Contents Part I Introduction Chapter l Computer-Aided Design MICROWA VE CIRCUITS AND of Microwave Circuits COMPUTER-AIDED DESIGN 3 1.1 EVOLUTION OF MICROWAVE CIRCUITS 3 1.2 COMPUTER-AIDED DESIGN APPROACH 8 1.3 OUTLINE OF THE BOOK 14 REFERENCES 19 Chapter 2 MICROWA VE NETWORK REPRESENTATION 25 2.1 ABCD PARAMETERS 26 2.2 SCATTERING PARAMETERS 30 2.2.1 Definition and Generai Propertìes 30 2.2.2 Relationship with Other Representations 33 K. C. Gupta (Author, Preface), 2.3 TRANSFER SCATTERING MATRIX' Ramesh Garg, REPRESENTATION 36 Rakesh Chadha (Preface) APPENDIX 2.1 ABCD, 8-AND T-MATRICES FOR SOME OF THE COMMONLY USED TWO-PORTS 39 REFERENCES 43 Part II .Modelling of Circuit Elements Chapter 3 CHARACTERIZATION OF TRANSMISSION STRUCTURES 47 3.1 COAXIALLINES 49 3.2 WAVEGUIDES 53 3.2.1 Rectangular Waveguides 53 3.2.2 Circular Waveguides 55 3.3 STRIPLINES 57 5.2 COAXIAL LINE DISCONTINUITIES 131 3.4 MICROSTRIP LINES 60 5.2.1 Capacitive Gaps in Coaxial Lines 131 3.5 SLOTLINES 65 5.2.2 Steps in Coaxial Lines 135 3.6 COPLANAR LINES 67 5.2.3 Capacitive Windows in Coaxial Lines 136 3.7 COUPLED STRIPLINES 72 5.2.4 T-Junction or Stub in Coaxial Lines 139 3.8 COUPLED MICROSTRIP LINES 76 5.3 RECTANGULAR WAVEGUIDE DISCONTINUITIES 139 REFERENCES 85 5.3.1 Posts in a W aveguide 139 5.3.2 Strips in Waveguides 149 Chapter-4 5.3.3 Diaphragms or Windows in Waveguides 152 5.3.4 Steps in Waveguides 159 SENSITIVITIES OF TRANSMISSION 5.3.5 Right-Angled Bends or Comers in Waveguides 166 STRUCTURES 91 5.3.6 T-junctions 169 4.1 INTRODUCTION 91 5.3.7 Circular and Elliptical Apertures in Waveguides 173 4.1.1 Definitions 91 4.1.2 Applications of Sensitivity Analysis 92 REFERENCES 177 4.1.3 Tolerance Analysis of Transmission Lines 93 4.2 COAXIAL LINES 95 Chapter 6 4.3 WAVEGUIDES 97 4.4 STRIPLINES AND MICROSTRIP LINES 100 CHARACTERIZATION OF 4.4.1 Striplines 100 DISCONTINUITIES -II 4.4.2 Microstrip Lines 102 Striplines and Microstrip Lines 179 4.5 SLOTLINES AND COPLANAR LINES 106 6.1 STRIPLINE DISCONTINUITIES 181 4.5.1 Slotlines 106 6.1.1 Open-End 183 4.5.2 Coplanar Lines 112 6.1.2 Round-Hole 184 6.1.3 Gap 185 4.6 COUPLED STRIPLINES AND 6.1.4 Step in Width 186 COUPLED MICROSTRIP LINES 117 6.1.5 Ben d 186 4.6.1 Coupled Striplines 117 6.1.6 T-junction 188 4.6.2 Coupled Microstrip Lines 120 6.2 MICROSTRIP DISCONTINUITIES 189 REFERENCES 126 6.2.1 Open-End 190 6.2.2 Gap 191 Chapter 5 6.2.3 Notch 192 6.2.4 Step in Width 192 CHARACTERIZATION OF 6.2.5 Right-angled Bend 195 DISCONTINUITIES-1 6.2.6 T-junction 195 Coaxial Lines and Waveguides 129 6.2.7 Cross-junction 197 5.1 INTRODUCTION 129 REFERENCES 200 Chapter 7 9.2 VARACTOR DIODES 267 9.3 PIN DIODES 268 LUMPED ELEMENTS IN 9.4 BIPOLAR TRANSISTORS AND MESFETS 269 MICROWA VE CIRCUITS 205 9.4.1 Bipolar Transistors 269 7.1 BASIC CONSIDERATIONS 205 9.4.2 MESFETs 277 7.2 DESIGN OF LUMPED ELEMENTS 206 9.5 GUNN AND IMPATT DIODES 285 7.2.1 Resistors and Inductors 207 9.5.1 Gunn Diodes 285 7.2.2 Capacitors 213 9.5.2 Impatt Diodes 289 7.3 MEASUREMENTSOFLUMPED REFERENCES 293 ELEMENTPARAMETERS 220 REFERENCES 226 Chapter 10 Chapter 8 MEASUREMENTTECHNIQUES FOR MODELLING 295 TWO-DIMENSIONAL PLANAR COMPONENTS 229 10.1 MICROWA VE NETWORK ANALYZERS 296 8.1 BASIC CONCEPTS 231 10.1.1 N etwork Analyzer 296 8.2 GREEN'S FUNCTION APPROACH 237 10.1.2 Automatic Network Analyzer 301 8.3 EVALUATION OF GREEN'S FUNCTIONS 242 10.1.3 Six-Port Network Analyzer 303 8.3.1 Method of Images 243 10.2 SYSTEM ERROR MEASUREMENT AND 8.3.2 Expansion of Green's Function in CORRECTION 307 Eigenfunctions 244 10.2.1 Generai Considerations 307 8.3.3 Green's Functions for V arious Configurations 247 10.2.2 One-port Device Measurements 311 8.4 SEGMENTATION AND DESEGMENTATION 251 10.2.3 Two-port Measurements 313 10.2.4 Three-port and Multiport Measurements 317 8.5 NUMERICAL METHODS FOR 10.2.5 Characterization of Connectors 319 ARBITRARY SHAPES 253 10.3 DATA REDUCTION TECHNIQUES 321 8.6 SCALING FOR PLANAR CIRCUITS 256 REFERENCES 323 8.6.1 Frequency Scaling 257 8.6.2 Impedance Scaling 257 8.6.3 Designs of Stripline Type Circuits Part III Analysis from a Microstrip Type Planar Circuit (and Vice Versa) 257 REFERENCES 259 Chapter 11 EVALUATION OF CIRCUIT PERFORMANCE 329 Chapter 9 11.1 CIRCUITS CONSISTING OF TWO-PORTS 330 MODELS FOR MICROWAV E 11.1.1 U se of Synimetry in the Circuit Analysis 330 SEMICONDUCTOR DEVICES 263 11.1.2 Analysis of Cascaded Two-Ports 333 11.1.3 Analysis of Arbitrarily Connected Two-Ports 337 9.1 SCHOTTKY-BARRIER AND POINT-CONTACT DIODES 264 11.2 ARBITRARILY CONNECTED NETWORKS 338 Chapter 13 11.2.1 Analysis Using Connection Scattering Matrix 338 TOLERANCE ANAL YSIS 407 11.2.2 Multiport Connection Method 341 13.1 WORST-CASE ANALYSIS 408 11.2.3 An Example 343 13.2 STATISTICALTOLERANCE ANALYSIS 423 11.2.4 Analysis by Subnetwork Growth Method 347 13.2.1 Method ofMoments 423 11.3 CIRCUITS CONSISTING OF TWO 13.2.2 Monte-Carlo Analysis 426 DIMENSIONAL PLANAR COMPONENTS 353 APPENDIX 13.1 SOME RESULTS 11.3.1 Segmentation Method 353 FROM PROBABILITY THEORY 11.3.2 Desegmentation Method 359 AND STATISTICS 427 REFERENCES 369 REFERENCES 430 Chapter 12 Chapter 14 SENSITIVITY ANALY SIS OF TIME DOMAIN ANALY SIS OF MICROWA VE CIRCUITS 371 MICROWA VE CIRCUITS 433 14.1 TRANSIENT ANALYSIS OF 12.1 FINITE DIFFERENCE METHOD 373 TRANSMISSION LINES 434 12.3 ADJOINT NETWORK METHOD 373 14.2 LAPLACE TRANSFORM METHOD 441 12.2.1 Tellegen's Theorem in Wa ve Variables 374 14.3 COMPANION M O DEL APPROACH 450 12.2.2 The Adjoint Network 376 14.4 STATE VARIABLE APPROACH 456 12.2.3 Comparison with the Direct Method 380 REFERENCES 460 12.2.4 Evaluation of Gradients for Subnetwork Growth Method 382 12.3 EVALUATION OF DIFFERENTIAL Chapter 15 SCA'ITERING MATRICES 388 MATRIX SOLUTION TECHNIQUES 463 12.3.1 Sensitivity Invariants for Scattering Matrices 388 15.1 GAUSSIAN ELIMINATION 464 12.3.2 Differential S-Matrices for Typical 15.2 PIVOTING 469 Components 392 15.3 L-U FACTORIZATION AND 12.4 AN EXAMPLE OF EVALUATION F -B SUBSTITUTION 470 OF SENSITIVITIES 393 15.3.1 L-U Decomposition 471 12.5 LARGE CHANGE SENSITIVITIES 397 15.3.2 Forward Elimination and Back Substitution 475 APPENDIX 12.1 DIFFERENTIAL -· 15.4 SPARSE MATRIX TECHNIQUES 478 SCA'ITERING MATRICES .. · • · 15.4.1 Reordering of E q uations 479 FOR SOME TYPICAL COMPONENTS 402 15.4.2 Data Structures for Reordering 486 REFERENCES 401 15.4.3 L-U Factorization and F-B Substitution 491 15.4.4 Remarks on Sparse Matrix Techniques 498 REFERENCES 499 Part IV Optimization 18.4 OPTIMIZATION OF LEAST SQUARE OBJECTIVE FUNCTIONS 572 REFERENCES 575 Chapter 16 INTRODUCTION TO OPTIMIZATION 505 Part V CAD Programs 16.1 BASIC CONCEPTS AND DEFINITIONS 507 16.2 OBJECTIVE FUNCTIONS FOR Chapter 19 CIRCUIT OPTIMIZATION 515 16.2.1 Generai Considerations 515 A MICROWA VE CIRCUIT ANALY SIS 16.2.2 Leastpth Approxirnation 516 PROGRAM (MCAP) 579 16.2.3 Minimax Approximation 518 19.1 PROGRAM DESCRIPTION 581 16.3 CONSTRAINTS 520 19.1.1 Flow Chart 581 16.3.1 Transformation of Constraints 520 19.1.2 Description of Subroutines 581 16.3.2 Penalty for·Constraint Violation 522 19.1.3 An Example 583 16.3.3 Sequential Unconstrained Minimization 19.2 INSTRUCTIONS FOR USERS 586 Technique 522 19.3 PROGRAM LISTING 594 16.4 ONE-DIMENSIONAL OPTIMIZATION REFERENCES 617 TECHNIQUES 523 16.4.1 Elimination Methods 524 Chapter 20 16.4.2 Interpolation Methods 530 REFERENCES 540 CAD PROGRAMS FOR MICROWA VE CIRCUITS 619 (contributed by Les Besser, Compact Engineering I ne., USA) Chapter 17 20.1 INTRODUCTION 619 20.2 INTEGRATED DESIGN AND DIRECT SEARCH OPTIMIZATION METHODS 543 MANUFA CTURING SYSTEM CONCEPT 620 17.1 PATIERN SEARCH METHODS 543 20.3 SUMMARY OF MICROWAVE CAD PROGRAMS 625 17.1.1 Hooke and Jeeves Methods 544 20.3.1 HANDY-COMPACT Circuit Analysis on 17.1.2 Powell's Method 546 HP-41C Handheld Calculator 626 17.1.3 Razor Search Method 548 20.3.2 MICRO-COMPACT Circuit Optimization on 17.2 ROTATING COORDINATES METHOD 550 HP-9845 BIT Desktop Computer 628 17.3 THE SIMPLEX METHOD 553 20.3.3 SUPER-COMPACT 632 20.3.4 Lumped Element Matching Synthesis with REFERENCES 561 AMPSYN 637 ; "f • . . .-.. , : ,. ·1 .. , 20.3.5 Transmission Line Matching N etwork Chapter 18 :~ Synthesis with CADSYN 639 •. 20.3.6 FILSYN 643 GRADIENT METHODS FOR OPTIMIZATION 563 REFERENCES 648 18.1 STEEPEST DESCENT METHOD . 563 18.2 GENERALIZED NEWTON-RAPHSON METHOD 566 IN DEX 651 , o" n A uTnnN.PT .RTCHER-POWELL METHOD 567 2 Microwave Network Representation A generai microwave circuit is a multiport network, as shown in Figure 2.1, which consists of severa! components connected by sections of transmission lines (or waveguides ). These networks may be characterized in terms of voltages and currents at various ports. This procedure is followed at lower frequencies and an impedance matrix, or an admittance matrix or a hybrid matrix representation is used [l]. At microwave frequencies, voltages an d currents are replaced by normalized wave variables, and the resulting scattering matrix formulation [ 2-5] is commonly used. A large variety of components used at microwave frequencies are two-port components with a single input port and a single output port. Severa! microwave circuits may be expressed as a cascaded combination of such two-port components. Analysis of these cir cuits becomes very convenient if the individuai two-ports are char acterized in terms Òf ABCD parameters [ 3]. The scattering matrix formulation is not very convenient for the analysis of cascaded two--· ports. However, there is another representation that uses wave vari ables and allows cascaded networks to be analyzed easÙy. -This is known as transfer scattering matrix formulation. The three types of representations mentioned above are reviewed in this èhapter. 26 COMPUTER-AIDED DESIGN Microwave Network Representation 27 -----------1;N ' ,~ " ' / " \ l \ / A GENERAL 1 (a) cr--{ l 1 1 NETWORK L---o o-----' / ..-----. 12 a i 1b .-----. \ ' 1-----<) ~-o--l ---~ / a b ----<>---L__j----< ',:r:r---------~~ (b) 2 Figure 2.2 (a) A two-port network: (b) A cascaded chain of two-port networks. Figure 2.1 A generai N-port network. (2.1) In addition to these three representations, conventional impedance and admittance matrices also find specific applications in microwave Note that i is shown to flow outward and becomes i of the next circuit design. In the two-dimensional microwave planar circuits 2 1 two-port network in a case ad ed c hai n as shown in Figure 2.2(b ). discussed in Chapter 8, the third dimension of the components Thus, if there are two networks "a" and "b" cascaded as shown in is assumed to be much smaller than the wavelength, and a volt- Figure 2. 3, w e h ave age could be conveniently defined in terms of the E-field along this direction. An impedance Green's function which defines volt - - age at any point in terms of the current at some other point is used to find the Z-matrix. The Y-matrix formulation is very con --+ venient when networks or some of their ports are connected in .~..,- - Aa Ba + ~ + Ab Bb ~ v2a v1b parallel. These matrices have also been used in time domain analy ;;.,._ - sis discussed in Chapter 14. Relationships for transformations Ca Da ~ cb Db ~ from Z- and Y-matrices to scattering matrix and vice-versa are de- scribed in Section 2.2. : - ,, ,._ j • :; a b 2.1 ABCD P ARAMETERS ABCD parameters for a two-port network, such as shown in Figure Figure 2.3 Cascade connection of 2 two-port networks. 2.2(a), àre defined by " · · .-; 28 COMPUTER-AIDED DESIGN Microwave Network Representation 29 j 1 =[ Ba][~''] The driving point impedance at the input port is given by v Aa (2.2) Z in = (A ZL + B) l (C ZL + D) (2.6) [ 11a Ca Da 2a where the load impedance ZL is.equal to v 1i . The driving point 2 2 impedance at the output port may be written as and also , Zout = (D Z + B) l (C Z + A) (2.7) 5 5 v'l =[ v'j =[Ab Bb]lv2b] where the source impedance Z is equal to (E v ) l i . The [ (2.3) 5 5 - 1 1 voltage transfer coefficient (from the source to the output port) ~2b 12a 1lb Cb Db becomes Combining (2.2) and (2.3), 'l (2.8) 1 =[ ~Ab The voltage reflection coefficient at the input port is given by v [v ] Aa Ba] Bj r . = (Z. - Z *) l (Z. + Z) (2.9) [ m m s m s 11a Ca Da lcb Db 12b whereas the voltage reflection coefficient at the output port may be written as =[A B]['b] * f out = (Zout - ZL ) l (Zout + ZL) · (2.10) (2.4) The ABCD matrices far some typical microwave circuit elements C D t 2b are listed in the Appendix 2.1 at the end of this chapter. These where [ABCD] in (2.4) is the product of two individuai ABCD parameters have been derived by applying Kirchhoff's laws andlor matrices. Generalizing this analysis for the case of N two-ports cas transmission line equations to the various networks. caded together, we get far the resultant matrix ABCD matrices exhibit the following characteristics: [: :}C J l: :J[: i) Far reciproca! networks, :J AD - BC = l . (2.5) ii) Far symmetrical networks (which remain unaltered when the l 2 N two ports are interchanged, we have A = D. Thus, a long chain of network elements may be analyzed simply Normalized ABCD Matrix by multiplying their individuai ABCD matrices to obtain the aver ABCD parameters as defined above may be normalized by dividing ali ABCD matrix. This representation provides an efficient way the parameter B and multiplying the parameter C by a reference im far the analysis of networks that can be represented as cascaded pedance Z . We may write the normalized ABCD matrix as two-ports. The analysis of cascaded two-ports is discussed in detail o in Chapter 11. ·· Various performance functions far a network may be obtained from (2.11) its ABCD matrix. Referring to Figure 2.2(a) we may derive the fol lowing relations. _ ..... 30 COMPUTER-AIDED DESIGN Microwave Network Representation 31 guide) connected to the nth port and Zon is the characteristic im It may be noted that in (2.11) all the four parameters are dimen sionless. Quite often, the normalizing impedance Z is the charac- pedance of the line (or waveguide). Knowledge of v n+ and vn - is o teristic impedance of the transmission lines connected to the two not required to evaluate coefficients of the scattering matrix. Re ports of the network. For example, consider a transmission line lationships between bn 'sand an 's fora two-port network may be section of length Q and an impedance Z interposed between two written as lines of impedance Z as shown in Figure 2.4. The normalized = (2.15) 0 hl sll al + sl2a2 ' ABCD matrix for this component may be written as an d z cos~Q zo sin~ Q hz = S 21 al + S 22a2 · (2.16) In generai, for an n-port network, we have [ABCD]n (2.12) b = S a (2.17) zo Z sin~Q cos~Q where S is an n x n matrix (2 x 2 for a two-port network). The ma trix S is known as the scattering matrix of the network. Normalized ABCD matrices for severa'! eiements in cascade can be The average power flowing into the port n may be evaiuated by muitipiied together to obtain the overall normalized ABCD matrix ' using (2.13) and (2.14). For this purpose, v n and in may be written provided the reference impedance Z is the same for all the eiements. 0 as (from transmission line theory.) An 2.2 SCATTERING P ARAMETERS vn vn + + vn = (an + b n ) ' (2.18) an d 2.2.1 Definition and Generai Properties l l The use of ABCD parameters at microwave frequencies is not very l n (v n + - v n- ) = VZ:::, (a n - b n ) . (2.19) convenient from the measurements point of view. Also, its main zon o n advantage of cascading network components does not hoid when Power flowing into the port n is given by the network consists of components with three or more ports and aw hmeonr et hgee ntoepraoii omgeyt hiso dd iofffe rreenptr.e sSenctaitntger minigc rmowatarvixe fnoertmwuoiraktsi.o n is Wn = -l2 Re(v nn1. * ) = -2l (an an * - b nbn * ) . (2.20) A scattering matrix represents the reiationship between variabies an (proportionai to the incoming wave at nth port) and variables This shows that Wn is eqau:a) l to the power carried into the port n by bn (proportional to the outgoing wave at nth port) defined in the the incident w ave (lh an less the power reflected back (1,/:z bn b: ). following manner Advantages of S-matrix Representation One of the advantages of S-matrices is the ease of measurement. The (2.13) ABCD parameters discussed earlier are nota convenient setto mea sure at microwave frequencies. At lower frequencies the measure (2.14) ments are carried out by using open circuit (or short circuit) termi where vn + and vn - represent voltages corresponding to the incom nations atone of the ports and measuring voltage and/or current ing an d the outgoing waves in the transmission line (or the wave- at the other port. At microwave frequencies an ideai open circuit 32 COMPUTER-AIDED DESIGN Microwave Network Representation 33 is extremely difficult to realize, and it is equally difficult to estab iii) Again for lossless passive networks, the power conservation lish the exact location of a short circuit. The S-parameter measure condition yields an orthogonality constraint given by ments, on the other hand, are carried out by terminating one or the other port with the normalizing impedance Z (usually 50 ohms). LN Quantities like the reflection coefficient and the transmission coef sns sn; = o (forali s, r = l, 2, . . . N, s =!= r) . (2.23) ficient, commonly measured at microwave frequencies, can be ex n=l pressed directly in terms òf the scattering parameters. This equation states that the inner product of any column of Another important advantage of the S-matrix approach emerges the scattering matrix with the complex conjugate of any other from the fact that the S-parameters are defined on the basis of column is zero. traveling waves and, unlike terminai currents or voltages, the wave The conditions (2.22) and (2.23) restrict the number of independ variables do not vary in magnitude along a lossless line. This allows x ent elements in an N N scattering matrix t o N(N + l )/2. A ma the S-parameters for an unknown device to be measured relatively trix that satisfies these two conditions is called a unitary matrix. far away from its physicallocation. The effect of the intermediate In terms of matrix operations these conditions imply length of the low loss line may be eliminated by modifying the phase of the measured data. This is equivalent to the shifting of (2.24) the reference planes. On the other han d, if the ABCD (or Z- or Y-) parameters were used, they would vary with distance not The above equation states that in the case of a lossless passive net only in phase but in magnitude also. work, the complex conjugate of the scattering matrix is equal to the inverse of its transpose. Some of the important characteristics of S-matrices are listed belo w: Examples of Scattering Matrices Examples of S-matrices for some of the most commonly used two port components are listed in Appendix 2.1 along with ABCD and i) For a reciprÒcal network the S-matrix is symmetrical, i.e. T-parameters for thes'e components. The S-matrix for a lossless s = st (2.21) transmission line (given in row l of Appendix 2.1) may be used to find the modified S-matrix far a network when the reference piane where the subscript t indicates the transpose of a matrix. for any of its ports is shifted down the transmission line (or the waveguide) connected to that port. If an additionalline length Q ii) For a lossless passive network is connected to the jth port, it can be shown that each of the coef N N ficients S.. or S .. (i. e. the matrix elements involving j) will be mul L l s .12 L s . s .* l (2.22) tipli.e d b;\h. e fa~tor e -i(3Q w h ile the coefficient Sn.. will be multi- m n1 m plied by e-J2f3Q • n=l n= l for ali i l, 2, ...N 2.2.2 Relationship with Other Representations , . ,. ..,.' •I..e1.Ù, ,t; he m. ~n·e~r: .p r·o' d•u ct of .a. n_y. ·c:o.lu_m~tn. 'o·f 'th e s\.ct a,t~t')e'ri.nNg m1 .a.. t~r-i.. x-·• '.·; ., ... Relationship Betiveè~' S-matrix and ABC;D matrix .. with its own conjugate equals unity. Equation (2.22) is a di-. As mentioned earlier S-parameters are no,l t .s.. u. itable- for analyzing cas-~. rect consequence of the power conservation property of a caded networks and therefore, for this application, their conversion lossle,ss pas_sive network. .. .,. ,, · · .. i. -