Book Contents
Circuit Design
Using Personal Computers
1. Introduction, 1
2. Some Fundamental Numerical Methods
2.1. Complex Four Functions, 7
2.2. Linear Systems of Equations, 8
2.2.1. The Gauss-Jordan Elimination
Method, 9
2.2.2. Linear Equations With Complex
Coefficients, 11
2.2.3. Linear Equations Summary, 12
2.3. Romberg Integration, 13
2.3.1 Trapezoidal Integration, 14
2.3.2. Repeated Linear Interpolation
and the Limit, 15
2.3.3. Romberg Integration Program,
17
2.3.4. Simpson's Integration Rule, 18
2.3.5. Summary of Integration, 18
2.4. Polynomial Minimax Approximation of Piecewise Linear
Functions, 19
2.4.1. Chebyshev Functions of the
First Kind, 19
2.4.2. Chebyshev Expansions, 20
2.4.3. Expansion Coefficients for
Piecewise Linear Functions, 21
2.4.4. A Minimax Approximation
Program, 22
2.4.5. Piecewise Linear Function
Approximation Summary, 24
2.5. Rational Polynomial LSE Approximation of Complex
Functions, 24
2.5.1. The Basis of Levy's Complex
Curve-Fitting Method, 26
2.5.2. Complex Curve-Fitting
Procedure, 27
2.5.3. Summary of Complex
Curve-Fitting by Rational Polynomials, 30
Problems, 30
3. Some Tools and Examples of Filter Synthesis, 34
3.1. Complex Zeros of Complex Polynomials, 34
3.1.1. Moore's Root Finder, 36
3. I.2. Synthetic Division, 37
3.1.3. Efficient Evaluation of a
Polynomial and Its Derivatives, 39
3. I.4. Root-Finder Program, 40
3.1.5. Polynomial Scaling, 42
3. I.6 Root-Finder Summary, 44
3.2. Polynomials From Complex Zeros and Products, 44
3.2.1. Polynomials From Complex
Zeros, 44
3.2.2. Polynomials From Products of
Polynomials, 45
3.2.3. Power Transfer, 46
3.2.4. Network Synthesis Polynomials,
47
3.2.5. Summary of Polynomials From
Zeros and Products, 49
3.3. Polynomial Addition and Subtraction of Parts, 49
3.3.1. Program for Addition and
Subtraction of Parts, 49
3.3.2. The ABCD Matrix of Rational
Polynomials, 50
3.3.3. Summary of Polynomial Addition
and Subtraction of Parts, 51
3.4. Continued Fraction Expansion, 51
3.4.1. Lowpass and Highpass
Expansions, 52
3.4.2. A Continued Fraction Expansion
Program, 52
3.4.3. Finding LC Values From ABCD
Polynomials, 54
3.4.4. Comments on Continued Fraction
Expansion, 56
3.5. Input Impedance Synthesis From its Real Part, 57
3.5.1. Synthesis Problem Statement,
58
3.5.2. Gewertz Procedure to Find RLC
Input Impedance, 58
3.5.3. Reactance Functions From
Impedance Functions, 60
3.5.4. Impedance Real-Part Synthesis
Summary, 61
3.6. Long Division and Partial Fraction Expansion, 62
3.6.1. Long Division, 62
3.6.2. A Partial Fraction Expansion
Program, 63
3.6.3. Summary of Partial Fraction
Expansion, 64
Problems, 65
4. Ladder Network Analysis 69
4.1. Recursive Ladder Method, 70
4.1.1. Ladder Nomenclature, 70
4.1.2. Complex Linear Update, 71
4.1.3. An Elementary Topology Code,
72
4.1.4. Ladder Analysis Program, 73
4.1.5. Branch Topology Levels and
Packing, 74
4.1.6. Recursive Ladder Analysis
Summary, 77
4.2. Embedded Two-Port Networks, 78
4.2.1. Some Chain Parameter
Properties, 78
4.2.2. Chain Parameters in Complex
Linear Updates, 80
4.2.3. Summary of Embedded Two-Port
Networks, 81
4.3. Uniform Transmission Lines, 81
4.3.1. Transmission Line ABCD
Parameters, 82
4.3.2. Lossy Transmission Line Stubs,
82
4.3.3. Lossy Transmission Lines in
Cascade, 83
4.3.4. Transmission Line Topology
Codes, 83
4.3.5. Transmission Line Summary, 85
4.4. Nonadjacent Node Bridging, 86
4.4.1. Derivation of Bridged-T Chain
Parameters, 86
4.4.2. A Group Delay Equalizer, 88
4.4.3. Interpolation of Nonadjacent
Node-Bridging Current, 89
4.4.4. Summary of Nonadjacent Node
Bridging, 91
4.5. Input and Transfer Network Responses, 91
4.5.1. Impedance and Power Response
Functions, 91
4.5.2. Scattering Parameters, 92
4.5.3. Wave Response Functions, 94
4.5.4. Conclusion to Network
Responses, 96
4.6. Time Response From Frequency
Response, 97
4.6.1. Real, Causal Fourier
Integrals, 98
4.6.2. Numerical Convolution of Time
Functions, 98
4.6.3. Time Response Summary, 100
4.7. Sensitivities, 101
4.7.1. Sensitivity Relationships, IO1
4.7.2. Approximate Sensitivity, 103
4.7.3. Exact Partial Derivatives by
Tellegen's Theorem, I04
4.7.4. Summary of Sensitivities, 109
Problems, 109
5. Gradient Optimization, 113
5.1. Quadratic Forms and Ellipsoids, I16
5.1.1. Quadratic Functions, 118
5.1.2. Gradients and Minima, I I9
5.1.3. Quadratic Forms and Graphics,
I21
5.1.4. Taylor Series, 123
5. I.5. Newton's Method, 125
5.1.6. Summary of Quadratic Forms and
Ellipsoids, 125
5.2. Conjugate Gradient Search, 126
5.2.1. Linear Search, 127
5.2.2. Elementary Search Schemes, 129
5.2.3. More Quadratic Function
Properties, I3 I
5.2.4. Fletcher-Reeves Conjugate
Gradient Search Directions, 134
5.2.5. Summary of Conjugate Gradient
Search, 136
5.3. Linear Search, 137
5.3.1. Slope in the Linear Search
Direction, 138
5.3.2. Finding the Order of Magnitude
of the First Step, 138
5.3.3. Extrapolation, Bounding, and
Interpolation, 139
5.3.4. Fletcher's Linear Search
Strategy, 141
5.3.5. Summary of Linear Searches,
143
5.4. The Fletcher-Reeves Optimizer, 143
5.4.1. Summary of Fletcher-Reeves
Strategy, I44
5.4.2. The BASIC Language Computer
Program, 145
5.4.3. The Rosenbrock Example, 145
5.4.4. Scaling, 149
5.4.5. Summary of the Fletcher-Reeves
Program, 149
5.5. Network Objective Functions, 150
5.5.1. Integral Error Functions, 150
5.5.2. Discrete Objective Functions,
151
5.5.3. Objective Function Gradient,
153
5.5.4. L-Section Optimization
Example, 153
5.5.5. Summary of Network Objective
Functions and Optimization, 155
5.6. Constraints, 157
5.6.1. Simple Constraints, 159
5.6.2. Barrier Functions for
Inequality Constraints, 162
5.6.3. Penalty Functions for Equality
Constraints, 164
5.6.4. Mixed Compound Function for
All Constraints, 165
5.6.5. Summary of Constraints, 166
5.7. Some Final Comments on Optimization, 166
Problems, 167
6. Impedance Matching,170
6.1. Narrow-band L, T, and Pi Networks, 172
6.1.1. Lossless Network Interface
Impedances, 173
6.1.2. Real Source and Real Load, 174
6.1.3. Series-Parallel Impedance
Conversions, 176
6.1.4. Complex Sources and/or Complex
Loads, 178
6.1.5. Graphic Methods, 180
6.1.6. Summary of L, T, and Pi
Matching, 182
6.2. Lossless Uniform Transmission Lines, 182
6.2.1. Input Impedance and Reflection
Coefficient, 183
6.2.2. Complex Sources and Complex
Loads, 185
6.2.3. Real Sources and Complex
Loads, 186
6.2.4. Real-to-Real Transmission Line
Matches, 187
6.2.5. Summary of Transmission Line
Matching, 189
6.3. Fano's Broadband-Matching Limitations, 189
6.3.1. Fano's Gain-Bandwidth-Integral
Limitations, 191
6.3.2. A Chebyshev Approximation of
the Ideal Response, 194
6.3.3. Optimally Matching a
Single-Reactance Load, 195
6.3.4. Summary of Fano's
Broadband-Matching Limitations, 198
6.4. Network Elements for Three Source Conditions, 200
6.4.1. Resistive Source Optimally
Matched to a Single-Reactance Load, 200
6.4.2. Complex Source and Complex
Load, 201
6.4.3. Reactive Source and Complex
Load, 203
6.4.4. Summary of Broadband Matching
Under Three Source Conditions, 204
6.5. Bandpass Network Transformations, 205
6.5.1. Lowpass-to-Bandpass
Transformations, 205
6.5.2. Frequency and Impedance
Scaling, 208
6.5.3. Norton Transformations, 209
6.5.4. Summary of Bandpass Network
Transformations, 211
6.6. Pseudobandpass Matching Networks, 212
6.6.1. A Pseudobandpass Frequency
Transformation, 2 13
6.6.2. Evaluation of Gain-Bandwidth
Integrals, 214
6.6.3. Network Synthesis Procedure,
216
6.6.4. Summary of Pseudobandpass
Matching, 217
6.7. Carlin's Broadband-Matching Method, 218
6.7.1. Piecewise Hilbert Transform, 2
19
6.7.2. Gain Objective Function With
Derivatives, 222
6.7.3. Optimization of the Piecewise
Resistance Function, 224
6.7.4. Rational Approximation and
Synthesis, 224
6.7.5. Summary of Carlin's
Broadband-Matching Method, 226
Problems, 227
7. Linear Amplifier Design Tools
7.1. Bilinear Transformations, 23 1
7.1.1. Determining Bilinear
Coefficients, 232
7.1.2. Generalized Smith Chart, 235
7.1.3. Summary of Bilinear
Transformations, 237
7.2. Impedance Mapping, 238
7.2.1. Three-Port to Two-Port
Conversion, 238
7.2.2. The Bilinear Theorem, 240
7.2.3. Mapping, 241
7.2.4. Summary of Impedance Mapping,
246
7.3. Two-Port Impedance and Power Models, 246
7.3.1. Output Power Paraboloid, 247
7.3.2. Input Admittance Plane, 248
7.3.3. Maximum Efficiency, 250
7.3.4. Conjugate Terminations, 252
7.3.5. Maximum Added Power, 254
7.3.6. Summary of Two-Port Impedance
and Power Models, 256
7.4. Bilateral Scattering Stability and Gain, 257
7.4. I. Changing S-Parameter Port
Normalization, 258
7.4.2. Stability, 259
7.4.3. Bilateral Gains and
Terminations, 263
7.4.4. Summary of Scattering
Stability and Gain, 267
7.5. Unilateral Scattering Gain, 267
7.5.1. Transducer Gain
Simplification, 268
7.5.2. Unilateral Figure of Merit,
268
7.5.3. Unilateral Gain Circles, 269
7.5.4. Summary of Unilateral
Scattering Gain, 271
Problems, 271
8. Direct-Coupled Filters, 273
8.1. Prototype Network, 276
8.1.1I. Prototype Resonators, 276
8.1.2. Ideal Inverters, 277
8.1.3. Prototype Network Selectivity,
279
8.1.4. Prototype Selectivity Graphs,
28 1
8.1.5. Summary of Prototype Network,
282
8.2. Designing with L and C Inverters, 283
8.2.1. Simple L and C Inverters, 283
8.2.2. Magnetically Coupled
Inverters, 284
8.2.3. An Accurate Stopband
Selectivity Estimate, 285
8.2.4. A Design Example, 286
8.2.5. Summary of Designing With
Simple L and C Inverters, 289
8.3. General Inverters, Resonators, and End Couplings, 290
8.3.1. Inverters in Admittance
Parameters, 290
8.3.2. Trap Inverters, 292
8.3.3. Dissipation Effects, 295
8.3.4. Equivalent Resonators, 291
8.3.5. End Coupling, 302
8.3.6. Summary of Inverters,
Resonators, and End Couplings, 303
8.4. Four Important Passband Shapes, 304
8.4.1. The Chebyshev Overcoupled
Response Shape, 305
8.4.2. The Butterworth Maximally Flat
Response Shape, 308
8.4.3. The Fano Undercoupled Response
Shape, 310
8.4.4. Comparison of Elliptic Family
Responses, 3 13
8.4.5. The Minimum-Loss Response
Shape, 314
8.4.6. Summary of Four Important
Passband Shapes, 319
8.5. Comments on a Design Procedure, 321
8.5.1. Design Flowchart, 321
8.5.2. Design Limitations, 323
8.5.3. Adjustment of Shunt-L Values,
323
8.5.4. Sensitivities, 324
8.5.5. Tuning, 325
8.5.6. Summary of Comments on a
Design Procedure, 327
8.6. A Complete Design Example, 327
8.6.I. Response Shapes, 328
8.6.2. Physical Data, 328
8.6.3. Pass Band, 328
8.6.4. Stop Bands, 328
8.6.5. Q Effects, 329
8.6.6. Design Limitations, 329
8.6.7. Minimum Shunt Inductance, 329
8.6.8. Prototype Ohmic Values, 330
8.6.9. Component Acceptability, 330
8.6.10. Shunt Inductance Adjustment,
330
8.6.11. Final Component Values, 330
8.6.12. Performance and Sensitivity
Analysis, 331
8.6.13. Design Adjustment, 332
Problems. 332
9. Other Direct Filter Design Methods, 335
9.1 Equal-Stub Admittance Filters, 336
9.1.1. Equal-Stub-Filter Development.
336
9.1.2. Equal-Stub-Filter Design
Procedure, 341
9.1.3. Variations for Printed-Circuit
Filters, 342
9.1.4. Summary of Equal-Stub
Admittance Filters, 344
9.2. Introduction to Cauer Elliptic Filters, 344
9.2.1. From Butterworth to Elliptic
Filter Functions, 345
9.2.2. Elliptic Filter Degree,
Attenuation, and Pole Frequencies, 348
9.2.3. The Four Types of Elliptic
Filters, 350
9.2.4. Summary of Introduction to
Cauer Elliptic Filters, 352
9.3. Doubly Terminated Elliptic Filters, 352
9.3.1. Input Impedance Relationships,
353
9.3.2. The Permutation Method to
Calculate Trap-Section Elements, 354
9.3.3. The Complete Permutation
Algorithm, 357
9.3.4. Symmetric Type-s Filter
Program, 358
9.3.5. Antimetric Type-a, Type-b, and
Type-c Filter Program, 360
9.3.6. Summary of Doubly Terminated
Elliptic Filters, 362
9.4. Some Lumped-Element Transformations, 363
9.4.1. Exact Transformations, 363
9.4.2. Trap Approximations, 366
9.4.3. Summary of Some Lumped-Element
Transformations, 369
9.5. Load Effects on Passive Networks, 369
9.5.1. Unit-Circle to Unit-Circle
Bilinear Mapping, 370
9.5.2. Power Bounds Between a Complex
Source and Loads, 372
9.5.3. Bounds on Input Impedance and
SWR, 374
9.5.4. Summary of Load Effects on
Passive Networks, 376
9.6. Invulnerable Filters, 377
9.6.1. Invulnerable Bridged-T
Network, 377
9.6.2. Three-Pole Invulnerable
Filter, 380
9.6.3. Summary of Invulnerable
Filters, 383
Problems, 383
Appendix A. HP-67/97 Programs
Appendix B. PET BASIC Programs
Appendix C. Derivation of the Fletcher-Reeves Scalar Multiplier
Appendix D. Linear Search Flowchart
Appendix E. Defined Complex Constants for Amplifier Scattering Analysis
Appendix F. Doubly Terminated Minimum-Loss Selectivity
Appendix G. Direct-Coupled-Filter Design Equations
Appendix H. Zverev's Tables of Equivalent Three- and Four-Element Networks
References
Index