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Inductance: Loop and Partial

ISBN: 978-0-470-46188-4
400 pages
December 2009, Wiley-IEEE Press
Inductance: Loop and Partial (0470461888) cover image


The only resource devoted Solely to Inductance

Inductance is an unprecedented text, thoroughly discussing "loop" inductance as well as the increasingly important "partial" inductance. These concepts and their proper calculation are crucial in designing modern high-speed digital systems. World-renowned leader in electromagnetics Clayton Paul provides the knowledge and tools necessary to understand and calculate inductance.

Unlike other texts, Inductance provides all the details about the derivations of the inductances of various inductors, as well as:

  • Fills the need for practical knowledge of partial inductance, which is essential to the prediction of power rail collapse and ground bounce problems in high-speed digital systems

  • Provides a needed refresher on the topics of magnetic fields

  • Addresses a missing link: the calculation of the values of the various physical constructions of inductors—both intentional inductors and unintentional inductors—from basic electromagnetic principles and laws

  • Features the detailed derivation of the loop and partial inductances of numerous configurations of current-carrying conductors

With the present and increasing emphasis on high-speed digital systems and high-frequency analog systems, it is imperative that system designers develop an intimate understanding of the concepts and methods in this book. Inductance is a much-needed textbook designed for senior and graduate-level engineering students, as well as a hands-on guide for working engineers and professionals engaged in the design of high-speed digital and high-frequency analog systems.

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Table of Contents


1 Introduction.

1.1 Historical Background.

1.2 Fundamental Concepts of Lumped Circuits.

1.3 Outline of the Book.

1.4 "Loop" Inductance vs. "Partial" Inductance.

2 Magnetic Fields of DC Currents (Steady Flow of Charge).

2.1 Magnetic Field Vectors and Properties of Materials.

2.2 Gauss’s Law for the Magnetic Field and the Surface Integral.

2.3 The Biot–Savart Law.

2.4 Ampére’s Law and the Line Integral.

2.5 Vector Magnetic Potential.

2.5.1 Leibnitz’s Rule: Differentiate Before You Integrate.

2.6 Determining the Inductance of a Current Loop:.

A Preliminary Discussion.

2.7 Energy Stored in the Magnetic Field.

2.8 The Method of Images.

2.9 Steady (DC) Currents Must Form Closed Loops.

3 Fields of Time-Varying Currents (Accelerated Charge).

3.1 Faraday’s Fundamental Law of Induction.

3.2 Ampère’s Law and Displacement Current.

3.3 Waves, Wavelength, Time Delay, and Electrical Dimensions.

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?.

3.5 Vector Magnetic Potential for Time-Varying Currents.

3.6 Conservation of Energy and Poynting’s Theorem.

3.7 Inductance of a Conducting Loop.

4 The Concept of "Loop" Inductance.

4.1 Self Inductance of a Current Loop from Faraday’s Law of Induction.

4.1.1 Rectangular Loop.

4.1.2 Circular Loop.

4.1.3 Coaxial Cable.

4.2 The Concept of Flux Linkages for Multiturn Loops.

4.2.1 Solenoid.

4.2.2 Toroid.

4.3 Loop Inductance Using the Vector Magnetic Potential.

4.3.1 Rectangular Loop.

4.3.2 Circular Loop.

4.4 Neumann Integral for Self and Mutual Inductances Between Current Loops.

4.4.1 Mutual Inductance Between Two Circular Loops.

4.4.2 Self Inductance of the Rectangular Loop.

4.4.3 Self Inductance of the Circular Loop.

4.5 Internal Inductance vs. External Inductance.

4.6 Use of Filamentary Currents and Current Redistribution Due to the Proximity Effect.

4.6.1 Two-Wire Transmission Line.

4.6.2 One Wire Above a Ground Plane.

4.7 Energy Storage Method for Computing Loop Inductance.

4.7.1 Internal Inductance of a Wire.

4.7.2 Two-Wire Transmission Line.

4.7.3 Coaxial Cable.

4.8 Loop Inductance Matrix for Coupled Current Loops.

4.8.1 Dot Convention.

4.8.2 Multiconductor Transmission Lines.

4.9 Loop Inductances of Printed Circuit Board Lands.

4.10 Summary of Methods for Computing Loop Inductance.

4.10.1 Mutual Inductance Between Two Rectangular Loops.

5 The Concept of "Partial" Inductance.

5.1 General Meaning of Partial Inductance.

5.2 Physical Meaning of Partial Inductance.

5.3 Self Partial Inductance of Wires.

5.4 Mutual Partial Inductance Between Parallel Wires.

5.5 Mutual Partial Inductance Between Parallel Wires that are Offset.

5.6 Mutual Partial Inductance Between Wires at an Angle to Each Other.

5.7 Numerical Values of Partial Inductances and Significance of Internal Inductance.

5.8 Constructing Lumped Equivalent Circuits with Partial Inductances.

6 Partial Inductances of Conductors of Rectangular Cross Section.

6.1 Formulation for the Computation of the Partial Inductances of PCB Lands.

6.2 Self Partial Inductance of PCB Lands.

6.3 Mutual Partial Inductance Between PCB Lands.

6.4 Concept of Geometric Mean Distance.

6.4.1 Geometrical Mean Distance Between a Shape and Itself and the Self Partial Inductance of a Shape.

6.4.2 Geometrical Mean Distance and Mutual Partial Inductance Between Two Shapes.

6.5 Computing the High-Frequency Partial Inductances of Lands and Numerical Methods.

7 "Loop" Inductance vs. "Partial" Inductance.

7.1 Loop Inductance vs. Partial Inductance: Intentional Inductors vs. Nonintentional Inductors.

7.2 To Compute "Loop" Inductance, the "Return Path" for the Current Must be Determined.

7.3 Generally, There is no Unique Return Path for all Frequencies, Thereby Complicating the Calculation of a "Loop" Inductance.

7.4 Computing the "Ground Bounce" and "Power Rail Collapse" of a Digital Power Distribution System Using "Loop" Inductances.

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?.

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?.

7.7 Modeling Vias on PCBs.

7.8 Modeling Pins in Connectors.

7.9 Net Self Inductance of Wires in Parallel and in Series.

7.10 Computation of Loop Inductances for Various Loop Shapes.

7.11 Final Example: Use of Loop and Partial Inductance to Solve a Problem.

Appendix A: Fundamental Concepts of Vectors.

A.1 Vectors and Coordinate Systems.

A.2 Line Integral.

A.3 Surface Integral.

A.4 Divergence.

A.4.1 Divergence Theorem.

A.5 Curl.

A.5.1 Stokes’s Theorem.

A.6 Gradient of a Scalar Field.

A.7 Important Vector Identities.

A.8 Cylindrical Coordinate System.

A.9 Spherical Coordinate System.

Table of Identities, Derivatives, and Integrals Used in this Book.

References and Further Readings.


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Author Information

Clayton R. Paul received his PhD in electrical engineering from Purdue University. He is the Sam Nunn Eminent Professor of Electrical and Computer Engineering at Mercer University in Macon, Georgia. Dr. Paul is also Emeritus (retired with distinction after 27 years on the faculty) Professor of Electrical Engineering at the University of Kentucky. He is the author of 15 textbooks on electrical engineering subjects and has published over 200 technical papers, the majority of which are in his primary research area of the electromagnetic compatibility (EMC) of electronic systems. Dr. Paul is a Life Fellow member of the Institute of Electrical and Electronics Engineers (IEEE) and an Honorary Life Member of the IEEE EMC Society. He received the prestigious 2005 IEEE Electromagnetics Award and the 2007 IEEE Undergraduate Teaching Award.
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