Inductance: Loop and partial

INDUCTANCE......Page 3
CONTENTS......Page 9
Preface......Page 13
1.1 Historical Background......Page 17
1.2 Fundamental Concepts of Lumped Circuits......Page 18
1.3 Outline of the Book......Page 23
1.4 “Loop” Inductance vs. “Partial” Inductance......Page 24
2.1 Magnetic Field Vectors and Properties of Materials......Page 29
2.2 Gauss’s Law for the Magnetic Field and the Surface Integral......Page 31
2.3 The Biot–Savart Law......Page 35
2.4 Ampère’s Law and the Line Integral......Page 50
2.5 Vector Magnetic Potential......Page 63
2.5.1 Leibnitz’s Rule: Differentiate Before You Integrate......Page 83
2.6 Determining the Inductance of a Current Loop: A Preliminary Discussion......Page 87
2.7 Energy Stored in the Magnetic Field......Page 95
2.8 The Method of Images......Page 96
2.9 Steady (DC) Currents Must Form Closed Loops......Page 99
3 Fields of Time-Varying Currents (Accelerated Charge)......Page 103
3.1 Faraday’s Fundamental Law of Induction......Page 104
3.2 Ampère’s Law and Displacement Current......Page 114
3.3 Waves, Wavelength, Time Delay, and Electrical Dimensions......Page 118
3.4 How Can Results Derived Using Static (DC) Voltages and Currents be Used in Problems Where the Voltages and Currents are Varying with Time?......Page 121
3.5 Vector Magnetic Potential for Time-Varying Currents......Page 123
3.6 Conservation of Energy and Poynting’s Theorem......Page 127
3.7 Inductance of a Conducting Loop......Page 129
4.1 Self Inductance of a Current Loop from Faraday’s Law of Induction......Page 133
4.1.1 Rectangular Loop......Page 137
4.1.2 Circular Loop......Page 142
4.1.3 Coaxial Cable......Page 146
4.2 The Concept of Flux Linkages for Multiturn Loops......Page 149
4.2.1 Solenoid......Page 150
4.2.2 Toroid......Page 153
4.3 Loop Inductance Using the Vector Magnetic Potential......Page 155
4.3.1 Rectangular Loop......Page 157
4.3.2 Circular Loop......Page 160
4.4 Neumann Integral for Self and Mutual Inductances Between Current Loops......Page 161
4.4.1 Mutual Inductance Between Two Circular Loops......Page 163
4.4.2 Self Inductance of the Rectangular Loop......Page 166
4.4.3 Self Inductance of the Circular Loop......Page 169
4.5 Internal Inductance vs. External Inductance......Page 171
4.6 Use of Filamentary Currents and Current Redistribution Due to the Proximity Effect......Page 174
4.6.1 Two-Wire Transmission Line......Page 175
4.6.2 One Wire Above a Ground Plane......Page 177
4.7 Energy Storage Method for Computing Loop Inductance......Page 179
4.7.1 Internal Inductance of a Wire......Page 180
4.7.3 Coaxial Cable......Page 181
4.8 Loop Inductance Matrix for Coupled Current Loops......Page 183
4.8.1 Dot Convention......Page 185
4.8.2 Multiconductor Transmission Lines......Page 187
4.9 Loop Inductances of Printed Circuit Board Lands......Page 195
4.10 Summary of Methods for Computing Loop Inductance......Page 198
4.10.1 Mutual Inductance Between Two Rectangular Loops......Page 200
5 The Concept of “Partial” Inductance......Page 211
5.1 General Meaning of Partial Inductance......Page 212
5.2 Physical Meaning of Partial Inductance......Page 217
5.3 Self Partial Inductance of Wires......Page 221
5.4 Mutual Partial Inductance Between Parallel Wires......Page 225
5.5 Mutual Partial Inductance Between Parallel Wires that are Offset......Page 229
5.6 Mutual Partial Inductance Between Wires at an Angle to Each Other......Page 240
5.7 Numerical Values of Partial Inductances and Significance of Internal Inductance......Page 255
5.8 Constructing Lumped Equivalent Circuits with Partial Inductances......Page 258
6 Partial Inductances of Conductors of Rectangular Cross Section......Page 262
6.1 Formulation for the Computation of the Partial Inductances of PCB Lands......Page 264
6.2 Self Partial Inductance of PCB Lands......Page 270
6.3 Mutual Partial Inductance Between PCB Lands......Page 278
6.4 Concept of Geometric Mean Distance......Page 282
6.4.1 Geometrical Mean Distance Between a Shape and Itself and the Self Partial Inductance of a Shape......Page 289
6.4.2 Geometrical Mean Distance and Mutual Partial Inductance Between Two Shapes......Page 301
6.5 Computing the High-Frequency Partial Inductances of Lands and Numerical Methods......Page 307
7.1 Loop Inductance vs. Partial Inductance: Intentional Inductors vs. Nonintentional Inductors......Page 323
7.2 To Compute “Loop” Inductance, the “Return Path” for the Current Must be Determined......Page 325
7.3 Generally, There is no Unique Return Path for all Frequencies, Thereby Complicating the Calculation of a “Loop” Inductance......Page 327
7.4 Computing the “Ground Bounce” and “Power Rail Collapse” of a Digital Power Distribution System Using “Loop” Inductances......Page 328
7.5 Where Should the “Loop” Inductance of the Closed Current Path be Placed When Developing a Lumped-Circuit Model of a Signal or Power Delivery Path?......Page 330
7.6 How Can a Lumped-Circuit Model of a Complicated System of a Large Number of Tightly Coupled Current Loops be Constructed Using “Loop” Inductance?......Page 333
7.7 Modeling Vias on PCBs......Page 334
7.8 Modeling Pins in Connectors......Page 336
7.9 Net Self Inductance of Wires in Parallel and in Series......Page 337
7.10 Computation of Loop Inductances for Various Loop Shapes......Page 340
7.11 Final Example: Use of Loop and Partial Inductance to Solve a Problem......Page 344
Appendix A: Fundamental Concepts of Vectors......Page 351
A.1 Vectors and Coordinate Systems......Page 352
A.2 Line Integral......Page 356
A.3 Surface Integral......Page 359
A.4 Divergence......Page 361
A.4.1 Divergence Theorem......Page 363
A.5 Curl......Page 366
A.5.1 Stokes’s Theorem......Page 369
A.6 Gradient of a Scalar Field......Page 370
A.7 Important Vector Identities......Page 373
A.8 Cylindrical Coordinate System......Page 374
A.9 Spherical Coordinate System......Page 378
Table of Identities, Derivatives, and Integrals Used in this Book......Page 383
References and Further Readings......Page 389
Index......Page 393