Laplace Transforms for Electronic Engineers (eBook)

Laplace Transforms for Electronic Engineers 4
Copyright Page 5
Table of Contents 6
CHAPTER 1. FUNCTIONS OF A COMPLEX VARIABLE 16
1.1. Introduction 16
1.2. Complex numbers 16
1.3. Complex planes 20
1.4. Relations between the z- and s-planes 23
1.5. Additional transformations between the z- and s-planes 24
1.6. Simplification of problems by transforming into the complex s-plane 27
1.7. Functions in the complex plane 30
1.8. Poles of complex functions 31
1.9. Zeros of complex functions 33
1.10. The pole-zero diagram 34
1.11. Integration along a curve in the s-plane 35
1.12. Integration around a pole 38
1.13. Integration around a path not containing a pole 40
1.14. Residues 42
1.15. Integration around two or more poles in the s-plane 47
1.16. Summary of Chapter I 50
CHAPTER 2. THE FOURIER SERIES AND INTEGRAL 52
2.1. The Fourier series 52
2.2. Exponential form of the Fourier series 55
2.3. The Fourier integral 61
2.4. The unit step function 65
2.5. The Fourier transform of the unit step function 66
2.6. Convergence factors 67
2.7. The complex Fourier integral transform 69
2.8. The Laplace transform 70
CHAPTER 3. THE LAPLACE TRANSFORMATION 72
3.1. Introduction 72
3.2. Transforms of constants 73
3.3. The Laplace transform of exponentials 74
3.4. The Laplace transform of imaginary exponents 75
3.5. The Laplace transform of trigonometric terms 75
3.6. The Laplace transform of hyperbolic functions 77
3.7. The Laplace transform of complex exponentials 79
3.8. Transforms of more complicated functions 79
3.9. Additional practice with sine waves 80
3.10. The Laplace transform of a derivative 81
3.11. The Laplace transform of an integral 83
CHAPTER 4. THE INVERSE LAPLACE TRANSFORMATION 87
4.1. Introduction 87
4.2. Functions of s from electronic networks 91
4.3. Functions of s involving simple poles 94
4.4. Functions of s involving both simple poles and zeros 98
4.5. Functions of s having higher order poles 101
CHAPTER 5. LAPLACE TRANSFORM THEOREMS 105
5.1. Introduction 105
5.2. Linear s-plane translation 105
5.3. Final value theorem 108
5.4. Initial value theorem 111
5.5. Real translation 113
5.6. Complex differentiation 116
5.7. Complex integration 118
5.8. Sectioning a function of time 120
5.9. The convolution theorem 123
5.10. Scale change theorem 127
5.11. Summary of Chapter V 129
CHAPTER 6. NETWORK ANALYSTS BY MEANS OF THE LAPLACE TRANSFORMATION 131
6.1. Introduction 131
6.2. Writing network equations for multiple loop circuits 132
6.3. Relay damping problems 134
6.4. The Wien-bridge oscillator 137
6.5. A phase-shift oscillator 141
6.6. Harmonic discrimination in a three-section phase shift oscillator 144
6.7. The R–C cathode follower oscillator 146
6.8. Odd and even functions of s 148
6.9. R–C voltage step-up networks 151
6.10. R–C oscillator, single section variable capacity 154
6.11. Active integrating and differentiating networks 157
6.12. Operational amplifiers 160
6.13. Charge amplifiers 162
6.14. Analysis of the charge amplifier 164
6.15. Reactive feedback voltage amplifiers 167
6.16. Analysis of 3-stage reactive feedback amplifier 171
6.17. Single section low-pass R–C filter 174
6.18. Two-section non-tapered R–C low-pass filter 175
6.19. Three-section non-tapered R–C low-pass filter 177
6.20. Iterative networks 179
6.21. Initial conditions in network parameters 182
6.22. Initial charge or voltage on condenser 182
6.23. Initial current in an inductance 184
6.24. Mutual inductance 188
CHAPTER 7. TRANSFORMS OF SPECIAL WAVESHAPES AND PULSES 194
7.1. Introduction 194
7.2. Laplace transform of a displaced step function 194
7.3. Transform of the dirac delta function 198
7.4. Derivatives of infinité slopes expressed as delta functions 199
7.5. Sampling another function with a delta function 200
7.6. Fourier coefficients ascertained by means of delta functions 201
7.7. The Laplace transform of a series of pulses 204
7.8. The Laplace transform of a general periodic wave 206
7.9. The Laplace transform of a single sawtooth pulse 209
7.10. Pulsed periodic functions 210
7.11. Transform of a displaced ramp function 213
CHAPTER 8. ELECTRONIC FILTERS 214
8.1. Introduction 214
8.2. Normalization of transfer functions 214
8.3. Low-pass filters 218
8.4. Maximally flat functions 220
8.5. Pole location for Butterworth functions 223
8.6. Synthesis of the third-order maximally flat function 226
8.7. High-pass maximally flat functions 228
8.8. Maximally flat band-pass filters 229
8.9. Design of a band-pass filter 230
8.10. The band-rejection filter 233
8.11. Matched low-pass filters 234
8.12. Magnitude and phase functions of "s" 236
8.13. Maximally flat time-delay filters 240
8.14. The linear-phase approximation 243
8.15. Bessel polynomials, or linear-phase filter design made easy 249
8.16. Finding the transfer function from a given magnitude 252
8.17. Tchebycheff and Legendre polynomial filters 254
8.18. Active nth order low-pass filters 257
8.19. Optimum n-section R–C filters for high voltage power supplies 260
CHAPTER 9. SPECIALIZED APPLICATIONS OF THE LAPLACE TRANSFORM 264
9.1. Functions of vs 264
9.2. Application of the impedance 1vs in oscillator design 267
9.3. Iterative networks 270
9.4. Transfer functions by tabular methods 273
9.5. Simplifications with the tabular method 276
9.6. Iterative networks, Pascal triangle method 280
9.7. Formulas for iterative network coefficients 286
9.8. Alternate approach to the Laplace integral 287
9.9. The Laplace integral used to sum infinite series into closed form 289
CHAPTER 10. SYNTHESIS OF TRANSFER FUNCTIONS BY MODELS 299
10.1. Introduction 299
10.2. Lossless network models 299
10.3. Odd and even parts of the transfer function 303
10.4. Synthesis of transfer impedances 304
10.5. Alternate method of transfer function synthesis 305
10.6. Conclusion 310
APPENDIX I 312
(a) Driving point transforms 312
(b) Transfer functions 314
(c) Active transfer functions 324
APPENDIX II: OPERATIONAL LAPLACE TRANSFORM PAIRS 328
APPENDIX III: TABLE OF LAPLACE TRANSFORM PAIRS 329
INDEX 360
OTHER TITLES IN THE SERIES 364