26] J. Kim, J. Moon, J. Kim, S. Boumaiza, and B. Kim, “A novel design method of highly efficient saturated power amplifier based on self-generated harmonic current,” in Proc. Eur. Microw. Conf., Oct. 2009, pp. 1082–1085. [27] A. Al Tanany, A. Sayed, and G. Boeck, “Design of F- power amplifier using GaN pHEMT for industrial application,” in German Microw. Conf., Mar. 2009, pp. 1–4. Joon Hyung Kim received the B.S. degree in electronics from Chonbuk National University, Chonju, Korea in 2001, and the M.S. degree in electronic engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2003. Since February 2003, he has been with the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea. His present research interests focus on the analysis of nonlinearities of microwave amplifiers, linearization techniques, and high efficiency improvement techniques of RF PAs.

Gweon Do Jo received the B.S. degree in electronics engineering and M.S. degree in mobile communications from Chonnam University, Gwang-ju, Korea, in 1991 and 1994, respectively, and the Ph.D. degree in computer and communication engineering from Chungbuk National University, Cheong-ju, Korea, in 2005. Since February 1994, he has been with the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea, where he has been engaged in the research and development of third- and fourthgeneration mobile communication systems and software-defined radio (SDR). He is currently the Principal Member of Engineering Staff with the Internet Research Division, ETRI. His current research interests include the design and performance of DPD for PAs.

Jung Hoon Oh received the B.S. degree in electronics from Kyungpook National University, Daegu, Korea in 1997, and the M.S. degree in electronic engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1999. He is currently with the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea. His current research interests include linearization techniques of RF PAs and field-programmable gate-array (FPGA) hardware implementation of DPD.

KIM et al.: MODELING AND DESIGN METHODOLOGY OF HIGH-EFFICIENCY CLASS-F AND CLASS-F- PAs

Young Hoon Kim received the B.S. degree in electronics from Sogang University, Seoul, Korea, in 1985, and the Ph.D. degree in electronic engineering from Colorado State University, Fort Collins, in 1998. In 1999, he joined the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea, where he has been involved in the area of WCDMA, WiMAX/WiBro, and LTE system development. His research interests are signal processing for digital communication systems and PA DPD.

Kwang Chun Lee received the B.S. and M.S. degrees in electronic engineering from Chungang University, Seoul, Korea, in 1986 and 1988, respectively. Since 1988, he has been with the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea. His main research interests include high-efficiency PA and RF technologies for mobile communication system design.

165

Jae Ho Jung received the B.S., M.S., and Ph.D. degree in electronics from Kyungbuk National University, Daegu, Korea in 1994, 1996, and 2004, respectively. He is currently with the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea. His current research interests focus on fourth-generation (4G) mobile communication systems, high-efficiency improvement base stations, and RF CMOS transceiver design.

166

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Design and Performance of a 600–720-GHz Sideband-Separating Receiver Using AlOx and AlN SIS Junctions F. P. Mena, J. W. Kooi, A. M. Baryshev, C. F. J. Lodewijk, T. Zijlstra, R. Hesper, G. Gerlofsma, T. M. Klapwijk, and W. Wild

Abstract—We present the design, modeling, construction, and characterization of a sideband separating heterodyne receiver that covers the frequency range from 600 to 720 GHz. The receiver has been constructed using waveguide technology in the split-block technique. The core of the mixer consists of a quadrature hybrid, two directional couplers to inject the local oscillator signal, two superconductor–insulator–superconductor (SIS) junctions, three signal-termination loads, and two planar IF/bandpass-filter/dc-bias circuits. The instrument that we have constructed presents excellent performance as demonstrated by two important figures of merit: receiver noise temperature and sideband ratio. Across the entire band, the uncorrected single-sideband noise temperature is below 500 K and reaches 190 K at the best operating point. The sideband ratio is greater than 10 dB over most of the frequency operating range. Superconducting junctions containing AlO - and AlN-tunnel barriers were tested. Index Terms— Mixers, superconducting integrated devices, superconducting microwave devices.

I. INTRODUCTION

T

HE ATACAMA Large Millimeter Array (ALMA) is the largest radio astronomical enterprise ever proposed. Currently, ALMA is under construction and is expected to be fully operational by 2012.1 When completed, each of its more than

Manuscript received February 11, 2010; revised July 20, 2010; accepted September 03, 2010. Date of publication November 29, 2010; date of current version January 12, 2011. This work was supported by NOVA, by the Netherlands Research School for Astronomy, and by the European Community’s Sixth Framework Programme under RadioNet R113CT 2003 5058187. The work of A. M. Baryshev was supported by the Netherlands Technology Foundation (STW) and The Netherlands Organisation for Scientific Research (NWO) VENI 08119. F. P. Mena is with the Netherlands Institute for Space Research (SRON) and the Kapteyn Institute, University of Groningen, 9747 AD Groningen, The Netherlands, and also with the Department of Electrical Engineering, Universidad de Chile, 2007 Santiago, Chile (e-mail: [email protected]). J. W. Kooi is with the Submillimeter Astrophysics Research Group, Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125 USA (e-mail: [email protected]). A. M. Baryshev, R. Hesper, G. Gerlofsma, and W. Wild are with the Netherlands Institute for Space Research (SRON) and the Kapteyn Institute, University of Groningen, 9747 AD Groningen, The Netherlands (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). C. F. J. Lodewijk, T. Zijlstra, and T. M. Klapwijk are with the Kavli Institute of Nanoscience, Delft University of Technology, 2628 Delft, The Netherlands (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2090417 1[Online].

Available: http://www.alma.nrao.edu.

60 constituent radio-telescopes will include ten heterodyne receivers covering the spectral windows allowed by the atmospheric transmission at ALMA’s construction site, the altiplanos of the northern Chilean Andes. In contrast to the sideband-separating receivers being developed at low frequencies [1]–[4], double-sideband (DSB) receivers are being developed for the highest two spectral windows (bands 9 [5] and 10 [6]). In the DSB detection mode, the signal to be detected (RF) is mixed, in a nonlinear device, with a well-determined reference signal called the local oscillator (LO). As a result, the RF . This signal is down converted to an IF, down conversion process allows further study of the signal in a more manageable frequency range (usually a few gigahertz). It is evident that DSB mixers cannot distinguish between signals at frequencies above or below the LO frequency signal, known as upper sidebands (USBs) and lower sidebands (LSBs), respectively. To circumvent this problem, it is possible to suppress one of the sidebands before it is fed into the mixer. This, however, requires extra instrumentation, such as a bandpass filter, in front of the mixer. A more frequency agile solution is a sideband-separating mixer, which produces two IF outputs corresponding to the two sidebands. As a tradeoff, this solution does require extra RF components inside the mixer, the design of which are outlined in this paper. The characteristic that sideband-separating mixers distinguish between the image and signal sidebands can be exploited in astronomy by providing enhanced atmospheric noise reduction when compared with DSB receiving techniques. When the astronomical spectral line of interest is located in one of the sidebands, atmospheric noise present in the image sideband is not folded with the signal sideband as would be the case with a DSB mixer, thus allowing for a lower system noise temperature. Despite this advantage, sideband-separating mixers have not been implemented in the highest-frequency bands of ALMA because of the small (waveguide) dimensions required for the RF components inside the mixer. However, advances in state-of-the-art micromachining technology now achieve the accuracies necessary to realize this development.2 In this paper we report on the design, modeling, construction, and characterization of a sideband-separating mixer that covers the frequency range from 600 to 720 GHz corresponding to band 2Radiometer Physics GmbH, Meckenheim, Germany. [Online]. Available: http://www.radiometer-physics.de

0018-9480/$26.00 © 2010 IEEE

MENA et al.: DESIGN AND PERFORMANCE OF 600–720-GHz SIDEBAND-SEPARATING RECEIVER

167

where

Fig. 1. Schematic of the selected sideband-separating configuration. The incoming RF signal (V +V ) is divided in a 90 hybrid resulting in two branches having the same amplitude, but differing 90 in phase. The branches are then combined with an LO signal and mixed into two nonlinear response devices (in our case, SIS junctions). The resulting down converted signals are fed into a second quadrature hybrid, after which the IF signals corresponding to the USB and LSB (v and v ) are obtained in the two separate IF output channels.

9 of ALMA. The receiver we have constructed has excellent performance, both in sensitivity and sideband rejection. Across the entire band, the uncorrected single-sideband (SSB) noise temperature is less than 500 K and reaches 190 K at the best operating point. In addition, the sideband ratio is greater than 10 dB over 95% of the frequency range. With those values, the mixer accommodates ALMA specifications [7] at most operating frequencies. Incremental progress reports of this work have been presented in several conference proceedings [8]–[11]. Here, we present a comprehensive summary and extend it in two ways: first, we have completed our simulations to understand the operation of the receiver and, secondly, we have tested the performance with a pair of AlN-barrier SIS junctions. Both can orient future work on improving the performance of this and similar receivers. II. DESIGN AND MODELING A. General Concept and Model From a variety of possible sideband-separating schemes, we have selected the configuration shown in Fig. 1. The RF signal to be detected can be represented as a superposition of signals , and below, , the LO frequency. This signal is above, coupled to a quadrature hybrid which splits the signal into two branches of (approximately) equal amplitude and 90 phase separation. Each branch is then coupled with the LO signal and mixed (multiplied) into a nonlinear device. The two resulting IF signals are coupled to a second 90 hybrid whose outputs are and . In a perfect scenario, these outputs contain the desired USBs and LSBs. If perfect terminating loads are assumed, this process can be described as

(1)

In these equations, , , and represent the transmission coefficient of nonlinear device after LO injection, and the transmission matrices of the RF and IF hybrids, respectively. This model will be used in Section II-D. Superconductor–insulator–superconductor (SIS) junctions, as detailed in Section II-C, are used as nonlinear mixing elements. Due to the intrinsic parasitic capacitance of SIS junctions, intermodulation products and higher harmonics are naturally suppressed. Therefore, a 90 RF hybrid can be used instead of a 180 hybrid, despite the latter having superior fundamental and intermodulation product suppression capabilities. Moreover, a 90 hybrid is simpler to construct and thus easier to implement at these high frequencies [12]. In the implementation of the mixer, we have opted for waveguide technology in the construction of the RF components (Section II-B) and planar stripline for the IF filtering and matching parts (Section II-E). The entire design is based on an analytical model [13], and verified by HFSS3 and Microwave Studio [14]. The dimensions of each RF component were adjusted for optimal performance in the 600–720-GHz range, while the IF components are optimized for 4–8 GHz. B. RF Components For implementation of the RF circuitry, we have selected the design presented in Fig. 2. Note that this figure represents the channels to be machined into the split mixer blocks. The coupling sections, shown in detail in Fig. 3(a), are based on a narrow-bandwidth split-block version developed for the ALMA project at lower frequencies [15]. Here, however, the waveguide widths in the hybrid and the LO couplers have been increased by 36.5% to maximize the thickness of the branch lines [12], thereby easing machining constraints. This width increase is the maximum limit before the mode gets excited at the high end of the frequency band. Each of these components was simulated and optimized using commercial software [14]. The geometric parameters are summarized in Table I. Some selected -parameters obtained from the design process are depicted as solid lines in Fig. 3(b). These show a rather flat response in our frequency window (600–720 GHz). Although several configurations have been proposed for the signal termination loads [15], [16], we have selected a rather novel and simple configuration, which is appropriate for the small dimensions involved in this work. The design, presented in Fig. 4(a), consists of a cavity at the end of the waveguide partially filled with an absorbing material. Since the longest dimensions of this geometry are parallel to the splitting plane of the block, this cavity and the filling material are relatively easy to machine compared to other geometries. Extensive simulations of this configuration have been presented elsewhere [8]. If Eccosorb MF112 [17] is used as absorbing material, the load shows 3Ansoft

Corporation, Pittsburgh, PA.

168

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 2. Proposed realization of the RF components schematically indicated in Fig. 1. These components are designed in full-height waveguide. In the design, 12% of the RF signal and 88% of the injected LO are routed to the terminating m and b m. loads. The transverse waveguide dimensions are a

= 310

= 145

( 2 )

Fig. 4. (a) Detailed view of the terminating load. A rectangular cavity a c with the same depth of the waveguide is machined at the end of the waveguide and partially filled with a wedged piece of absorbing material. For dimensions see Table II. (b) Simulated reflection coefficient of the load in two different m (dashed line) and c m (solid line). cases, c

=0

= 300

TABLE II GEOMETRIC PARAMETERS OF THE TERMINATION LOAD SHOWN IN FIG. 4

Fig. 3. (a) Detailed top views of the 90 RF hybrid (left), LO coupler (center), and LO power splitter (right). Note that in the hybrid and LO coupler, the width a of the waveguide is increased in the proximity of the branch lines to maximize the spacing s and s . All components are drawn to scale. For dimensions see Table I. (b) Calculated S -parameters between the input and output as designed (solid lines) and as measured (dashed lines) .

( )

TABLE I WAVEGUIDE DIMENSIONS OF FIG. 3

excellent performance, as demonstrated by the reflection coefficient [see Fig. 4(b)]. When the cavity is terminated at the point

where the wedged part ends , large resonances appear. They can be easily damped by adding extra absorbing material. We have constructed the termination loads with m as they provide the maximum damping [8]. Moreover, it has also been shown that this configuration is reasonably robust as various mounting errors have little influence in the overall performance. The combined RF-LO signal is coupled to the thin-film tuning structure of the SIS junction via a radial-probe-based full-height waveguide-to-microstrip transition, as shown in detail in Fig. 5(a). In this transition, the waveguide backshort has a radius of 35 m and is spatially located 59 m from the edge of the quartz substrate. The overall performance of the coupling is improved by adding a capacitive step in front of the substrate [18]. The use of an “across-the-waveguide” probe configuration facilitates biasing the junction and extraction of the IF signal. Care must be taken, however, in the way the opposite sides of the waveguide are connected as the meandering transmission line is prone to excite resonances [19]. The whole structure was simulated and optimized with commercial software [14]. Table III summarizes the obtained dimensions. In Fig. 5(b), we present the calculated coupling efficiency, and input return loss, between the waveguide and the throat of the radial probe. C. SIS Junction and RF Tuning Structure At the beginning of this project, we were faced with the option of selecting Nb/AlO Nb or Nb/AlN/Nb junctions for our

MENA et al.: DESIGN AND PERFORMANCE OF 600–720-GHz SIDEBAND-SEPARATING RECEIVER

Fig. 5. (a) Detailed views of the waveguide-to-microstrip transition and of the RF tuning structure. The combined RF-and-LO signal is brought into the SIS junction via a radial probe and RF matching network. The superconducting RF choke is used to bring out the IF signal from the SIS junction and to provide bias. The most important dimensions are given in Table III and the material properties in Table VI. (b) Coupling efficiency (S ) and return loss (S ) between the waveguide input and the tip of the radial probe (tuning structure and SIS junction not included in this simulation).

TABLE III GEOMETRIC PARAMETERS OF THE WAVEGUIDE-TO-SIS TRANSITION SHOWN IN FIG. 5

169

in our discussion of the experimental results as this specificity is reflected in the design of the planar IF filter of Section II-E. The other important design selection we made is the choice of a single junction approach. The reasons for this choice are twofold. It facilitates higher uniformity, important to the balance in the sideband separation mixer, and secondly, it allows to suppress more easily the Josephson currents through the tunnel barrier. Following [22] and [23], it was decided to terminate the to achieve the required SIS junction into a load of 8-GHz IF bandwidth [7]. Expecting 10- normal state resiskA/cm , the tance AlN-barrier tunnel junctions IF termination impedance at the junction was designed to (Section II-E). However, due to initial fabrication be 20 difficulties, lower current density AlO -barrier tunnel junctions kA/cm were used instead for much of the measurement campaign. These mixing devices are similar to the ALMA band 9 SIS junctions, which are designed to directly , thereby contact a 50- SMA IF output connector providing a nonoptimal IF termination. The geometric capacitance of SIS junctions is another matter, the reactance being significant at the RF frequencies we concern ourselves with. Combined with the LO-pumped quantum susceptance [24], it shunts the relatively constant impedance of the radial probe waveguide-to-microstrip transition of Fig. 5. To optimize coupling efficiency to the waveguide over an as large as possible RF bandwidth, we have employed, for both the AlO and AlN SIS junction designs, an “end-loaded” stub [25] radial probe matching network. The geometric parameters (inset Fig. 6) were tuned to get good coverage from 600 to 720 GHz [25]. For the AlO and AlN junction designs, the dimensions of the RF matching network are provided in Table IV. The calculated transmission (thick solid line) overlaid on the measured direct detection response is shown in Fig. 6. D. Sideband Ratio An important parameter characterizing a sideband-separating mixer is the so-called sideband ratio. Following (1) and the inset of Fig. 7, the sideband ratio for a given channel is defined as the ratio between the gains coming from the two input ports, i.e., (2) In a perfect sideband separating mixer, these ratios are infinity. However, imperfection of the different components reduce this value. This quantity can be modeled by considering the different transmission coefficients of RF and IF hybrids and SIS junctions given in (1). These coefficients can be written as

design. On one hand, SIS junctions with AlN tunnel barriers and high current density have intrinsically better properties [20], but the process of fabrication was still under development at the Delft University of Technology, Delft, The Netherlands, although with excellent perspectives [21]. On the other hand, the fabrication process of Nb/AlO Nb junctions was more mature as evidenced by our successful experience with the development of DSB receivers for band 9 of ALMA [5]. As a compromise, we baselined AlN junctions for our design with the hope that AlO junctions could also be tested inside of the same structure, albeit with nonoptimized parameters. This is an important detail

where and represent the amplitude imbalance of the hybrids, their phase imbalances, and the conversion gain of the junc-

170

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 7. Calculated sideband ratio assuming different amplitude and phase imbalances of the IF hybrid. The horizontal dotted line represents the ALMA specifications as described in the text. For the calculations, we have assumed a mixer imbalance (G ) of 0.25 dB and the chip misalignment phase error (') of 5 . Inset: definition of gains between the different ports necessary to define the sideband ratios. In our model, both sideband ratios are equal.

Fig. 6. (a) Thin-film tuning structure between the SIS junction and the tip of the radial probe. Dimensions are given in Table IV. The same kind of structure was used with the AlN junctions. Note that, when necessary, L has been meandered to fit in the actual fabricated device. (b) Calculated (includes the waveguide transition) and measured direct detection response of the fabricated AlO SIS junctions. Typical error bars of the experiment are also shown. Vertical dashed lines show the band 9 range.

TABLE IV DIMENSIONS (IN MICROMETERS) OF THE RF TUNING STRUCTURE BASED ON A 40- WAVEGUIDE TRANSITION IMPEDANCE. THE LAYOUT IS SHOWN IN THE INSET OF FIG. 6

tion(s), and the phase error introduced by chip misalignment in the waveguide. Given these coefficients, the sideband ratios can be rewritten as

(3) Here, is the product of the mixer gain imbalance and the imbalance of the two hybrids . The total phase error is accounted for by . In Fig. 7, we show the sideband ratio calculated according to (3) for different values of and . For these calculations, we have set dB, the mixer phase error (due to mounting of the individual SIS junctions) , with and derived from the calculated transmission coefficients of the RF hybrid and the coupling between the radial probe and the SIS junction. If a perfect IF hybrid is assumed, a sideband ratio larger than 20 dB is expected across the entire band. This sets the upper limit for the performance of the present mixer. A more realistic

situation is obtained if some imbalance is added. This results in a degraded sideband ratio, as shown in Fig. 7. E. Planar IF Filter To facilitate reliability, modeling, and repeatability, we have opted for a planar IF design [see Fig. 8(a)]. This is a compact unit containing the IF match, dc-break, bias tee, and electromagnetic interference (EMI) filter. At the heart of the structure there is a pair of parallel coupled suspended transmission lines [26]. By removing part of the ground underneath the transmission lines, as indicated in Fig. 8(a), a 3–9-GHz bandpass filter is formed. The advantage of such planar structure has been demonstrated and used in various astronomical instruments [12], [27]. An added advantage of the planar approach is homogeneity, as this helps minimize the differential phase error at the mixer IF output. The IF circuit, one per SIS mixer, is mounted in cavities milled out in the split block [see Fig. 9(b)] and connected to the junction via bond wires. The entire IF structure has been designed in Microwave Office [13] and HFSS and employs as substrate material alumina with a height of 635 m. The dimensions of the coupled lines (Table V) were optimized to cover the 4–8-GHz frequency range. In the design of the IF matching network, we have taken into account the combined geometric and thin-film microstrip capacitance of the junction, RF choke, wire bond inductance that connects the SIS chip to the IF board, and SIS normal resistance . The calculated total parasitic capacitance is 307 fF, with the inductance of the bond wire determined by its length. In the modeling, we have assumed a bond wire length of 0.3–0.4 mm. Following the discussion in Section II-C, was taken to be 10 , considering availability of AlN junctions [21]. The results of the simulations are summarized in Fig. 8(b). For comparison, we also present the transmission of only the IF planar circuit and of the complete system with two different lengths of bond wires. The wirebond inductance and combined parasitic junction capacitance act to transform the IF impedance ( in Fig. 8), thereby reducing the available IF bandwidth. For a 0.9-mm wirebond length, the highest IF

MENA et al.: DESIGN AND PERFORMANCE OF 600–720-GHz SIDEBAND-SEPARATING RECEIVER

171

Fig. 8. (a) Realization of the planar-stripline IF system. The connection to the SIS junction is accomplished via a (inductive) bond-wire. The dashed line shows the positioning of a cutout bellow the substrate needed for the 3–9-GHz bandpass filter. (b) Calculated transmission between the input and output ports of the IF structure. Inset: complex impedances, from 4 to 8 GHz, at the input of the IF board (Z ) and at the SIS tunnel junction (Z ). The Smith chart is normalized to 50 . See text for details.

frequency is approximately 7 GHz, whereas for a 0.4-mm wirebond length, this limit is pushed to about 8.75 GHz. Note also that the apparent 1-dB loss in coupling efficiency is the result of absorption loss in the alumina substrate loss tangent . At cryogenic temperatures, the substrate loss is expected to reduce to about 0.3 dB. III. CONSTRUCTION A. Waveguide Block We have constructed the mixer in a split-block configuration, as demonstrated in Fig. 9. Conventional machining was used for the large structures and computer numerically controlled (CNC) micromachining for the small RF features. Both parts of the block were made of copper, which were gold plated afterwards to a thickness of approximately 2 m. The fabricated unit is rather compact 8 2 3 cm , it contains all the RF components, the IF filtering board, the dc bias circuit, and the magnetic probes needed to suppress the Josephson currents in the SIS junctions. A closer inspection of the fabricated block shows that all the waveguides and cavities are approximately 5 m wider than designed. The reason appears to be etching of the copper block during the gold plating process. However, the erosion is rather uniform through the entire block. To determine the influence of wider waveguide dimensions, we have repeated the simulation process with the measured dimensions (dashed lines in Fig. 3). It is clear that our design is reasonably robust as long as the symmetry of the RF components is maintained. We also like to point out that a rather recent fabrication

Fig. 9. Constructed sideband-separating block and its different components. (a) Final assembly of the block. (b) Upper half-block showing the defluxing magnets, cavities for the IF board, and the RF components. (c) Close up of the different RF components mounted in the upper block. Note that the SIS junctions and the IF board are mounted in the bottom block, which is not shown here.

TABLE V PARAMETERS OF THE IF-DC BLOCK/BANDPASS FILTER SHOWN IN FIG. 8

process based on photolithography can achieve the small details presented here with an accuracy of less than 1 m [28]. If this method is combined with conventional CNC machining to implement the larger details of the block, it would permit in the future a more reliable fabrication process and its application towards even higher frequencies and/or array receivers [11]. B. SIS Junctions The SIS devices are fabricated on a 200- m-thick quartz substrate. First, a sacrificial Nb monitor layer is deposited, followed

172

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

(out of a sector containing 20 identical junctions) in Fig. 10. The relevant SIS parameters are provided in Table VI. For the depicted AlO -barrier SIS junctions, the subgap-to-normal state resistance ratio is 20, and the gap smear 40 V. Two of the junctions were selected and inserted in the described sidebandseparating block. After mounting, one of the devices showed an increased normal state resistance of 1 . This can be attributed to the silver epoxy used to make the ground contact. The fabrication process of AlN junctions provided lower yield, but did otherwise also produce junctions with excellent properties (Table VI). I–V plots of the two selected junctions, before mounting, are also shown in Fig. 10 for comparison. An important difference is the absence of the above-the-gap proximity knee, characteristic of AlO junctions. Fig. 10. I–V curves of eight different AlO (black thin lines) and two AlN (thick gray lines) before mounting them on the split block. Inset: cross section of the SIS junction and tuning structure (not drawn to scale). TABLE VI SIS PARAMETERS

B. Band Coverage The direct response, as function of frequency, of both SIS junctions contained in our receiver has been measured using a homemade Fourier-transform spectrometer. The measurements were performed with the junctions mounted inside the mixer block and with the test signal fed through the RF port. The results for AlO junctions are presented in Fig. 6. Both junctions present good band coverage and are in good agreement with the predicted response. The agreement is obtained despite the fabrication issues discussed in Section III-A, as those errors do not significantly effect the impedances at the probe tip. C. Sideband Ratio

by an optically defined trilayer of Nb/Al/AlO Nb. The thickness of the Al layer is 5–7 nm, the top and bottom niobium layers are 100-nm thick. Junctions are defined by e-beam lithography using negative SAL601 resist and etched in a SF O reactive ion etch (RIE) plasma. The AlO layer acts as an etch stop for the groundplane. The junction resist pattern is subsequently used as a lift-off mask for a dielectric layer of SiO . An Nb/Au top layer is deposited to define the top wire and contact pads. The Au is wet etched in a KI/I2 solution using an optically defined window. Finally, using an e-beam defined top wire mask pattern, the top-layer of Nb is etched in a SF O RIE plasma, which finishes the fabrication process. The inset of Fig. 10 shows a schematic cross section of the fabricated device. In Table VI, the parameters of the SIS junctions and RF matching network are given. An identical process is used for junctions with AlN tunnel barriers, except for the barrier growth itself. The AlN layer is grown using a plasma source with a remote inductively coupled plasma at 4 10 mbar. The plasma source is at a distance of 10 cm. The inductively coupled plasma source power (ICP) power set point is 550 W. A nitridation time of 17 min resulted in 23.8-kA/cm junctions. IV. EXPERIMENT, RESULTS, AND DISCUSSION A. I–V Curves The AlO SIS fabrication process provides high yield and good reproducibility, as demonstrated by the eight I–V curves

The sideband ratios of the two IF outputs, when AlO junctions were inserted, were measured using the experimental set up shown schematically in Fig. 11. This configuration allows the measurement of the rejection ratios without knowledge of the RF signal levels. The IF response to broadband RF noise sources at two temperatures, however, has to be measured [29]. We have used a commercial microwave absorber4 to insert the noise sources at ambient and liquid-nitrogen temperatures. Fig. 12 shows the measured sideband ratio of every channel as function of RF and IF frequency. ALMA science specifications determine that sideband-separating channels shall provide at least 10-dB image sideband suppression [7]. In our case, it is found that 95% of the operational bandwidth exceeds 10 dB. The obtained sideband ratios are in close agreement with the modeling prediction given in the gray line of Fig. 7 if amplitude and phase mismatches of 1.5 dB and 5 in the IF hybrid are considered. This is consistent with the experimental values obtained for the IF hybrid at 77 K, which shows an imbalance in excess of 1 dB and 4 in phase across the 4–8-GHz band.5 Note that the commercial hybrid6 has been optimized for operation at ambient temperature. To investigate the effect of an improved IF hybrid, we have repeated the calculations assuming an imbalance of 0.5 dB and 3 . In that case, an improvement of 3–5 dB is expected (dashed line of Fig. 7) across the whole band. In the event of a perfect IF hybrid, the sideband ratio is found to be 4Eccosorb AN-72, Emmerson & Cumming, Randolph, MA, 2007. [Online]. Available: http://www.eccosorb.com 5F. P. Mena and A. Baryshev, Netherlands Institute for Space Research, Groningen, The Netherlands, unpublished results. 6ATMh915, ATM Inc., Patchogue, NY, 2007. [Online]. Available: http://www.atmmicrowave.com

MENA et al.: DESIGN AND PERFORMANCE OF 600–720-GHz SIDEBAND-SEPARATING RECEIVER

173

Fig. 11. Schematic of the experimental setup used to determine sideband ratios. The signals from two different sources are brought into the sideband separating mixer (2SB) by means of a combination of lenses and mirrors (L1, L2, M1, M2, and M3). The intensities of the sources can be regulated with the metal grids G1 and G2, respectively. A hot/cold load is also fed into the mixer using a chopper which selects between them. If the beam splitter (BS) is removed, the same configuration can be used to determine noise temperatures. The figure also shows how the IF chain (hybrid, isolators, amplifiers, and switch) were connected during the experiments.

Fig. 13. Measured heterodyne response of the LSB channel of the receiver mounted: (a) with AlO junctions at an LO frequency of 648 GHz and (b), with AlN junctions at an LO frequency of 630 GHz. The corresponding DSB receiver noise temperature are 260 and 250 K. Notice that for these measurements, both junctions were biased simultaneously at the same voltage. For reference, the as0:6) curves are shown. sociated pumped (



Fig. 12. Sideband ratios of the two IF outputs at different LO frequencies when AlO junctions are used. ALMA specification, as described in the text, are summarized by the horizontal dashed line. More than 95% of the operation points comply with ALMA specification.

larger than 20 dB. It is expected that such a design may soon be realizable by direct digital processing of the IF. D. Heterodyne Response The experimental setup of Fig. 11 can be easily modified to measure the heterodyne response of the system. By removing the beam splitter, i.e., the RF insertion, the setup reduces to the conventional variable load method. As an example of these measurements, we show in Fig. 13 the heterodyne response measured at the LSB channel of the receiver when containing AlO and AlN junctions. To calculate the noise temperatures, we have used the Callen and Welton formulation [30], [31] to determine the temperatures of the noise sources, but have not taken into account the finite reflectivity of the absorber (estimated to be less than 1%). However, the induced errors due to this imperfection is only a few degrees, which is much smaller than the measurement uncertainty, calculated to be 13 K in the frequency range of interest [32].

We can gain further insight in the behavior of the mixer by studying the LO-pumped I–V curves of Fig. 13. We have done this analysis for AlO . From a fit, we calculate the LO pumping parameter and the RF junction admittance [24]. With this information, and considering the superconducting RF tuning structure shunted by a 80-fF geometric junction capacitance, we are able to de-embed the RF junction impedance to the . This radial probe waveguide transition reference plane impedance is plotted in the inset of Fig. 14 together with the “as built” waveguide-to-microstrip transition impedance . For optimal coupling , which, given a locus of , provides a coupling efficiency of 63% ( 2 dB) over the 600–720-GHz ALMA band 9 frequency band. Please note that the AlO barrier SIS design is based on a 40waveguide transition impedance locus. The RF coupling loss due to the difference between the “theoretical” versus “actual” probe impedance is 0.75 dB. E. Noise Temperature The variable wideband black-body-radiator load method presented in the previous section allows to determine the noise temperature at the two IF channels, . It has to be noted that the IF power response measured in this way contains contributions from both sidebands and that the resulting noise temperature corresponds to a DSB quantity [29]. The pure SSB noise temperature is obtained from the DSB quantity via [29] (4)

174

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 14. Calculated coupling between the LO-pumped AlO -barrier SIS junction (2.1 mV) and the radial probe waveguide-to-microstrip transition using actual constructed mixer block dimensions. Inset: complex impedance between 600–720 GHz of the radial probe (“as measured”) and of the LO pumped SIS junction referenced to the throat of the radial probe waveguide transition (Z ) with  = eV = h! = 0:6.

where the appropriate sideband ratio has to be used. Note, however, that with a sideband ratio in excess of 10 dB, this correction amounts to less than 10%. The uncorrected SSB noise temperatures calculated in such a way for different pumping frequencies are presented in Fig. 15. Regarding this quantity, must not ALMA specifications for band 9 requires that exceed 335 K over the 80% range of the nominal bandwidth that has the best performance, whereas 500 K may not be exceeded at any frequency within the nominal bandwidth [7]. The latter specification is presented in the same figure as a dashed horizontal line. From our data, we see that the performance is excellent, 50% of the points are below 335 K and 90% below 500 K. However, in an actual ALMA receiver setup, the noise temperature may be expected to be reduced by approximately 30 K via the use of a cooled (90 K) multiplier. With such reduction, our sideband-separating mixer would be completely compliant with the ALMA specifications. Some caution is advised when comparing the presented SSB results with, for example, the official ALMA Band-9 DBS mixer [33] and HIFI Band-1 and Band-2 mixers [34] on the Herschel Space Observatory [35]. In the case of the ALMA Band-9 DBS mixer, the average measured receiver noise temperature across the band is 100 K. This is however with a cooled (90 K) laststage multiplier. Comparison to the HIFI mixers is complicated by the use of a cooled LO (80 K), lack of infrared (IR) blocking, and vacuum windows (the calibration loads are in a vacuum at 4 and 90 K), and the fact that ALMA B9 (600–720 GHz) covers the upper part of HIFI mixer Band-1 and the lower part of HIFI mixer Band-2. In the 600–720-GHz frequency range, the HIFI in flight DSB sensitivity ranges from 85 to 140 K. The insets of Fig. 15 shows a typical dependence of the receiver noise temperature as function of the IF frequency. First, let us concentrate on the inset of Fig. 15(a) where the IF response of both sideband channels for AlO is shown. In this case, the noise temperature is rather flat up to a frequency of 7 GHz after which a rise of 120 K is apparent. This behavior can be understood from our simulations presented in Fig. 8. As explained in Section II-E, the original circuit board

Fig. 15. Uncorrected SSB noise temperatures. (a) Noise temperatures at the two IF outputs with the use of AlO junctions. (b) Comparison of the noise temperatures, at the same IF channel (LSB), for both AlO - and AlN-barrier junctions. The insets present the data as a function of IF frequency for an RF frequency of 648 GHz (i.e., f = 642 GHz and f = 654 GHz). In this scale, the error bars have the size of the symbols used. As described in the text, both types of junctions were not perfectly matched with the IF circuit and, therefore, the average noise temperatures are similar in both cases. However, the IF response is different as the length of the connecting bond wires was different (0.9 mm for the AlO junctions and 0.4 mm for AlN junctions). The bond wires used with the AlN junctions, being shorter, resulted in a flatter behavior of the noise temperature. See also Fig. 8.

design was based a 0.3–0.4-mm IF wirebond length. However, it was discovered that a wirebond contact length of 0.9 mm was employed. Unfortunately, the higher than expected wirebond inductance resonates with the parasitic junction capacitance, thin-film RF matching network and RF choke, as shown in Fig. 8(b). Having understood the problem, we have made modifications to the sideband-separating mixer block that allow placing shorter wire bonds. Measurements with high current density AlN-barrier SIS junctions and a 0.4-mm IF contact confirm that the resonance has indeed been pushed out of the IF band, as it can be seen in the inset of Fig. 15(b). F. Noise Temperature Contributions The receiver noise temperature at an specific output channel contains contributions from the optics in front of the mixer, loss from the sideband-separating mixer itself, and from the IF chain connected to the mixer. This breakdown is given by (5)

MENA et al.: DESIGN AND PERFORMANCE OF 600–720-GHz SIDEBAND-SEPARATING RECEIVER

TABLE VII MEASURED AND CALCULATED RECEIVER PARAMETERS WITH AlO -BARRIER JUNCTIONS AT f GHz AND V V : mV

= 648

=

=21

175

load termination, and on chip integrated IF matching network [27]. In this scenario, we estimate an improvement of 34 K in the measured SSB noise temperature of Fig. 15. Ohmic loss in the waveguide structure, estimated from our measurements, is in the order of 0.75–1.5 dB, and loss in the IF matching network/output-hybrid will remain. Only improved micromaching or electroplating techniques are liable to reduce the front-end loss. Loss simulations with perfect waveguide walls and with a conductivity equal to that of gold with a residual-resistivity ratio of 10 (corresponding to cooling to liquid-helium temperature) indicate a minimum front-end loss of 0.37 dB. V. CONCLUSION

Corrected for IF reflection. Including ohmic loss in RF and IF.

where and are the gain and noise temperature of the and the conversion gain optics in front of the mixer, and intrinsic noise temperature of the mixer, and the noise temperature of the IF chain. These quantities were calculated as described below, and are compiled in Table VII. In front of the mixer’s RF port, a window and two heat filters are placed. The window is an antireflection-coated quartz window constructed by QMC7 following a design that has an optimized transmission to cover the frequency range of band 9 [36]. Moreover, to maximize the hold time of the cryostat, heat filters [37] were placed at the two thermal shields of the cryostat (77 and 4 K, respectively). Provided their respective transmission [38] and physical temperature, the noise temperature of the optical system is calculated to be 9 K. To determine the contribution from the IF chain, we use the characteristics of the AlO junctions above the gap voltage as a calibrated shot noise source [39]. An important point to consider is that the power measured in this way contains contributions from both junctions as the signals are mixed in the 3-dB 90 IF hybrid

In this paper, we have presented the design, modeling, and construction of a sideband-separating mixer that covers the frequency range of ALMA band 9 (600–720 GHz). A full test of the mixer and receiver performance has been presented. It was found that the performance is excellent when compared with ALMA specifications. More than 95% of the operational bandwidth is above the specified sideband ratio (10 dB) and below the specified noise temperature (500 K). By analyses, the receiver is found to be fully compliant if a cooled (90 K) multiplier could have been used. Moreover, it was demonstrated that a better IF match and an improved RF match to the waveguide will make it easier to meet the ALMA specifications. ACKNOWLEDGMENT The authors wish to thank B. Jackson and C. Major, both with the Netherlands Institute for Space Research (SRON), Groningen, The Netherlands, for reading early versions of this paper’s manuscript, M. Bekema, SRON, for her diligent help in soldering various components in the mixer and the cryostat, and K. Kaiser, SRON, for his help in machining the cryostat components. REFERENCES

(6) Given the good reproducibility of our junctions (see Fig. 10), we have assumed that both junctions contribute equal to the total power, except for the noise generated by the extra 1- contact resistance of one of the devices. This extra noise can be estimated by biasing one of the junctions at a particular bias voltage and measuring the generated power as function of the bias voltage of the other junction, then the role of the junctions is interchanged. The difference between them gives the extra noise, which is subtracted from the shot noise analysis of the data shown in Fig. 13. The result is an estimated IF noise temperature (corrected for mismatch reflections) of 8 K for both the USB and LSB output ports. The mixer gain and intrinsic noise temperature were also calculated and are presented in Table VII [39], [40]. In future designs, it is not unrealistic to expect an improvement in the mixer conversion gain of up to 1.5 dB with a more optimized SIS junction RF coupling design, better matched IF 7QMC Instrum. Ltd., Cardiff, U.K., 2007. [Online]. Available: http://www. terahertz.co.uk

[1] S.-K. Pan, A. R. Kerr, M. W. Pospiezalski, E. F. Lauria, W. K. Crady, N. Horner, Jr., S. Srikanth, E. Bryerton, K. Saini, S. M. X. Claude, C. C. Chin, P. Dindo, G. Rodrigues, D. Derdall, J. Z. Zhang, and A. W. Lichtenberger, “A fixed-tuned SIS mixer with ultra-wide-band IF and quantum-limited sensitivity for ALMA band 3 (84–116 GHz) receivers,” in Proc. 15th Int. Space Terahertz Technol. Symp., 2004, pp. 62–69. [2] A. Navarrini, B. Lazareff, D. Billon-Pierron, and I. Peron, “Design and characterization of 225–370 GHz DSB and 250–360 GHz SSB full height waveguide SIS mixers,” in Proc. 13th Int. Space Terahertz Technol. Symp., 2002, pp. 33–40. [3] D. Maier, S. Devoluy, M. Schicke, and K. F. Schuster, “230 GHz SSB SIS mixer for band 3 of the new generation receivers for the Plateau de Bure interferometer,” in Proc. 16th Int. Space Terahertz Technol. Symp., 2005, Art. ID S03-02. [4] D. Maier, A. Barbier, B. Lazareff, and K. F. Schuster, “The Alma band 7 mixer,” in Proc. 16th Int. Space Terahertz Technol. Symp., 2005, Art. ID S08-02. [5] R. Hesper, B. D. Jackson, A. M. Baryshev, J. Adema, K. Wielinga, M. Kroug, T. Zijlstra, G. Gerlofsma, M. Bekema, K. Keizer, H. Schaeffer, J. Barkhof, F. P. Mena, A. Koops, R. Rivas, T. M. Klapwijk, and W. Wild, “Design and development of a 600–720 GHz receiver cartridge for ALMA band 9,” in Proc. 16th Int. Space Terahertz Technol. Symp., 2005, pp. 110–115. [6] W. Shan, S. Shi, T. Matsunaga, M. Takizawa, A. Endo, T. Noguchi, and Y. Uzawa, “Design and development of SIS mixers for ALMA band 10,” IEEE Trans. Appl. Superconduct., vol. 17, no. 2, pp. 363–366, Jun. 2007.

176

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

[7] W. Wild and J. Payne, “ALMA specification: Specifications for the ALMA front end assembly,” NRAO, Charlottesville, VA, Dec. 2000. [Online]. Available: http://www.cv.nrao.edu/~awootten/mmaimcal/ asac/ALMA_rx_specs_V10.pdf [8] F. P. Mena and A. Baryshev, “Design and simulation of a waveguide load for ALMA-band 9,” NRAO, Charlottesville, VA, ALMA Memo 513, Jan. 2005. [Online]. Available: http://science.nrao.edu/alma/aboutALMA/Technology/ALMA_Memo_Series/main_alma_memo_series.shtml [9] F. P. Mena, J. W. Kooi, A. M. Baryshev, C. F. J. Lodewijk, R. Hesper, W. Wild, and T. M. Klapwijk, “An SIS-based sideband-separating heterodyne mixer optimized for the 600 to 720 GHz band,” J. Phys. (Conf. Series), vol. 97, 2008, Art. ID 012331, presented at the 8th Eur. App. Superconduct. Conf.. [10] F. P. Mena, J. W. Kooi, A. M. Baryshev, C. F. J. Lodewijk, R. Hesper, W. Wild, and T. M. Klapwijk, “Construction of a side-bandseparating heterodyne mixer for band 9 of ALMA,” presented at the 18th Int. Space Terahertz Technol. Symp., Pasadena, CA, Mar. 21–23, 2007. [11] F. P. Mena, J. W. Kooi, A. M. Baryshev, C. F. J. Lodewijk, T. M. Klapwijk, W. Wild, V. Desmaris, D. Meledin, A. Pavolotsky, and V. Belitsky, “RF performance of a 600–720 GHz sideband-separating mixer with all-copper micromachined waveguide mixer block,” presented at the 19th Int. Space Terahertz Technol. Symp., Groningen, The Netherlands, Apr. 28–30, 2008. [12] J. W. Kooi, A. Kovács, B. Bumble, G. Chattopadhyay, M. L. Edgar, S. Kaye, R. LeDuc, J. Zmuidzinas, and T. G. Phillips, “Heterodyne instrumentation upgrade at the caltech submillimeter observatory,” in Proc. SPIE Millimeter and submillimeter Detectors for Astronomy II Conf., 2004, vol. 5498, pp. 332–348. [13] Microwave Office. Appl. Wave Res. Inc., El Segundo, CA. [14] Microwave Studio CST, Darmstadt, Germany, 2006. [Online]. Available: http://www.cst.com/ [15] S. M. X. Claude and C. T. Cunningham, “Design of a sideband-separating balanced SIS mixer based on waveguide hybrids,” NRAO, Charlottesville, VA, ALMA Memo 316, Sep. 2000. [Online]. Available: http://science.nrao.edu/alma/aboutALMA/Technology/ALMA_Memo_Series/main_alma_memo_series.shtml [16] A. R. Kerr, H. Moseley, E. Wollack, W. Grammer, G. Reiland, R. Henry, and K. P. Stewart, “MF-112 and MF-116: Compact waveguide loads and FTS measurements at room temperature and 5 K,” NRAO, Charlottesville, VA, ALMA Memo 494, May 2004. [Online]. Available: http://science.nrao.edu/alma/aboutALMA/Technology/ALMA_Memo_Series/main_alma_memo_series.shtml [17] G. A. Ediss, A. R. Kerr, H. Moseley, and K. P. Stewart, “FTS measurements of Eccosorb MF112 at room temperature and 5 K from 300 GHz to 2.4 THz,” NRAO, Charlottesville, VA, ALMA Memo 273, Sep. 1999. [Online]. Available: http://science.nrao.edu/alma/aboutALMA/ Technology/ALMA_Memo_Series/main_alma_memo_series.shtml [18] J. W. Kooi, G. Chattopadhyay, S. Withington, F. Rice, J. Zmuidzinas, C. Walker, and G. Yassin, “A full-height waveguide to thin-film microstrip transition with exceptional RF bandwidth and coupling efficiency,” Int. J. Infrared Millim. Waves, vol. 24, no. 3, pp. 261–284, Mar. 2003. [19] C. Risacher, V. Vassilev, A. Pavolotsky, and V. Belitsky, “Waveguide-to-microstrip transition with integrated bias-T,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 7, p. 262, Jul. 2003. [20] J. Kawamura, D. Miller, J. Chen, J. Zmuidzinas, B. Bumble, H. G. LeDuc, and J. A. Stern, “Very high-current-density Nb/AlN/Nb tunnel junctions for low-noise submillimeter mixers,” Appl. Phys. Lett., vol. 76, pp. 2119–2121, Apr. 2000. [21] T. Zijlstra, C. F. J. Lodewijk, N. Vercruyssen, F. D. Tichelaar, D. N. Loudkov, and T. M. Klapwijk, “Epitaxial aluminum nitride tunnel barriers grown by nitridation with a plasma source,” Appl. Phys. Lett., vol. 91, Dec. 2007, Art. ID 233102. [22] A. R. Kerr and S.-K. Pan, “Some recent developments in the design of SIS mixers,” Int. J. Infrared Millim. Waves, vol. 11, no. 10, pp. 1169–1187, Oct. 1990. [23] J. W. Kooi, F. Rice, G. Chattopadhyay, S. Sundarum, and S. Weinreb and T. G Phillips, “Regarding the IF output conductance of SIS tunnel junctions and the integration with cryogenic InP MMIC amplifiers,” in Proc. 10th Int. Space Terahertz Technol. Symp. Conf., 1999, pp. 100–101. [24] J. R. Tucker and M. J. Feldman, “Quantum detection at millimeter wavelengths,” Rev. Mod. Phys., vol. 57, pp. 1055–1113, Oct. 1985.

[25] C. F. J. Lodewijk, M. Kroug, T. M. Klapwijk, F. P. Mena, A. M. Baryshev, and W. Wild, “Improved design for low noise Nb SIS devices for band 9 of ALMA (600–720 GHz),” in Proc. 16th Int. Space Terahertz Technol. Symp. Conf., 2005, pp. 42–45. [26] W. Menzel, L. Zhu, K. Wu, and F. Bögelsack, “On the design of novel compact broadband planar filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 364–370, Feb. 2003. [27] J. W. Kooi, A. Kovács, M. C. Sumner, G. Chattopadhyay, R. Ceria, D. Miller, B. Bumble, H. G. LeDuc, J. A. Stern, and T. G. Phillips, “A 275–425 GHz tunerless waveguide receiver based on AlN-barrier SIS Technology,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 10, pp. 2086–2096, Oct. 2007. [28] V. Desmaris, D. Meledin, A. Pavolotsky, R. Monje, and V. Belitsky, “All-metal micromachining for the fabrication of sub-millimetre and THz waveguide components and circuits,” J. Micromech. Microeng., vol. 18, 2008, Art. ID 095004. [29] A. R. Kerr, S.-K. Pan, and J. E. Effland, “Sideband calibration of millimeter-wave receivers,” NRAO, Charlottesville, VA, ALMA Memo 357, Mar. 2001. [Online]. Available: http://science.nrao.edu/alma/aboutALMA/Technology/ALMA_Memo_Series/main_alma_memo_series.shtml [30] H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev., vol. 83, no. 1, pp. 34–40, Jul. 1951. [31] A. R. Kerr, “Suggestions for revised definitions of noise quantities, including quantum effects,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 3, pp. 325–329, Mar. 1999. [32] Working Group 1, Joint Committee for Guides in Metrology, “Evaluation of measurement data—Guide to the expression of uncertainty in measurement,” JCGM 100:2008, 2008. [Online]. Available: http://www.bipm.org/utils/common/documents/jcgm/ JCGM_100_2008_E.pdf [33] A. M. Baryshev, R. Hesper, F. P. Mena, B. D. Jackson, J. Adema, J. Barkhof, W. Wild, M. Candotti, M. Whale, C. Lodewijk, D. Loudkov, T. Zijlstra, and T. M. Klapwijk, “Performance of ALMA band 9 receiver series,” presented at the 18th Int. Space Terahertz Technol. Symp., Pasadena, CA, Mar. 21–23, 2007. [34] T. de Graauw, F. P. Helmich, T. G. Phillips, J. Stutzki, E. Caux, N. D. Whyborn, P. Dieleman, P. R. Roelfsema, H. Aarts, R. Assendorp, R. Bachiller, W. Baechtold, A. Barcia, D. A. Beintema, V. Belitsky, A. O. Benz, R. Bieber, A. Boogert, C. Borys, B. Bumble, P. Caïs, M. Caris, P. Cerulli-Irelli, G. Chattopadhyay, S. Cherednichenko, M. Ciechanowicz, O. Coeur-Joly, C. Comito, A. Cros, A. de Jonge, G. de Lange, B. Delforges, Y. Delorme, T. den Boggende, J.-M. Desbat, C. Diez-González, A. M. Di Giorgio, L. Dubbeldam, K. Edwards, M. Eggens, N. Erickson, J. Evers, M. Fich, T. Finn, B. Franke, T. Gaier, C. Gal, J. R. Gao, J.-D. Gallego, S. Gauffre, J. J. Gill, S. Glenz, H. Golstein, H. Goulooze, T. Gunsing, R. Güsten, P. Hartogh, W. A. Hatch, R. Higgins, E. C. Honingh, R. Huisman, B. D. Jackson, H. Jacobs, K. Jacobs, C. Jarchow, H. Javadi, W. Jellema, M. Justen, A. Karpov, C. Kasemann, J. Kawamura, G. Keizer, D. Kester, T. M. Klapwijk, T. Klein, E. Kollberg, J. W. Kooi, P.-P. Kooiman, B. Kopf, M. Krause, J.-M. Krieg, C. Kramer, B. Kruizenga, T. Kuhn, W. Laauwen, R. Lai, B. Larsson, H. G. Leduc, C. Leinz, R. H. Lin, R. Liseau, G. S. Liu, A. Loose, I. Lpez-Fernandez, S. Lord, W. Luinge, A. Marston, J. Martín-Pintado, A. Maestrini, F. W. Maiwald, C. McCoey, I. Mehdi, A. Megej, M. Melchior, L. Meinsma, H. Merkel, M. Michalska, C. Monstein, D. Moratschke, P. Morris, H. Muller, J. A. Murphy, A. Naber, E. Natale, W. Nowosielski, F. Nuzzolo, M. Olberg, M. Olbrich, R. Orfei, P. Orleanski, V. Ossenkopf, T. Peacock, J. C. Pearson, I. Peron, S. Phillip-May, L. Piazzo, P. Planesas, M. Rataj, L. Ravera, C. Risacher, M. Salez, L. A. Samoska, P. Saraceno, R. Schieder, E. Schlecht, F. Schlöder, F. Schmülling, M. Schultz, K. Schuster, O. Siebertz, H. Smit, R. Szczerba, R. Shipman, E. Steinmetz, J. A. Stern, M. Stokroos, R. Teipen, D. Teyssier, T. Tils, N. Trappe, C. van Baaren, B.-J. van Leeuwen, H. van de Stadt, H. Visser, K. J. Wildeman, C. K. Wafelbakker, J. S. Ward, P. Wesselius, W. Wild, S. Wulff, H.-J. Wunsch, X. Tielens, P. Zaal, H. Zirath, J. Zmuidzinas, and F. Zwart, “The Herschel-heterodyne Instrument for the far-infrared (HIFI),” Astron. Astrophys., vol. 518, 2010, Art. ID L6. [35] G. L. Pilbratt, J. R. Riedinger, T. Passvogel, G. Crone, D. Doyle, U. Gageur, A. M. Heras, C. Jewell, L. Metcalfe, S. Ott, and M. Schmidt, “Herschel space observatory. An ESA facility for far-infrared and submillimetre astronomy,” Astron. Astrophys., vol. 518, 2010, Art. ID L1.

MENA et al.: DESIGN AND PERFORMANCE OF 600–720-GHz SIDEBAND-SEPARATING RECEIVER

[36] D. Koller, A. R. Kerr, and G. A. Ediss, “Proposed quartz vacuum window designs for ALMA bands 3–10,” NRAO, Charlottesville, VA, ALMA Memo 397, Jun. 2003. [Online]. Available: http://science.nrao.edu/alma/aboutALMA/Technology/ALMA_Memo_Series/main_alma_memo_series.shtml [37] “GORE-TEX GR sheet gasketing, according DIN 28091, TF-0-0,” Gore, Newark, DE. [Online]. Available: http://www.gore.com [38] A. M. Baryshev, M. Candotti, and N. A. Trappe, “Cross-polarization characterization of GORE-TEX slabs at band 9 frequencies,” NRAO, Charlottesville, VA, ALMA Memo 551, Jun. 2006. [Online]. Available: http://science.nrao.edu/alma/aboutALMA/Technology/ALMA_Memo_Series/main_alma_memo_series.shtml [39] D. P. Woody, R. E. Miller, and M. J. Wengler, “85–115-GHz receivers for radio astronomy,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 2, pp. 90–95, Feb. 1985. [40] M. J. Wengler and D. P. Woody, “Quantum noise in heterodyne detection,” IEEE J. Quantum Electron., vol. QE-23, no. 5, pp. 613–622, May 1987.

177

J. W. Kooi, photograph and biography not available at time of publication.

A. M. Baryshev, photograph and biography not available at time of publication.

C. F. J. Lodewijk, photograph and biography not available at time of publication.

T. Zijlstra, photograph and biography not available at time of publication.

R. Hesper, photograph and biography not available at time of publication.

G. Gerlofsma, photograph and biography not available at time of publication. F. P. Mena received the B.S. degree in physics from the Escuela Politécnica Nacional, Quito, Ecuador, in 1994, and the M.S. and Ph.D. degrees in physics from the University of Groningen, Groningen, The Netherlands, in 2000 and 2004, respectively. In 2004, he joined the Netherlands Institute for Space Research (SRON), Groningen, The Netherlands, as an Instrument Scientist with the Low Energy Division. Since 2008, he has been an Assistant Professor with the Department of Electrical Engineering, Universidad de Chile, Santiago, Chile.

T. M. Klapwijk, photograph and biography not available at time of publication.

W. Wild, photograph and biography not available at time of publication.

178

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

A Calibration Method for RF and Microwave Noise Sources Leonid Belostotski, Member, IEEE

Abstract—Uncertainties in noise-figure and noise-parameter measurements depend to a large degree on the accuracy with which noise sources are calibrated. In some cases, the uncertainties are similar to measured quantities. This paper discusses a method for calibration of RF and microwave noise sources against a reference noise source. This method takes into account noise parameters of the receiver used for noise power measurements and includes temperature corrections to account for both drifts in the ambient temperature and heat transfer between the receiver and noise sources. Experimental results demonstrate the feasibility of the proposed method. As estimated with a Monte Carlo analysis, a 2 uncertainty of 0.046 dB for a 5-dB excess-noise-ratio (ENR) noise source is obtained with the proposed method. This uncertainty is approximately one third of ENR uncertainties reported for commercial noise sources. Index Terms—Calibration, excess noise ratio (ENR), noise figure, noise parameters, noise source.

reported through the noise source’s excess noise ratio (ENR) defined1 as (1) K is the reference temperature. A noise-power where measurement system (“receiver” in this paper) that uses noise figure analyzers (NFAs) such as Agilent’s N8975A reports for Agilent SNS-series noise sources. Having ENRs, , supplied with a noise source and reported by the receiver, is calculated from (1). In a standard noise-figure measurement system, a single automated noise source is used. This noise source is turned ON (Hot) and OFF (Cold) by the receiver. With the noise source connected to the input of the receiver, the receiver measures two and , noise power levels where is the noise power added by the receiver. Having measured , the receiver calculates a Y-factor (2)

I. INTRODUCTION from which the noise factor of the receiver itself is found from ESIGNS of communication systems and instrumentation equipment often require measurements of noise factors (figures) and noise parameters for characterization of low noise amplifiers (LNAs) and other system subcircuits. These measurements require noise sources that generate two different but known levels of noise power. One of these power levels is usually generated by a 50- termination at the ambient temperature and is well known. For standard noise sources, this “Cold,” , i.e., the noise source is OFF, power level is where is the Cold noise temperature of the noise source, is the Boltzmann constant, and is the measurement bandwidth of the receiver. The other level, i.e., the Hot power level generated when the noise source is ON with being the Hot noise temperature, is usually not know as accurately, is determined during a noise source calibration, and is

D

Manuscript received June 14, 2010; revised August 24, 2010; accepted September 25, 2010. Date of publication October 28, 2010; date of current version January 12, 2011. This work was supported by the University of Calgary, the Natural Sciences and Engineering Research Council of Canada, Alberta Innovates - Technology Futures, NSERC’s Special Research Opportunity Grant Program, the Alberta Provincial Government’s iCORE program, the Dominion Radio Astrophysical Observatory, National Research Council, Herzberg Institute of Astrophysics, and CMC Microsystems. The author is with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada T2N1N4 (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2086066

(3) Since often , (3) reveals that any uncertainties in the ENR, , will affect . Similarly, uncertainties in ENR affect the noise factors of devices under test (DUTs). In addition, as shown in [2], these uncertainties in ENR have a very significant contribution to systematic errors in extraction of noise parameters. Due to difficulties associated with calibrations of noise sources, ENRs supplied are usually known to within 0.15 dB of their true values and are only reported for a few cardinal frequencies. While 0.15-dB uncertainty is not very significant when relatively high noise figures are measured, with the advances of modern semiconductor processes, LNA noise figures of less than 1 dB have been reported in the literature [3]–[14]. All of these results are subject to some ENR uncertainty. This uncertainty is sometimes similar in size to the measured quantities as demonstrated in [2]. In some specialized laboratories, high-accuracy cryogenic terminations are used to generate the other well-known noise power levels and to act as noise sources. These noise power levels are related to physical temperatures of the cryogenic 1There is an inconsistency when it comes to definitions of the ENR. The definition in (1) is adopted in this work and is recommended in [1]. However, in some works T is replaced with T . The main reason (1) is adopted in this paper is because equipment used in experimental analysis in Section IV uses this definition.

0018-9480/$26.00 © 2010 IEEE

BELOSTOTSKI: A CALIBRATION METHOD FOR RF AND MICROWAVE NOISE SOURCES

terminations and are well known. However, cryogenic noise sources are not automated, i.e., they cannot be easily used in noise parameter extraction systems, and they require highly skilled personnel to operate them properly. In this paper, a method of calibrating automated noise sources against an accurate cryogenic noise source is discussed. The method takes into account practical difficulties associated with drifts in the ambient temperature and heat transfer between various devices in the measurement system. This procedure transfers, with some degradation, the precision of a cryogenic noise source to an automated noise source that can then be used within systems that require repeated toggling of noise sources ON and OFF such as in noise parameter measurement systems [2]. After a review of the conventional calibration procedure, this paper explains theoretical concepts behind the proposed noise source calibration. The proposed theory is verified by experimental calibration of four Agilent SNS N4000A noise sources against a reference cryogenic noise source. Appendix A explains symbols and notations used in this paper.

179

Fig. 1. Graphical representation of the measurement system. A 3-dB attenuator is added at the input of the receiver to improve its input impedance.

noise source ENR values and the noise-source calibration results. In addition to correcting for impedance issues discussed , this paper adabove and effects of the impedances on dresses the effect of changes in the ambient temperature on the noise-source calibration and presents a set of expressions needed for calibration of conventional noise sources. III. PROPOSED NOISE-SOURCE CALIBRATION PROCEDURE

II. REVIEW OF NOISE-SOURCE CALIBRATION METHODS The standard noise-source calibration technique is summarized in [15]. This noise-source calibration, i.e., noise-source ENR determination, is performed by comparing receiver noise factors measured with a reference noise source with a known , to the receiver noise factors measured with a noise ENR, . source under calibration (NSUC) with an unknown ENR, The receiver noise factor is assumed to be constant during is found from both measurements. Then using (3), (4) In (4), and are measured by the receiver as discussed in Section I with the reference noise source and the NSUC, reand are the Cold noise temperatures of the spectively, and reference noise source and the NSUC, respectively. This standard calibration method assumes that the reference noise source and the NSUC have identical impedances. In practice, this assumption is not correct. Noise sources have different impedances and their impedances change when the noise sources are depends on its signal-source toggled from ON to OFF. Since impedance, depends on both noise sources and their states. For the same reasons, power loss due to mismatches between the receiver input impedance and noise source impedances is also noise source dependent. The mismatch problem can be dealt with by inclusion of isolators, which are narrow band at lower frequencies, as addressed in [16], [17]. However, the former problem of dependence on noise source impedances still remains and is addressed in [18], where a procedure is described to properly account for changes in when noise sources have different impedances. [18] makes an assumption that variations in the ambient temperature are not affecting the noise temperatures of the receiver and the NSUC. In standard laboratory settings, the ambient temperature exhibits drifts which, as is demonstrated in this work, affect the

To continue with the theoretical description of the proposed noise-source calibration method, it is assumed that the following quantities have been measured and/or are known at each frequency of interest: the reflection coefficient of the re, all input reflection coefficients (admittances) of all ceiver, noise sources used in measurements, ’s ( ’s), reflection coefficients (admittances) of passive mismatched loads, ’s ( ’s), Cold temperatures of noise sources, ’s, the ambient temperature, , and both Cold and Hot noise temperatures of the reference noise source, . A general representation of the measurement system used in this work is shown in Fig. 1. Step 1 (Measurements With a Reference Noise Source): A reference noise source generates at least two known noise power levels: Hot and Cold. Using the receiver,2 combinations of the noise powers emitted from the reference noise source and the noise due to the receiver itself are measured. The relationships between measured noise powers by the receiver and the noise temperatures of the receiver and the noise source can be represented as [2]

(5)

where the receiver transducer gain

is [2]

(6)

2When using NFAs as receivers, one must be careful to fix their radio frequency (RF) attenuations and, if available, their intermediate frequency (IF) attenuations for all measurements otherwise the receiver gain may not be consistent from one measurement to next. Note these levels may be different for different frequencies.

180

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

In (6), takes one of two values ( or ) depending on the state of the noise source connected to the receiver input, is a noise source mismatch factor that can be calcuis the unlated from quantities readily measurable, and known portion of the receiver transducer gain that depends on the inner-workings of the receiver, i.e., the receiver -parameters and the reflection coefficient of the receiver load . In (5), are the receiver input-referred noise temperatures referenced to and are expressed in terms of the receiver noise parameters and noise-source admittances as

(7)

where indicates the real part. In (7), is the minimum is the equivalent noise renoise temperature of the receiver, is the receiver optimum admitsistance of the receiver, and tance for minimum noise. scales in (5) from to the ambient temperature [2]. This scaling assumes a linear dependence of the receiver noise temperature on the ambient temperature over a relatively small range of change in [2]. The two equations in (5) are not enough to solve for five un, and and knowns thus more measurements are required. Step 2 (Measurements With an Unknown Noise Source): With an NSUC connected to the receiver, measurements in Step 1 are repeated. As in Step 1, two equations

(8)

describe relationships between the measured noise powers and the noise temperatures of the NSUC and the receiver. In (8), is in addition to the unknowns from (5), a new unknown introduced. This unknown is the subject of this work. Together, (5) and (8) give four equations but the number of unknowns is now six; therefore, additional measurements are necessary to find these unknowns. Note, it is possible to measure two additional unknown noise sources to increase the number of equations and unknowns to eight each. However, because impedances of noise sources are very similar to each other and close to 50 , there is not enough spread in impedance constellations produced by the noise sources to extract , , and accurately [19]. Therefore, a few measurements with mismatched passive standards are recommended and discussed next. Step 3 (Measurements With Mismatched Loads): To increase the number of equations without generating any additional unknowns and to improve the impedance constellation presented to the receiver, mismatched passive loads with reflection coefficients ’s are employed [18], [20] provided that their physical temperatures, , can be measured. Similarly to (5), with each

additional mismatched passive load, an additional equation relating power reported by the receiver and the noise temperature of the passive load is

(9) Two mismatched passive loads bring the total number of independent equations in (5), (8), and (9) to equal the number of unknowns. A larger number of loads, whose reflection coefficients are both well characterized and provide adequate Smith Chart coverage, will improve the extraction of the unknown parameters [19]. Measurements presented later in Section IV produced poor results at some frequencies when only two mismatched loads were used. Three or more of such loads appear to provide accurate results. Step 4 (Least-Squares Fit): Once the number of equations in (5), (8), and (9) equals or exceeds the number of unknowns in these equations, a least-squares approach can be used to find the unknowns. All individual equations in (5), (8), and (9) are of the same form and can be rewritten as

(10) where subscript individually references all of the is the total measurements performed in Steps 1–3 and number of measurements. (10) can be further rewritten as

(11) . Depending on which noise source where (7) is used for in (10) is either known (i.e., those the subscript identifies, of the reference noise source, ’s, passive loads, ’s, and Cold temperatures of NSUCs, ’s) or unknown (i.e., those of the Hot temperatures, ’s, of NSUCs). Thus, in (11), is replaced by , where are known noise temperatures, are unknown noise temperatures, and can take values of either (1, 0) or (0, 1) depending whether is known or not, respectively. Next, a system of equations linear in new parameters is created from (11) by expanding the work in [21] to include and . First, the set of new variables ( and ) is defined: (12) (13) (14) (15) (16) (17)

BELOSTOTSKI: A CALIBRATION METHOD FOR RF AND MICROWAVE NOISE SOURCES

where and is the number of unknown noise temperatures. Then, with these new variables (11) becomes

181

Step 5 (ENR Correction): With the Hot temperatures of ’s, obtained in Step 4, the calibrated NSUC ENR, NSUCs, referenced to the conventional 296 K ambient temperature,4 is found from (24)

(18) where . Note that when noise source temperatures are known, and, thus, , whereas for NSUC’s and are found by solving (18) as Hot temperatures discussed next. The least-squares solution of (18) is found by minimizing the sum of the squared residuals3:

where is used instead of since the noise source is no longer “unknown.” Expression (24) is new and is a modified version of (1) in which the extra term in brackets accounts for increase in with increase in . This modification is discussed in Appendix B. IV. EXPERIMENTAL APPLICATIONS AND VERIFICATIONS OF THE PROPOSED METHOD A. Experimental Applications

(19) which in a matrix notation is equivalent to solving (20) where , , and are shown in (21)–(23) at the bottom of this is a transpose of . Once the solution to (20) page and is found, then the unknown , and ’s, i.e., ’s, are determined from (12)–(16), in particular ’s, with (16) and (17) the unknown Hot noise temperatures, of NSUCs are found. 3The

accuracy of the receiver noise parameter extraction can be improved by M =P [22]. scaling (19) by w

=(

)

The calibration procedure described in Section III was performed experimentally and is presented in this section. For the reference noise source, a Maury Microwaves noise calibration system MT7208J was used. This system has three 50- terminations: at liquid nitrogen temperature, at 370 K, and at the room temperature. In the results presented below, four different Agilent N4000A noise sources5 were calibrated at once. Four mismatched loads, created by terminating a 3-dB attenuator and a 1-dB attenuator each with an open circuit and a short circuit, were used. The ambient temperature was monitored throughout 4It is a standard practice to calibrate noise sources at 296 K ambient temperature [23] while noise figure measurements and ENR calibrations are still K. This convention is adopted in this work. referenced to T 5For readers’ reference, the serial numbers of the noise source #1 to #4 are: MY44420986, MY44420107, MY44420504, and MY44420729, respectively.

= 290

(21) .. .

(22)

.. .

(23) ..

.

182

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE I MEASUREMENT UNCERTAINTIES ADDED TO EVALUATE NOISE PARAMETER EXTRACTION METHODS BASED ON THE INSTRUMENT SPECIFICATIONS NOTED IN THE REFERENCE COLUMN

Fig. 2. Comparison of the ENRs given by the manufacturer, E , and corrected ENRs using (a) the proposed method, E , and (b) the standard approach in (4), E . Measurements were performed in the frequency range of a mid-band receiver intended for the Square Kilometer Array radio telescope [24]. 2 uncertainty errors supplied with the noise source ENRs are shown with error bars where available.

the measurements. Since the physical temperatures of the receiver and the NSUCs are not the same, there is a possibility of heat transfer between the two devices, which was observed in this work. Additional measurements described in Appendix C had to be performed to determine the unknown temperature of the receiver due to the heat transfer. In addition, the physical temperatures of the mismatched loads were also increased by the transfer of heat from the receiver. These were measured to improve the noise-source calibration accuracy. Fig. 2(a) shows calibrated ENRs determined using the pro, as compared to ENRs supplied with the posed method, . It is interesting to note that noise source #2 noise sources, had its ENR specified significantly away from what its value should have been. In Fig. 2(b), ENRs calculated with the standard method, , in (4) are also shown for comparison. ’s ’s as the standard calibration does not are less “smooth” than account for the effects of mismatches between the receiver and the noise source impedances, even though as shown in Fig. 1 a 3-dB attenuator was inserted at the receiver input to keep its return loss larger than 18 dB. Fig. 2 shows that ’s are mostly within ENR uncertainties supplied with noise-source calibration reports for ’s, with the exception of noise source #2.

Fig. 3. 2 error bars for corrected ENRs, E .

To estimate how accurate the proposed calibration method is, a Monte Carlo analysis was performed. Similarly to the work in [2], measurement uncertainties shown in Table I are randomly added to all measured quantities and these are fed to the proposed ENR calibration algorithm. After repeating this process 1000 times, a standard deviation6 of 0.023 dB in was obtained. The error bars are added in Fig. 3 to ’s, already shown in Fig. 2(a), to illustrate their relative sizes. Note, measurements presented in this work were performed in an RF shielded chamber but still suffered from interference at 0.7, 1.3, and 1.5 GHz from the measurement equipment itself. B. Calibration Verifications During measurements required for the noise-source calibration, the receiver reported its own noise figures calculated with 6If only one noise source is calibrated with just three mismatched loads the standard deviation is 0.027 dB.

BELOSTOTSKI: A CALIBRATION METHOD FOR RF AND MICROWAVE NOISE SOURCES

183

Fig. 5. Comparison of receiver noise figures reported by the NFA, NF, the noise figures calculated based on the receiver noise parameters and the noise source temperatures obtained with the proposed method, NF . In this figure the ambient temperature correction is not applied.

Fig. 4. Comparison of receiver noise figures reported by the NFA, NF, and the noise figures calculated based on the receiver noise parameters and the noise source temperatures obtained with (a) the proposed method, NF , and (b) the standard method in (4), NF .

(3) and based on ’s. The proposed noise-source calibration procedure is designed to determine the unknown noise temperatures of noise sources. As a verification of the noise-source calibration, these newly found noise temperatures and the receiver noise parameters extracted during the calibration process were used to calculate the noise figures that the receiver reported. None or little difference between the calculated noise figures and noise figures reported by the receiver itself indicates that the noise temperatures of the noise sources and the receiver noise parameters have been determined accurately. The results are shown in Fig. 4(a), where a close agreement between calculated and reported noise figures is observed. For the four noise sources, the standard deviation of differences between the measured and the calculated noise figures based on calibrated ’s is 0.009 dB. The noise-source noise temperatures, obtained from ’s calibrated with (4), give a 0.05-dB standard deviation of differences between the resultant noise figures and those measured by the receiver (see Fig. 4(b)). It is interesting to observe that noise source #2 produces noise figures that are more than 0.5 dB away from what the other noise sources produce. This was expected based on the results in Fig. 2 and is due to erroneous ENRs supplied with the noise source. It is important to highlight that the noise figures calculated with noise temperatures derived from ’s are very close to those found with ’s. These noise figures are dependent on the receiver noise

Fig. 6. Noise figures of an amplifier measured with the NFA and three noise sources. For each noise source three different ENRs were used, i.e., ENRs supplied with the noise source, the calibrated ENRs with the procedure from this work, and ENRs calibrated with the standard calibration procedure.

parameters but calculations of ’s do not require these noise parameters. The close agreement between the noise figures indicates that the noise parameters were extracted accurately and suggests that the proposed noise-source calibration is also accurate. To test of this further, the ambient temperature correcterms in (18) were set to 290 K tion was removed, i.e., all making the calibration procedure similar to that in [18], and data in Fig. 4(a) were recalculated. This resulted in a significant difference, observed in Fig. 5, between all measured noise confirming figures and those calculated based on extracted that changes in the ambient temperature affect the accuracy of the noise source calibration and they should be measured to improve the ENR calibration. For the purpose of the calibration verification, noise figures of an amplifier were measured with three7 of the four calibrated noise sources. The resultant noise figures, shown in Fig. 6, ob’s show maxtained with the three noise sources and their imum difference of less than 0.03 dB, mostly due to differences 7Noise

source #2 was not available.

184

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

the agreement between the noise figures improves as shown in Fig. 7 as expected. In addition, the noise source #4 was sent to the Radiometer Group at the California Institute of Technology for a cross-verification against their cryogenic noise source [29]. There, the cryogenic noise source and the noise source #4 were used to measure noise temperatures of an LNA. Measurement results, shown in Fig. 8, demonstrate a good agreement between the noise temperatures measured with the cryogenic noise source is used. The visible discrepand the noise source #4 when ancies are at 0.7 GHz, 1.3 GHz, and 1.5 GHz where interference ’s. The other discrepancies are subject to difcontaminated ferences in noise source impedances and their effect on the LNA noise temperatures and the amount of delivered noise power. V. CONCLUSION Fig. 7. Noise figures of an amplifier measured by the NFA and three noise sources. By using ENRs calibrated with the procedure described in this work, amplifier noise figures were measured with noise sources #1, #3, and #4. These ENRs were calibrated with T used instead of T in the ENR definition of (1) and the measurements were repeated. In addition, noise figure measurements with the noise source #1 and its ENR calibrated at NPL are also shown.

Fig. 8. Noise source calibration verification using a liquid nitrogen cryogenic noise source at the California Institute of Technology [29]. The LNA noise temperatures shown were measured with the noise source #4. Both E (found in this work) and E (supplied with the noise source) were used to calculate the noise temperatures. The other set of curves shows four repeated measurements of the LNA noise temperatures with the cryogenic noise source.

in the noise-source impedances affecting the amplifier noise fig’s are used for calculations of the noise figures, ures. When ’s the maximum difference is less than 0.05 dB. With the the noise figures are different by as much as 0.18 dB. In Fig. 6, noise figures obtained with the noise source #1 and its are ’s. for this noise close to the noise figures calculated with source were measured directly by the National Physical Labo’s ratories (NPL) and are known with better accuracy then for the other two noise sources. A good agreement between the noise figures calculated with ’s and the noise figures measured with the noise source #1 shows that the presented calibration approach is viable. The main difference between the noise insource #1 noise figures is mainly due to the NPL using stead of in (1). If the same definition is used in this work,

This paper discussed a method of calibration of RF and microwave noise sources. The proposed calibration scheme takes into account the noise parameters of the receiver, ambient temperature effects, and the heat transfer effects on the measurements. The calibration was verified experimentally. LNA noise figures measured with the ENRs, obtained with the proposed calibration, were also successfully compared against the noise figures measured by the Radiometer Group at the California Institute of Technology [29]. In addition, a relationship between the Hot noise temperature and the Cold noise temperature of a noise source was investigated experimentally to further improve the calibration accuuncertainty in the caliracy. Using a Monte Carlo analysis brated ENRs is estimated to be 0.046 dB which is about a third of uncertainties in commercial noise source ENRs. It is also shown that the standard calibration procedure produces a sufficiently accurate estimate of the noise source ENRs, provided the impedance mismatch between noise sources and the receiver is not significant. The standard procedure is easier to perform and, therefore, can be used when slightly less accurate ENR values are satisfactory for their intended applications.

APPENDIX A SYMBOLS AND NOTATIONS A. Subscripts and Superscripts • Subscript “a” identifies ambient temperatures. • Superscripts “c” and “h” identify Cold (i.e., OFF) and Hot (i.e., ON) states of a noise source, respectively. • Subscript “g” identifies quantities Given by the noise source manufacturers. • Subscript “x” identifies quantities pertaining to an unknown noise source. • Subscript “off” identifies measurements conducted when a noise source is turned OFF for a long period of time. • Subscript “p” identifies quantities pertaining to passive mismatched loads. • Subscript “r” identifies quantities pertaining to a reference noise source.

BELOSTOTSKI: A CALIBRATION METHOD FOR RF AND MICROWAVE NOISE SOURCES

185

• Subscript or superscript “rec” identifies quantities pertaining to the receiver. • Subscript “tp” identifies quantities obtained with the procedures described in This Paper. • Subscript “sp” identifies quantities obtained with Standard Procedures. B. Symbols • • • • • • • • • • • • • • • • • • •

: ENR of a noise source. : bandwidth for noise measurements. : noise factor. : transducer gain. : internal gain of the receiver. : reflection coefficient. : Boltzmann constant. : mismatch between the receiver and a noise source. : power measured by the receiver. : equivalent noise resistance of the receiver. : ratio of receiver and noise source thermal resistances. : temperatures. : reference temperature of 290 K. : minimum noise temperature of the receiver. : admittance, conductance and susceptance, respectively. : receiver optimum admittance for minimum noise. : Y-factors. , and : temporary variables used in the least-squares fit. and : parameters used in the least-squares fit. They take values of 1 or 0, and indicate whether measured data correspond to a known or an unknown noise source temperature, respectively.

APPENDIX B ENR CORRECTION Equation (1) assumes that when the Cold temperature of a noise source increases by some amount, the Hot temperature increases by the same amount. This Appendix investigates this assumption. Since in a standard measurement environment it is difficult to control ambient temperatures, there are some temperature drifts that potentially affect noise powers generated by noise sources. The Cold temperatures of the noise sources track the ambient temperature fairly accurately with some offset. To analyze how Hot temperatures behave when Cold temperatures change, a noise source, connected to the receiver input, is warmed up and and are measured by the receiver. First, with the noise source at the ambient temperature, the input-referred noise tem, is obtained by using the standard perature of the receiver, Y-factor technique [30]. Then as the noise source is warmed , the Hot noiseup, for each Cold noise-source temperature, source temperatures are obtained from (25)

Fig. 9. Dependence of T on T sources at five frequencies.

for two Agilent’s 5-dB ENR N4000A noise

where (25) is derived by expressing temperatures

and

in terms of

(26) and since the receiver is not affected by the localized temperature increase of the noise source is constant and (25) follows. Fig. 9 shows results and demonstrates that increases at approximately twice8 the rate of . Note that (25) is only approximately correct as the second order effects such as impedance variations of the noise source with temperature are not taken into account. Based on results in Fig. 9, a noise K, the ambient temperature at which source ENR for noise sources are generally characterized, is found from (24). In (24), the term in brackets accounts for the increase in above 296 K and together with removes the increase in due to an ambient-temperature drift.

APPENDIX C RECEIVER PHYSICAL TEMPERATURE The noise sources calibrated in this work warm up themselves to temperatures approximately 5 C higher than the ambient temperature. These noise sources are connected to the receiver, which operates at its own physical temperature. This results in heat transfer between the two devices. While the temperature of the noise source is measured by the noise source itself and is reported by the receiver, the physical temperature of the receiver itself is not known, but it is different from that when the receiver is connected to other devices. In the least-squares solution of (18), the receiver noise temperature is referenced to the ambient which is measured. Since the heat transfer from temperature the noise source affects the temperature of the receiver, this can be modeled by a change in the ambient temperature. The rest of this Appendix deals with the determination of this unknown . apparent ambient temperature, 8Since during this work only 5-dB ENR Agilent’s N4000A noise sources were used, the conclusion regarding the temperature dependence of T on T only applies to these noise sources.

186

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Fig. 10. Measured and estimated T

.

A test was conducted to estimate the temperature change of the receiver due to heat transfer from a noise source. In this test, a noise source was connected to the receiver and the hot were measured by the receiver. and cold noise powers, of the noise source were The ambient temperature and the recorded. The noise source was then turned OFF for a long time , was measured by the receiver. At the and the noise power, same time the receiver reported the physical temperature of the , which is the same as the physical temperature noise source of the receiver itself since the noise source was OFF and connected to the receiver for a long time so that its temperature had time to stabilize to that of the receiver. From the first measurement, by using the conventional noise factor method in (3) the is derived from which the noise noise factor of the receiver, temperature of the receiver is (C.1) During the first measurement discussed in this Appendix, the is measured (C.2)

Fig. 11. Standard deviation of the differences between measured noise figures and those calculated based on calibrated temperatures of noise sources as a func. tion of r

is a ratio of thermal resistances associated with the where receiver and the noise source. is estimated from fitting data . estimated with in Fig. 10 to (C.6) to obtain and (C.6) is also shown in Fig. 10. Equation (C.6) dethis scribes how much the temperature of the receiver rises relative to the ambient temperature due to the noise source temperature. obtained in this Appendix is used in Section IV to calculate the apparent ambient temperature of the receiver. It is important to note that the results of this section are very sensitive to the actual measurement setup, the receiver configuration selected for measurements, and the noise sources being used. In some situations, for example if the receiver is equipped with a temperature may be insensitive to noise controller or a large heat sink, source physical temperatures and then . found in this Appendix is To confirm that the value of correct, data in Fig. 4 were recalculated for a few different values . Standard deviations of differences between the noise of figures measured and calculated based on the proposed noise are shown in Fig. 11 and source calibration as function of by the location of the smallest standard deviation confirm the . findings in this Appendix that

From the second measurement ACKNOWLEDGMENT

(C.3) (C.4) where

. From (C.2) and (C.4)

The author would like to thank Z. Zhao for performing some of the measurements reported in this paper, and Dr. S. Weinreb and H. Mani from the Radiometer Group at the California Institute of Technology for conducting cross-verification measurements.

(C.5) REFERENCES Fig. 10 shows the unknown measured as discussed in this Appendix. The measurement data shown in Fig. 10 are for a given ambient temperature. As the ambient temperature drifts, and the cold temperature of the noise source. To so does capture this behavior can be expressed as (C.6)

[1] Agilent Technologies, Inc., Agilent Fundamentals of RF and Microwave Noise Figure Measurements, Application Note 57-1, Oct. 2006. [2] L. Belostotski and J. W. Haslett, “Evaluation of tuner-based noise-parameter extraction methods for very low noise amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 1, pp. 236–250, Jan. 2010. [3] G. Gramegna, M. Paparo, P. G. Erratico, and P. De Vita, “A sub-1-dB NF or-2.3-kV ESD-protected 900-MHz CMOS LNA,” IEEE J. SolidState Circuits, vol. 36, no. 7, pp. 1010–1017, Jul. 2001.

+

BELOSTOTSKI: A CALIBRATION METHOD FOR RF AND MICROWAVE NOISE SOURCES

[4] P. Leroux, J. Janssens, and M. Steyaert, “A 0.8-dB NF ESD-protected 9-mW CMOS LNA operating at 1.23 GHz [for GPS receiver],” IEEE J. Solid-State Circuits, vol. 37, no. 6, pp. 760–765, Jun. 2002. [5] D. J. Cassan and J. R. Long, “A 1-V transformer-feedback low-noise amplifier for 5-GHz wireless LAN in 0.18-m CMOS,” IEEE J. SolidState Circuits, vol. 38, no. 3, pp. 427–435, Mar. 2003. [6] J. Xu, B. Woestenburg, J. G. B. de Vaate, and W. A. Serdijn, “GaAs 0.5 dB NF dual-loop negative-feedback broadband low-noise amplifier IC,” Electron. Lett., vol. 41, no. 14, pp. 780–782, Jul. 7, 2005. [7] L. Belostotski, J. W. Haslett, and B. Veidt, “Wide-band CMOS low noise amplifier for applications in radio astronomy,” in IEEE Int. Symp. Circuits and Systems, Kos, May 21–24, 2006, pp. 1347–1350. [8] E. A. M. Klumperink, Q. Zhang, G. J. M. Wienk, R. Witvers, J. G. B. de Vaate, B. Woestenburg, and B. Nauta, “Achieving wideband sub-1 dB noise figure and high gain with MOSFETs if input power matching is not required,” in IEEE Radio Frequency Integrated Circuits Symp., Honolulu, HI, Jun. 3–5, 2007, pp. 673–676. [9] S. Weinreb, J. Bardin, and H. Mani, “Design of cryogenic SiGe lownoise amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 11, pp. 2306–2312, Nov. 2007. [10] L. Belostotski and J. W. Haslett, “Wide band room temperature 0.35-dB noise figure LNA in 90-nm bulk CMOS,” in IEEE Radio and Wireless Symp., Long Beach, CA, Jan. 7–11, 2007, pp. 221–224. [11] L. Belostotski and J. W. Haslett, “Noise figure optimization of wide-band inductively-degenerated CMOS LNAs,” in IEEE Midwest Symp. Circuits and Systems, Montreal, QC, Canada, Aug. 5–8, 2007, pp. 1002–1005. [12] L. Belostotski and J. W. Haslett, “Sub-0.2 dB noise figure wide-band room-temperature CMOS LNA with non-50 signal-source impedance,” IEEE J. Solid-State Circuits, vol. 42, no. 11, pp. 2492–2502, Nov. 2007. [13] L. Belostotski and J. W. Haslett, “Two-port noise figure optimization of source-degenerated cascode CMOS LNAs,” Analog Integr. Circuits Signal Process., vol. 55, no. 2, pp. 125–137, May 2008. [14] L. Belostotski and J. W. Haslett, “A technique for differential noise figure measurement of differential LNAs,” IEEE Trans. Instrum. Meas., vol. 57, no. 7, pp. 1298–1303, Jul. 2008. [15] Agilent Technologies, Inc., Noise Source Calibration: Using the Agilent N8975A Noise Figure Analyzer and the N2002A Noise Source Test Set (Product Note), Dec. 1, 2007. [16] M. W. Sinclair, “Untuned systems for the calibration of electrical noise sources,” in IEE Colloq. What’s New in Microwave Measurements, London, U.K., Dec. 6, 1990, pp. 7/1–7/5. [17] C. A. Grosvenor, J. Randa, and R. L. Billinger, “Design and testing of NFRad—A new noise measurement system,” Natl. Inst. Standards Technol., Boulder, CO, Tech. Rep. 1518, Mar. 2000. [18] G. Williams, “Source mismatch effects in coaxial noise source calibration,” Meas. Sci. Technol., vol. 2, no. 8, pp. 751–756, Aug. 1991.

187

[19] A. C. Davidson, B. W. Leake, and E. Strid, “Accuracy improvements in microwave noise parameter measurements,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1973–1978, Dec. 1989. [20] G. L. Williams, “Measuring amplifier noise on a noise source calibration radiometer,” IEEE Trans. Instrum. Meas., vol. 44, no. 2, pp. 340–342, Apr. 1995. [21] R. Lane, “The determination of device noise parameters,” Proc. IEEE, vol. 57, no. 8, pp. 1461–1462, Aug. 1969. [22] L. Escotte, R. Plana, and J. Graffeuil, “Evaluation of noise parameter extraction methods,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 3, pp. 382–387, Mar. 1993. [23] Agilent Technologies, Inc., Smart Noise Sources SNS Series: Operating and Service Guide (N4000A, N4001A, and N4002A), Mar. 2001. [24] P. Dewdney, P. Hall, R. Schilizzi, and T. Lazio, “The square kilometre array,” Proc. IEEE, vol. 97, no. 8, pp. 1482–1496, Aug. 2009. [25] D. K. Rytting, “Network analyzer accuracy overview,” in Proc. 58th ARFTG Conf., 2001, vol. 40, pp. 1–13. [26] Agilent Technologies, Inc., Agilent N8973A, N8974A, N8975A NFA Series Noise Figure Analyzers: Data Sheet, Nov. 5, 2007. [27] Maury Microwave Corp., Noise Calibration System: Coaxial—Three Loads (Technical Data 4E-009), Jun. 10, 1997. [28] J. Randa, “Uncertainties in NIST noise-temperature measurements,” Natl. Inst. Standards Technol., Tech. Rep. 1502, Mar. 1998. [29] Liquid Nitrogen Noise Standard “LN7” on Loan From National Radio Astronomy Observatory, , Charlottesville, VA, USA. This standard was built in 1983 by Dr. S. Weinreb, now at the California Institute of Technology. [30] Agilent Technologies, Inc., Noise Figure Measurement Accuracy—The Y-Factor Method, Application Note 57-2 Oct. 2004. Leonid Belostotski (S’97–M’01) received the B.Sc. and M.Sc. degrees in electrical engineering from the University of Alberta, Edmonton, AB, Canada, in 1997 and 2000, respectively, and the Ph.D. degree from the University of Calgary, Calgary, AB, Canada, in 2007. A large portion of his M.Sc. thesis program was spent at the Dominion Radio Astrophysical Observatory, NRC, Penticton, BC, Canada, where he designed and prototyped a distance measurement and phase synchronization system for the Canadian Large Adaptive Reflector telescope. After graduation, he worked at Murandi Communications Ltd., as an RF Engineer designing devices for high volume consumer applications and low-volume high-performance devices for the James Clerk Maxwell Telescope in Hawaii. He started working on the Ph.D. degree at the University of Calgary, in September 2004 researching low-noise amplifiers for the SKA radio telescope. In 2007, he was appointed Assistant Professor at the University of Calgary.

188

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

A Compact Variable-Temperature Broadband Series-Resistor Calibration Nathan D. Orloff, Jordi Mateu, Member, IEEE, Arkadiusz Lewandowski, Member, IEEE, Eduard Rocas, Student Member, IEEE, Josh King, Dazhen Gu, Member, IEEE, Xiaoli Lu, Carlos Collado, Member, IEEE, Ichiro Takeuchi, and James C. Booth

Abstract—We present a broadband on-wafer calibration from 45 MHz to 40 GHz for variable temperature measurements, which requires three standards: a thru, reflect, and series resistor. At room temperature, the maximum error of this technique, compared to a benchmark nine-standard multiline thru-reflect-line (TRL) method, is comparable to the repeatability of the benchmark calibration. The series-resistor standard is modeled as a lumped-element -network, which is described by four frequency-independent parameters. We show that the model is stable over three weeks, and compare the calibration to the multiline TRL method as a function of time. The approach is then demonstrated at variable temperature, where the model parameters are extracted at 300 K and at variable temperatures down to 20 K, in order to determine their temperature dependence. The resulting technique, valid over the temperature range from 300 to 20 K, reduced the total footprint of the calibration standards by a factor of 17 and the measurement time by a factor of 3. Index Terms—Calibration, cryogenic, error correction, microwave, scattering parameters, series resistor, temperature.

I. INTRODUCTION

M

EASUREMENT-BASED circuit modeling of devices is an important design tool for the development of microwave components and systems. Since typical measurement and test systems possess their own frequency-dependent response, accurate modeling and characterization of devices over a broad range of frequencies is challenging without effective correction techniques. A variety of calibration techniques have been developed to address this issue, and are routinely applied to coaxial and on-wafer measurements. Among them, multiline Manuscript received April 28, 2010; revised September 21, 2010; accepted October 04, 2010. Date of publication December 10, 2010; date of current version January 12, 2011. This work was supported by the National Science Foundation (NSF) under Grant DMR-0520471 (MRSEC) and Grant ARO W911NF-07-1-0410. N. D. Orloff and I. Takeuchi are with the Department of Physics and the Department of Materials Science and Engineering, University of Maryland at College Park, College Park, MD 20742 USA (e-mail: [email protected]; [email protected]). J. Mateu, E. Rocas, and C. Collado are with the Centre Tecnològic de Telecomunicacions de Catalunya, Universitat Politecnica de Catalunya, Barcelona 08860, Spain (e-mail: [email protected]; [email protected]; [email protected]). A. Lewandowski, D. Gu, X. Lu, and J. C. Booth are with the National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: areklew@nist. gov; [email protected]; [email protected]; [email protected]). J. King is with the Business Department, University of Colorado at Boulder, Boulder, CO 80302 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2091200

thru-reflect-line (TRL) [1], [2] is considered to be one of the most accurate because of its basis in circuit theory, and is often used as a benchmark for new calibrations [3], [4]. Apart from the multiline TRL technique, Padmanabhan et al. [5] recently showed that on-wafer open-short-load-thru (OSLT) calibrations can have comparable accuracy to the multiline TRL approach provided that the shunt-resistor loads are adequately modeled to account for their high-frequency response. Moreover, the development of a compact OSLT calibration greatly reduced the footprint and total measurement time needed to accurately correct room temperature broadband measurements to 110 GHz. Temperature-dependent experiments are ubiquitous in scientific literature as they often elucidate fundamental thermodynamics and temperature-dependent properties of materials, including conductors and dielectrics. For cryogenic experiments, the relatively large number of standards needed to correct broadband microwave measurements by the multiline TRL method, as compared to OSLT, implies longer measurement times and limits the number of measurements that can be achieved in a single cool-down. Above room temperature, there are also instances where we must limit the period during which the sample and measurement system are exposed to heat, an example of which would be protein measurements [6]. There are many applications of accurate on-wafer calibrations at temperatures other than room temperature, including materials science engineering applications for thin films [7], characterization biological fluids [8], and cryogenic characterization of various components for communications applications [9], [10]. Designing calibration standards with predictable temperature dependent properties will help reduce measurement time, while also reducing the footprint of the calibration standards, for variable temperature applications. We have chosen the series-resistor calibration [11] to achieve our goal of designing a standard with a predictable temperature dependence because it requires one less standard than an OSLT, which further decreases the footprint of the calibration devices on the wafer. In addition to the series-resistor standard, the calibration technique employed here also uses a thru and a symmetric short-circuit reflect. Experiments have also shown that the accuracy of the series-resistor calibration can be comparable to the repeatability of multiline TRL [11] at room temperature. In what follows, we first describe our fabrication procedure for the series-resistor standard, which is a resistive strip embedded in the center conductor of a coplanar waveguide (CPW) [12] [see Fig. 1(a)]. We explain how to use the series resistor to extract the characteristic impedance of the CPW

0018-9480/$26.00 © 2010 IEEE

ORLOFF et al.: COMPACT VARIABLE-TEMPERATURE BROADBAND SERIES-RESISTOR CALIBRATION

Fig. 1. (a) Representative Pd Au series-resistor standard. The resistive strip is embedded in a gold CPW with a ground plane of width (g ), center conductor of width (w ), gap of width (s), and resistive strip. The three hash marks on each ground plane are alignment marks for the ground–signal–ground probes. (b) Base electrode layer is fabricated by depositing a 20-nm titanium adhesion layer followed by 330 nm of gold. (c) 1-nm titanium seed layer is used to imAu pads. (d) 8.5 nm of Pd Au are deprove adhesion of the Pd posited for the resistor layer. The resistor layer consists of a 10 m (width) 10 m (length) strip which is connected on either side to two large pads, 100 m 50 m. (e) Electrode layer is electrically connected to the resistor by a 50-nm-thick gold counter electrode layer. The result is a 10 m 10 m strip embedded in the center of a CPW that is the same length on either side as half the thru standard.

2

2

2

in which it is embedded, and compare it to the characteristic impedance obtained from a more common technique using a shunt resistor. We next describe the modeling procedure used to fit the measured complex -parameters of the series resistor, which enables us to use it as a calibration standard. We subsequently compare a series-resistor calibration with a multiline TRL calibration at room temperature, and show that the worst case deviations between the two calibrations are comparable to the repeatability error of the multiline TRL calibration. We then demonstrate that the lumped-element circuit parameters used to describe the series resistor are stable over three weeks, and develop a description of the temperature dependence of the lumped-element circuit parameters down to 20 K. We compare the series-resistor calibrations based on a model using parameters with a simplified temperature dependence to series-resistor calibrations where the model is extracted at each measurement temperature and show that this error is also comparable to the repeatability of the benchmark calibration. II. FABRICATION The series-resistor artifact is the central feature of this calibration technique. The choice of metals used along with the dimensions of the resistive strip make it useful for a broad range of frequencies and affects the temperature dependence of its response. Fig. 1(a) shows a representative 10 m 10 m series-resistor standard embedded at the center of a gold CPW of mm. The gold CPW has a 100 m wide center length conductor ( ), a 10 m wide gap ( ), and 250 m wide ground planes ( ) fabricated on a 0.5-mm-thick low-loss polycrystalline resistive quartz substrate. The composition of the Pd Au strip was inferred from the composition of the target. We chose alloy as the resistive material because it has a weak Pd Au linear temperature dependence from 4 to 500 K [13]. We patterned the devices with conventional photo-lithographic techniques and deposited the metal layers with electron-beam evaporation.

189

In Fig. 1(b), we show the base electrode layer. We patterned this layer with approximately 1 m thick lift-off-resist, overcoated using 1 m thick image resist. The lift-off-resist process ensures well-defined devices and edges for thick conductors [14]. We used electron-beam evaporation to deposit a 20-nm titanium adhesion layer, followed by 330 nm of gold. The remaining layers are sequentially patterned with only 1- m image resist for each layer. Before depositing each layer in Fig. 1, we use an Argon plasma to clean the surface, which helps to ensure clean metal–metal interfaces. For the resistor layer, a 1-nm titanium layer [see Fig. 1(c)] is deposited and defines the 100- m-wide by 50- m-long pads to promote the adhesion of the resistor and counter-electrode resistor layer shown in layers. The 8.5-nm-thick Pd Au Fig. 1(d), includes the 10 m 10 m resistive strip and the 100- m wide by 50- m-long contact pads. The resistor layer is electrically connected to the electrode layer by depositing 50 nm of gold over the conductors and pads in Fig. 1(e). The entire fabrication process for the series resistor (also used to fabricate separate shunt resistors) presented here required four total deposition layers. In addition to the series-resistor standard, we also fabricated the devices to complete the benchmark multiline TRL calibration in the base electrode layer. These nine devices included a set of seven CPWs with lengths of mm mm mm mm mm mm mm , a symmetric short circuit reflect standard, and the series-resistor standard. The reflect standard consisted of a 0.210-mm-long transmission line on each port centered about the termination. For all of the following measurements, we calibrated the network analyzer first with a first-tier coaxial OSLT calibration with a six-position sliding load. We found that this improved the accuracy of the low-frequency response of the on-wafer calibrations and it removed the requirement to directly measure the reflection coefficients of the loads internal to the vector-network analyzer. III. DETERMINATION OF THE CPW REFERENCE IMPEDANCE Before we can develop the series-resistor calibration, we need to determine the characteristic impedance of the CPW devices that are used in the benchmark multiline TRL calibration. This is typically done using a set of shunt resistors; however, we show that a single series resistor can be used to acquire the same information. The ability to transform the calibration reference impedance from the characteristic impedance of the CPW to a known reference impedance is valuable for simplifying the modeling of a given device. It is also necessary to transform the reference impedance of the calibration to a real value to compare different is required to achieve quanticalibrations because tative comparison of different calibrations [4]. In the case of the multiline TRL, the calibration is relative to the character, which is neither istic impedance of the transmission line real, nor constant as a function of frequency for normal metal transmission lines. Hence, multiline TRL calibrations need to be transformed to a known reference impedance for calibration comparison.

190

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Williams and Marks [15] showed that a shunt load could be used to approximate for transmission lines fabricated on lowloss substrates. We can also develop a similar approach using the series resistor, where we make the same assumptions as in [15]. One assumption is that at low frequencies, the impedance can be approximated by dc resistance of the resistive load . The dc resistance of the strip is deterof the strip mined experimentally from the difference between two-port dc measurements of the series resistor and the thru. For low-loss substrates, such as that employed here, we can assume and , where and are the capacitance and conductance per unit length of the CPW transmission lines and are constant as a function of frequency. Thus, the characteristic impedance for transmission lines on a low-loss substrate , where is the comcan be simplified as plex propagation constant. We can obtain by either analyzing the corrected -parameters of a transmission line of known length or from the multiline TRL method. We can later to translate the reference planes of the resulting caliuse bration. For the shunt-resistor standard discussed in [5] and [15], the device has two resistive strips, one on port 1 and the other on port 2. In this case, both shunt resistors are needed for the twoport OSLT calibration, otherwise the wafer must be rotated a full 180 so that the device can be measured at both ports. An added benefit of the series-resistor approach is that instead of one reflection coefficient corresponding to a single standard (as is the case for the shunt resistor), all four -parameters of the series resistor can be used to extract the capacitance per unit length from a measurement of a single resistive strip. of the CPW from the series To obtain an estimate for the resistor, we start with the simple model for the -parameters of a series load (1) and make the same definitions and approximations as before. into (1) and , we can Inserting then solve for (2) (3) Once we obtain (2) and (3) from the reflection and transmission coefficients of a single series resistor, we determine by taking the low-frequency limit, which yields for estimates of . Note that (2) and (3) are derived using an oversimplified model of the frequency dependence for the series resistor, where we . assume that, at low frequency, obtained by the series and shunt We demonstrated that the approaches are equivalent by fabricating both shunt and seriesresistor standards in identical CPW transmission lines on polycrystalline quartz substrates. The devices were corrected with multiline TRL calibrations relative to the CPW characteristic . We then used (2) and (3) for the series reimpedance sistor and a similar equation for the shunt resistors [15] to eval-

Fig. 2. Comparison of the effective capacitance per unit length extracted from the series-resistor and shunt resistor standards. The red curve (in online version) extracted from the series-resistor transmission coefficients shows the C of the corrected S -parameters, and the blue line (in online version) shows the from the reflection coefficients. The black line is C extracted from C the shunt resistor on port 1, and green (in online version) (C ) for port 2. ,C ,C , C is determined by taking the low-frequency limit of C and C .

with determined by the multiline TRL algorithm. uate Fig. 2 shows as calculated from the two approaches. In Fig. 2, the extracted from the shunt resistors, one on each port, are shown as the solid green (in online version) and black lines. value extracted from (2) using the reflection coefficient The data for series resistor, averaged together, is shown as the solid red line; the corresponding value calculated from (3) using the transmission coefficient data, averaged together, is shown as the solid blue line (in online version). We then use the median value to estimate at zero frquency and obbelow 1 GHz of tain a value of 1.05 pF/cm. This value is consistent with the capacitance per unit length obtained from finite-element simulations of the cross-sectional geometry of these CPW transmission lines on quartz. We can then obtain the reference impedance of . Once we the CPW from the expression , we can transform the reference impedance have obtained to 50 . IV. MODELING THE SERIES RESISTOR In Section IV, we used the approximation that to estimate the of the CPW transmission line from measurements of the series resistor, but we need a more accurate to use the series resistor as a calibration stanmodel of dard at high frequencies. As a first step toward obtaining a better model of the series resistor, we correct a measurement of the series resistor with multiline TRL calibration (making use of the technique from the previous section to transform to a reference impedance of 50 ) with the calibration reference planes transobtained lated to either end of the resistive strip using the from the multiline TRL algorithm. We then transform these corrected -parameters of the series resistor to an admittance matrix, which we model by a -network. The series-admittance term describes the response of the , resistive strip, and we parameterize this by the resistance , and capacitance , which are assumed to inductance be frequency independent. In general, there is a shunt admitbetween the strip and ground plane of the tance transmission line; however, for these experiments, we found that

ORLOFF et al.: COMPACT VARIABLE-TEMPERATURE BROADBAND SERIES-RESISTOR CALIBRATION

191

Fig. 3. Equivalent lumped-element circuit model for the series-resistor standard. The series components of the circuit include the resistance (R ), inductance (L ), and capacitance (C ). The shunt element is Y = i!C , where C is the capacitance to ground.

Fig. 4. Comparison of the real part (blue in online proofs) and imaginary part (red in online proofs) of the admittance, Y , of the series-resistor standard corrected with the multiline TRL calibration. The modeled Y (black) was calculated with the series resistor lumped element model parameters of R = 56:86 , L = 21:90 pH, and C = 4:52 fF.

including this term did not affect the overall error. Fig. 3 shows the lumped-element model of the series resistor, with the shunt admittance, , and the series admittance, , where (4) We then obtain initial values for , , and in the following manner. A series capacitor, which is identical to the series resistor, except that it has no resistive strip, is used to obtain an estimate for . Using (4) and this value for , we estimated with a linear fit as a function of frequency. was approximated by the dc resistance. We construct the modeled -param, , and . We then minieters using these estimates for mized the difference between the modeled and measured -parameters (summed over all frequencies) by varying , , and using the Levenburg–Marquadt algorithm [16]. We define the error to be , where are the -parameters of the series resistor corrected with are the -parameters of the model multiline TRL and , , and . In Fig. 4, we calculated with the parameters show the real part (blue in online version) and imaginary part (red in online version) of the series admittance corrected with multiline TRL calibration along with the resulting lumped-element model (black). For the data shown in Fig. 4, we obtain , pH, and fF. In practice, the optimized values differ from our initial estimates by at most approximately 10%. From Fig. 4, it is clear that the series-resistor model captures the basic frequency-dependent response of the series resistor, but it cannot explain the fine features such

Fig. 5. (a) Maximum error in S -parameters between two successive multiline TRL calibrations is shown as the blue line (in online version) (TRL-TRL). The black line shows the maximum error for two successive calibrations with the series-resistor standard (SR-SR). In red (in online version), we show the maximum error between the series-resistor calibration and multiline TRL (SR-TRL). , in S -parameters for CPWs 1.000- and 11.570-mm-long (b) Difference, S corrected with multiline TRL and with the series-resistor calibration.

as the small peak around 1 GHz. At present, it is unclear if this is an artifact in the multiline TRL calibration, series-resistor standard, or measurement system; however, future experiments will address this issue in greater detail. V. COMPARISON WITH MULTILINE TRL Once we have adequately modeled the frequency-dependent response of the series resistor, we can use this model to perform a calibration using measurements of the series resistor, thru, and symmetric short-circuit reflect and evaluate the accuracy of the series-resistor calibration relative to the benchmark multiline TRL calibration. For the -parameters of an arbitrary passive device corrected using two different calibrations relative to the same real-valued reference impedance, a matrix corresponding to the maximum errors in each -parameter can be calculated by the approach outlined in [3] and [4]. Although the error matrix carries information in all four elements, the worst case maxis often used as the metric for imum error comparing calibrations. We use this metric in what follows to compare calibration techniques, like those shown in Fig. 5. for a series of calibrations Fig. 5(a) shows comparing the series resistor to subsequent series resistor (black), multiline TRL to subsequent multiline TRL (blue in online version), and multiline TRL to series resistor (red in online version). The maximum error between successive multiline TRL calibrations (blue in online version) shows a gradual

192

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

TABLE I SERIES-RESISTOR CIRCUIT PARAMETERS VARIATION OVER A THREE-WEEK PERIOD

increase below 500 MHz and above 4 GHz. The TRL–TRL maximum error demonstrates the repeatability of the benchmark calibration. The successive series-resistor calibrations (SR-SR) show similar repeatability at high frequencies to the multiline TRL calibration, and show that good repeatability is maintained for very low frequencies. We have also compared the series resistor to TRL (red in online version), which is larger than the repeatability between the successive multiline TRL and successive series-resistor calibrations. At low frequencies, the increase in the maximum error follows the repeatability of the TRL calibration. comparable to the maximum error at high frequency in the repeatability of both calibration techniques and the maximum error is below 2% for the majority of the fre, quency regime. The measured error in the -parameters, is shown in Fig. 5(b) for specific devices (CPW of lengths , as defined 1.000 and 11.570 mm) where we compute in the previous section, but we take the difference between the measurements corrected with the multiline TRL calibration and the measurements corrected with the series-resistor calibration. We demonstrated the stability of this technique by repeating both the series resistor and benchmark multiline TRL calibrations each week for a three-week period. In Table I, we report , , and from Fig. 3 extracted over a the parameters values are comparable to the exthree-week period. The to within 1%. The other circuit parameters showed tracted increased variation from measurement to measurement, but represent a small contribution to frequency-dependent impedance of the series resistor. In Fig. 6(a), we compare the series-resistor calibration to a subsequent multiline TRL calibration using the model parameters extracted on Week 1 over a period of three weeks (Week 1, Week 2, Week 3). Fig. 6(a) shows that the lumped element circuit parameters used to describe the series resistor are stable over relatively long times. In Fig. 6(b), we show the maximum error between series-resistor calibrations using the model from Week 1 and the models extracted on Week 2 and Week 3. The long-term stability of the circuit parameters used to characterize the series resistor is an important issue. Within the conditions of our experiments, we have seen no evidence of degradation in the device response and variation in modelling parameters over a period of approximately one year. VI. VARIABLE TEMPERATURE In order to quantify the applicability of the series-resistor calibration over variable temperature, we repeated the analysis described in Section V for measurements at different temperatures, from room temperature down to 20 K. We then examined the temperature-dependent behavior of , , and . We have performed our temperature-dependent measurements with a cryogenic probe station, which maintains the measurement devices and microwave probes under vacuum. The substrate is

Fig. 6. (a) Maximum error between the series-resistor calibration and the multiline TRL calibration, using the same series-resistor model. Each series resistor calibration was calculated using the model from Week 1. The lumped-element : ,L : pH, circuit parameters for the series resistor are R : fF. The red (in online version), black, and blue (in online verand C sion) lines correspond to Week 1, Week 2, and Week 3, respectively. (b) Maximum error between series-resistor calibrations extracted on Week 2 and Week 3 compared to calibrations using the model extracted on Week 1.

= 4 55

= 56 94

= 21 90

TABLE II SERIES-RESISTOR CIRCUIT PARAMETERS AT VARYING TEMPERATURE

physically, electrically, and thermally connected to a platform with integrated heaters and thermometers that is cooled with a cold finger, which allows us to control the sample and probe temperatures to within 0.01 K. At each temperature, we performed the benchmark multiline TRL calibration described in Section I and then measured the series-resistor standard. fit Table II summarizes the model parameters , , and to the data corrected with the multiline TRL calibration at each temperature (as described in Section IV). The capacitance per unit length of the CPW, , was 1.05 pF/cm, and varied by less that 1% as a function of temperature from 300 down to 20 K. values extracted from the series resistor Fig. 7 shows the

ORLOFF et al.: COMPACT VARIABLE-TEMPERATURE BROADBAND SERIES-RESISTOR CALIBRATION

193

Fig. 7. Temperature dependence of R lumped-element circuit parameter of the series-resistor standard, which has been extracted at each temperature with are measurements corrected with the multiline TRL calibration. R and R shown as discrete red circles and blue (in online version) squares, respectively. R is fit by the solid black line and R is fit by the dashed black line. The equation for the solid black line is R (T) = 51:27 + 0:02[ 1 K ] 1 T and the dashed black line is R (T) = 50:98 + 0:02[ 1 K ] 1 T.

over the measured temperature range. The extracted from the series resistor can be linearly fit as a function of temperature T K T. The dashed black line in by T Fig. 7 is the fit to the dc resistance, K T. and showed comparatively weak temperature and away from the value extracted dependence. Varying at 300 K did not significantly affect the magnitude or frequency dependence of the maximum error. Given the small changes in the model parameters as a function of temperature, we explored the possible use of the room-temperature model parameters to correct low-temperature measurement data. The dashed lines in Fig. 8(a) are the maximum error between a series-resistor calibration using the 300-K model and one where the model parameters are extracted at the given temperature. For clarity, we show only 20 K (red in online version), 100 K (blue in online version), and 180 K (black) data sets in Fig. 8(a). This figure shows, as expected, that significant errors are generated by applying a room-temperature series-resistor calibration to data measured at lower temperatures. To improve the accuracy of the low-temperature calibration without extracting the series-resistor model at each individual temperature, we can fit the temperature dependence of the model T . In Fig. 8(a), the solid lines show parameters, such as the maximum error between a series-resistor calibration at the given temperature and one where the linear fit used is to calcuT , while and values from the 300 K are used. late Fig. 8(b) shows the solid lines from Fig. 8(a) in greater detail. The solid gray line in Fig. 8(b) is the maximum error beT evaluated tween the modeled temperature dependence of at 300 K and model extracted at 300 K. The solid gray line il, and therefore, has no lustrates the effect of only changing and are the same as those frequency dependence because extracted at 300 K without extracting the model at each temperature. Fig. 8 demonstrates that by simply allowing for temperature T we can dramatically reduce the maximum dependence of error for any temperature between 300–20 K. We also calculated the maximum error at each temperature between the series

Fig. 8. (a) Dashed lines compare a series-resistor calibration using the model extracted at 300 K compared to a calibration using the model extracted at 20 K (red in online version), 180 K (blue in online version), and 300 K (black). The model extracted at each temperature was calibrated with a multiline TRL calibration conducted at that temperature. The solid lines show the maximum error between to series-resistor calibrations using the model extracted at a given temperature compared to one using the temperature dependent model for R (T) and the 300-K values for L and C . (b) Solid lines from (a) are shown in more detail with the maximum error for the 300-K calibration in light gray using the predicted temperature dependence of R (T) and the actual extracted value.

resistor and multiline TRL calibrations and found it to be similar in magnitude and frequency dependence to that shown in Fig. 5(a). We can apply these results to create a compact calibration set for use from 45 MHz to 40 GHz at variable stable temperatures. Our compact variable-temperature calibration set includes a series resistor, thru, and symmetric short circuit reflect. The series resistor can be modeled after an initial characterization at room temperature that includes a correction with a brenchmark multiline TRL calibration. Once the series resistor is characterized at room temperature, dc measurements of the series resistance T , and as a function of temperature can be used to obtain the compact series-resistor calibration can be applied at variable static temperatures to correct broadband microwave measurements. Also, based on the frequency response of the series resistor, this technique can be extended to lower frequencies, where it is similar to the line-reflect-match technique [17]. VII. CONCLUSION We have demonstrated a broadband on-wafer calibration technique based on a series-resistor artifact that decreased the total area required for the calibration standards by a factor of 17 compared to a nine-standard multiline TRL calibration.

194

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

By parameterizing the response of the series resistor with a simple model, we obtained a maximum error comparable to the repeatability of the benchmark multiline TRL calibration for room temperature calibrations. We showed that the parameters extracted using this simple model were relatively stable over a period of three weeks. We also found that the series-resistor calibration showed improved repeatability compared to a nine standard multiline TRL calibration at frequencies below 1 GHz, and the technique can easily be extended to even lower frequencies. In addition to greatly reducing the footprint and measurement time, the series resistor could also be used to extract the characteristic impedance of transmission lines on low-loss substrates. We demonstrated that the predictable temperature response of the series-resistor artifact enabled compact broadband on-wafer calibrations that can be applied over a wide range of temperatures with a minimal increase in maximum error. ACKNOWLEDGMENT The authors thank Y. Wang, National Institute of Standards and Technology (NIST), Boulder, CO, T. M. Wallis, NIST, D. F. Williams, NIST, U. Arz, Physikalisch-Technische Bundesanstalt, Braunschweig, Germany, and D. K. Walker, NIST, for their critical review of this paper’s manuscript and valuable contributions to this work. The authors also thank D. LeGovlan, NIST, for his help in performing the coaxial OSLT calibrations. REFERENCES [1] R. B. Marks, “A multiline method of network analyzer calibration,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1205–1215, Jul. 1991. [2] D. F. Williams, J. C. M. Wang, and U. Arz, “An optimal vector-network-analyzer calibration algorithm,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2391–2401, Dec. 2003. [3] D. F. Williams, R. B. Marks, and A. Davidson, “Comparison of on-wafer calibrations,” in Proc. 38th ARFTG Conf. Dig.–Winter, Dec. 1991, vol. 20, pp. 68–81. [4] R. B. Marks, J. A. Jargon, and J. R. Juroshek, “Calibration comparison method for vector network analyzers,” in Proc. 48th ARFTG Conf. Dig. –Fall, Dec. 1996, vol. 30, pp. 38–45. [5] S. Padmanabhan, L. Dunleavy, J. E. Daniel, A. Rodriguez, and P. L. Kirby, “Broadband space conservative on-wafer network analyzer calibrations with more complex load and thru models,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 9, pp. 3583–3593, Sep. 2006. [6] D. Rees and A. Robertson, “Some thermodynamic implications for the thermostability of proteins,” Protein Sci., vol. 10, no. 6, pp. 1187–1194, Jun. 2001. [7] N. Orloff, W. Tian, C. Fennie, C. Lee, D. Gu, J. Mateu, X. Xi, K. Rabe, D. Schlom, I. Takeuchi, and J. Booth, “Broadband Dielectric SpecTi O (n = 1; 2; 3) Thin troscopy of Ruddlesden-Popper Sr Films,” Appl. Phys. Lett., vol. 94, no. 4, Jan. 2009, Art. ID 042908 (3 pp). [8] J. C. Booth, N. D. Orloff, J. Mateu, M. Janezic, M. Rinehart, and J. A. Beall, “Quantitative permittivity measurements of nanoliter liquid volumes in microfluidic channels to 40 GHz,” IEEE Trans. Instrum. Meas., vol. 59, no. 12, pp. 3279–3288, Dec. 2010. [9] H. Meschede, R. Reuter, J. Albers, J. Kraus, D. Peters, W. Brockerhoff, F.-J. Tegude, M. Bode, J. Schubert, and W. Zander, “On-wafer microwave measurement setup for investigations on HEMTs and high-T c superconductors at cryogenic temperatures down to 20 K,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 12, pp. 2325–2331, Dec. 1992. [10] J. Laskar, J. J. Bautista, M. Nishimoto, M. Hamai, and R. Lai, “Development of accurate on-wafer, cryogenic characterization techniques,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 7, pp. 1178–1183, Jul. 1996. [11] D. F. Williams and D. K. Walker, “Series-resistor calibration,” in Proc. 50th ARFTG Conf. Dig.–Fall, Dec. 1997, vol. 32, pp. 131–137.

[12] C. P. Wen, “Coplanar waveguide: A surface strip transmission line suitable for nonreciprocal gyromagnetic device applications,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 12, pp. 1087–1090, Dec. 1969. [13] J. Conybeare, “The resistance of palladium and palladium–gold alloys,” Proc. Phys. Soc., vol. 49, pp. 29–37, Jan. 1937. [14] Y. Chen, K. Peng, and Z. Cui, “A lift-off process for high resolution patterns using PMMA/LOR resist stack,” Microelectron. Eng., vol. 73–74, pp. 278–281, 2004. [15] D. F. Williams and R. B. Marks, “Transmission line capacitance measurement,” IEEE Microw. Guided Wave Lett., vol. 1, no. 9, pp. 243–245, Sep. 1991. [16] D. W. Marquardt, “An algortihm for least-squares estimation of nonlinear paramters,” J. Soc. Indus. Appl. Math., vol. 11, no. 2, pp. 431–441, 1963. [17] D. F. Williams and R. B. Marks, “LRM probe-tip calibrations using nonideal standards,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 2, pp. 466–469, Feb. 1995.

Nathan D. Orloff was born in Columbia, SC, on August 10, 1981. He received the B.S. degree in physics (with high honors) from the University of Maryland at College Park, in 2004, and is currently working toward the Ph.D. degree at the University of Maryland at College Park. His doctoral thesis concerns the study and extraction of microwave properties of ferroelectrics and fluids. Mr. Orloff was the recipient of the 2004 Martin Monroe Undergraduate Research Award, the 2006 CMPS Award for Excellence for Teaching Assistants, and 2010 Michael J. Pelczar Award for Excellence in Graduate Education.

Jordi Mateu (M’03) received the Telecommunication Engineering and Ph.D. degrees from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1999 and 2003, respectively. He is currently an Associate Professor with UPC, and an Associate Researcher with Centre Tecnològic de Telecomunicacions de Catalunya, UPC. In Summer 2001, he was a Visiting Researcher with Superconductor Technologies Inc., Santa Barbara, CA. Since September 2004, he has held several Guest Researcher appointments with the National Institute of Standards and Technology (NIST), Boulder, CO, where he was a Fulbright Research Fellow from 2005 to 2006. From 2003 to 2005, he was a Part-Time Assistant Professor with the Universitat Autonoma de Barcelona.

Arkadiusz Lewandowski (M’09) received the M.Sc. degree in electrical engineering and Ph.D. degree in microwave measurements from the Warsaw University of Technology, Warsaw, Poland, in 2001 and 2009, respectively. In 2002, he joined the Institute of Electronics Systems, Warsaw University of Technology. Since 2004 he has been a Guest Researcher with the National Institute of Standards and Technology (NIST), Boulder, CO, where he is engaged in the development of uncertainty analysis and calibration methods for coaxial and on-wafer VNA measurements. Mr. Lewandowski was the recipient of Best Paper Award of the International Microwave Conference MIKON 2008, Wroclaw, Poland, and the 2005 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Graduate Fellowship Award in 2002.

Eduard Rocas (S’06) was born in Palafrugell, Catalonia, Spain, in 1982. He received the Telecommunication Engineering degree from the Universitat Polit ècnica de Catalunya (UPC), Barcelona, Spain, in 2005, and is currently working toward the Ph.D. degree at the UPC. Mr. Rocas was the recipient of a Formación del Profesorado Universitario grant and a Formación de Personal Investigador grant.

Josh King was born in Denver, CO, in 1988. He is currently working toward the B.A. degree in business from the University of Colorado at Boulder.

Dazhen Gu (S’01–M’08) received the Ph.D. degree in electrical engineering from the University of Massachusetts–Amherst, in 2007.

ORLOFF et al.: COMPACT VARIABLE-TEMPERATURE BROADBAND SERIES-RESISTOR CALIBRATION

Since November 2003, he has been with the Electromagnetics Division, National Institute of Standards and Technology (NIST), Boulder, CO. In May 2007, he joined the Microwave Measurement Services Project, where he has been involved in microwave metrology, in particular thermal noise measurements and instrumentation, and fabrication of on-wafer devices for ultra-wideband material calibration and nanowire characterization.

195

Ichiro Takeuchi received the B.S. degree in physics from the California Institute of Technology, Pasadena, in 1987, and the Ph.D. degree in physics from the University of Maryland at College Park, in 1996. From 1996 to 1998, he was a Postdoctoral Fellow with the Lawrence Berkeley National Laboratory. Since 1999, he has been on the faculty of the Department of Materials Science and Engineering, University of Maryland at College Park, where he is currently a Professor. His research focuses on the development and applications of the combinatorial approach to electronic and magnetic materials, multilayer thin-film devices, and scanning probe microscopes.

Xiaoli Lu, photograph and biography not available at time of publication.

Carlos Collado (A’02–M’03) received the Telecommunication Engineering and Ph.D. degrees from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1995 and 2001, respectively, and the M.S. degree in biomedical engineering from UPC, in 2002. In 1998, he joined the faculty of UPC, and he became an Associate Professor in 2005. From 2005 to 2008, he was Vice-Dean of the Technical School of Castelldefels, where he is responsible for tthe telecommunication and aeronautic engineering degrees. He was a Visiting Researcher with the University of California at Irvine in 2004 and with the National Institute of Standard and Technology (NIST) from 2009 to 2010. His primary research interests include microwave devices and systems.

James C. Booth received the B.A. degree in physics from the University of Virginia, Blacksburg, in 1989, and the Ph.D. degree in physics from the University of Maryland at College Park, in 1996. Since 1996, he has been a Physicist with the National Institute of Standards and Technology (NIST), Boulder, CO, originally as a National Research Council (NRC) Post-Doctoral Research Associate (1996–1998) and currently as a Staff Scientist. His research with NIST is focused on exploring the microwave properties of new electronic materials and devices including ferroelectric, magnetoelectric, and superconducting thin films, as well as developing experimental platforms integrating microfluidic and microelectronic components for RF and microwave frequency characterization of liquid and biological samples.

196

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Flicker Noise Effects in Noise Adding Radiometers Jonathan J. Lynch and Robert G. Nagele

Abstract—This paper presents an analysis of noise adding radiometers that includes the effects of flicker, or 1 , noise which often limits radiometer performance. The 1 noise processes within the detector are affected by the modulation of the detector voltage as a result of the injected noise power and cause the overall 1 noise to increase. The functional dependence of the 1 noise parameters on degree of source modulation is determined experimentally. The primary result of this paper is a concise mathematical formula for the noise equivalent temperature difference of a noise adding radiometer explicitly showing the dependences on the radiometer parameters, as well as detector voltages when the source is on and off. From this formula, we derive the optimum on and off voltage ratio that maximizes sensitivity. Index Terms—Flicker noise, noise adding radiometer, passive millimeter-wave imaging.

I. INTRODUCTION ASSIVE millimeter-wave imaging is beginning to gain traction in consumer and military markets for weapons screening, theft prevention, border control, contraband detection, and all weather navigation [1]–[7]. In contrast to X-ray and radar, passive millimeter-wave imaging requires no transmitted energy, which is advantageous in a health-conscious society, as well as in military applications that require clandestine operation. Passive millimeter-wave images also suffer less from spectral reflection, known as glint, that often obscures objects and makes them difficult to detect or identify. The main drawback of passive millimeter-wave imaging is the low level of blackbody radiation present at typical operating frequencies (e.g., 94 GHz) that requires highly sensitive radiometers at the system front end. Radiometric sensitivity (also known as radiometric resolution) is measured as the minimum detectable change in temperature and referred to as the noise equivalent temperature difference (NETD). Since many weapon detection systems must operate in low-contrast environments, such as indoors, sensor NETD is typically required to be lower than 0.5 K. Since sensor NETD is one of the primary limitations of camera performance, methods for increasing sensitivity are of great interest to manufacturers of millimeter-wave imaging systems for security screening applications. Thermal sensitivity (NETD) is often considered separately from radiometer stability, the latter being limited by various

P

Manuscript received July 27, 2009; revised April 14, 2010; accepted September 15, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. The authors are with HRL Laboratories LLC, Malibu, CA 90265 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2090709

noise inand drift processes within the sensor. However, creases NETD to some degree in all radiometers, even those with stabilization circuitry, effectively reducing thermal sensitivity by increasing the random fluctuations of the sensor output. For low-cost radiometers used in security screening systems, output stabilization circuitry may be nonexistent, or of minimal complexity, so quantifying the degradation of NETD due to noise is important for system design and optimization. A number of effective methods have been developed to stabilize the radiometer output using signal modulation followed by synchronous detection, including Dicke switching, noise injection [8], phase switching, and optical chopping [9]–[12]. However, for commercial security screening applications, the complexity of the sensor array must be kept sufficiently low to ensure a cost-effective system. Any method that requires additional RF hardware within each sensor will likely increase system cost significantly. Optical chopping is a potential solution, but simultaneously chopping an entire array is not trivial and does add cost, and the performance improvement is limited by the mechanical scan rate of modulator. Of all the approaches to improve radiometer sensitivity, the so-called “noise adding” radiometer approach has the potential to improve sensitivity while adding minimal RF hardware to an imaging system [13]. Noise adding radiometers operate by periodically injecting noise into the radiometer front end. By comparing the scene temperature to that of the stable noise source, one forms an estimate of scene temperature that is not gain fluctuations within the radegraded by unavoidable diometer front end. The addition of noise can be accomplished radiatively across an entire array of sensors, requiring no modification to the sensors and requiring only one noise source per system. To optimize radiometer performance, one must understand the functional dependence of NETD on all of the radiometer panoise is a significant limitation to radiometer senrameters. sitivity and must be included in any useful sensor model. The periodic injection of noise modulates the detector output voltage, noise processes within the dealtering the behavior of the tector. Although components other than the detector contribute noise (e.g., low-noise amplifier (LNA) gain fluctuations), measurements performed at HRL Laboratories LLC, Malibu, CA (not yet published) of our millimeter-wave imaging chipset [14] indicate that the detector is the dominant contributor of noise. Although noise adding radiometers are described extensively in the literature [15], to our knowledge, a detailed analysis of the effects of detector voltage modulation in a noise adding radiometer as a function of the injected signal strength, including noise sources, has not been presented. In this paper, we develop a mathematical model for a noise adding radiometer and experimentally determine the functional dependence of

0018-9480/$26.00 © 2010 IEEE

LYNCH AND NAGELE: FLICKER NOISE EFFECTS IN NOISE ADDING RADIOMETERS

197

is the radiometric bandwidth (Hz). The mean (V/W), and output voltage when the source is on (called the “on” voltage) includes additional noise power from the source (3)

Fig. 1. Block diagram of noise adding radiometer experimental setup. The source noise is produced by an LNA with input terminated and is injected using a directional coupler. A variable attenuator allows adjustment of the injected power. A dynamic signal analyzer (DSA) digitizes and records the output.

noise, and therefore, NETD, on the degree of detector voltage modulation. The result is a formula for NETD that is a function of the detector “on” and “off” voltages, as well as other parameters such as RF bandwidth, duty cycle, modulation frequency, integration time, and frame period. The availability of this formula, as well as of the methods presented for experimentally characterizing radiometers, will allow system performance to be optimized within the myriad constraints of a given application.

with effective source power (W) referred to the radiometer input. Fluctuations about the mean values occur from downconverted RF noise, detector thermal noise, low frequency gain fluctuations (from the LNA and the detector), and possibly additional detector noise generated by the modulation of its output voltage. The downconverted RF and added thermal noise processes for a radiometer with mean output voltage , bandwidth, are white with a one-sided PSD given and detector resistance by [14] (4) In the last expression, we simplified the notation by combining the two white noise contributions to form an “effective” bandwidth and will use this effective bandwidth throughout the remainder of this paper without explicit indication. The LNA gain and detector responsivity fluctuations are behavior [16]. Writing the known to have (approximately) time-dependent gain-responsivity product as

II. NETD OF A NOISE ADDING RADIOMETER A. Radiometer Noise Modeling

(5)

In this section, we derive the power spectral density (PSD) of the output of a typical millimeter-wave radiometer, and in doing so introduce the notation that will be utilized throughout this paper. For constant scene temperature, the output of a preamplified radiometer is a steady dc (mean) value with random fluctuations (1) where we assume the “fractional fluctuation” is a zero mean stationary random process. Throughout this paper, we will use the lower case to signify time dependent variables and the lower case Greek delta ( ) to indicate the fractional fluctuation. We will also assume the radiometer utilizes a zero bias detector. The noise adding radiometer, represented by the block diagram in Fig. 1, periodically injects RF noise, generated here using an LNA with terminated input, into the radiometer at a rate (for simplicity, we assume equal on and off times). noise We assume the noise source has significantly lower than that of the radiometer, an assumption justified by the fact noise is significantly greater than that conthat detector tributed by the LNA. The mean value of the output when the source is off (called the “off” voltage) can be estimated from the radiometer parameters, as described in [14] (2) is the scene temperature, is 290 K, is the LNA where gain, is the LNA noise factor, is the detector responsivity

with mean values and , we see that the fractional fluctuations from the LNA and detector add directly. We assume the PSDs of these (fractional) independent noise processes have the same form (6) To simplify the resulting expressions, we will assume the exponent parameter is unity throughout this paper. An exponent close to, but not equal to, unity may be included using techniques similar to those described in [14]. In the following, it is not necessary to distinguish the source of the gain fluctuations so the LNA and detector contributions will be combined to form noise factor . a single Using (2) and (4)–(6), we find that the PSD for the fractional output voltage noise process in the absence of source power noise is due purely to gain flucmodulation, assuming the tuations, is given by (7) This result shows that temporal gain fluctuations produce a PSD for the fractional noise process that is independent of mean output voltage. We will see that modulating the injected source power alters the behavior of the detector during both the paon and off times and produces a dependence of the rameter on the mean detector voltage. In the following, we will develop a mathematical representation for the PSD including

198

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

source modulation and will experimentally determine its functional form. B. NETD Estimate for a Noise Adding Radiometer In this section, we define the NETD for the noise adding radiometer and derive an expression showing its dependence on the on- and off-voltages, as well as the co- and cross-correlations of the output voltage during the on and off time periods. Following the well-established “noise adding” radiometer approach, we estimate the scene temperature (within a proportionality constant) in a way that tends to cancel out gain fluctuations

on- and off-periods as well as for the PSD corresponding to the covariance between the on- and off-periods have the same gen, but with different values of . We eral functional form believe the agreement obtained between theory and experiment justifies this assumption. We form the scene temperature estimate (8) by gated integration of the output during intervals when the source is on and off to produce a series of output samples, then averaging of these samples to form “integrated” on and off voltages for the th time period

(8) producing a first-order fractional fluctuation equal to (9) In general, NETD is defined as the ratio of the root mean square (rms) value of the output fluctuations to the “thermal sen, sitivity” (distinguished from radiometric sensitivity) where is the mean output voltage and is the scene temperature. For a zero bias radiometer with no injected source power, (10)

(13) The parameter is a “duty cycle” of the time gate relative to the switching period. A typical radiometer output with modulated injected power is shown in Fig. 2, explicitly indicating the gating and integration parameters. With the above notation, the results of Appendix A show that NETD is given by

where the brackets denote ensemble average, and we have defined an equivalent system input noise temperature for the zero bias radiometer (see [14] for details). Since the denominator in (8) is independent of scene temperature, the is proportional to , thus, for the noise mean value of adding radiometer, (14)

(11) is the mean value of the estimate (8). Using (9) and where (11), we express NETD in terms of the output voltage

where and are related to the on- and off-periods, the parameter cording to

parameters during the , and the duty cycle ac-

(12) For typical radiometers used in -band weapon detection systems, is in the 600–1000-K range. From (12), we see that (primarily determined by the low NETD depends on low LNA noise figure), , and, when noise is the dominant noise contribution in the unstabilized sensor, strong correlation between the output voltages during the on and off periods.

(15)

The dependence on duty cycle is through the function

C. Explicit Formulas for NETD In this section we show explicitly how NETD depends on the radiometer parameters, duty cycle, and switching rate. The analysis, whose details are included in Appendix A, proceeds by deriving the variance of the output voltage during the separate gating periods and , as well as for the correlation between the on- and off-voltages, and utilizes (12) to determine NETD. In the analysis we assume that the PSD during the

(16) As explained in the Appendix, describes the behavior of the cross correlation of the noise processes between the on and off periods, the Fourier transform of which is assumed to exhibit a spectral decay. The NETD expression (14) is similar to that of an imperfect Dicke switched radiometer where

LYNCH AND NAGELE: FLICKER NOISE EFFECTS IN NOISE ADDING RADIOMETERS

199

TABLE I RADIOMETER PERFORMANCE PARAMETERS

Fig. 2. (a) Plot of a typical radiometer output with injected modulated source power. The switching period is T , the time gating duty cycle is D , and the time gate boundaries are indicated by diamonds. (b) Same plot over an extended time period, showing the integration time (P switching periods) and frame period (N integration times). These plots include a video amp with a voltage gain of 100.

is proportional to the fluctuation preceding the switch. We will see in Section III that for low values of injected power so that and fundamental term becomes negligible. However, the detector voltage modulation produced by large values of injected signal power, necessary for low NETD, create differences between and that increase the fundamental noise contribution. The last term under the radical in (14) is the contribution from the aliased replicas of the modulated noise spectrum. Thus, modulation of the detector output voltage due to the periodically injected power alters the noise processes within the detector and produce a dependence of the parameters and on the voltages and . We will see that this leads to an increase in NETD as the injected noise power is increased. The functional dependence of the parameters on injected noise power must be determined experimentally, and in Section III, we describe how these parameters may be estimated by measurements of the noise adding radiometer output. III. EXPERIMENTAL DETERMINATION OF

PARAMETERS

In this section, we describe the experimental setup and data collection for a noise adding radiometer corresponding to Fig. 1. The purpose of these measurements is to deduce the depenparameters and on the amount dence of the

of source power applied to the radiometer during the “on” time. The primary result of this paper is a formula for NETD explicitly showing the dependence on the on- and off-voltages, including due to modulation of the detector output. the increase in To determine this functional relationship, we constructed the setup shown in Fig. 1 that consists of a radiometer of the type described in [14] connected to an RF source through a directional coupler. The source consists of an input-terminated RF amplifier with its output power adjusted through a variable waveguide noise contribution of attenuator. As mentioned above, the this amplifier is significantly lower than that of the radiometer. The source power is modulated by switching the drain bias on and off at a 960-Hz rate (chosen for various reasons of convenience). The radiometer output is amplified by a low-noise op amp, passed through a single-pole low-pass filter, and digitized by a dynamic signal analyzer (DSA). The data samples are then processed to extract estimates of various radiometer parameters as a function of the on/off output voltage ratio. The radiometer consists of a waveguide input that feeds an InP high electron-mobility transistor (HEMT) LNA that, in turn, feeds a zero bias backward tunnel diode detector. This radiometer was characterized using the methods described in [14] and the resulting performance parameters are listed in . Table I. The measurements were made with The ratio of the first two values in the table gives an equivalent system input noise temperature of 1247 K. This radiometer , primarily due to the inserexhibits a higher than typical tion loss of the directional coupler, as well as a higher than typical LNA noise figure. The effective RF bandwidth includes the thermal noise contributed by the detector and the video amplinoise factor listed in the table is not normalized to fier. The . The NETD value assumes an integration time of 9.6 ms and a frame period of 0.24 s, consistent with the parameters used below for the case of injected source power. The video amplifier was implemented using a low-noise op amp (Linear Technology LT1028) configured for noninverting voltage gain of 100. The low-pass filter is a single pole R–C network with a cutoff frequency of about 25 kHz. The equivalent input noise resistance of this video amp is 1500 and its noise contribution is negligible. Both the radiometer and source amplifier were biased from battery supplies to minimize 60-Hz interference. The amplified video output was digitized using a Stanford Research SR785 DSA that samples at a rate of 262.1 kHz. Fig. 2 shows a plot of a typical video output. The gated integration was

200

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

performed by averaging the data samples over the individual time gates, which is equivalent to the integrations (13) due to the antialiasing filters within the DSA. The duty cycle was set to reject extraneous contributions from the at or below transient response of the antialiasing and low-pass filters near the switching boundaries. The effect of this transient response is to increase the estimate of the white noise, thereby decreasing the effective RF bandwidth. Multiple gated integration values were then averaged to form filtered estimates of the on and off voltages over a “frame” pegated integrariod, consistent with (13). We chose tion values for each estimate, giving an integration time (not ms, and formed a frame including duty cycle) of such averaged samples giving a frame period of of s. Multiple data sets were recorded with the source attenuation at various settings to produce voltage ratios ranging from one to about six. -K paFor each data set, the effective RF bandwidths and rameters during the on and off periods were estimated using the following method. For a robust estimate of the RF bandwidth, the variance of the video output was computed over a single gate period and multiple such values were averaged over periods ) (for this bandwidth estimate

Fig. 3. RF bandwidth during the on and off periods for various source power levels. The decrease in f is likely due to the high-frequency rolloff of the at source LNA that reduces its effective bandwidth. The decrease in f higher source power levels is primarily due to quantization noise that increases with increasing V .

1

1

The computed value was 25.1 kHz and this value was then used paramein subsequent estimates of the RF bandwidths and ters using (18). Equation (18) (applied separately to the on and off periods) together with (50) and (53) (which utilize averages noise over longer time periods, thereby providing better parameter estimates) may be solved simultaneously for the RF and , and the -K parameters bandwidths, and . The parameter can be determined by setting (57) equal to the estimated value

(17)

where s is the sampling period of the A/D conis the gate period. Using the methods of verter and Section II, we evaluate (17) as an integral over the PSD of the Hz output voltage process

(18) is the equivalent video bandwidth, set priThe quantity marily by the low-pass filter that follows the video amp. We determine this value using the DSA in spectrum analyzer mode to measure the PSD of the output voltage with the radiometer LNA turned off. This provides a direct measure of the detector thermal noise and video amp contributions. We also recorded the time samples of the output (again with the LNA off) and computed the mean square fluctuations using (17). Since the noise contribution is negligibly low, we detervideo amp mine the video bandwidth using (19)

(20) Fig. 3 contains plots of estimates of RF bandwidths for the on and off periods. For (i.e., no source power), the bandwidths are equal, but as the voltage ratio increases both decreases fairly bandwidths decrease. The plot shows that quickly with increased source power, indicating that the bandwidth of the noise source is less than that of the radiometer. This is consistent with the measured (on-wafer) gain of the source LNA, which rolls off more quickly above 95 GHz than that of the radiometer LNA. As the source power is increased, the effective bandwidth of the entire channel approaches the bandwidth of the source, which appears to be about 18 GHz. The plot also decreases at higher voltage ratios. This effect shows that may be partially caused by an excessively wide time gate that does not reject the filter transient response near the switching minimizes that times, but our conservative choice of effect. The primary reason for the decrease seen in the plot is quantization noise from the A/D conversion. Since we digitize the entire switching waveform, the voltage step increases proportionally to the full-scale range needed to accommodate large . This effect can be eliminated using analog time values of

LYNCH AND NAGELE: FLICKER NOISE EFFECTS IN NOISE ADDING RADIOMETERS

201

Fig. 5. NETD computed directly from measured data (solid lines with markers) as compared to analytical model (dashed lines) for various values of duty cycle. The model assumes RF bandwidth values from Fig. 3 and 1=f K parameters as given by best fit lines in Fig. 4. The bottom-most solid line is the NETD predicted by noise-adding radiometer models that neglect 1=f noise.

Based on these results, we arrive at a closed-form expression for the NETD of a noise-adding radiometer as a function of the noise on and off voltages, including the voltage-dependent

Fig. 4. Normalized 1=f parameters. (a) Parameter K lies approximately midway between K and K , and the average value (dashed line) is approximately constant with voltage ratio and has a value K = 5:28 1 10 . (b) Parameter K has an approximately linear dependence with voltage ratio. The dashed line shows the best fit line with slope K = 0:123 1 10 , and a = 1:09 was chosen so that the best fit line equals zero at V = V .

gating to digitize the on and off periods separately, thereby reducing the required full-scale range. parameters as functions of the Fig. 4(a) is a plot of the voltage ratio. These parameters have been normalized to acexponent. The equations we used to count for a nonunity generate the plots in Figs. 3–5 accommodate the nonunity exponent using the technique described in [14], but these details have been omitted here for the sake of brevity. One can see that is approximately equal to the average of and , thus we define an average parameter (21) that is largely independent of voltage ratio. This allows us to separate out the duty cycle dependence of in (59)

(23) with given by (22). A plot of NETD computed directly from measured radiometer data is shown in Fig. 5 (solid lines with markers), as compared to the analytical model (23) (dashed lines) for duty and a theoretical value for cycle values of assuming no degradation from switching transients. For the model we used the RF bandwidths indicated in Fig. 3, and indicated in Fig. 4, but extrapolated as well as to different duty cycle values using (22) and (23). The plot in Fig. 5 shows agreement within about 10% between measured and modeled values. The bottom most curve on the plot is the noise, showing that both result obtained by neglecting noise and duty cycle have significant effects on sensor performance. One of the main results of this paper is that NETD for a noise adding radiometer does not decrease monotonically as the innoise that arises jected power is increased due to additional from the detector voltage modulation. From (23), we find that the optimal voltage ratio is given by

(22) The average value of over the plotted range of voltage . ratios, indicated by the dashed line, is to be approximately linear Fig. 4(b) shows the parameter with the voltage ratio.

(24)

202

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Using the measured data with , the optimum voltage ratio is 6.0, assuming an RF bandwidth of 23 GHz (valid for large voltage ratios). This produces a minimum NETD value of (25) with a numerical value of 0.38 K. This value of NETD could be made lower by increasing the source modulation rate. From (24) and (25), we see that the modulation rate should be increased so that (26)

Note that this definition depends on the number of integrations over a time period typically called a frame in mechanically scanned systems. The mean square fractional fluctuation is estimated over integration periods (i.e., a frame) in the standard way [17] (28) Because the rms fluctuations are typically much smaller than the mean, the denominator in (27) can be considered constant to first order, giving the variance of the fractional fluctuations

which has a value of 523 Hz using our measured data. Increasing the modulation rate beyond our value of 960 Hz would reduce NETD closer to the optimum. In the limit of infinite modulation rate, the minimum achievable NETD is 0.34 K for this noise adding radiometer. (29) IV. CONCLUSION Noise adding radiometer performance is ultimately limited noise characteristics, as by the radiometer bandwidth and described by expressions derived in this paper. Injected source noise processes in a modulation alters the behavior of the way that increases the contributions as the injected noise power increases. This functional dependence was determined experimentally and has been explicitly described by the formulas presented here. The relative simplicity of the formulas allows the radiometer performance to be optimized within the constraints of a particular system design. Additionally, the noise pamethods used to extract the RF bandwidth and rameters can be used to accurately characterize radiometers in the presence of injected source modulation. The results of this analysis and measured data show that the NETD for a typical radiometer with a zero bias detector can be cut in half using the noise adding technique as compared to the unstabilized case, and by radiatively coupling the injected signal across arrays of sensors this approach will improve sensor performance without integrating additional RF components into each of the array sensors. The drawback of this approach as compared to other methods of increasing sensitivity, such as Dicke switching, is the need for relatively high levels of injected power. However, the advantage of the noise adding approach is the minimal hardware complexity, attained by radiatively coupling the injected signal across arrays of sensors.

Using this approach, the quantities under the radical in (12) can be expressed as

(30) Beginning with the first expression in (30), we evaluate the terms in the double summation using (13)

(31) APPENDIX A DERIVATION OF NETD To facilitate the analysis, we define fractional fluctuation variables for the th integration period as

where and . The autocovariance function for the random process can be related to the (one-sided) PSD using the Wiener–Khinchin theorem

(27)

(32)

LYNCH AND NAGELE: FLICKER NOISE EFFECTS IN NOISE ADDING RADIOMETERS

where we introduced a normalized frequency variable . Evaluating the time integrals gives

203

the well-known result for the white noise contribution to the variance (39)

(33) Substituting (33) into the first of (30) and evaluating the summations, we arrive at an expression for the variance as an integral over the PSD

(34) The factors in the integrand reflect the various steps of the estimation process for the scene temperature. The PSD is low-pass filtered by the second factor, produced by gated integration. The third factor is additional filtering that reflects the “boxcar” averaging of gated samples and defines the integration time of the radiometer. The last factor is a result of computing the variance about the mean over a finite observation (frame) period and produces a zero at the origin, limiting the potentially infinite energy process and ensuring a finite result for available from the measurements of finite duration. Expressing the PSD (7) in terms of the normalized frequency variable, (35) Integrating the white noise component first, we insert the first term of (35) into (33)

is the actual integration time for the output The quantity voltage, including the duty cycle of the on/off gating. In the litis usually referred to as the integration erature, the quantity time with the duty cycle factor included as a separate factor of 2. contribution, we substitute the second term Treating the of (35) into (34)

(40) The last factor in the integrand represents a set of periodic passbands with center frequencies near multiples of the modulation frequency, corresponding to for . The periodic nature of this filter results from averaging discrete are aliases gated samples, and the passbands that occur for . This can be explicitly shown by of the “fundamental” at expanding the cosecant squared function in partial fractions

(41) (corresponding Confining ourselves to the practical case to a high modulation frequency ), each passband contributes to the integral near its center frequency, but little outside that , the passbands lie outside the region of passband. For integration and their small contributions will be neglected. term in (41) is The

(36) (42)

The integral can be evaluated using the cosine transform for for (37)

Integrals of this type can be evaluated by expanding the sine functions into a sum of cosine functions and integrating term by term using

which results in (38) is the Kronicker delta function. Inserting (38) into where the first equation in (30) and carrying out the summations gives

(43)

204

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

where is Euler’s constant. Expression (43) was obtained by repeated integration by parts and using the approximation

. The sum in (49) is a function of duty cycle that for we express in closed form using the approximate formula (51) with the result

(44) Evaluating (42) using (43), we discover that all the terms containing cancel out, as we expect since (42) is integrable at , and the result is (45) accurate to the first order in used the approximation

and

(52) Carrying out the analysis for the on voltage produces a similar result

. To obtain (45), we

(46)

(53)

and neglected terms on the order of and higher. The result (45) is understandable when one considers the inintegrated tegrand of (42), which consists of the function over a bandpass filter function. For , the function is close to unity over the passband and can be neglected. The lower passband edge is the (normalized) bandwidth , which is , and the upper band edge is of the bandwidth of , which is . Thus, we find that the integral (42) can be approximated as

We now turn to the cross correlation between the on and off voltages. The on and off white noise processes do not contribute to the cross correlation since the source white noise is uncorrelated with the sensor white noise, and the sensor white noise contributions during the “on” and “off” periods are uncorrelated [a result that can be shown using (37)]. To evaluate the third equation in (30), we first compute the quantity

(47) We utilize this approach for the more general case when the exponent differs (slightly) from unity, although these results are not presented here for the sake of brevity. in (41), we change variables to shift the passbands For to , and approximate the noise power as constant over the filter passbands, giving

(54) where we have defined a cross correlation function . Following the same procedure, we arrive at an expression similar to (34)

(48) The last integral can be evaluated using , resulting in

(55) (49)

Combining (39), (45), and (49) gives the variance

(50)

the only difference being the additional cosine factor in the integrand due to the half cycle time delay between the on and off voltage samples. We assume the “cross” PSD has the same form as the other processes (56)

LYNCH AND NAGELE: FLICKER NOISE EFFECTS IN NOISE ADDING RADIOMETERS

Substituting (56) into (55) and evaluating, we arrive at the following expression for the cross correlation :

(57)

Finally, inserting (50), (53), and (57) into (12) gives

(58) where we have defined the following

noise parameters:

(59)

and

is given by (52). REFERENCES [1] R. N. Anderton, R. Appleby, J. E. Beale, P. R. Coward, and S. Price, “Security scanning at 94 GHz,” Proc. SPIE 6211, pp. 1–7, May 2006, Art. ID 62110C. [2] G. V. Tryon, “Millimeter wave case study of operational deployments: Retail, airport, military, courthouse, and customs,” Proc. SPIE 6948, pp. 1–16, Apr. 2008, Art. ID 694802. [3] D. A. Wikner, “Passive millimeter-wave imagery of helicopter obstacles in a sand environment,” Proc. SPIE 6211, pp. 1–8, May 2006, 621103. [4] N. M. Vaidya and T. Williams, “A novel approach to automatic threat detection in MMW imagery of people scanned in portals,” Proc. SPIE 6948, pp. 1–10, Apr. 2008, Art. ID 69480F. [5] C. A. Martin, C. E. G. Gonzalez, V. G. Kolinko, and J. A. Lovberg, “Rapid passive MMW security screening portal,” Proc. SPIE 6948, pp. 1–9, Apr. 2008, Art. ID 69480J. [6] A. Luukanen, L. Grönberg, T. Haarnoja, P. Helistö, K. Kataja, M. Leivo, A. Rautiainen, J. Penttilä, J. E. Bjarnason, C. R. Dietlein, M. D. Ramirez, and E. N. Grossman, “Passive THz imaging system for stand-off identification of concealed objects: Results from a turn-key 16 pixel imager,” Proc. SPIE 6948, pp. 1–9, Apr. 2008, Art. ID 69480O.

205

[7] C. Mann, “A compact real time passive terahertz imager,” Proc. SPIE 6211, pp. 1–5, May 2006, Art. ID 62110E. [8] A. B. Tanner, W. J. Wilson, and F. A. Pellerano, “Development of a high stability L-band radiometer for ocean salinity measurements,” in Proc. IEEE Int. Geosci. Remote Sens. Symp., Jul. 2003, vol. 2, pp. 1238–1240. [9] M. E. Tiuri, “Radio astronomy receivers,” IEEE Trans. Antennas Propag., vol. AP-12, no. 7, pp. 930–938, Dec. 1964. [10] R. H. Dicke, “The measurement of thermal radiation at microwave frequencies,” Rev. Sci. Instrum., vol. 17, pp. 268–275, Jul. 1946. [11] W. N. Hardy, K. W. Gray, and A. W. Love, “An S band radiometer design with high absolute precision,” IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 4, pp. 382–390, Apr. 1974. [12] T. Gaier, P. Kangaslahti, A. Tanner, B. Lambrigtsen, S. Brown, M. Seiffert, D. Dawson, S. Weinreb, W. J. Wilson, B. Lim, C. Ruf, and J. Piepmeier, “Millimeter-wave array receivers for remote sensing,” Proc. SPIE 6410, pp. 11–8, May 2006, Art. ID 64100G. [13] P. E. Batelaan, R. M. Goldstein, and C. T. Stelzried, “Improved noise adding radiometer for microwave receivers,” JPL, Pasadena, CA, NASA Tech. Brief 73-10345, Aug. 1973. [14] J. J. Lynch, H. P. Moyer, J. H. Schaffner, Y. Royter, M. Sokolich, B. Hughes, Y. J. Yoon, and J. N. Schulman, “Passive millimeter-wave imaging module with preamplified zero-bias detection,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 7, pp. 1592–1600, Jul. 2008. [15] F. T. Ulaby, R. K. Moore, and A. K. Fung, “Radiometer systems,” in Microwave Remote Sensing: Active and Passive. Norwood, MA: Artech House, 1981, vol. I, ch. 6, sec. 12, pp. 391–392. [16] N. C. Jarosik, “Measurements of the low-frequency-gain fluctuations of a 30-GHz high-electron-mobility-transistor cryogenic amplifier,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 2, pp. 193–197, Feb. 1996. [17] C. W. Helstrom, “Multiple random variables,” in Probability and Stochastic Processes for Engineers. New York: Macmillan, 1984, ch. 4, sec. 6, pp. 219–222. Jonathan J. Lynch received the B.S. degree, M.S. degree, and Ph.D. degree in the area of quasi-optical power combining for continuous wave and pulsed millimeter-wave sources from the University of California at Santa Barbara, in 1987, 1992, and 1995, respectively. Since 1995, he has been with HRL Laboratories LLC., Malibu, CA, where he is currently a Senior Scientist and Manager of the Microwave Technology Department, Microelectronic Laboratory. His areas of expertise include microwave and millimeter-wave antennas, filters, waveguide circuits, and radiometers, as well as nonlinear components and subsystems, such as synchronized microwave oscillators and quasioptical power combining.

Robert G. Nagele received the B.S. degree in electrical and computer engineering from the School of Engineering, Rutgers University, New Brunswick, NJ, in 2003, and the M.S. degree from the University of Maryland at College Park, in 2005. He is currently a Research Staff Member with HRL Laboratories LLC, Malibu, CA. Prior to joining HRL in 2008, he spent several years with the High Power Microwave Section, Naval Research Laboratory, Washington, DC. His research has focused on microwave coupling theory, electromagnetic compatibility analysis, impulse generator subsystem design, and millimeter-wave imaging.

206

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Letters Corrections to “Editorial”

Corrections to “Wideband IF-Integrated Terahertz HEB Mixers: Modeling and Characterization”

D. Williams and A. Mortazawi

Fernando Rodriguez-Morales, K. Sigfrid Yngvesson, and Dazhen Gu In the above paper [1], Jen-Tsai Kuo’s affiliation was listed as National Chiao Tung University, Hsinchu, Taiwan. His affiliation has since changed. His updated biography is listed below.

Some errors have been detected in the above paper [1]. First, there is an incorrect sign in the denominator of the second factor of [1, eq. (2)]. The correct expression for the small-signal impedance according to the standard model should be

C 10C 1+

Z (!) = R0

REFERENCES [1] D. Williams and A. Mortazawi, “Editorial,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 8, pp. 2073–2076, Aug. 2010.

 j! 1 + C  1 + j! 10C 1+

:

Second, “low0frequency” in the first paragraph of the right column, is page 1145, should read “low-frequency.” Third, the variable incorrectly defined in [1, eq. (14)] as “the effective bandwidth of the receiver.” Instead, should be defined as “the bandwidth around the intermediate frequency of interest, over which the measurement is made.” As further clarification on this part, we note that for the measurements reported, is set by the 200-MHz passband of the tunable IF backend, as mentioned in the above paper [1,Sec. III-B.1]. Finally, [1, eq. (19)] should have appeared as

B

B

B

Jen-Tsai Kuo (S’88–M’92–SM’03) was with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan. Since February, 2010, he has been a Professor with the Department of Electronic Engineering, Chang Gung University, Taoyuan, Taiwan. His research interests include analysis and design of microwave integrated circuits and numerical techniques in electromagnetics.

PIFout jhot = Grec kB (2Thot + 2TR;DSB (f )) PIFout jcold = Grec kB (2Tcold + 2TR;DSB (f )) and the expression for the total (single-sideband) receiver gain should be written as

Grec

jhot 0 PIFout jcold Grec = PIFout 2k B(Thot 0 Tcold ): To conclude this letter, we would like to point out that the above errors were merely typographic and do not affect any other equations or the results or the discussions presented in [1].

REFERENCES [1] F. Rodriguez-Morales, K. S. Yngvesson, and D. Gu, “Wideband IF-integrated terahertz HEB mixers: Modeling and characterization,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 5, pp. 1140–1150, May 2010.

Manuscript received September 29, 2010; revised September 29, 2010; accepted September 29, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. D. Williams is with the National Institute of Standards and Technology (NIST), Boulder, CO 80305 USA. A. Mortazawi is with the Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor, Ann Arbor, MI 48109-2122 USA. Digital Object Identifier 10.1109/TMTT.2010.2090692

Manuscript received July 01, 2010; accepted July 23, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. F. Rodriguez-Morales was with the Department of Electrical and Computer Engineering, University of Massachusetts at Amherst, Amherst, MA 01003 USA. He is now with the Center for Remote Sensing of Ice Sheets, University of Kansas, Lawrence, KS 66045 USA (e-mail: [email protected]). K. S. Yngvesson is with the Department of Electrical and Computer Engineering, University of Massachusetts at Amherst, Amherst, MA 01003 USA. D. Gu is with the National Institute of Standards and Technology, Boulder, CO 80305 USA. Digital Object Identifier 10.1109/TMTT.2010.2090355

0018-9480/$26.00 © 2010 IEEE

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 1, JANUARY 2011

Corrections to “Impedance-Transforming Symmetric and Asymmetric DC Blocks”

REFERENCES [1] H.-R. Ahn and T. Itoh, “Impedance-transforming symmetric and asymmetric DC blocks,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 9, pp. 2463–2474, Sep. 2010.

Hee-Ran Ahn and Tatsuo Itoh There is a publication error in the above paper [1, Fig. 8]. The correct Fig. 8 is presented here.

Fig. 8. Frequency responses of symmetric Chebyshev dc blocks. (a) S . (b) S S .

=

S

207

=

Manuscript received October 07, 2010; accepted October 12, 2010. Date of publication December 03, 2010; date of current version January 12, 2011. The authors are with the Department of Electrical Engineering, University of California at Los Angeles (UCLA), Los Angeles, CA 90095 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2090360

0018-9480/$26.00 © 2010 IEEE

Digital Object Identifier 10.1109/TMTT.2010.2103872

Digital Object Identifier 10.1109/TMTT.2011.2104550

Digital Object Identifier 10.1109/TMTT.2011.2104555

Digital Object Identifier 10.1109/TMTT.2011.2104554

EDITORIAL BOARD Editor-in-Chief: GEORGE E. PONCHAK Associate Editors: H. ZIRATH, W. VAN MOER, J.-S. RIEH, Q. XUE, L. ZHU, K. J. CHEN, M. YU, C.-W. TANG, B. NAUWELAERS, J. PAPAPOLYMEROU, N. S. BARKER, C. D. SARRIS, C. FUMEAUX, D. HEO

P. Aaen A. Abbaspour-Tamijani A. Abbosh D. Abbott A. Abdipour M. Abe M. Abegaonkar R. Abhari A. Abramowicz M. Acar L. Accatino R. Achar E. Ackerman J. Adam K. Agawa M. Ahmad H.-R. Ahn B. Ai M. Aikawa J. Aikio C. Aitchison M. Akaike T. Akin S. Aksoy I. Aksun A. Akyurtlu G. Ala L. Albasha A. Alexanian W. Ali-Ahmad F. Alimenti R. Allam K. Allen A. Alphones A. Alu A. Álvarez-Melcon A. Al-Zayed S. Amari H. Amasuga R. Amaya H. An D. Anagnostou M. Andersen K. Andersson M. Ando Y. Ando P. Andreani M. Andrés W. Andress K. Ang C. Angell I. Angelov Y. Antar G. Antonini H. Aoki V. Aparin F. Apollonio R. Araneo J. Archer F. Ares F. Ariaei T. Arima M. Armendariz L. Arnaut F. Arndt E. Artal H. Arthaber F. Aryanfar U. Arz M. Asai Y. Asano A. Asensio-Lopez K. Ashby H. Ashoka A. Atalar A. Atia S. Auster I. Awai A. Aydiner M. Ayza K. Azadet R. Azaro A. Babakhani P. Baccarelli M. Baginski I. Bahl S. Bajpai J. Baker-Jarvis B. Bakkaloglu M. Bakr A. Baladin C. Balanis S. Balasubramaniam J. Balbastre J. Ball P. Balsara Q. Balzano A. Banai S. Banba R. Bansal D. Barataud A. Barbosa F. Bardati I. Bardi J. Bardin A. Barel S. Barker F. Barnes J. Barr G. Bartolucci R. Bashirullan S. Bastioli A. Basu B. Bates R. Baxley Y. Bayram J.-B. Bégueret N. Behdad F. Belgacem H. Bell D. Belot J. Benedikt T. Berceli C. Berland M. Berroth G. Bertin E. Bertran A. Bessemoulin M. Beurden A. Bevilacqua A. Beyer M. Bialkowski

E. Biebl P. Bienstman S. Bila D. Blackham R. Blaikie M. Blank P. Blockley P. Blondy P. Blount D. Boccoli G. Boeck L. Boglione R. Boix G. Bonaguide F. Bonani G. Bonmassar O. Boos B. Borges V. Boria-Esbert O. Boric-Lubecke A. Borji S. Borm J. Bornemann W. Bosch R. Bosisio H. Boss G. Botta N. Boulejfen S. Boumaiza J. Bouny C. Boyd C. Bozler M. Bozzi R. Bradley D. Braess N. Braithwaite M. Brandolini G. Branner T. Brazil J. Breitbarth M. Bressan K. Breuer B. Bridges D. Bridges J. Brinkhoff E. Brown S. Brozovich E. Bryerton D. Budimir G. Burdge P. Burghignoli N. Buris C. C. Galup-Montoro B. Cabon P. Cabral L. Cabria C. Caloz C. Camacho-Peñalosa V. Camarchia E. Camargo R. Cameron M. Camiade C. Campbell M. Campovecchio F. Canavero A. Cangellaris A. Cantoni C. Cao F. Capolino F. Cappelluti G. Carchon J. Carmo K. Carr F. Carrez R. Carrillo-Ramirez P. Carro R. Carter N. Carvalho P. Casas R. Castello J. Catala M. Cavagnaro R. Caverly D. Cavigia J. Cazaux M. Celuch Z. Cendes D. Chadha M. Chae S. Chakraborty C. Chan C. Chang H. Chang K. Chang S. Chang T. Chang W. Chang E. Channabasappa H. Chapell W. Chappell C. Charles M. Chatras I. Chatterjee G. Chattopadhyay S. Chaudhuri S. Chebolu A. Cheldavi A. Chen C. Chen H. Chen J. Chen K. Chen M. Chen N. Chen S. Chen Y. Chen Z. Chen Z.-N. Chen H. Cheng K. Cheng M. Cheng Y. Cheng C. Cheon C. Chi M. Chia Y. Chiang J. Chiao A. Chin K. Chin H. Chiou Y. Chiou C. Chiu

H. Chiu A. Chizh C. Cho K. Cho T. Cho A. Choffrut C. Choi J. Choi W. Choi C. Chong M. Chongcheawchamnan C. Chou D. Choudhury E. Chow Y. Chow C. Christodoulou C. Christopoulos Q. Chu T. Chu H. Chuang M. Chuang Y. Chun S. Chung Y. Chung D. Chye A. Cidronali T. Cisco C. Cismaru O. Civi S. Clavijo M. Clénet D. Cogan P. Colantonio M. Cole J. Coleman J. Collantes R. Collin C. Collins B. Colpitts R. Compton G. Conciauro M. Condon D. Consonni A. Constanzo M. Converse K. Cools F. Cooray I. Corbella A. Costanzo S. Cotton C. Courtney G. Coutts J. Cowles J. Craninckx C. Crespo-Cadenas J. Cressler S. Cripps T. Crowe J. Cruz T. Cui E. Cullens T. Cunha W. Curtice J. Dabrowski W. Dai G. Dambrine P. Dankov F. Danneville I. Darwazeh A. Darwish N. Das M. Davidovich L. Davis D. Dawn J. Dawson H. Dayal F. De Flaviis D. De Zutter B. Deal A. Dearn J. Deen M. Dehan C. Dehollain C. Deibele G. Dejean M. DeLisio N. Deltimple S. Demir V. Demir J. Deng A. Dengi T. Denidni W. DeRaedt H. Deshpande Y. Deval R. Dey T. Dhaene L. Diaz A. Diaz-Morcillo L. Ding M. Dionigi C. Diskus A. Djordjevi T. Djordjevic J. Dobrowolski H. Dogan S. Donati X. Dong A. Dounavis P. Draxler R. Drayton A. Dreher J. Drewniak J. Duchamp A. Duffy L. Dunleavy J. Dunsmore S. Durden L. Dussopt C. Duvanaud J. East J. Ebel K. Eccleston I. Ederra R. Egri I. Ehrenberg N. Ehsan T. Eibert H. Eisele W. Eisenstadt G. Eleftheriades

F. Ellinger G. Ellis T. Ellis M. El-Nozahi M. Elsbury S. Elschner M. El-Shenawee T. Enoki K. Entesari L. Epp I. Erdin O. Ergul T. Eriksson C. Ernst D. Erricolo I. Eshrah M. Essaaidi H. Esteban C. Eswarappa W. Eyssa A. Ezzeddine C. Fager M. Fahmi Y. Fan D. Fang M. Farina A. Fathy M. Faulkner P. Fay A. Fazzi E. Fear P. Fedorenko D. Feld Y. Feng A. Feresidis A. Fernandez T. Fernandez M. Fernández-Barciela M. Ferndahl F. Fernez P. Ferrari E. Ferre-Pikal A. Ferrero M. Ferriss H. Fetterman J. Fiedziuszko S. Fiedziuszko G. Fikioris J. Fikioris I. Filanovsky F. Filicori D. Filipovic R. Fletcher B. Floyd H. Foltz N. Fong B. Fornberg F. Fortes K. Foster P. Foster P. Franzon A. Frappe J. Freire M. Freire A. Freundorfer F. Frezza I. Frigyes R. Frye J. Fu O. Fu R. Fujimoto O. Fujiwara C. Fumeaux C. Furse V. Fusco D. Gabbay E. Gad M. Gadringer N. Gagnon J. Gajadharsing A. Gala C. Galbraith B. Galwas J. Gambini A. Gameiro O. Gandhi B. Gao J. Gao S. Gao C. Gaquiere H. Garbe J. Garcia M. Garcia P. Garcia-Ducar F. Garcia-Vidal K. Gard P. Gardner P. Garland P. Gaudo J. Gautier S. Gedney B. Geelen F. Gekat B. Geller R. Genov A. Georgiadis N. Georgieva J. Gerdes W. Gerhard S. Gevorgian H. Ghali M. Ghanevati F. Ghannouchi K. Gharaibeh R. Gharpurey G. Ghione M. Ghovanloo F. Giannini A. Gibson I. Gil P. Gilabert B. Gimeno D. Ginste A. Goacher E. Godshalk A. Goel C. Goldsmith M. Golio M. Golosovsky R. Gómez-García A. Goncharenko X. Gong

R. Gonzalo S. Goodnick S. Gopalsami A. Gopinath A. Görür K. Gosalia M. Gouker K. Goverdhanam W. Grabherr J. Graffeuil L. Gragnani J. Grahn J. Grajal V. Granatstein A. Grbic A. Grebennikov I. Gresham A. Griol D. Grischowsky S. Grivet-Talocia E. Grossman S. Gruszczynski T. Grzegorczyk S. Guenneau T. Guerrero S. Gunnarsson J. Guo Y. Guo C. Gupta M. Gupta R. Gupta R. Gutmann W. Gwarek R. Habash S. Hadjiloucas D. Haemmerich M. Hagmann S. Hagness A. Halappa P. Hale D. Ham E. Hamidi O. Hammi H. Han T. Hancock A. Hanke G. Hanson Y. Hao Z. Hao R. Harjani L. Harle H. Harris P. Harrison O. Hartin J. Hasch H. Hashemi K. Hashimoto J. Haslett G. Hau S. Hauptmann L. Hayden L. He Y. He R. Heath E. Hegazi G. Hegazi S. Heinen W. Heinrich G. Heiter M. Hella R. Henderson F. Henkel B. Henning D. Heo K. Herrick F. Herzel J. Hesler J. Hesthaven K. Hettak H. Heuermann P. Heydari A. Hietala A. Higgins A. Hirata J. Hirokawa M. Ho K. Hoffmann R. Hoffmann E. Holzman V. Hombach J. Hong S. Hong W. Hong K. Honjo G. Hopkins Y. Horii J. Horng T.-S. Horng J. Horton K. Hosoya M. Hotta J. Hoversten J. Howard M. Høyerby H. Hsieh L. Hsieh C. Hsu H. Hsu J. Hsu C. Hsue R. Hu C. Huang F. Huang H. Huang P. Huang T. Huang J. Hubert W. Huei A. Hülsmann A. Hung C. Hung J. Hung I. Hunter I. Huynen H. Hwang J. Hwang K. Hwang R. Hwang G. Iannaccone K. Ikossi M. Isaksson T. Ishizaki

Digital Object Identifier 10.1109/TMTT.2010.2103870

S. Islam M. Ito K. Itoh T. Itoh Y. Itoh A. Ittipiboon F. Ivanek D. Iverson M. Iwamoto D. Jablonski D. Jachowski C. Jackson D. Jackson R. Jackson A. Jacob K. Jacobs S. Jacobsen D. Jaeger J. Jaeger S. Jagannathan N. Jain G. James M. Janezic S. Jang M. Jankovic D. Jansen L. Jansson H. Jantunen H. Jardon-Aguilar J. Jargon N. Jarosik B. Jarry P. Jarry A. Jastrzebski B. Jemison W. Jemison S. Jeng A. Jenkins S. Jeon D. Jeong J. Jeong Y. Jeong A. Jerng T. Jerse T. Jiang X. Jiang G. Jianjun D. Jiao J. Jin J. M. Jin J. Joe T. Johnson B. Jokanovic U. Jordan K. Joshin J. Joubert S. Jung T. Kaho S. Kanamaluru K. Kanaya S. Kang P. Kangaslahti B. Kapilevich I. Karanasiou M. Karim T. Kataoka A. Katz R. Kaul R. Kaunisto T. Kawai S. Kawasaki M. Kazimierczuk L. Kempel P. Kenington P. Kennedy A. Kerr D. Kettle A. Khalil W. Khalil S. Khang A. Khanifar A. Khanna R. Khazaka J. Khoja S. Kiaei J. Kiang B. Kim C. Kim D. Kim H. Kim I. Kim J. Kim S. Kim T. Kim W. Kim N. Kinayman R. King N. Kinzie S. Kirchoefer A. Kirilenko M. Kishihara T. Kitazawa J. Kitchen T. Klapwijk E. Klumperink D. Klymyshyn L. Knockaert R. Knoechel M. Koch K. Koh N. Kolias J. Komiak A. Komijani G. Kompa A. Konanur A. Konczykowska H. Kondoh B. Kopp B. Kormanyos J. Korvink P. Kosmas Y. Kotsuka S. Koziel A. Kozyrev V. Krishnamurthy H. Krishnaswamy C. Krowne J. Krupka D. Kryger H. Ku H. Kubo A. Kucar

A. Kucharski C. Kudsia A. Kudymov D. Kuester B. Kuhn W. Kuhn T. Kuki A. Kumar J. Kuno C. Kuo J.-T. Kuo H. Kurebayashi F. Kuroki L. Kushner S. Kusunoki D. Kuylenstierna Y. Kwon G. Kyriacou A. Lacaita J. Lamb P. Lampariello U. Langmann T. Larsen L. Larson J. Laskar C. Lau K. Lau A. Lauer D. Lautru P. Lavrador A. Lavrinenko A. Lazaro G. Lazzi R. Lech B. Lee C.-H. Lee C. Lee H. Lee J. Lee J.-H. Lee K. Lee R. Lee S. Lee T. Lee Y. Lee D. Leenaerts Z. Lei G. Leizerovich K. Leong Y. Leong R. Leoni C. Ler G. Leuzzi B. Levitas R. Levy C. Li L. Li M. Li X. Li L. Lianming C. Liao S. Liao D. Lie E. Lima E. Limiti F. Lin J. Lin K. Lin T. Lin Y. Lin S. Lindenmeier A. Lindner F. Ling D. Linkhart P. Linnér D. Linten D. Linton D. Lippens F. Little V. Litvinov C. Liu H. Liu J. Liu K. Liu Q. Liu S. Liu Y. Liu Z. Liu A. Llewandowski O. Llopis I. Lo L. Locht A. Loke K. Lonngren T. Lopetegi N. Lopez U. Lott G. Lovat D. Lovelace Z. Low C. Lu L. Lu S. Lu Y. Lu V. Lubecke S. Lucyszyn D. Ludwig N. Luhmann M. Lui J. Luy G. Lyons A. M. Niknejad K. Ma Z. Ma S. Maas P. Maccarini G. Macchiarella J. Machac B. Machiels M. Madihian A. Madjar G. Magerl S. Magierowski R. Mahmoudi I. Maio F. Maiwald A. Majedi H. Majedi M. Majewski M. Makimoto R. Makinen D. Malocha J. Manges

R. Mansour D. Manstretta J. Mao S. Mao F. Maradei A. Margomenos D. Markovic E. Márquez-Segura J. Martens F. Martin E. Martini K. Maruhashi J. Marzo D. Masotti A. Massa G. Massa F. Mastri J. Mateu A. Matsushima M. Mattes G. Matthaei K. Mayaram M. Mayer U. Mayer W. Mayer J. Mazeau S. Mazumder A. Mazzanti G. Mazzarella K. McCarthy G. McDonald I. McGregor M. McKinley J. McLean D. McQuiddy A. Mediano F. Medina M. Megahed I. Mehdi K. Mehrany A. Melcon R. Melville F. Mena D. Mencarelli C. Meng R. Menozzi W. Menzel P. Mercier B. Merkl F. Mesa R. Metaxas A. Metzger P. Meyer P. Mezzanotte E. Michielsen A. Mickelson D. Miller P. Millot J. Mingo F. Miranda D. Mirshekar A. Mirzaei S. Mitilineos R. Miyamoto K. Mizuno J. Modelski W. Moer M. Moghaddam A. Mohammadi S. Mohammadi A. Mohammadian P. Mohseni E. Moldovan M. Mollazadeh M. Mongiardo P. Monteiro J. Montejo-Garai G. Montoro J. Monzó-Cabrera J. Morente T. Morf D. Morgan M. Morgan A. Morini A. Morris J. Morsey A. Mortazawi M. Moussa M. Mrozowski Q. Mu J.-E. Mueller J. Muldavin K. Murata S.-S. Myoung M. Myslinski B. Nabet V. Nair K. Naishadham Y. Nakasha M. Nakatsugawa M. Nakhla J.-C. Nallatamby I. Nam S. Nam J. Nanzer T. Narhi A. Nashashibi A. Natarajan J. Nath A. Navarrini J. Navarro J. Nebus R. Negra J. Neilson B. Nelson P. Nepa A. Neri H. Newman G. Ng D. Ngo E. Ngoya C. Nguyen E. Nicol A. Nicolet S. Nicolson E. Niehenke M. Nielsen K. Nikita P. Nikitin N. Nikolova M. Nisenoff K. Nishikawa T. Nishino

G. Niu B. Noori C. Nordquist B. Notaros K. Noujeim D. Novak I. Novak G. Nusinovich K. O I. Obeid J. Obregon R. O’Dea M. O’Droma M. Odyniec J.-E. Oh T. Ohira E. Öjefors H. Okazaki V. Okhmatovski A. Oki M. Okumura G. Olbrich S. Olson F. Olyslager A. Omar K. Onodera B.-L. Ooi S. Ootaka H. Oraizi G. Orengo A. Orlandi R. Orta J. Ortega-Gonzalez S. Ortiz S. Otaka B. Otis K. Ozdemir T. Ozdemir O. Ozlem P. Paco R. Paknys S. Pal Y. Palaskas D. Palmer S. Pamarti G.-W. Pan S.-K. Pan A. Panariello K. Pance J. Papapolymerou S. Parisi C.-S. Park E. Park J.-S. Park M.-J. Park S. Park W. Park A. Parker T. Parker D. Pasquet M. Pastorino H. Pau S. Paulotto A. Pavio D. Pavlidis W. Pearson J.-C. Pedro S. Peik S. Pellerano G. Pelosi M. Pelosi D. Pelz R. Pengelly J. Pereda F. Pereira A. Perennec B. Perlman D. Peroulis L. Perregrini K. Per-Simon M. Persson M. Petelin A. Peterson A. Petosa O. Peverini U. Pfeiffer A.-V. Pham J. Phillips H. Pickett M. Pieraccini L. Pierantoni B. Pillans S. Pinel Z. Ping M. Pirola S. Pisa G. Pisano D. Pissoort D. Plant C. Plett J. Plumridge C. Pobanz A. Poddar F. Podevin R. Pogorzelski G. Ponchak A. Poon D. Popovic Z. Popovic J. Portilla M. Pospieszalski A. Pothier K. Pourvoyeur J. Powell H. Powen R. Prabhu L. Pradell S. Prasad D. Prather A. Priou S. Pruvost Y. Qian R. Qiang J. Qiu T. Quach X. Quan R. Quay C. Queck C. Quendo R. Quéré F. Quesada F. Raab V. Radisic

M. Raffetto A. Raffo T. Rahkonen R. Raich A. Raisanen O. Ramahi M. Ramdani R. Ranson P. Rantakari L. Ranzani P. Ratajczak H. Rategh C. Rauscher J. Rautio T. Rautio B. Rawat J. Rayas-Sanchez G. Rebeiz J. Rebollar M. Reddy J. Reid R. Reid J. Reina-Tosina S. Reising B. Rembold K. Remley R. Renaut S. Rengarajan D. Resca P. Reynaert S. Reynolds A. Rezazadeh E. Rezek S. Ricci A. Riddle L. Rienzo D. Ritter E. Rius J. Rizk V. Rizzoli M. Roberg I. Robertson P. Roblin A. Roden C. Rodenbeck W. Rodriguez F. Rodriguez-Morales M. Rodwell A. Rofougaran R. Rogers H. Rogier U. Rohde V. Rokhlin Y. Rolain J.-M. Rollin R. Romanofsky S. Romisch G. Romo Y. Rong D. Rönnow D. Root N. Rorsman M. Rosario L. Roselli A. Rosen U. Rosenberg M. Rosker T. Roste F. Rotella E. Rothwell R. Rotman P. Rovati J. Roy L. Roy M. Roy T. Rozzi T. Rubaek J. Rubio D. Rudolph M. Rudolph A. Ruehli C. Ruppel A. Rydberg J. Ryynänen C. Saavedra F. Sabath K. Sachse B. Sadler N. Safari A. Safarian A. Safavi-Naeini A. Safwat P. Saha K. Saito I. Sakagami S. Sakhnenko T. Samaras J. Sambles C. Samori A. Sanada J. Sanchez S. Sancho K. Sano A. Santarelli H. Santos S. Sanyal K. Sarabandi T. Sarkar C. Sarris H. Sato P. Saunier M. Sawan H. Sayadian A. Sayeed W. Scanlon E. Schamiloglu J. Schellenberg M. Schindler E. Schlecht E. Schmidhammer L.-P. Schmidt S. Schmidt D. Schmitt F.-J. Schmueckle J. Schoebel D. Schreurs D. Schrijver A. Schuchinsky P. Schuh L. Schulwitz K. Schünemann J. Schutt-Aine

J. Scott F. Sechi K. Sellal V. Semenov E. Semouchkina K.-S. Seo J. Sercu A. Serebryannikov J. Sevic O. Sevimli F. Seyfert L. Shafai A. Shameli O. Shanaa Z. Shao I. Shapir A. Sharma S. Sharma J. Sharp D. Sheen T. Shen Z. Shen Y. Shestopalov J. Shi Y.-Q. Shi H. Shigematsu Y. Shih H. Shin S. Shin S.-H. Shin N. Shino W. Shiroma S. Shitov K. Shu D. Shyroki D. Sievenpiper C. Silva D. Silveira M. Silveirinha K. Silvonen W. Simbuerger G. Simin R. Simons C. Simovsky J. Simpson V. Simulik D. Simunic H. Singh D. Sinnott Z. Sipus C. Siviero H. Sjöland M. Slazar-Palma R. Sloan P. Smith C. Snowden R. V. Snyder M. Sobhy A. Sodagar N. Sokal K. Solbach J. Sombrin Y.-K. Song R. Sorrentino A. Soury E. Sovero J. Sowers R. Sperlich B. Spielman K. Stadius P. Staecker D. Staiculescu D. Stancil A. Stancu A. Stanitzki S. Stapleton J. Staudinger P. Stauffer B. Stec D. Steenson P. Steenson M. Steer G. Stegmayer J. Stenarson B. Stengel K. Stephan C. Stevens N. Stevens M. Steyaert J. Stiens I. Stievano S. Stitzer M. Straayer B. Strassner A. Street W. Struble M. Stubbs M. Stuchly B. Stupfel A. Suárez G. Subramanyam T. Sudo N. Suematsu T. Suetsugu C. Sullivan F. Sullivan A. Sulyman N. Sun S. Sun X. Sun R. Sutton K. Suzuki J. Svacina M. Swaminathan D. Swanson B. Szendrenyi W. Tabbara A. Taflove Y. Tajima T. Takagi M. Takahashi I. Takenaka T. Takenaka V. Talanov S. Talisa K.-W. Tam B. Tan E. Tan J. Tan T. Tanaka C.-W. Tang W.-C. Tang

X.-H. Tang T. Taris R. Tascone P. Tasker J. Taub J. Tauritz V. Tavares S. Taylor D. Teeter R. Temkin M. Tentzeris V. Teppati J.-P. Teyssier N. Thakor H. Thal J. Tham M. Thumm M. Tiebout E. Tiiliharju M.-R. Tofighi P. Tognolatti T. Toifl T. Tokumitsu A. Tombak A. Topa E. Topsakal H. Torres-Silva G. Town S. Tretyakov R. Trew P. Troyk C. Trueman A. Truitt C.-M. Tsai Z.-M. Tsai J. Tsalamengas C.-H. Tseng T. Tsiboukis J. Tsui M. Tsutsumi S. H.-L. Tu W.-H. Tu N. Tufillaro V. Turin G. Twomey C.-K. Tzuang T. Ueda V. Urick K. U-Yen N. Uzunoglu T. Vähä-Heikkilä R. Vahldieck A. Valdovinos G. Vandenbosch K. Vanhille D. Vanhoenacker-Janvier G. Vannini L. Vardapetyan G. Vasilescu C. Vaucher J. Vaz L. Vegni G. Vendelin S. Verdeyme M. Vérez A. Verma J. Verspecht P. Vial H.-O. Vickes A. Victor L. Vietzorreck C. Vittoria S. Vitusevich R. Voelker S. Voinigescu J. Volakis A. Vorst M. Vossiek M. Vouvakis B. Vowinkel L. Vreede K. Vryssas C. Wagner B. Waldmann P. Waldow A. Walker P. Wambacq S. Wane B.-Z. Wang C. Wang C.-F. Wang C.-J. Wang E. Wang F. Wang H. Wang J. Wang K.-C. Wang N. Wang X. Wang Y. Wang Y.-H. Wang Z.-G. Wang C. Ward J. Ward W. Wattanapanitch J. Webb D. Webster R. Webster S. Wedge J. Weem X. Wei D. Weide R. Weigel R. Weikle C. Weil T. Weiland D. Weile S. Weinreb M. Weiss S. Weiss T. Weller C. Wen G. Wen S. Wentworth D. Wentzloff R. Wenzel J. Whelehan J. Whitaker J. White J. Wiart M. Wickert

A. Wiesbauer J. Wight D. Willems B. Willemsen D. Williams A. Williamson J. Wilson J. Wiltse T. Winkel K. Wise D. Wisell M. Wolf E. Wollack G. Wollenberg F. Wong K. Wong M. Wong S. Wong K. Woo J. Wood G. Woods D. Woolard C. Wu J.-M. Wu K.-L. Wu K. Wu L. Wu R.-B. Wu T. Wu T.-L. Wu R. Wylde T. Wysocki M. Xia S. Xiang J. Xiao Y. Xiao C. Xie J. Xu S. Xu Q. Xue M. Yagoub T. Yakabe A. Yakovlev K. Yamamoto K. Yamauchi W. Yan C.-L. Yang F. Yang N. Yang X. Yang Y. Yang Z. Yang F. Yanovsky H.-W. Yao J. Yao A. Yarovoy Y. Yashchyshyn K. Yashiro K. Yasumoto J. Yau S. Ye J. Yeh K.-S. Yeo S.-P. Yeo K.-W. Yeom L.-K. Yeung W.-Y. Yin X.-S. Yin S. Yngvesson D. Yongsheng D. Yoo H.-J. Yoo J.-G. Yook E. Yoon J.-B. Yoon R. York S. Yoshikado A. Young B. Young D. Young P. Young W. Young H.-K. Yu M. Yu P. Yu R. Yu W. Yu Y. Yu M. Yuan M. Yuce S.-W. Yun F. Zabini J. Zaeytijd K. Zaki P. Zampardi J. Zapata L. Zappelli C. Zelley P. Zhai C. Zhang F. Zhang G. Zhang H. Zhang J. Zhang N. Zhang Q.-J. Zhang R. Zhang Y. Zhang A.-P. Zhao Y.-J. Zhao Y. Zhao Y. Zheng Q. Zhiguo H. Zhou A. Zhu L. Zhu N.-H. Zhu X. Zhu J. Zhuang H. Zirath