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Case 1 (row ~t (al N" 3 -1 '" 25 + 1- 1- '" 245 3 + 65 Y 35 +45 1252 +1 Ih) N = 4 z '" 25 + -.L -.L 1- = 1205 4 + 365 2 + 1 3s+45+5560s3 +85 '~ 2 4 T_3__T~ Case 2 (low pa55l (a) N 3 z-t "25 + 1- -.L '" 245 3 + 65 35+ 45 12s2 +1 (b) N = 4 111 1205 4 +365 2 +1 y"25+ 3S +45 +SS= 60s3 +85 Case 3 (high pa55) (a) N 3 --H 1 + 185 2 2 _ 1 1 1 y-l~ Y 1 = 25 + l73S + 1/45 = 25 + 245 3 ~3 ~J Ih) N = 4 ---H 1 1 1 ~l -t 385 2 + 1205 4 2 4 1 ,~ z = 2S + 1/35 + 1/45 + 1/55 '" 25 + 645 3 ~3 Il ~5 Case 4 (high pa5s) ta) N 3 z-l = ~5 + l/~S + 1/~5 '" 25 + 245 3 ' + 185 2 3 Z-l~ Il ~2 ~4 Ih) N = 4 1 1 1 1 1 + 385 2 + 1205 4 3 y= y~ 25 + 1/35 + 1/45 + 1/55 = 25 + 645 3 Figure 3.5. ~2 Some continued fraction expansions. Il ~4 ~1 53
L _ ~J 54 Some Tools and Examples ofFilter Synthesis be senseless. Run Program B3-5, answering "YES" to indicate that the first element is a series L, because (3.72) is considered an impedance function. However, note that the same programmed solution applies to the network in Figure 3.5 for Case 2, N = 4, if (3.72) is considered a two-port input admittance function (with a short-circuit load). In that case, an open circuit after the 5-farad inductor would be senseless. The reader is urged to run Program B3-5 for all the possible combinations shown in Figure 3.5. Program B3-5 can also be used to determine whether all roots of a polynomial (ao+a,s+'" +ans n ) are in the left-half ~ plane, Le., whether the polynomial is "Hurwitz." If any of the continued fraction expansion coefficients are negative or zero, or if the program fails with a Hdivide by zero" error, then the polynomial was not HHurwitz." The polynomials being tested in this way are not rational, but are just the sum of all terms in (3.75) or (3.76). 3.43. Finding LC Values From ABCD Polynomials. It should'be clear from the last section that two-port networks subjected to open- or short·circuit port conditions are relevant to the synthesis procedure. Equations using ABCD parameters to describe two-port networks were introduced in Section 3.3.2. Two more of the infinite set of such descriptions are now introduced, based on the port voltages and currents and terminal conditions. Consider the equations based on Figure 3.4: VI=I1Zl1+12Z12' (3.77) V,= I,z" + I,Z22' (3.78) These characterize any two-port network, lossless or not. It is important to understand what the coefficients mean. For example, z" is V,/I, when 1,=0, as seen from (3.78). 1,=0 says that the output port is terminated by an open circuit. These two equations are known as the open-circuit impedance parameters because both independent variables are the port currents. Look at z" another way: it is the output voltage into an open circuit when the input current is I ampere. A similar characterization is based on short-circuit terminal conditions where 1,=V'Yll+V'Y12' (3.79) 1,=V,Y21+V,Y22' (3.80) Now, for example, Y21 is the current entering port 2 in Figure 3.4, carried by a short-circuit load, for I volt applied across the input port. It is convenient to write the open- and short-circuit equation systems in matrix notation: V=ZI (3.81 ) I=YV (3.82) It is well known that matrix Y is the inverse of matrix Z and vice versa; doing this algebra provides relationships between z and y parameters.
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Wile,.· Interactenee ..A. .. 1 18
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Circuit Design U sing Personal Comp
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To my late parents, Tommy and Brown
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viii Preface computer. Excessive th
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Contents l. Introduction 1 2. Some
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Contetlts xiii 4.3.4. Transmission
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~----- ---- Contents xv 6.6. Pseudo
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-------------------------- Contents
Page 17 and 18: -- ------- 2 Introduction viewpoint
Page 19 and 20: 4 Introduction Chapters Four and Fi
Page 21 and 22: 6 Introduction the selectivity effe
Page 23 and 24: 8 Some Fundamenurl Numerical Method
Page 25 and 26: 10 Some Fundamental Numerical Metho
Page 28 and 29: --------------------- Romberg Integ
Page 30 and 31: and it is convenient to rearrange (
Page 32 and 33: Romberg Integranon 17 Truncation Er
Page 34 and 35: ---~-- - - -- ----- - -------------
Page 36 and 37: Polynomial Minimox Approximation of
Page 38 and 39: Polynomial Minimax Approximation of
Page 40 and 41: Rational Polynomial LSE Approximati
Page 42 and 43: ---~~--_._--- ------ ID(w)£(w)1 2
Page 44 and 45: Rational Polynomial LSE Approximati
Page 46 and 47: Problems 31 2.6. If a,Z+a2 w = -a'-
Page 48 and 49: sense by the sum of first-kind Cheb
Page 50 and 51: Complex Zeros of Complex Polynomial
Page 52 and 53: Complex Zeros ofComplex Polynomials
Page 54 and 55: which are equal to the polynomial C
Page 56 and 57: Complex Zeros of Complex Polynomial
Page 58 and 59: Table 3.3. Complex Zeros of Complex
Page 60 and 61: Polynomials From Complex Zeros and
Page 62 and 63: Polynomials From Complex Zeros and
Page 64 and 65: Polynomial Addition and Subtraction
Page 66 and 67: Continued Fraction Expansion 51 Not
Page 70 and 71: ---- ------------------------- Cont
Page 72 and 73: ----~------- Input ImpedafJce $ytrt
Page 74 and 75: Input Impedance Synthesis From Its
Page 76 and 77: ------- - ------------- ---- Input
Page 78 and 79: Long Division and Partial Fraction
Page 80 and 81: Problems 6S magnitude. The program
Page 82 and 83: - Problems 67 3.12. Given the resis
Page 84 and 85: .Chapter Four Ladder Network Analys
Page 86 and 87: - -- - -------- Recun;"" ludder Met
Page 88 and 89: Recursive Ladder Method 73 This alw
Page 90 and 91: --------------- Example a: 3,325,20
Page 92 and 93: Recursive Ladder Method 77 This las
Page 94 and 95: - - - -------------- Embedded T>vo-
Page 96 and 97: - - - - - --- - - - ---- Unifonn Tr
Page 98 and 99: --- - .._----- Unifonn T/'Qnsmissio
Page 100 and 101: --------------- Uniform Transmissio
Page 102 and 103: ------- --- -----------_. - ------
Page 104 and 105: Table 4.4. 6.d L1 6.dc, 6.d L2 6.d
Page 106 and 107: ------- - _. - - Input and Transfer
Page 108 and 109: Input and Transfer Network Response
Page 110 and 111: Input and Transfer Network Response
Page 112 and 113: ----------------------------- Time
Page 114 and 115: ----- - - h(-rJ h(2.6.Tl o I II ---
Page 116 and 117: -------- Sensitivities 101 the corr
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formula: Sensitivities 103 dZ""A,Z
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Sensitivities 105 obeying at least
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and its derivative with respect to
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Problems 109 branch immittance; the
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Problems 111 Solve by using (4.40)
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Chapter Five Gradient Optimization
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Gradient Optimization 115 Figure 5.
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-- ------------------ Quadratic Fon
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Quadratic Fonns and Ellipsoids 119
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Quadratic Forms and EUipsoids 121 F
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Quadratic Fonm and Ellipsoids 123 x
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Quadratic Forms and Ellipsoids 115
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Conjugate Gradient Search 127 choic
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-_.. ---------- Conjugate Gradient
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Conjugate Gradient Search 131 x, Fi
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-- -~--- Conjugate Gradient Search
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Conjugate Gradient Search 135 Hxl =
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Linear Search 137 the variable metr
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Then, the minimum function value in
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- ------- Linear Search 141 subtrac
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The F/etcher- Reeves Optimizer 143
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The netcher- Reeves Optimizer 145 c
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Table 5.4. Typical Output for the R
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The Fletcher- Reeves Optimizer 149
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Network Objective Functions 151 and
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-------- - - Network Objecti"" Func
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Network Objective Functions 155 Tab
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-- ------------ Constraints 157 tra
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- - - - - -------------- Constraint
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Table 5.7. Objective Subroutine lor
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-70. Constrainis 163 315-325). For
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Constraints 165 In fact, one applic
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--------~ - - ---------------------
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(b) Problems 169 Find the diagonal
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Impedance Matching 171 I, z, ---+__
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Narrow·Band L, T, and Pi Networks
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- - - --------------- + '" R, . " N
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NalTO...Band L, T, and Pi Networks
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Narrow-Bond L, T, and Pi Networks ]
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Narrow-BaruJ L, T, aruJ Pi Networks
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Loss/ess Unifonn Transmission Lines
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------- - -------------------------
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----------- and (6.34) can be writt
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Fano's Broodband-Matching Limitatio
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Fano's Broadband-Matching Limitatio
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----------------- Fano's Broadband-
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Fono's Broadband·Matching Limitati
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Green's recursive element formula i
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f 9, Figure 6.24. 92
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Bandpass Network Transformations 20
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Bandpass Network Trans/onnotions 20
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50 II 2.6548 16.594 Bandpass Networ
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BaruJpass Network TransJof1lUltions
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Pseudobandpass Matching Networks 21
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I InlPJ =In Pseudobandpass Matching
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Pseudobandpass Matching Networks 21
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----- -- Carlin's Broodband-Matchin
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---- -------------------- expressio
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Car/in's Broadband.Matching Method
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Carlin's Broadband-Matching Methad
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Prohle"" 227 Third, an objective fu
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Problems 229 6.16. Program Equation
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Bilinear Tronsfonnatl'ons 231 I, ~-
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Bi/inear Transjormations 233 (w" w"
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Bilinear Transfonnations 235 accomp
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Bilinear Transfonnations 237 circle
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Impedance Mapping 239 1 3 al~ ~a3 [
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Impedance Mapping 241 r-- Z, s" I,
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---------- Impedan£e Mapping 243 T
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Impedance Mapping 245 in Figure 7.7
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Two-Port Impedance and Power Models
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---------------- This may be put in
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----~-------~--- - - --------------
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-------- -- - -------------- and th
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Two-Port Impedance and Power Models
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Billlteral Scattering Stability and
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Bilateral Scattering Stability and
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Page 278 and 279:
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--- -------- Bilateral Scattering S
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Unilateral Scattering Gain 267 the
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Unilateral Scattering Gain 269 The
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Prohle"" 271 _merit is u=0.08; by (
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Chapter Eight Direct-Coupled Filter
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- -- ~-------~- -------------- Dire
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Prototype Network 277 and Wo is the
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Prototype Network 279 element value
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Prototype Network 281 8./.4. Protot
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Designing with Lande Inverters 283
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Designing with L and C Inverters 28
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Designilrg with Lande Inverters Sup
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Designing with L and C Inverters 28
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General Inverters, Resonators, and
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Genera/Inverters, Resonators, and E
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conductance of the Kth resonator is
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Table 8.2. General Inverters, Reson
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General Inveners, Resonators, and E
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Four Importunt Pussband Shapes 305
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Four Important Passband Shapes 307
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FOIl' Important Possband Shopes 309
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Page 328 and 329:
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Fou, Important Passband Shapes 317
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Four lmporlunt Pussband Shupes 319
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Comments on a Design Procedure 321
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-----~------------ A Complete Desig
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A Complete Design Example 329 DB3 =
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A Complete Design Example 331 R,,(l
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------- Problems 333 For Z=O+jX, fi
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------------- Chapter Nine Other Di
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, I II r L;.. I I . C, ~ ~ Wo ell :
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• e/2 e/2 Figure 9.3. A typical e
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Another trigonometric identity can
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o---iS~-----, I '--------;::==-----
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Introduction to Cauer Elliptic Filt
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Introduction to Cauer Elliptic Filt
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40 20 140 12 11 130 10 120 110 9 10
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00 Introduction to Cauer Elliptic F
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Doubly Termi/lllted Elliptic Filter
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Doubly TermillQted Elliptic Filters
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Doubly Terminated Elliptic Filters
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Doubly Terminated Elliptic FilteN 3
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Doubly TerrnilUlted Elliptic Filten
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Some Lumped.Element Transformations
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-lin '" 1 +jOn : c, 3L, L, I ) r So
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1 1CT (T Some Lumped.Element Tronsj
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Load Effects on Passive Networks 36
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Invulnerable Filters 377 9.6. Invul
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Invulnerable Filters 379 20 t lmin
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Invulnerable Filters 381 40 t L mil
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Problems. 383' corresponding number
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Problems 385 9.11. A passive networ
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Program AS-I. Swain's Snrface See E
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------- -------- - _. _. - Program
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----- _. - ------------- - Program
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- - - _._------------------ &61 TAN
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Program A6-4. Norton Transformation
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861 862 863 86' 865 866 B6? 868 869
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Program A7·2. Three-Port to Two·P
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15B .LBU Cue & 5to 3x3 Elements J5J
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e61 STOB 32 degrees 115 RCL9 a3 deg
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Program A7-4. Maximally Efficient G
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169 CHS Register Assignments: 17e e
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84e 841 842 843 844 845 846 847 848
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Program A8-2. Doubly Termiuated Min
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----- - - - _. - _.--------- - -- _
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-------_._--- ------- Program A9-1.
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Appendix B PET BASIC Programs Progr
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Program B2-2A. Gauss-Jordan Solutio
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- - -------------------------------
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Program B2-4B. Polynomial Minimax A
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Program B2-5. Levy's Matrix Coeffic
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----------- _. - - - Flowchart for
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Program B3-2. Polynomials From Comp
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--------- - - Program 83-4. Polynom
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Program 83-7. Partial Fraction Expa
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Flowchart for Ladder Analysis Progr
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Program B5-1. The Fletcher-Reeves O
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Program B5-2. L-Seetion Optimizatio
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Flowchart for Matching Program B6-1
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Program B6-3. Levy Matching to Resi
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---------- -- Program 86-5. Hilbert
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--~-- - ---------- Program B9-1. El
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---------- - -_.-------------------
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3260 TO~(TO+W)/(l-TO'N) 3270 B(l,=B
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Appendix D Linear Search Flowchart*
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Appendix E Defined Complex Constan
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Appendix F _ Doubly Terminated Mini
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Appendix G _ Direct-Coupled Filter
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461 (G.22) (G.23) (G.24) A,= (G.25)
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G.4.2. Resonator Asymptote Slopes D
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Appendix G 465 except for overcoupl
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----------_._-_.- ---- - - -_.. - A
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Appendix H _ Zverev's Tables ofEqui
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------ 6(0) 6(b) L _ (L,C,- L,C,)'
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L, L, c. " 12(a) C\=W C 2 =X L\=Y L
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Appendix H 475 Simplified Notations
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References 477 Box, M. J., D. Davie
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--------- Johnson, L. W" References
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References 481 Van Valkenburg, M. E
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_ 484 Author Index Noble, B.. 120.
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486 Subject Index second kind, 32.
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488 Subject Index proper, 62 quadra
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490 Subject Index equivalent: bandp
Page 506 and 507:
1 1 _ 492 Subject Index I I Reflect
Page 508 and 509:
494 Subject Index length. electrica
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