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----- - - h(-rJ h(2.6.Tl o I II ---t---- 1 t'" 2 .::>or -2D.T 1,1 Ihl o h(t - Tl o t'" 2 AT L T o 2M "I Idl I 11A". = Fol2 MI Y, I 1 1 1 I Yo o lei ( '" 2 tlr Figure 4.17. Graphical interpretation of the convolution integral (or an exponential impulse response. (a) Impulse response; (b) folding; (c) shifting; (d) driving function; (e) product to be integrated. [From Ley, 1970.] 99
100 £Odder Network AlUllysis multiplying, and integrating the proper functions. Figure 4.17a is the impulse response function shown with values for Tat 0, ilT, and 2ilT. It is folded by its negative argument in Figure 4.17b, then shifted by the amount 2ilT in Figure 4.17c. The unit-step excitation shown in Figure 4.17d (an arbitrary choice) multiplies the shifted function according to (4.74), with the resulting integrand in Figure 4.17e. This area is the output function Fo at time t=2il1, the shift interval. The second application of Simpson's rule is the convolution process illustrated in Figure 4.17. Integration of Figure 4.17e uses the three samples fo= h(2ilT)F;(0), f, =h(ilT)F;(ilT), (4.75) f 2 = h(O)F;(2 ilT), according to the integrand in (4.74). Then the integral estimate by (2.32) is F o (2ilT)"" ~T (f o +4f, +f 2 ). (4.76) This result can be compared to a general expression by Ley (1970): where Fo(kilT)= ~T (f o +4f,+2f 2 + .. · +2f o _ 2 +4f o _,+f o )' (4.77) f,=h(t-T)F;(T), t=kilT; k=2,4,6, ..., (4.78) 'f=n.6.'T; 0=0,1,2,... ,k. The algorithm calls for a choice of k, the even number of integration intervals, and letting n vary from 0 to k to obtain the output time response Fo(kilT). The reader is urged to write the algebraic expressions in (4.77) and (4.78) for k = 2 to confirm the (4.76) case shown in Figure 4.17e. Lines 480-620 in Program B4-2 accomplish this numerical convolution. The unit-step excitation is computed in lines 410-430 and provides an estimate of the exact step response g(t) = l-e-'. For engineering accuracy, about 100 frequency samples and limiting are required. (4.79) 40-dB band 4.6.3. Time Response Summary. Simpson's rule for numerical integration has been employed for both the Fourier and convolution integrals. The Fourier integral can be evaluated over a finite range for band-limited response functions. Furthermore, its integrand is the product of the system transfer function's real part and the cosine function when the system has a real impulse response that is zero in negative time. It does not take long to compute and save 100 frequency response samples for fairly complicated ladder networks. These are used to compute and save
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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
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-- ------- 2 Introduction viewpoint
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4 Introduction Chapters Four and Fi
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6 Introduction the selectivity effe
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8 Some Fundamenurl Numerical Method
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10 Some Fundamental Numerical Metho
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--------------------- Romberg Integ
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and it is convenient to rearrange (
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Romberg Integranon 17 Truncation Er
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---~-- - - -- ----- - -------------
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Polynomial Minimox Approximation of
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Polynomial Minimax Approximation of
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Rational Polynomial LSE Approximati
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---~~--_._--- ------ ID(w)£(w)1 2
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Rational Polynomial LSE Approximati
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Problems 31 2.6. If a,Z+a2 w = -a'-
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sense by the sum of first-kind Cheb
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Complex Zeros of Complex Polynomial
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Complex Zeros ofComplex Polynomials
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which are equal to the polynomial C
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Complex Zeros of Complex Polynomial
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Table 3.3. Complex Zeros of Complex
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Polynomials From Complex Zeros and
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Polynomials From Complex Zeros and
Page 64 and 65: Polynomial Addition and Subtraction
Page 66 and 67: Continued Fraction Expansion 51 Not
Page 68 and 69: Case 1 (row ~t (al N" 3 -1 '" 25 +
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 116 and 117: -------- Sensitivities 101 the corr
Page 118 and 119: formula: Sensitivities 103 dZ""A,Z
Page 120 and 121: Sensitivities 105 obeying at least
Page 122 and 123: and its derivative with respect to
Page 124 and 125: Problems 109 branch immittance; the
Page 126 and 127: Problems 111 Solve by using (4.40)
Page 128 and 129: Chapter Five Gradient Optimization
Page 130 and 131: Gradient Optimization 115 Figure 5.
Page 132 and 133: -- ------------------ Quadratic Fon
Page 134 and 135: Quadratic Fonns and Ellipsoids 119
Page 136 and 137: Quadratic Forms and EUipsoids 121 F
Page 138 and 139: Quadratic Fonm and Ellipsoids 123 x
Page 140 and 141: Quadratic Forms and Ellipsoids 115
Page 142 and 143: Conjugate Gradient Search 127 choic
Page 144 and 145: -_.. ---------- Conjugate Gradient
Page 146 and 147: Conjugate Gradient Search 131 x, Fi
Page 148 and 149: -- -~--- Conjugate Gradient Search
Page 150 and 151: Conjugate Gradient Search 135 Hxl =
Page 152 and 153: Linear Search 137 the variable metr
Page 154 and 155: Then, the minimum function value in
Page 156 and 157: - ------- Linear Search 141 subtrac
Page 158 and 159: The F/etcher- Reeves Optimizer 143
Page 160 and 161: The netcher- Reeves Optimizer 145 c
Page 162 and 163: 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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--- -------- 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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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
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1 1 _ 492 Subject Index I I Reflect
Page 508 and 509:
494 Subject Index length. electrica
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