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Bilinear Transfonnations 235 accomplished easily on desktop computers; the interested reader is referred to Kajfez (1975). 7.1.2. Generalized Smith Chart. Using methods described by Churchill (1960, pp. 76-77), it is a straightforward matter to show that every bilinear transformation of the closed right-half Z plane onto a closed unit circle must have the form (7.15) where f3 is real and Re(Z,)>O. The last requirement is especially emphasized. The exponential term merely rotates the unit-circle image and will henceforth be dropped. The generalized Smith chart maps impedances according to Z-Z, Z-jX,-R, p=--.=., (7.16) Z+Z, Z-JX,+R, where Z=R+jX and Z,=R,+jX,. Clearly, (7.16) could be normalized to R, by division in the numerator and denominator in the fashion of the ordinary Smith chart relationship given previously in (6.22). The obvious remaining difference is the term jX,. A little thought shows that it may be combined into a new reactance component (X - Xc> instead of the usual X component. The generalized Smith chart then represents the ordinary chart with center Z, and constant reactance lines (X - X,). One practical application concerns power transfer from a complex source to a complex load, as discussed in Section 3.2.3. Thus (7.16) is exactly comparable to (3.46). Note that whether Z, appears in the numerator or in the denominator is a matter of arbitrary definition. It is convenient here to represent the chart center as Ze- It is also important to define the generalized reflection coefficient in admittance form, as follows: Y,-Y p= Y'+ Y . (7.17) , The generalized Smith chart no longer allows substitution of Z= I/Y in order to change from an impedance to an admittance basis. This does not change (7.16) into (7.17) unless Z,= I/Y, is real. Example 7.1. Consider a complex source connected directly to a complex load, as in Figure 3.3 in Section 3.2.3. Suppose that Z,= 25 - j50 ohms and ZL is defined as causing a 2 : I standing-wave ratio (SWR) on a 50-ohm transmission line. What is the range of power delivered to the load relative to the power available from the source? The solution will be obtained graphically here and analytically in Section 9.5. The procedure will be to select three or more impedance points on a 2: I SWR circle from a Smith chart normalized to 50 ohms. Then these points will be plotted on a generalized Smith chart normalized to the conjugate of the source impedance in accordance with the
236 Linear Amplijier Design Tools Table 7.4. Impedance Data for 2: 1 SWR Renormalized for Example 7.1 Z 2: I SWR Z wrt a 25 ohms Z wrt 25 + j50 ohms 100+jO 25+jO 42.5+j32.5 42.5-j32.5 "With respect to. 4+jO J +jO 1.70+j 1.30 1.70-j 1.30 4-j2 J -j2 1.70-jO.70 1.70-j3.30 generalized reflection coefficient in (3.46). Thus the reflection magnitude extreme values can be determined graphically and applied in (3.47) to find the answer. In this case, the complex normalizing impedance is Z, = 25 + j50 ohms. Four convenient 2: I SWR impedance points with respect to 50 ohms are shown in Table 7.4 unnormalized, normalized wrt 25 ohms, and normalized wrt 25 + j50 ohms. These points are plotted on the generalized Smith chart normalized to Z,=50+jO and Z,=25+j50 in Figure 7.2, and the required +' x = +0.5 +2 x = -0.5 -2 1.0 0.8 0.6 0.4 0.2 -1 I o 0.2 0.4 I I I 0.6 0.8 1.0 Figure 7.2. Generalized Smith chart with Zc=25+j50 ohms and a 2: I SWR circle with respect to 50 ohms.
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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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Page 202 and 203: ----------- and (6.34) can be writt
Page 204 and 205: Fano's Broodband-Matching Limitatio
Page 206 and 207: Fano's Broadband-Matching Limitatio
Page 208 and 209: ----------------- Fano's Broadband-
Page 214 and 215: Fono's Broadband·Matching Limitati
Page 216 and 217: Green's recursive element formula i
Page 218 and 219: f 9, Figure 6.24. 92
Page 220 and 221: Bandpass Network Transformations 20
Page 222 and 223: Bandpass Network Trans/onnotions 20
Page 224 and 225: 50 II 2.6548 16.594 Bandpass Networ
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Page 228 and 229: Pseudobandpass Matching Networks 21
Page 230 and 231: I InlPJ =In Pseudobandpass Matching
Page 232 and 233: Pseudobandpass Matching Networks 21
Page 234 and 235: ----- -- Carlin's Broodband-Matchin
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Page 238 and 239: Car/in's Broadband.Matching Method
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Page 242 and 243: Prohle"" 227 Third, an objective fu
Page 244 and 245: Problems 229 6.16. Program Equation
Page 246 and 247: Bilinear Tronsfonnatl'ons 231 I, ~-
Page 248 and 249: Bi/inear Transjormations 233 (w" w"
Page 252 and 253: Bilinear Transfonnations 237 circle
Page 254 and 255: Impedance Mapping 239 1 3 al~ ~a3 [
Page 256 and 257: Impedance Mapping 241 r-- Z, s" I,
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
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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
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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
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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
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494 Subject Index length. electrica
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