FOREWORD ..................................................... xiii
PREFACE ........................................................ xv
CHAPTER 1
Classical approach to transient analysis ........................ 1
1.1. Introduction ............................................... 1
1.2. Appearance of transients in electrical circuits ............ 2
1.3. Differential equations describing electrical circuits ...... 4
1.3.1. Exponential solution of a simple differential
equation ................................................... 7
1.4. Natural and forced responses .............................. 11
1.5. Characteristic equation and its determination ............. 14
1.6. Roots of the characteristic equation and different kinds
of transient responses .................................... 21
1.6.1. First order characteristic equation ................ 21
1.6.2. Second order characteristic equation ............... 22
1.7. Independent and dependent initial conditions .............. 26
1.7.1. Two switching laws (rules) ......................... 26
(a) First switching law (rule) ..................... 26
(b) Second switching law (rule) .................... 27
1.7.2. Methods of finding independent initial
conditions ......................................... 29
1.7.3. Methods of finding dependent initial conditions .... 31
1.7.4. Generalized initial conditions ..................... 35
(a) Circuits containing capacitances ............... 35
(b) Circuits containing inductances ................ 39
1.8. Methods of finding integration constants .................. 44
CHAPTER 2
Transient response of basic circuits ........................... 49
2.1. Introduction .............................................. 49
2.2. The five steps of solving problems in transient
analysis .................................................. 49
2.3. RL circuits ............................................... 51
2.3.1. RL circuits under d.c. supply ...................... 51
2.3.2. RL circuits under a.c. supply ...................... 62
2.3.3. Applying the continuous flux linkage law to
L-circuits ......................................... 72
2.4. RC circuits ............................................... 80
2.4.1. Discharging and charging a capacitor ............... 80
2.4.2. RC circuits under d.c. supply ...................... 82
2.4.3. RC circuits under a.c. supply ...................... 88
2.4.4. Applying the continuous charge law to C-circuits ... 95
2.5. The application of the unit-step forcing function ........ 101
2.6. Superposition principle in transient analysis ............ 105
2.7. RLC circuits ............................................. 110
2.7.1. RLC circuits under d.c. supply .................... 110
(a) Series connected RLC circuits ................. 113
(b) Parallel connected RLC circuits ............... 118
(c) Natural response by two nonzero initial
conditions .................................... 120
2.7.2. RLC circuits under a.c. supply .................... 131
2.7.3. Transients in RLC resonant circuits ............... 135
(a) Switching on a resonant RLC circuit to an
a.c. source ................................... 136
(b) Resonance at the fundamental (first)
harmonic ...................................... 139
(c) Frequency deviation in resonant circuits ...... 140
(d) Resonance at multiple frequencies ............. 141
2.7.4. Switching off in RLC circuits ..................... 143
(a) Interruptions in a resonant circuit fed from
an a.c. source ................................ 147
CHAPTER 3
Transient analyses using the Laplace transform techniques ..... 155
3.1. Introduction ............................................. 155
3.2. Definition of the Laplace transform ...................... 156
3.3. Laplace transform of some simple time functions .......... 157
3.3.1. Unit-step function ................................ 157
3.3.2. Unit-impulse function ............................. 158
3.3.3. Exponential function .............................. 159
3.3.4. Ramp function ..................................... 159
3.4. Basic theorems of the Laplace transform .................. 159
3.4.1. Linearity theorem ................................. 160
3.4.2. Time differentiation theorem ...................... 161
3.4.3. Time integration theorem .......................... 163
3.4.4. Time-shift theorem ................................ 165
3.4.5. Complex frequency-shift property .................. 169
3.4.6. Scaling in the frequency domain ................... 170
3.4.7. Differentiation and integration in the frequency
domain ............................................ 171
3.5. The initial-value and final-value theorems ............... 172
3.6. The convolution theorem .................................. 176
3.6.1. Duhamel's integral ................................ 179
3.7. Inverse transform and partial fraction expansions ........ 180
3.7.1. Method of equating coefficients ................... 182
(a) Simple poles .................................. 182
(b) Multiple poles ................................ 183
3.7.2. Heaviside's expansion theorem ..................... 184
(a) Simple poles .................................. 184
(b) Multiple poles ................................ 185
(c) Complex poles ................................. 186
3.8. Circuit analysis with the Laplace transform .............. 188
3.8.1. Zero initial conditions ........................... 190
3.8.2. Non-zero initial conditions ....................... 193
3.8.3. Transient and steady-state responses .............. 197
3.8.4. Response to sinusoidal functions .................. 200
3.8.5. Thevenin and Norton equivalent circuits ........... 203
3.8.6. The transients in magnetically coupled circuits ... 207
CHAPTER 4
Transient analysis using the Fourier transform ................ 213
4.1. Introduction ............................................. 213
4.2. The inter-relationship between the transient behavior
of electrical circuits and their spectral properties ..... 214
4.3. The Fourier transform .................................... 215
4.3.1. The definition of the Fourier transform ........... 215
4.3.2. Relationship between a discrete and continuous
spectra ........................................... 223
4.3.3. Symmetry properties of the Fourier transform ...... 226
(a) An even function of t ......................... 226
(b) An odd function of t .......................... 227
(c) A non-symmetrical function (neither even nor
odd) .......................................... 228
4.3.4. Energy characteristics of a continuous spectrum ... 228
4.3.5. The comparison between Fourier and Laplace
transforms ........................................ 231
4.4. Some properties of the Fourier transform ................. 232
(a) Property of linearity ......................... 232
(b) Differentiation properties .................... 232
(c) Integration properties ........................ 233
(d) Scaling properties ............................ 234
(e) Shifting properties ........................... 234
(f) Interchanging t and ω properties .............. 235
4.5. Some important transform pairs ........................... 237
4.5.1. Unit-impulse (delta) function ..................... 238
4.5.2. Unit-step function ................................ 241
4.5.3. Decreasing sinusoid ............................... 244
4.5.4. Saw-tooth unit pulse .............................. 244
4.5.5. Periodic time function ............................ 246
4.6. Convolution integral in the time domain and its Fourier
transform ................................................ 247
4.7. Circuit analysis with the Fourier transform .............. 250
4.7.1. Ohm's and Kirchhoff's laws with the Fourier
transform ......................................... 252
4.7.2. Inversion of the Fourier transform using the
residues of complex functions ..................... 252
4.7.3. Approximate transient analysis with the Fourier
transform ......................................... 258
(a) Method of trapezoids .......................... 259
CHAPTER 5
Transient analysis using state variables ...................... 265
5.1. Introduction ............................................. 265
5.2. The concept of state variables ........................... 266
5.3. Order of complexity of a network ......................... 270
5.4. State equations and trajectory ........................... 272
5.5. Basic considerations in writing state equations .......... 276
5.5.1. Fundamental cut-set and loop matrixes ............. 276
5.5.2. "Proper tree" method for writing state
equations ........................................ 283
5.6. A systematic method for writing a state equation based
on circuit matrix representation ......................... 287
5.7. Complete solution of the state matrix equation ........... 294
5.7.1. The natural solution .............................. 294
5.7.2. Matrix exponential ................................ 295
5.7.3. The particular solution ........................... 296
5.8. Basic considerations in determining functions of a
matrix ................................................... 297
5.8.1. Characteristic equation and eigenvalues ........... 298
5.8.2. The Caley-Hamilton theorem ........................ 299
(a) Distinct eigenvalues .......................... 302
(b) Multiple eigenvalues .......................... 308
(c) Complex eigenvalues ........................... 311
5.8.3. Lagrange interpolation formula .................... 313
5.9. Evaluating the matrix exponential by Laplace
transforms ............................................... 314
CHAPTER 6
Transients in three-phase systems ............................. 319
6.1. Introduction ............................................. 319
6.2. Short-circuit transients in power systems ................ 320
6.2.1. Base quantities and per-unit conversion in
three-phase circuits .............................. 321
6.2.2. Equivalent circuits and their simplification ...... 327
(a) Series and parallel connections ............... 327
(b) Delta-star (and vice-versa) transformation .... 328
(c) Using symmetrical properties of a network ..... 330
6.2.3. The superposition principle in transient
analysis .......................................... 330
6.3. Short-circuiting in a simple circuit ..................... 333
6.4. Switching transformers ................................... 339
6.4.1. Short-circuiting of power transformers ............ 339
6.4.2. Current inrush by switching on transformers ....... 345
6.5. Short-circuiting of synchronous machines ................. 346
6.5.1. Two-axis representation of a synchronous
generator ......................................... 347
6.5.2. Steady-state short-circuit of synchronous
machines .......................................... 350
(a) Short-circuit ratio (SCR) of a synchronous
generator ..................................... 351
(b) Graphical solution ............................ 356
(c) Influence of the load ......................... 364
(d) Approximate solution by linearization of the
OCC ........................................... 365
(e) Calculation of steady-state short-circuit
currents in complicated power networks ........ 368
6.5.3. Transient performance of a synchronous
generator ......................................... 370
(a) Transient EMF, transient reactance and time
constant ...................................... 370
(b) Transient effects of the damper windings:
subtransient EMF, subtransient reactance and
time constant ................................. 379
(c) Transient behavior of a synchronous
generator with AVR ............................ 385
(d) Peak values of a short-circuit current ........ 387
6.6. Short-circuit analysis in interconnected (large)
networks ................................................. 394
6.6.1. Simple computation of short-circuit currents ...... 399
6.6.2. Short-circuit power ............................... 400
6.7. Method of symmetrical components for unbalanced fault
analysis ................................................. 404
6.7.1. Principle of symmetrical components ............... 405
(a) Positive-, negative-, and zero-sequence
systems ....................................... 405
(b) Sequence impedances ........................... 411
6.7.2. Using symmetrical components for unbalanced
three-phase system analysis ....................... 431
6.7.3. Power in terms of symmetrical components .......... 449
6.8. Transient overvoltages in power systems .................. 451
6.8.1. Switching surges .................................. 452
6.8.2. Multiple oscillations ............................. 459
CHAPTER 7
Transient behavior of transmission lines ...................... 465
7.1. Introduction ............................................. 465
7.2. The differential equations of TL and their solution ...... 465
7.3. Travelling-wave property in a transmission line .......... 469
7.4. Wave formations in TL at their connections ............... 472
7.4.1. Connecting the TL to a d.c./a.c. voltage source ... 473
7.4.2. Connecting the TL to load ......................... 475
7.4.3. A common method of determining travelling waves
by any kind of connection ......................... 478
7.5. Wave reflections in transmission lines ................... 480
7.5.1. Line terminated in resistance ..................... 482
7.5.2. Open- and short-circuit line termination .......... 485
7.5.3. Junction of two lines ............................. 486
7.5.4. Capacitance connected at the junction of two
lines ............................................. 487
7.6. Successive reflections of waves .......................... 493
7.6.1. Lattice diagram ................................... 494
7.6.2. Bergeron diagram .................................. 496
7.6.3. Non-linear resistive terminations ................. 499
7.7. Laplace transform analysis of transients in
transmission lines ....................................... 500
7.7.1. Loss-less LC line ................................. 504
7.7.2. Line terminated in capacitance .................... 504
7.7.3. A solution as a sum of delayed waves .............. 506
7.8. Line with only LG or CR parameters ....................... 511
7.8.1. Underground cable ................................. 512
CHAPTER 8
Static and dynamic stability of power systems ................. 517
8.1. Introduction ............................................. 517
8.2. Definition of stability .................................. 517
8.3. Steady-state stability ................................... 518
8.3.1. Power-transfer characteristic ..................... 518
8.3.2. Swing equation and criterion of stability ......... 524
8.4. Transient stability ...................................... 529
8.4.1. Equal-area criterion .............................. 533
8.5. Reduction to a simple system ............................. 537
8.6. Stability of loads and voltage collapse .................. 540
APPENDIX I .................................................... 545
APPENDIX II ................................................... 549
APPENDIX III .................................................. 551
INDEX ......................................................... 559
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