All the authors of the book are professors at Moscow colleges.

Georgy V. Zeveke is professor at the Moscow Power Engineering Institute. He is the author of more than thirty publications on protective relaying, automatic control, and circuit analysis.

Pyotr A. lonkin is professor and head of the electrical engineering physics department at the same institute. He has penned over sixty publications on circuit analysis, network graph theory, and circuit topology, and also a number of books on applied electricity and general electrical engineering.

Anatoly V. Netushil is professor and head of the automation department at the same institute. He has written over a hundred

CONTENTS

DIRECT-CURRENT CIRCUITS

Chapter One. GENERAL OUTLINE OF LINEAR CIRCUITS AND

LINEAR-CIRCUIT ANALYSIS 23

1-1. Circuit Elements and Parameters 23 1-2. Ideal Sources. Series and Parallel Equivalent Circuits 26 1-3. Ohm's Law for a Circuit Branch with In E.M.F. . 30 1-4. Potential Distribution Around a Single-Loop Network 32 1-5. Power Relations in a Single-Loop Network 34 1-6. Application of Kirchhoff's Laws to Multi-Loop Networks 37 1-7. The Method of Node Voltages 44 1-8. The Method of Loop (or Mesh) Currents 50 1-9. The Superposition Principle 56 1-10. The Property of Reciprocity 59 1-11. Driving-Point and Transfer Conductances 60 1-12. Compensation Theorem 65 1-13. Linear Relations Between Voltages and 66 Currents 1-14. Incremental Voltage-Current Relationships 70

Chapter Two. LINEAR CIRCUIT TRANSFORMATIONS 72

2-1. General Remarks 72 2-2. Conversion of Series-Parallel Configurations of Passive and Active Branches. Star-Delta

Transformation 73

2-3. General Star-Mesh Conversion 83

2-4. Parallel-Generator Theorem 86

2-5. Source Transformation 88

2-6. Duality in Electrical Networks 92

Chapter Three. ONE-PORT AND TWO-PORT NETWORKS 97

3-1. General Remarks on One-Port Networks 97 3-2. Application of Thevenin's Theorem to

Network Analysis 99 3-3. Energy Transfer from an Active to a Passive

l^J

3-4. Two-Ports and Their Basic Equations 109

3-5. Finding the ABCD Parameters'of a Network llf1

3-6. Operation of Two-Ports under Load 114

3-7. Equivalent Circuits for Two-Ports 117 3-8. Basic Equations and Equivalent Circuits

for Active Two-Ports ' 120

Chapter Four. NON-LINEAR ELECTRIC CIRCUITS 123

4-1. Elements and Elementary Equivalent Circuits 123 4-2. Graphical Solution of Single-Loop Non-Linear Circuits 128 4-3. Graphical Solution of Parallel Non-Linear Circuits 129 4-4. Graphical Solution of Series-Parallel Non-Linear. Circuits 130' 4-5. Application of Voltage-Source Equivalent Circuits to Non-Linear Circuit Analysis 131 4-6. Volt-Ampere Characteristics of Active 134 Non-Linear One-Ports

4-7. Illustrative Solutions of Multi-Loop Non-Linear Circuits 136-

4-8. Application of Thevenin's Theorem to Solution of Linear and Non-Linear Circuits 141 4-9. Solution of Multi-Loop Non-Linear Circuits by Iteration (Successive Approximation) 145

Chapter Five. MAGNETIC CIRCUITS 153-

5-1. Definitions and Laws 153

5-2. Analysis of Undivided Magnetic Circuits 161 5-3. Analysis and Synthesis of Divided Magnetic

Circuits 164 5-4. A Toroidal Magnetic Circuit with an Air

Gap 170

5-5- A Composite Magnetic Circuit Containing 173-

a Permanent Magnet

«

Part Two ALTERNATING-CURRENT CIRCUITS

Chapter Six. SINUSOIDAL CURRENTS AND VOLTAGES 177

6-1. Basic Definitions 177

6-2. The A.C. Generator 179

6-3. Sinusoidal Current 180 6-4. -R.M.S. (Effective) Values of Current and

Voltage 182

6-5. Representation of Sinusoidal Time Functions by Vectors and Complex Numbers 183 6-6. Addition of Sinusoidal Time Functions 186

Chapter Seven. BASIC CONCEPTS OF A.C. CIRCUITS 188

7-1. Circuits and Their Representation 188 7-2. Resistances, Inductances and Conductances

in Series 192

7-3. Impedance 196 7-4. Phase Displacement Between Voltage and

Current ' 198 7-5. Resistance, Inductance and Capacitance

in Parallel 201

7-6. Admittance 202

7-7. Passive One-Ports 205

7-8. Power 207 7-9. Power in Resistive, Inductive and Capaci-

tive Circuits 212

7-10. Energy Relations in an A.C. Circuit 21 5 7-11. The Sign of Power and the Direction of

Energy Transfer 216 7-12. Determination of the Parameters of a Passive One-Port with an Ammeter, Voltmeter and

Wattmeter 219

i 7-13. Maximum Power Transfer 220

7-14. Skin Effect. Proximity Effect 221 v 7-15. Parameters and Equivalent Circuits of

Capacitors 223 7-16. Parameters and Equivalent Circuits of

Inductors and Resistor Coils 224

Chapter Eight. ANALYSIS OF CIRCUITS CARRYING SINEWAVE

CURRENTS 226 8-1. Applicability of D.C. Circuit Analysis to

A.C. Circuits 226

8-2. Loads in Series 227

8-3. Loads in Parallel 229

8-4. Loads in Series-Parallel 231

8-5. Composite Multi-Mesh Circuits 233

8-6. Topographic Diagrams 236

8-7. Passive Two-Ports 239

Chapter Nine. RESONANCE IN ELECTRIC CIRCUITS 241

9-1. Resonance in a Single-Mesh Circuit 241 9-2. Frequency Response Characteristics of a

Single-Mesh Circuit 243

9-3. Resonance in a Parallel Circuit 247

9-4. Frequency Characteristics of a Parallel

Resonance Circuit • 250

9-5. Resonance in Multi-Mesh Circuits 251

Chapter Ten. CIRCUITS WITH MUTUAL INDUCTANCE 253 10-1. Mutual Inductance 253 10-2. E.M.F. of Mutual Induction 255 10-3. Series Connection of Mutual Inductance 258 10-4. Parallel Connection of Mutual Inductance 260 10-5. Calculation of Inductively Coupled Multi-Mesh Circuits 261 10-6. Equivalent Circuits of Mutual Inductance 264 10-7. Energy Transfer Between Inductively Coupled Circuits 266 10-8. The Air-Core Transformer 270 10-9. The Ideal Transformer 273 10-10. Ideal-Transformer Equivalent Circuits for Two-Ports 276 10-11. Equivalent Circuits of an Iron-Cored Transformer 277 10-12. The Auto-Transformer 280 10-13. Calculation of Circuits Containing Transformers 280

Chapter Eleven. LINEAR PERFORMANCE OF CIRCUITS CONTAINING VALVES AND SEMICONDUCTOR DEVICES 283 11-1. The Vacuum Triode 283 11-2. Parameters of a Vacuum Triode 284 11-3. Equivalent Circuits of the Vacuum Triode 287 11-4. Transistors 292 11-5. Equivalent Circuits for Transistors 295

Chapter Twelve. LOCUS DIAGRAMS 297 12-1. Straight-Line and Circle Equations in Complex Form 297 12-2. Circle Diagrams for a Single-Mesh Network and an Active One-Port 299 12-3. Circle Diagrams for Any Multi-Mesh Networks 305 12-4. Circle Diagrams for Active and Passive Two-Ports 307

Chapter Thirteen. THREE-PHASE CIRCUITS 312

13-1. Polyphase Sources and Polyphase Circuits 312

13-2. Phase Intel-linkage 315

13-3. Interlinkage of Transformer Windings 318

13-4. Symmetrical Loading of a Three-Phase

Circuit 320 13-5. Some Properties of Three-Phase Circuits

with Various Forms of Phase Interlinkage 323 13-6. Calculation of Symmetrical Three-Phase

Circuits 329 13-7. Calculation of Unsymmetrical Three-Phase

Circuits under Static Load 331

13-8. Load-Phase Voltages in Special Cases 334

13-9. Equivalent Circuits of Three-Phase Lines 336

13-10. Faults to Earth in Three-Phase Circuits 340 with Unearthed Neutral Points

13-11. Three-Phase Power Measurement 342

13-12. The Rotating Magnetic Field 344

13-13. Theory of Induction and Synchronous Motors 346

Chapter Fourteen. THE METHOD OF SYMMETRICAL COMPONENTS 348 14-1. Symmetrical Components of a Three-Phase System of Quantities 348 14-2. ^Some -Properties of Three-Phase Circuits with Regard to Symmetrical Components of Currents and Voltages 351 14-3. Impedance of a Symmetrical Three-Phase Circuit to Phase-Sequence Currents 355 14-4. Determination of Currents in Balanced Circuits 360 14-5. Phase-Sequence Voltages and Currents in Unbalanced Three-Phase Networks 361 14-6. Calculation of Networks of Unbalanced Impedances 362 14-7. Calculation of Lines with an Unsymmetrical Section 368

Chapter Fifteen. DISTRIBUTED-PARAMETER NETWORKS 372

15-1. Currents and Voltages in Long Lines 372

15-2. The Line Constants and Equations 373 15-3. Steady-State Conditions in a Uniform

Line 374 15-4. Equations of a Uniform Line in Terms of

Hyperbolic Functions 380

15-5. Derived Characteristics of a Uniform Line 383

15-6. Input Impedance of a Transmission Line 387

15-7. Reflection 399

15-8. Matched Termination of a Line 391

15-9. Distortionless Lines 395

15-10. Open-Circuit, Short-Circuit and Loaded

Termination of a Line 398

15-11. Dissipationless Lines 404

15-12. Standing Waves 407

15-13. The Transmission Line as a Two-Port 416 15-14. Characteristic Impedance and Propagation

Constant of a Symmetrical Two-Port 420 15-15. Image Impedances and Transfer Constant

of an Unsymmetrical Two-Port 423

15-16. Ladder Networks 426

Chapter Sixteen. NON-SINUSOIDAL ANALYSIS 429

16-1. Introduction 429

16-2. Fourier Series 432

16-3. Peak, R.M.S. and Mean Values of Non-Sinusoidal Periodic Voltages and Currents 438 16-4. The Form Factor. The Peak Factor. The Distortion Factor 440

16-5. Non-Sinusoidal Waves with a Periodic Envelope 443

16-6. R.M.S. Values of Voltages and Currents with Periodic Envelopes 447

16-7. Calculation of Circuits Carrying Non-Sinusoidal Periodic Voltages and Currents 448 16-8. Resonance in the Case of Non-Sinusoidal Voltages and Currents 454 16-9. Power with Non-Sinusoidal Currents and Voltages 457 16-10. Harmonics in Three-Phase Systems 459

Chapter Seventeen. ELECTRICAL WAVE FILTERS 4?4

17-1. Introduction 464

17-2. The Low-Pass Filter 466

17-3. The High-Pass Filter 474

17-4. The Band-Pass Filter 476

17-5. The Band-Stop Filter 482

Part Three TRANSIENTS IN LINEAR CIRCUITS

Chapter Eighteen. THE CLASSICAL METHOD OF TRANSIENT ANALYSIS OF LUMPED-PARAMETER NETWORKS 485

18-1. Origin of Transients. Continuity Conditions 485 j 18-2. Transient Response. Forced and Free j Terms 486 j

18-3. Source-Free Response of an RL Network 489 18-4. Forced Response of an RL Network to a

D.C. Voltage Drive 492 18-5. Forced Response of an RL Network to

a Sine-Wave Voltage Drive 494

18-6. Source-Free Response of an RC Network 497 18-7. Forced Response of an RC Network te

a D.C. Drive 499 18-8. Forced Response of an RC Network to

a Sine-Wave Voltage Drive 502

18-10. An Overdamped Transient Response 505

18-11. Critically Damped Transient Response 508

18-12. Underdamped (Oscillatory) Response 509 18-13. Forced Response of an RLC Series Network

to a D.C. Voltage Drive 513 18-14. The General Case of Transient Analysis

by the Classical Method 516 18-15. Response to a'Continually Varying Drive.

The Duhamel Integral 523

18-16. Drives with Discontinuities 526

Chapter Nineteen. THE OPERATIONAL METHOD OF TRANSIENT ANALYSIS OF LUMPED-PARAMETER NETWORKS 532

19-1. Introduction 532

19-2. Laplace Transforms of the Simplest Time Functions 534

19-3. Laplace Transforms of Derivatives and Integrals 535

19-4. Heaviside Expansion Theorem 637

19-5. Derivation of the Partial-Fraction Expansion from the Inverse Laplace Transform 540 19-6. Ohm's and Kirchhoff's Laws in Operational Form 543 19-7. Operational (Transform) Equivalent Circuits 549 19-8. Transient Response of Circuits with Mutual Inductance 555 19-9. Transient Problems with Non-Zero Initial Conditions 560

19-10. Determination of Free Currents from

Their Transforms

Chapter Twenty. FOURIER TRANSFORM METHOD 565

20-1. The Fourier Transform 565 20-2. Ohm's and Kirchhoff's Laws and xrans-

formed Equivalent Circuits for Fourier Analysis 569

CONTENTS

20-3. Determination of Time Response from Frequency Response (Floyd's Trapezoidal Approximation Method) 574

20-4. Conversion of Fourier Transforms into Laplace Transforms 583

20-5. Comparison of Various Transient Analysis Methods 585

20-6. The Transfer Function of Minimum-Phase Two-Ports 587

Chapter

Twenty-One. TRANSIENT ANALYSIS OF DISTRIBUT-

ED-PARAMETER NETWORKS 593

21-1. Transients in Distributed-Parameter Networks 593 21-2. The General Solution of Uniform Line Equations 593 > 21-3. Surges on Transmission Lines 59& 21-4. General Cases of Surges Due to Switching 602 21-5. Reflection of a Square Wave from the End of the Line 604 21-6. A General Method for Determination of Rejected Surges 607 21-7. Qualitative Investigation of Transients in Lines Containing Lumped Capacitances and Inductances 613 21-8. Multiple Reflections of Steep-Fronted Surges from Resistor Termination 615 21-9. Wandering Surges 620

Part Four

PROPERTIES AND CALCULATION OF A.C. NON-LINEAR NETWORKS

Chapter

Twenty-Two. INTRODUCTION 625

22-1. Non-Linear Elements in A.C. Circuits 625

22-2. Effects of Non-Linearity 627

22-3. Methods of A.C. Non-Linear Analysis 628

Chapter

Twenty-Three. SINGLE-FREQUENCY NETWORKS 632

23-1. Introduction 632 23-2. Valve Action of Non-Linear Resistances 632

23-3. Simple Rectifier Circuits 635

23-4. Circuit Containing an Electric ArCj an

Inductance and a Resistance 641

23-5. Current and Voltage Waveforms in Reactive Non-Linear Circuits 643 23-6. Frequency Triplers 648 23-7. Current and Voltage Waveforms in Circuits Containing Thermal Resistors 651 23-8. Equivalent Non-Linear Elements 653 23-9. Practical Iron-Cored Inductors 654 23-10. Current Through an Iron-Cored Inductor 660 23-11. Solution of Quazi-Linear Magnetic Circuits 663 23-12. Ferro-Resonance 666 23-13. Ferro-Resonant Voltage Regulator 670-

1

Chapter Twenty-Four. NETWORKS WITH MULTI-FREQUENCY

SOURCES 672

24-1. Introduction 672

24-2 Diodes in D.C.-A.C. Networks 672

24-3. Saturable Reactors 675

24-4. The Static Frequency Doubler. The Para-metron 679

24-5. The Effect of D.C. Voltage on the Alternating Current Component in Circuits with Lag-Free Non-Linear Resistances 686 24-6. Modulated Oscillations in Non-Linear Circuits 688 24-7. Effect of a D.C. Component on an A.C. Component in Non-Linear Inductive Circuits 690 24-8. Magnetic Amplifier 692

Chapter

Twenty-Five TRANSIENTS IN NON-LINEAR CIRCUITS 695 25-1 Introduction 695

25-2. Application of a D.C. Voltage to an Iron-Cored Inductor 696 25-3. Transient Response of an Iron-Cored Inductor to a Sine-Wave Voltage Excitation 702 25-4. Presentation of Transient Responses on a Phase Plane 707 25-5. Oscillatory Discharge of a Capacitor Through a Non-Linear Inductance 713

Chapter

Twenty-Six. SELF-SUSTAINED OSCILLATIONS 715

26-1. Non-Linear Elements with a Negative Slope in the V-I Characteristic 715

26-2. Stability of Non-Linear Circuits