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General Inverters, Resonators, and End Couplings 291 Inverter ,--------l 2 0__1 : ,-'''-1-+--1-'-/~o Yl1 1 Y21 Y21 I Yn 0-0_1_-;..1_1 I 1 L- ~ 1---;---1_.L-I~o Figure 8.13. A reciprocal short·circuit~parameter equivalent circuit when Y12 =Y21. The dashed lines enclose the inverter portion of the network. number. A sufficient condition for Y21 being an imaginary number is that the network be lossless. In that case, the maximum possible efficiency is unity, and (7.64) shows that stability factor K = 1. Then the definition of K in (7.57) reveals that K = I when gil = 0 = gZ2 and YZI =jB, thus proving the sufficient condition. Note that susceptances YII and Y" in Figure 8.13 become part of the synchronous reactance in their respective resonators, as seen from the particu. lar case in Figure 8.12b. The frequency behavior of the resulting resonators will be equated to the ideal LC resonator in Section 8.3.4. Example 8. 2 Iden tify the inverter in a 3D-degree length of a lO0-0hm lossless transmission line, and resonate each end to make a direct-coupled filter between the 50-ohm terminations. The ABCD parameters of a transmission line in (4.13)-(4.15) can be converted to Y parameters according to (3.86); YIl= -jYocotO, and Y21= +jYo/sinO. Therefore, inverter YoI2 =Yo/sinO and half of each resonator must be composed of admittance -jYocotO. Figure 8.14 shows the equivalent circuit in impedance terms. The equivalent of a shorted-stub transmission line at each end of the inverter must be resonated to make a direct-coupled filter. In this example, ZOIZ =50 ohms, and the shunt reactances associated with Yll and Y" are +j57.74 ohms. Capacitors on either end having the negative of the latter value are required, as shown in Figure 8.15. Programs A6-1 and A6-2 may be used to confirm the impedances noted in Figure 8.15. It can be shown that the voltages across the capacitors have a :_}t", ,--------------, I Z012 "Zosine I i 0 ", )\-----L...i-;1'--,t,,~< Figure 8.14. I at W o I L J An equivalent circuit for a lossless transmission line having Zo ohms and (J length.
292 r_'l° Direct-Coupled Filters ~ ,-,.OW )f--J----' w o ), then the trap will appear to be an inductance at tune frequency W o where the inverter design is accomplished. The susceptance of a parallel LC network resonant at W n is available from (8.1) and (8.2): B(w)=wnC w n --;; , ( W W n ) (8.28) and w n = l/-ILC. The trap susceptance at W=Wo is thus Wo Wn ) w n (,,)0 B(wO)=wnC( --- , (8.29) and the adjacent resonators must absorb this equivalent negative inductance, as explained in Section 8.2.4. A useful conversion of (8.28) is obtained by using the frequency term within the parentheses in (8.29) to both multiply and divide (8.28). After some algebraic manipulation, (8.28) can be restated as: B(w) = _ B(wo)~ [ 1- (w
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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
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Polynomial Addition and Subtraction
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Continued Fraction Expansion 51 Not
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Case 1 (row ~t (al N" 3 -1 '" 25 +
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---- ------------------------- Cont
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----~------- Input ImpedafJce $ytrt
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Input Impedance Synthesis From Its
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------- - ------------- ---- Input
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Long Division and Partial Fraction
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Problems 6S magnitude. The program
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- Problems 67 3.12. Given the resis
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.Chapter Four Ladder Network Analys
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- -- - -------- Recun;"" ludder Met
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Recursive Ladder Method 73 This alw
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--------------- Example a: 3,325,20
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Recursive Ladder Method 77 This las
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- - - -------------- Embedded T>vo-
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- - - - - --- - - - ---- Unifonn Tr
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--- - .._----- Unifonn T/'Qnsmissio
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--------------- Uniform Transmissio
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------- --- -----------_. - ------
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Table 4.4. 6.d L1 6.dc, 6.d L2 6.d
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------- - _. - - Input and Transfer
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Input and Transfer Network Response
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Input and Transfer Network Response
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----------------------------- Time
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----- - - h(-rJ h(2.6.Tl o I II ---
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-------- 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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Page 258 and 259: ---------- Impedan£e Mapping 243 T
Page 260 and 261: Impedance Mapping 245 in Figure 7.7
Page 262 and 263: Two-Port Impedance and Power Models
Page 264 and 265: ---------------- This may be put in
Page 266 and 267: ----~-------~--- - - --------------
Page 268 and 269: -------- -- - -------------- and th
Page 270 and 271: Two-Port Impedance and Power Models
Page 272 and 273: Billlteral Scattering Stability and
Page 274 and 275: Bilateral Scattering Stability and
Page 280 and 281: --- -------- Bilateral Scattering S
Page 282 and 283: Unilateral Scattering Gain 267 the
Page 284 and 285: Unilateral Scattering Gain 269 The
Page 286 and 287: Prohle"" 271 _merit is u=0.08; by (
Page 288 and 289: Chapter Eight Direct-Coupled Filter
Page 290 and 291: - -- ~-------~- -------------- Dire
Page 292 and 293: Prototype Network 277 and Wo is the
Page 294 and 295: Prototype Network 279 element value
Page 296 and 297: Prototype Network 281 8./.4. Protot
Page 298 and 299: Designing with Lande Inverters 283
Page 300 and 301: Designing with L and C Inverters 28
Page 302 and 303: Designilrg with Lande Inverters Sup
Page 304 and 305: Designing with L and C Inverters 28
Page 308 and 309: General Inverters, Resonators, and
Page 310 and 311: Genera/Inverters, Resonators, and E
Page 312 and 313: conductance of the Kth resonator is
Page 314 and 315: General Inverters, Resonators, and
Page 316 and 317: Table 8.2. General Inverters, Reson
Page 318 and 319: General Inveners, Resonators, and E
Page 320 and 321: Four Importunt Pussband Shapes 305
Page 322 and 323: Four Important Passband Shapes 307
Page 324 and 325: FOIl' Important Possband Shopes 309
Page 332 and 333: Fou, Important Passband Shapes 317
Page 334 and 335: Four lmporlunt Pussband Shupes 319
Page 336 and 337: Comments on a Design Procedure 321
Page 342 and 343: -----~------------ A Complete Desig
Page 344 and 345: A Complete Design Example 329 DB3 =
Page 346 and 347: A Complete Design Example 331 R,,(l
Page 348 and 349: ------- Problems 333 For Z=O+jX, fi
Page 350 and 351: ------------- Chapter Nine Other Di
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Page 354 and 355: • 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
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494 Subject Index length. electrica
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