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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A unique text on the theory and design fundaments of inductors and transformers, updated with more coverage on the optimization of magnetic devices and many new design examples
The first edition is popular among a very broad audience of readers in different areas of engineering and science. This book covers the theory and design techniques of the major types of high-frequency power inductors and transformers for a variety of applications, including switching-mode power supplies (SMPS) and resonant dc-to-ac power inverters and dc-to-dc power converters. It describes eddy-current phenomena (such as skin and proximity effects), high-frequency magnetic materials, core saturation, core losses, complex permeability, high-frequency winding resistance, winding power losses, optimization of winding conductors, integrated inductors and transformers, PCB inductors, self-capacitances, self-resonant frequency, core utilization factor area product method, and design techniques and procedures of power inductors and transformers. These components are commonly used in modern power conversion applications. The material in this book has been class-tested over many years in the author’s own courses at Wright State University, which have a high enrolment of about a hundred graduate students per term. The book presents the growing area of magnetic component research in a textbook form, covering the foundations for analysing and designing magnetic devices specifically at high-frequencies. Integrated inductors are described, and the Self-capacitance of inductors and transformers is examined. This new edition adds information on the optimization of magnetic components (Chapter 5). Chapter 2 has been expanded to provide better coverage of core losses and complex permeability, and Chapter 9 has more in-depth coverage of self-capacitances and self-resonant frequency of inductors. There is a more rigorous treatment of many concepts in all chapters. Updated end-of-chapter problems aid the readers’ learning process, with an online solutions manual available for use in the classroom.
- Provides physics-based descriptions and models of discrete inductors and transformers as well as integrated magnetic devices
- New coverage on the optimization of magnetic devices, updated information on core losses and complex permeability, and more in-depth coverage of self-capacitances and self-resonant frequency of inductors
- Many new design examples and end-of-chapter problems for the reader to test their learning
- Presents the most up-to-date and important references in the field
- Updated solutions manual, now available through a companion website
An up to date resource for Post-graduates and professors working in electrical and computer engineering. Research students in power electronics. Practising design engineers of power electronics circuits and RF (radio-frequency) power amplifiers, senior undergraduates in electrical and computer engineering, and R & D staff.
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Veröffentlichungsjahr: 2013
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Table of Contents
Title Page
Copyright
Dedication
Preface
About the Author
List of Symbols
Chapter 1: Fundamentals of Magnetic Devices
1.1 Introduction
1.2 Fields
1.3 Magnetic Relationships
1.4 Magnetic Circuits
1.5 Magnetic Laws
1.6 Eddy Currents
1.7 Core Saturation
1.8 Inductance
1.9 Air Gap in Magnetic Core
1.10 Fringing Flux
1.11 Inductance of Strip Transmission Line
1.12 Inductance of Coaxial Cable
1.13 Inductance of Two-Wire Transmission Line
1.14 Magnetic Energy and Magnetic Energy Density
1.15 Self-Resonant Frequency
1.16 Quality Factor of Inductors
1.17 Classification of Power Losses in Magnetic Components
1.18 Noninductive Coils
1.19 Summary
1.20 References
1.21 Review Questions
1.22 Problems
Chapter 2: Magnetic Cores
2.1 Introduction
2.2 Properties of Magnetic Materials
2.3 Magnetic Dipoles
2.4 Magnetic Domains
2.5 Curie Temperature
2.6 Magnetic Susceptibility and Permeability
2.7 Linear, Isotropic, and Homogeneous Magnetic Materials
2.8 Magnetic Materials
2.9 Hysteresis
2.10 Low-Frequency Core Permeability
2.11 Core Geometries
2.12 Ferromagnetic Core Materials
2.13 Superconductors
2.14 Hysteresis Loss
2.15 Eddy-Current Core Loss
2.16 Steinmetz Empirical Equation for Total Core Loss
2.17 Core Losses for Nonsinusoidal Inductor Current
2.18 Complex Permeability of Magnetic Materials
2.19 Cooling of Magnetic Cores
2.20 Summary
2.21 References
2.22 Review Questions
2.23 Problems
Chapter 3: Skin Effect
3.1 Introduction
3.2 Resistivity of Conductors
3.3 Skin Depth
3.4 AC-to-DC Winding Resistance Ratio
3.5 Skin Effect in Long Single Round Conductor
3.6 Current Density in Single Round Conductor
3.7 Magnetic Field Intensity for Round Wire
3.8 Other Methods of Determining the Round Wire Inductance
3.9 Power Loss Density in Round Conductor
3.10 Skin Effect in Single Rectangular Plate
3.11 Skin Effect in Rectangular Foil Conductor Placed Over Ideal Core
3.12 Summary
3.13 Appendix
3.14 References
3.15 Review Questions
3.16 Problems
Chapter 4: Proximity Effect
4.1 Introduction
4.2 Orthogonality of Skin and Proximity Effects
4.3 Proximity Effect in Two Parallel Round Conductors
4.4 Proximity Effect in Coaxial Cable
4.5 Proximity and Skin Effects in Two Parallel Plates
4.6 Antiproximity and Skin Effects in Two Parallel Plates
4.7 Proximity Effect in Open-Circuit Conductor
4.8 Proximity Effect in Multiple-Layer Inductor
4.9 Self-Proximity Effect in Rectangular Conductors
4.10 Summary
4.11 Appendix
4.12 References
4.13 Review Questions
4.14 Problems
Chapter 5: Winding Resistance at High Frequencies
5.1 Introduction
5.2 Eddy Currents
5.3 Magnetic Field Intensity in Multilayer Foil Inductors
5.4 Current Density in Multilayer Foil Inductors
5.5 Winding Power Loss Density in Individual Foil Layers
5.6 Complex Winding Power in nth Layer
5.7 Winding Resistance of Individual Foil Layers
5.8 Orthogonality of Skin and Proximity for Individual Foil Layers
5.9 Optimum Thickness of Individual Foil Layers
5.10 Winding Inductance of Individual Layers
5.11 Power Loss in All Layers
5.12 Impedance of Foil Winding
5.13 Resistance of Foil Winding
5.14 Dowell's Equation
5.15 Approximation of Dowell's Equation
5.16 Winding AC Resistance with Uniform Foil Thickness
5.17 Transformation of Foil Conductor to Rectangular, Square, and Round Conductors
5.18 Winding AC Resistance of Rectangular Conductor
5.19 Winding Resistance of Square Wire
5.20 Winding Resistance of Round Wire
5.21 Inductance
5.22 Solution for Round Conductor Winding in Cylindrical Coordinates
5.23 Litz Wire
5.24 Winding Power Loss for Inductor Current with Harmonics
5.25 Winding Power Loss of Foil Inductors Conducting DC and Harmonic Currents
5.26 Winding Power Loss of Round Wire Inductors Conducting DC and Harmonic Currents
5.27 Effective Winding Resistance for Nonsinusoidal Inductor Current
5.28 Thermal Effects on Winding Resistance
5.29 Thermal Model of Inductors
5.30 Summary
5.31 Appendix
5.32 References
5.33 Review Questions
5.34 Problems
Chapter 6: Laminated Cores
6.1 Introduction
6.2 Low-Frequency Eddy-Current Laminated Core Loss
6.3 Comparison of Solid and Laminated Cores
6.4 Alternative Solution for Low-Frequency Eddy-Current Core Loss
6.5 General Solution for Eddy-Current Laminated Core Loss
6.6 Summary
6.7 References
6.8 Review Questions
6.9 Problems
Chapter 7: Transformers
7.1 Introduction
7.2 Transformer Construction
7.3 Ideal Transformer
7.4 Voltage Polarities and Current Directions in Transformers
7.5 Nonideal Transformers
7.6 Neumann's Formula for Mutual Inductance
7.7 Mutual Inductance
7.8 Magnetizing Inductance
7.9 Coupling Coefficient
7.10 Leakage Inductance
7.11 Dot Convention
7.12 Series-Aiding and Series-Opposing Connections
7.13 Equivalent T Network
7.14 Energy Stored in Coupled Inductors
7.15 High-Frequency Transformer Model
7.16 Stray Capacitances
7.17 Transformer Efficiency
7.18 Transformers with Gapped Cores
7.19 Multiple-Winding Transformers
7.20 Autotransformers
7.21 Measurements of Transformer Inductances
7.22 Noninterleaved Windings
7.23 Interleaved Windings
7.24 Wireless Energy Transfer
7.25 AC Current Transformers
7.26 Saturable Reactors
7.27 Transformer Winding Power Losses with Harmonics
7.28 Thermal Model of Transformers
7.29 Summary
7.30 References
7.31 Review Questions
7.32 Problems
Chapter 8: Integrated Inductors
8.1 Introduction
8.2 Skin Effect
8.3 Resistance of Rectangular Trace with Skin Effect
8.4 Inductance of Straight Rectangular Trace
8.5 Inductance of Rectangular Trace with Skin Effect
8.6 Construction of Integrated Inductors
8.7 Meander Inductors
8.8 Inductance of Straight Round Conductor
8.9 Inductance of Circular Round Wire Loop
8.10 Inductance of Two-Parallel Wire Loop
8.11 Inductance of Rectangle of Round Wire
8.12 Inductance of Polygon Round Wire Loop
8.13 Bondwire Inductors
8.14 Single-Turn Planar Inductor
8.15 Inductance of Planar Square Loop
8.16 Planar Spiral Inductors
8.17 Multimetal Spiral Inductors
8.18 Planar Transformers
8.19 MEMS Inductors
8.20 Inductance of Coaxial Cable
8.21 Inductance of Two-Wire Transmission Line
8.22 Eddy Currents in Integrated Inductors
8.23 Model of RF-Integrated Inductors
8.24 PCB Inductors
8.25 Summary
8.26 References
8.27 Review Questions
8.28 Problems
Chapter 9: Self-Capacitance
9.1 Introduction
9.2 High-Frequency Inductor Model
9.3 Self-Capacitance Components
9.4 Capacitance of Parallel-Plate Capacitor
9.5 Self-Capacitance of Foil Winding Inductors
9.6 Capacitance of Two Parallel Round Conductors
9.7 Capacitance of Round Conductor and Parallel Conducting Plane
9.8 Capacitance of Straight Parallel Wire Pair Over Ground
9.9 Capacitance Between Two Parallel Straight Round Conductors with Uniform Charge Density
9.10 Capacitance of Cylindrical Capacitor
9.11 Self-Capacitance of Single-Layer Inductors
9.12 Self-Capacitance of Multilayer Inductors
9.13 Self-Capacitance of Single-Layer Inductors
9.14 -to-Y Transformation of Capacitors
9.15 Overall Self-Capacitance of Single-Layer Inductor with Core
9.16 Measurement of Self-Capacitance
9.17 Inductor Impedance
9.18 Summary
9.19 References
9.20 Review Questions
9.21 Problems
Chapter 10: Design of Inductors
10.1 Introduction
10.2 Magnet Wire
10.3 Wire Insulation
10.4 Restrictions on Inductors
10.5 Window Utilization Factor
10.6 Temperature Rise of Inductors
10.7 Mean Turn Length of Inductors
10.8 Area Product Method
10.9 Design of AC Inductors
10.10 Inductor Design for Buck Converter in CCM
10.11 Inductor Design for Buck Converter in DCM Using Method
10.12 Core Geometry Coefficient Method
10.13 Inductor Design for Buck Converter in CCM Using Method
10.14 Inductor Design for Buck Converter in DCM Using Method
10.15 Summary
10.16 References
10.17 Review Questions
10.18 Problems
Chapter 11: Design of Transformers
11.1 Introduction
11.2 Area Product Method
11.3 Optimum Flux Density
11.4 Area Product for Sinusoidal Voltages
11.5 Transformer Design for Flyback Converter in CCM
11.6 Transformer Design for Flyback Converter in DCM
11.7 Geometrical Coefficient Method
11.8 Transformer Design for Flyback Converter in CCM Using Method
11.9 Transformer Design for Flyback Converter in DCM Using Method
11.10 Summary
11.11 References
11.12 Review Questions
11.13 Problems
Appendix A: Physical Constants
Appendix B: Maxwell's Equations
Answers to Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Index
© 2014, John Wiley & Sons, Ltd
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Library of Congress Cataloging-in-Publication Data
Kazimierczuk, Marian.
High-frequency magnetic components / Marian K. Kazimierczuk. – Second edition.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-71779-0 (hardback)
1. Electromagnetic devices. 2. Magnetic devices. I. Title.
TK7872.M25K395 2009
621.3815 – dc23
2013026579
A catalogue record for this book is available from the British Library.
ISBN 9781118717790
To my Father
Preface
This book is about high-frequency magnetic power devices: high-frequency power inductors and high-frequency power transformers. It is intended as a textbook at the senior and graduate levels for students majoring in electrical engineering and as a reference for practicing engineers in the areas of power electronics and radiofrequency (RF) power amplifiers as well as other branches of physical sciences. Power electronics has evolved as a major enabling technology, which is used to efficiently convert energy from one form to another. The purpose of this book is to provide foundations for the analysis and design of high-frequency power magnetic devices: inductors and transformers. Magnetic components have a broad variety of applications across many diverse industries, such as energy conversion from one from to another (DC–DC, AC–DC, DC–AC, and AC–AC), switch-mode power supplies (SMPS), resonant inverters and converters, laptops, radio transmitters, uninterruptable power supplies (UPS), power factor correction (PFC), solar and wind renewable energy circuits, distributed energy generation systems (microgrids), automotive power electronics in hybrid and electric vehicles, battery chargers, wireless (or contactless) power transfer, energy harvesting, electrical machines, portable electronic devices, chokes, active power filters, electromagnetic interference (EMI)/radiofrequency interference (RFI) filters, RF noise suppressors, oscillators, energy storage systems, aviation power systems, induction heating, electronic ballasts, light-emitting diode (LED) lighting, magnetic sensors, ferrous metal detectors, fuel cell power supplies, medical equipment, implantable medical devices, and current measurement probes.
This book addresses the skin and proximity effects on winding and core losses in magnetic components at high frequencies. Magnetic components have often a large size and weight, are lossy, and have low-power density. Special topics in this book include optimization of the size of winding conductors, integrated inductors, analysis of the self-capacitance of inductors and transformers, temperature effects on the performance of magnetic components, and high-frequency physics-based models of inductors and transformers. The International System (SI) of Units are used in this book. All quantities are expressed in units of the meter–kilogram–second (MKS) system. The second edition of this book is a thoroughly revised and updated textbook and includes new research results and advances in magnetic device technology.
Introduction to Physical Constants and Maxwell's Equations is given in Appendices A and B, respectively.
This textbook assumes that the student is familiar with electromagnetic fields, general circuit analysis techniques, calculus, and vector algebra. These courses cover the fundamental laws of physics, such as Faraday's law, Ampère's law, Gauss's law, Lenz's law, Ohm's law, Joule's law, Poynting's theorem, and Maxwell's equations. There is sufficient material in this textbook for a one-semester course. Complete solutions for all problems are included in the Solutions Manual, which is available from the publisher for those instructors who adopt the textbook for their courses.
I am pleased to express my gratitude to Dr Nisha Kondrath for MATLAB® figures, Dr Dakshina Murthy-Bellur for his help in developing the design examples of inductors and transformers, Dr Rafal Wojda for his contributions to optimization of winding conductors and for MATLAB® figures, Dr Gregory Kozlowski for his help with Bessel functions and analysis of a single round conductor, and Dr Hiroo Sekiya for creative discussions. I am deeply indebted to the students, reviewers, scientists, industrial engineers, and other readers for valuable feedback, suggestions, and corrections. Most of these suggestions have been incorporated in the second edition of this book.
Throughout the entire course of this project, the support provided by John Wiley & Sons was excellent. I wish to express my sincere thanks to Laura Bell, Assistant Editor; Richard Davies, Senior Project Editor; and Peter Mitchell, Publisher. It has been a real pleasure working with them. Finally, my thanks goes to my family for their patience, understanding, and support throughout the endeavor.
The author would welcome and greatly appreciate suggestions and corrections from the readers for improvements in the technical content as well as the presentation style.
Marian K. Kazimierczuk
About the Author
Marian K. Kazimierczuk is a Robert J. Kegerreis, Distinguished Professor of Electrical Engineering at Wright State University, Dayton, Ohio, United States. He received the MS, PhD, and DSc degrees from Warsaw University of Technology, Department of Electronics, Warsaw, Poland. He is the author of six books, over 170 archival refereed journal papers, over 200 conference papers, and seven patents.
Professor Kazimierczuk is a Fellow of the IEEE. He served as the Chair of the Technical Committee of Power Systems and Power Electronics Circuits, IEEE Circuits and Systems Society. He served on the Technical Program Committees of the IEEE International Symposium on Circuits and Systems (ISCAS) and the IEEE Midwest Symposium on Circuits and Systems. He also served as an associate editor of the IEEE Transactions on Circuits and Systems, Part I, Regular Papers, IEEE Transactions on Industrial Electronics, International Journal of Circuit Theory and Applications, and Journal of Circuits, Systems, and Computers, and as a guest editor of the IEEE Transactions on Power Electronics. He was an IEEE Distinguished Lecturer.
Professor Kazimierczuk received Presidential Award for Outstanding Faculty Member at Wright State University in 1995. He was Brage Golding Distinguished Professor of Research at Wright State University in 1996–2000. He received the Trustees' Award from Wright State University for Faculty Excellence in 2004. He received the Outstanding Teaching Award from the American Society for Engineering Education (ASEE) in 2008. He was also honored with the Excellence in Research Award, Excellence in Teaching Awards, and Excellence in Professional Service Award in the College of Engineering and Computer Science, Wright State University.
His research interests are in power electronics, including pulse-width modulated (PWM) DC–DC power converters, resonant DC–DC power converters, modeling and controls, radio frequency (RF) high-efficiency power amplifiers and oscillators, high-frequency magnetic devices, semiconductor power devices, renewable energy sources, and evanescent microwave microscopy.
Professor Kazimierczuk is the author or co-author of Resonant Power Converters (Second Edition), Pulse-Width Modulated DC-DC Power Converters, RF Power Amplifiers, Electronic Devices, A Design Approach, and Laboratory Manual to Accompany Electronic Devices, A Design Approach.
List of Symbols
A
Magnetic vector potential
Cross-sectional area of core
Cross-sectional area of winding cell
Cross-sectional area of copper
Specific inductance of core, inductance factor
Cross-sectional area of winding bare wire
Outer cross-sectional area of winding wire
Cross-sectional area of strand bare wire
Outer cross-sectional area of strand wire
Area product of core
Surface area of inductor or transformer
a
Unity vector
Unity vector normal to a surface
B
Magnetic flux density
DC component of magnetic flux density
Amplitude of the AC component of magnetic flux density
Residual flux density, remnant magnetization, and remnance
Peak value of total magnetic flux density ()
Saturation flux density
Bandwidth
b
Breadth of winding
Breadth of bobbin winding window
C
Capacitance
Equivalent series-resonant capacitance
D
Electric flux density, DC component of on-duty cycle of switch
Minimum DC component of on-duty cycle of switch
Maximum DC component of on-duty cycle of switch
Inner diameter of toroidal core
Outer diameter of toroidal core
Dwell duty cycle
Diameter of winding bare wire
Outer diameter of insulated winding wire
Diameter of bare strand wire
Outer diameter of insulated strand wire
E
Electric field intensity
Electromotive force
F
Force
Fringing factor
Air gap factor
AC resistance factor
Harmonic AC resistance factor
Harmonic AC resistance factor of primary winding
Harmonic AC resistance factor of secondary winding
f
Frequency
Self-resonant frequency of inductor
Switching frequency
Upper 3-dB frequency
Lower 3-dB frequency
H
Magnetic field intensity
Coercive force
h
Thickness of winding conductor
Height of winding window of bobbin
DC input current of converter
Average or DC current through inductor
L
Maximum current through inductor
L
rms current through inductor
L
Peak total current through inductor
L
rms value of
n
th harmonic of inductor current
DC output current of converter
rms value of current
i
Maximum value of DC load current
Minimum value of DC load current
i
Current
Inductor current
Current through magnetizing inductance
Current through primary winding
Current through secondary winding
J
Conduction current density
Displacement current density
Amplitude of current density
rms value of current density
Air factor
Bobbin factor
Air and wire insulation factor
Edge factor
Core geometry coefficient
Waveform factor
Wire insulation factor
Window utilization factor
k
Coupling coefficient, complex propagation constant
Magnetic path length (MPL)
Length of air gap
Mean turn length (MTL), length of turn
Length of winding wire
Length of primary winding wire
Length of secondary winding wire
L
Inductance
Leakage inductance
Magnetizing inductance of transformer
M
Mutual inductance
DC voltage transfer function of converter
Orbital magnetic moment
Spin magnetic moment
N
Number of turns
Number of layers
Number of layers of primary winding
Number of layers of secondary winding
Number of turns of primary winding
Number of turns of secondary winding
n
Transformer primary-to-secondary turns ratio
n
Unity vector normal to a surface
P
Power
Core loss
Core and winding power loss
Winding power loss
Winding DC power loss
Primary winding power loss
Primary winding DC power loss
Primary winding DC power loss with strands
Secondary winding power loss
Secondary winding DC power loss
Secondary winding DC power loss with strands
Output power of converter or amplifier
Core power loss per unit volume
Total apparent power
p
Winding pitch
Q
Quality factor
Loaded quality factor of resonant circuit at resonant frequency
Quality factor of inductor
Quality factor of magnetic material
R
Resistance
Core series equivalent resistance
Load resistance
Maximum load resistance
Minimum load resistance
Winding resistance
Winding DC resistance
Primary winding DC resistance
Primary strand winding DC resistance
Primary strand winding DC resistance
Secondary strand winding DC resistance
r
Radius
Equivalent series resistance (ESR) of inductor
Poynting vector
Number of strands
Number of strands in primary winding
Number of strands in secondary winding
T
Switching period
Curie temperature
Torque
t
Time
Core volume
rms value of voltage
v
v
Voltage, velocity
Drift velocity
Voltage across inductance
L
AC component of output voltage
Core window area
Core window area of primary winding
Core window area of secondary winding
Bobbin cross-sectional area
Magnetic energy stored in inductor or transformer
w
Energy density
Magnetic energy density
Z
Impedance
X
Parallel reactance
x
Series reactance
Permeance
Reluctance
Regulation of transformer
Complex propagation constant
Peak-to-peak value of inductor ripple current
Temperature rise
Skin depth
Skin depth in core
Loss angle of a magnetic material
Skin depth in winding conductor
Skin depth in winding conductor at
n
th harmonic
Flux linkage
Flux linkage produced by current
Flux linkage produced by current
Flux linkage with winding 2 produced by current
Flux linkage with winding 1 produced by current
Permittivity of free space
Relative permittivity
Efficiency, porosity factor
Efficiency of transformer
Permeability
Permeability of free space
Parallel complex permeability
Relative permeability
Relative core permeability
Series complex permeability
Real part of series complex permeability
Imaginary part of series complex permeability
Series complex relative permeability
Real part of series complex relative permeability
Imaginary part of series complex relative permeability
Magnetic flux
Phase angle of
n
th harmonic of inductor current
Voltage utilization factor
Resistivity
Resistivity of core material
Copper resistivity
Resistivity of winding conductor
Conductivity
Magnetic susceptibility
Series complex magnetic susceptibility
Real part of series complex magnetic susceptibility
Imaginary part of series complex magnetic susceptibility
Surface power loss density
w
Angular frequency
Self-resonant angular frequency
Chapter 1
Fundamentals of Magnetic Devices
1.1 Introduction
Many electronic circuits require the use of inductors and transformers 1–60. These are usually the largest, heaviest, and most expensive components in a circuit. They are defined by their electromagnetic (EM) behavior. The main feature of an inductor is its ability to store magnetic energy in the form of a magnetic field. The important feature of a transformer is its ability to couple magnetic fluxes of different windings and transfer AC energy from the input to the output through the magnetic field. The amount of energy transferred is determined by the operating frequency, flux density, and temperature. Transformers are used to change the AC voltage and current levels as well as to provide DC isolation while transmitting AC signals. They can combine energy from many AC sources by the addition of the magnetic flux and deliver the energy from all the inputs to one or multiple outputs simultaneously. The magnetic components are very important in power electronics and other areas of electrical engineering. Power losses in inductors and transformers are due to DC current flow, AC current flow, and associated skin and proximity effects in windings, as well as due to eddy currents and hysteresis in magnetic cores. In addition, there are dielectric losses in materials used to insulate the core and the windings. Failure mechanisms in magnetic components are mostly due to excessive temperature rise. Therefore, these devices should satisfy both magnetic requirements and thermal limitations.
In this chapter, fundamental physical phenomena and fundamental physics laws of electromagnetism, quantities, and units of the magnetic theory are reviewed. Magnetic relationships are given and an equation for the inductance is derived. The nature is governed by a set of laws. A subset of these laws are the physics EM laws. The origin of the magnetic field is discussed. It is shown that moving charges are sources of the magnetic field. Hysteresis and eddy-current losses are studied. There are two kinds of eddy-current effects: skin effect and proximity effect. Both of these effects cause nonuniform distribution of the current density in conductors and increase the conductor AC resistance at high frequencies. A classification of winding and core losses is given.
1.2 Fields
A field is defined as a spatial distribution of a quantity everywhere in a region. There are two categories of fields: scalar fields and vector fields. A scalar field is a set of scalars assigned at individual points in space. A scalar quantity has a magnitude only. Examples of scalar fields are time, temperature, humidity, pressure, mass, sound intensity, altitude of a terrain, energy, power density, electrical charge density, and electrical potential. The scalar field may be described by a real or a complex function. The intensity of a scalar field may be represented graphically by different colors or undirected field lines. A higher density of the field lines indicates a stronger field in the area.
A vector field is a set of vectors assigned at every point in space. A vector quantity has both magnitude and direction. Examples of vector fields are velocity v, the Earth's gravitational force field F, electric current density field J, magnetic field intensity H, and magnetic flux density B. The vector field may be represented graphically by directed field lines. The density of field lines indicates the field intensity, and the direction of field lines indicates the direction of the vector at each point. In general, fields are functions of position and time, for example, . The rate of change of a scalar field with distance is a vector.
1.3 Magnetic Relationships
The magnetic field is characterized by magnetomotive force (MMF) , magnetic field intensity H, magnetic flux density B, magnetic flux , and magnetic flux linkage .
1.3.1 Magnetomotive Force
An inductor with N turns carrying an AC current i produces the MMF, which is also called the magnetomotance. The MMF is given by
1.1
Its descriptive unit is ampere-turns (t). However, the approved SI unit of the MMF is the ampere (A), where electrons/s. The MMF is a source in magnetic circuits. The magnetic flux is forced to flow in a magnetic circuit by the MMF , driving a magnetic circuit. Every time another complete turn with thecurrent i is added, the result of the integration increases by the current i.
The MMF between any two points and produced by a magnetic field H is determined by a line integral of the magnetic field intensity H present between these two points
1.2
where is the incremental vector at a point located on the path l and . The MMF depends only on the endpoints, and it is independent of the path between points and . Any path can be chosen. If the path is broken up into segments parallel and perpendicular to H, only parallel segments contribute to . The contributions from the perpendicular segments are zero.
For a uniform magnetic field and parallel to path l, the MMF is given by
1.3
Thus,
1.4
The MMF forces a magnetic flux to flow.
The MMF is analogous to the electromotive force (EMF) V. It is a potential difference between any two points and . field E between any two points and is equal to the line integral of the electric field E between these two points along any path
1.5
The result is independent of the integration path. For a uniform electric field E and parallel to path l, the EMF is
1.6
The EMF forces a current to flow. It is the work per unit charge (J/C).
1.3.2 Magnetic Field Intensity
The magnetic field intensity (or magnetic field strength) is defined as the MMF per unit length
1.7
where l is the inductor length and N is the number of turns. Magnetic fields are produced by moving charges. Therefore, magnetic field intensity H is directly proportional to the amount of current i and the number of turns per unit length . If a conductor conducts current i (which a moving charge), it produces a magnetic field H. Thus, the source of the magnetic field H is a conductor carrying a current i. The magnetic field intensity H is a vector field. It is described by a magnitude and a direction at any given point. The lines of magnetic field H always form closed loops. By Ampère's law, the magnetic field produced by a straight conductor carrying current i is given by
1.8
The magnetic field intensity H is directly proportional to current i and inversely proportional to the radial distance from the conductor r. The Earth's magnetic field intensity is approximately T.
1.3.3 Magnetic Flux
The amount of the magnetic flux passing through an open surface S is determined by a surface integral of the magnetic flux density B
1.9
where n is the unit vector normal to the incremental surface area at a given position, is the incremental surface vector normal to the local surface at a given position, and . The magnetic flux is a scalar. The unit of the magnetic flux is Weber.
If the magnetic flux density B is uniform and forms an angle with the vector perpendicular to the surface S, the amount of the magnetic flux passing through the surface S is
1.10
If the magnetic flux density B is uniform and perpendicular to the surface S, the angle between vectors B and is and the amount of the magnetic flux passing through the surface S is
1.11
If the magnetic flux density B is parallel to the surface S, the angle between vectors B and is and the amount of the magnetic flux passing through the surface S is
1.12
For an inductor, the amount of the magnetic flux may be increased by increasing the surface area of a single turn A, the number of turns in the layer , and the number of layers . Hence, , where is the total number of turns.
The direction of a magnetic flux density B is determined by the right-hand rule (RHR). This rule states that if the fingers of the right hand encircle a coil in the direction of the current i, the thumb indicates the direction of the magnetic flux density B produced by the current i, or if the fingers of the right hand encircle a conductor in the direction of the magnetic flux density B, the thumb indicated the direction of the current i. The magnetic flux lines are always continuous and closed loops.
1.3.4 Magnetic Flux Density
The magnetic flux density, or induction, is the magnetic flux per unit area given by
1.13
The unit of magnetic flux density B is Tesla. The magnetic flux density is a vector field and it can be represented by magnetic lines. The density of the magnetic lines indicates the magnetic flux density B, and the direction of the magnetic lines indicates the direction of the magnetic flux density at a given point. Every magnet has two poles: south and north. Magnetic monopoles do not exist. Magnetic lines always flow from south to north pole inside the magnet, and from north to south pole outside the magnet.
The relationship between the magnetic flux density B and the magnetic field intensity H is given by
1.14
where the permeability of free space is
1.15
is the permeability, is the relative permeability (i.e., relative to that of free space), and is the length of the core. Physical constants are given in Appendix A. For free space, insulators, and nonmagnetic materials, . For diamagnetics such as copper, lead, silver, and gold, . However, for ferromagnetic materials such as iron, cobalt, nickel, and their alloys, and it can be as high as 100 000. The permeability is the measure of the ability of a material to conduct magnetic flux . It describes how easily a material can be magnetized. For a large value of , a small current i produces a large magnetic flux density B. The magnetic flux takes the path of the highest permeability.
The magnetic flux density field is a vector field. For example, the vector of the magnetic flux density produced by a straight conductor carrying current i is given by
1.16
For ferromagnetic materials, the relationship between B and H is nonlinear because the relative permeability depends on the magnetic field intensity H. Figure 1.1 shows simplified plots of the magnetic flux density B as a function of the magnetic field intensity H for air-core inductors (straight line) and for ferromagnetic core inductors. The straight line describes the air-core inductor and has a slope for all values of H. These inductors are linear. The piecewise linear approximation corresponds to the ferromagnetic core inductors, where is the saturation magnetic flux density and is the magnetic field intensity corresponding to . At low values of the magnetic flux density , the relative permeability is high and the slope of the B–H curve is also high. For , the core saturates and , reducing the slope of the B–H curve to .
Figure 1.1 Simplified plots of magnetic flux density B as a function of magnetic field intensity H for air-core inductors (straight line) and ferromagnetic core inductors (piecewise linear approximation)
The total peak magnetic flux density , which in general consists of both the DC component and the amplitude of AC component , should be lower than the saturation flux density of a magnetic core at the highest operating temperature
1.17
The DC component of the magnetic flux density is caused by the DC component of the inductor current
1.18
The amplitude of the AC component of the magnetic flux density corresponds to the amplitude of the AC component of the inductor current
1.19
Hence, the peak value of the magnetic flux density can be written as
1.20
where . The saturation flux density decreases with temperature. For ferrites, may decrease by a factor of 2 as the temperature increases from C to C. The amplitude of the magnetic flux density is limited either by core saturation or by core losses.
1.3.5 Magnetic Flux Linkage
The magnetic flux linkage is the sum of the flux enclosed by each turn of the wire wound around the core
1.21
For the uniform magnetic flux density, the magnetic flux linkage is the magnetic flux linking N turns and is described by
1.22
where is the core reluctance and is the effective area through which the magnetic flux passes. Equation (1.22) is analogous to Ohm's law and the equation for the capacitor charge . The unit of the flux linkage is Wb·turn. For sinusoidal waveforms, the relationship among the amplitudes is
1.23
The change in the magnetic linkage can be expressed as
1.24
1.4 Magnetic Circuits
1.4.1 Reluctance
The reluctance is the resistance of the core to the flow of the magnetic flux . It opposes the magnetic flux flow, in the same way as the resistance opposes the electric current flow. An element of a magnetic circuit can be called a reluctor. The concept of the reluctance is illustrated in Fig. 1.2. The reluctance of a linear, isotropic, and homogeneous magnetic material is given by
1.25
where is the cross-sectional area of the core (i.e., the area through which the magnetic flux flows) and is the mean magnetic path length (MPL), which is the mean length of the closed path that the magnetic flux flows around a magnetic circuit. The reluctance is directly proportional to the length of the magnetic path and is inversely proportional to the cross-sectional area through which the magnetic flux flows. The permeance of a basic magnetic circuit element is
1.26
When the number of turns , . The reluctance is the magnetic resistance because it opposes the establishment and the flow of a magnetic flux in a medium. A poor conductor of the magnetic flux has a high reluctance and a low permeance. Magnetic Ohm's law is expressed as
1.27
Magnetic flux always takes the path with the highest permeability .
Figure 1.2 Reluctance. (a) Basic magnetic circuit element conducting magnetic flux . (b) Equivalent magnetic circuit
In general, the magnetic circuit is the space in which the magnetic flux flows around the coil(s). Figure 1.3 shows an example of a magnetic circuit. The reluctance in magnetic circuits is analogous to the resistance R in electric circuits. Likewise, the permeance in magnetic circuits is analogous to the conductance in electric circuits. Therefore, magnetic circuits described by the equation can be solved in a similar manner as electric circuits described by Ohm's law , where , , , , B, , and , correspond to I, V, R, G, J, Q, and , respectively. For example, the reluctances can be connected in series or in parallel. In addition, the reluctance is analogous to the electric resistance and the magnetic flux density is analogous to the current density . Table 1.1 lists analogous magnetic and electric quantities.
Figure 1.3 Magnetic circuit. (a) An inductor composed of a core and a winding. (b) Equivalent magnetic circuit
Table 1.1 Analogy between magnetic and electric quantities
Magnetic quantity
Electric quantity
V
I
H
E
B
J
R
G
Q
L
C
1.4.2 Magnetic KVL
Physical structures, which are made of magnetic devices, such as inductors and transformers, can be analyzed just like electric circuits. The magnetic law, analogous to Kirchhoff's voltage law (KVL), states that the sum of the MMFs and the magnetic potential differences around the closed magnetic loop is zero
1.28
For instance, an inductor with a simple core having an air gap as illustrated in Fig. 1.4 is given by
1.29
where the reluctance of the core is
1.30
the reluctance of the air gap is
1.31
and it is assumed that . This means that the fringing flux in neglected. If , the magnetic flux is confined to the magnetic material, reducing the leakage flux. The ratio of the air-gap reluctance to the core reluctance is
1.32
The reluctance of the air gap is much higher than the reluctance of the core if .
Figure 1.4 Magnetic circuit illustrating the magnetic KVL. (a) An inductor composed of a core with an air gap and a winding. (b) Equivalent magnetic circuit
The magnetic potential difference between points a and b is
1.33
where is the reluctance between points a and b.
1.4.3 Magnetic Flux Continuity
The continuity of the magnetic flux law states that the net magnetic flux through any closed surface is always zero
1.34
or the net magnetic flux entering and exiting the node is zero
1.35
This law is analogous to Kirchhoff's current law (KCL) introduced by Gauss and can be called Kirchhoff's flux law (KFL). Figure 1.5 illustrates the continuity of the magnetic flux law. For example, when three core legs meet at a node,
1.36
which can be expressed by
1.37
If all the three legs of the core have windings, then we have
1.38
Usually, most of the magnetic flux is confined inside an inductor, for example, for an inductor with a toroidal core. The magnetic flux outside an inductor is called the leakage flux.
Figure 1.5 Magnetic circuit illustrating the continuity of the magnetic flux for EE core. (a) An inductor composed of a core and a winding. (b) Equivalent magnetic circuit
1.5 Magnetic Laws
1.5.1 Ampère's Law
Ampère1 discovered the relationship between current and the magnetic field intensity. Ampère's law relates the magnetic field intensity H inside a closed loop to the current passing through the loop. A magnetic field can be produced by a current and a current can be produced by a magnetic field. Ampère's law is illustrated in Fig. 1.6. A magnetic field is present around a current-carrying conductor or conductors. The integral form of Ampère's circuital law, or simply Ampère's law, (1826) describes the relationship between the (conduction, convection, and/or displacement) current and the magnetic field produced by this current. It states that the closed line integral of the magnetic field intensity H around a closed path (Amperian contour) C (2D or 3D) is equal to the total current enclosed by that path and passing through the interior of the closed path bounding the open surface S
1.39
where is the vector length element pointing in the direction of the Amperian path C and J is the conduction (or drift) and convection current density. The current enclosed by the path C is given by the surface integral of the normal component J over the open surface S. The surface integral of the current density J is equal to the current I flowing through the surface S. In other words, the integrated magnetic field intensity around a closed loop C is equal to the electric current passing through the loop. The surface integral of J is the current flowing through the open surface S. The conduction current is caused by the movement of electrons originating from the outermost shells of atoms. When conduction current flows, the atoms of medium normally do not move. The convection current is caused by the movement of electrically charged medium.
Figure 1.6 Illustration of Ampère's law
For example, consider a long, straight, round conductor that carries current I. The line integral about a circular path of radius r centered on the axis of the round wire is equal to the product of the circumference and the magnetic field intensity
1.40
yielding the magnetic field intensity
1.41
Thus, the magnetic field decreases in the radial direction away from the conductor.
For an inductor with N turns, Ampère's law is
1.42
Ampère's law in the discrete form can be expressed as
1.43
For example, Ampère's law for an inductor with an air gap is given by
1.44
If the current density J is uniform and perpendicular to the surface S,
1.45
The current density J in winding conductors of magnetic components used in power electronics is usually in the range 0.1–10 A/. The displacement current is neglected in (1.39). The generalized Ampère's law by adding the displacement current constitutes one of Maxwell's equations. This is known as Maxwell's correction to Ampère's law.
Ampère's law is useful when there is a high degree of symmetry in the arrangement of conductors and it can be easily applied in problems with symmetrical current distribution. For example, the magnetic field produced by an infinitely long wire conducting a current I outside the wire is
1.46
Ampère's law is a special case of Biot–Savart's law.
1.47
1.48
1.49
1.50
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.60
1.61
1.62
1.63
1.64
1.65
1.66
1.5.2 Faraday's Law
A time-varying current produces a magnetic field, and a time-varying magnetic field can produce an electric current. In 1820, a Danish scientist Oersted2 showed that a current-carrying conductor produces a magnetic field, which can affect a compass magnetic needle. He connected electricity and magnetism. Ampère measured that this magnetic field intensity is linearly related to the current, which produces it. In 1831, the English experimentalist Michael Faraday3 discovered that a current can be produced by an alternating magnetic field and that a time-varying magnetic field can induce a voltage, or an EMF, in an adjacentcircuit. This voltage is proportional to the rate of change of magnetic flux linkage , or magnetic flux , or current i, producing the magnetic field.
Faraday's law (1831), also known as Faraday's law of induction, states that a time-varying magnetic flux passing through a closed stationary loop, such as an inductor turn, generates a voltage in the loop and for a linear inductor is expressed by
1.67
This voltage, in turn, may produce a current . The voltage is proportional to the rate of change of the magnetic linkage , or to the rate of change of the magnetic flux density and the effective area through which the flux is passing. The inductance L relates the induced voltage to the current . The voltage across the terminals of an inductor L is proportional to the time rate of change of the current in the inductor and the inductance L. If the inductor current is constant, the voltage across an ideal inductor is zero. The inductor behaves as a short circuit for DC current. The inductor current cannot change instantaneously. Figure 1.8 shows an equivalent circuit of an ideal inductor. The inductor is replaced by a dependent voltage source controlled by .
Figure 1.8 Equivalent circuit of an ideal inductor. (a) Inductor. (b) Equivalent circuit of an inductor in the form of dependent voltage source controlled by the rate of change of the inductor current
The voltage between the terminals of a single turn of an inductor is
1.68
Hence, the total voltage across the inductor consisting of N identical turns is
1.69
Since ,
1.70
yielding the current in an inductor
1.71
For sinusoidal waveforms, the derivative can be replaced by and differential equations may be replaced by algebraic equations. A phasor is a complex representation of the magnitude, phase, and space of a sinusoidal waveform. The phasor is not dependent on time. A graphical representation of a phasor is known as a phasor diagram. Faraday's law in phasor form can be expressed as
1.72
The sinusoidal inductor current legs the sinusoidal inductor voltage by .
The impedance of a lossless inductive component in terms of phasors of sinusoidal inductor current and voltage is
1.73
where . The impedance of lossy inductive components in terms of phasors is
1.74
For nonlinear, time-varying inductors, the relationships are
1.75
and
1.76
where
1.77
In summary, a time-varying electric current produces magnetic fields , , and by Ampère's law. In turn, the magnetic field produces a voltage by Faraday's law. This process can be reversed. A voltage produces a magnetic fields , , and , which produced electric current .
1.5.3 Lenz's Law
Lenz4 discovered the relationship between the direction of the induced current and the change in the magnetic flux. Lenz's law (1834) states that the EMF induced by an applied time-varying magnetic flux has such a direction that induces current in the closed loop, which in turn induces a magnetic flux that tends to oppose the change in the applied flux , as illustrated in Fig. 1.9. If the applied magnetic flux increases, the induced current produces an opposing flux . If the applied magnetic flux decreases, the induced current produces an aiding flux . The induced magnetic flux always opposes the inducing (applied) magnetic flux . If increases, the induced current produces an opposing flux . If decreases, the induced current produces an aiding magnetic flux . The direction of the induced current with respect of the induced magnetic field is determined by the RHR.
Figure 1.9 Illustration of Lenz's law generating eddy currents. The applied time-varying magnetic flux induces eddy current , which in turn generates induced flux that opposes changes in the applied flux
If a time-varying magnetic field is applied to a conducting loop (e.g., an inductor turn), a current is induced in such a direction as to oppose the change in the magnetic flux enclosed by the loop. The induced currents flowing in closed loops are called eddy currents. Eddy currents occur when a conductor is subjected to time-varying magnetic field(s). In accordance with Lenz's law, the eddy currents produce their own magnetic field(s) to oppose the original field.
The effects of eddy currents on winding conductors and magnetic cores are nonuniform current distribution, increased effective resistance, increased power loss, and reduced internal inductance. If the resistivity of a conductor was zero (as in a perfect conductor), eddy-current loops would be generated with such a magnitude and phase to exactly cancel the applied magnetic field. A perfect conductor would oppose any change in externally applied magnetic field. Circulating eddy currents would be induced to oppose any buildup of the magnetic field in the conductor. In general, nature opposes to everything we want to do.
1.5.4 Volt–Second Balance
Faraday's law is , yielding . Hence,
1.78
For periodic waveforms in steady state,
1.79
This equation is called a volt–second balance, which states that the total area enclosed by the inductor voltage waveform is zero for steady state. As a result, the area enclosed by the inductor voltage waveform above zero must be equal to the area enclosed by the inductor voltage waveform below zero for steady state. The volt–second balance can be expressed by
1.80
which gives
1.81
This can be written as .
1.5.5 Ohm's Law
Materials resist the flow of electric charge. The physical property of materials to resist current flow is known as resistivity. Therefore, a sample of a material resists the flow of electric current. This property is known as resistance. Ohm5 discovered that the voltage across a resistor is directly proportional to its current and is constant, called resistance. Microscopic Ohm's law describes the relationship between the conduction current density J and the electric field intensity E. The conduction current is caused by the movement of electrons. Conductors exhibit the presence of many free (conduction or valence) electrons, from the outermost atom shells of a conducting medium. These free electrons are in random constant motion in different directions in a zigzag fashion due to thermal excitation. The average electron thermal energy per one degree of freedom is andthe average thermal energy of an electron in three dimensions is . At the collision, the electron kinetic energy is equal to the thermal energy
1.82
where is the Boltzmann's constant and is the rest mass of a free electron. The thermal velocity of electrons between collisions is
1.83
In good conductors, mobile free electrons drift through a lattice of positive ions encountering frequent collisions with the atomic lattice. If the electric field E in a conductor is zero, the net charge movement over a large volume (compared with atomic dimensions) is zero, resulting in zero net current. If an electric field E is applied to a conductor, a Coulomb's force F is exerted on an electron with charge
1.84
According to Newton's second law, the acceleration of electrons between collisions is
1.85
where is the mass of electron. If the electric field intensity E is constant, then the average drift velocity of electrons increases linearly with time
1.86
The average drift velocity is directly proportional to the electric field intensity Efor low values of E and saturates at high value of E. Electrons are involved in collisions with thermally vibrating lattice structure and other electrons. As the electron accelerates due to electric field, the velocity increases. When the electron collides with an atom, it loses most or all of its energy. Then, the electron begins to accelerate due to electric field E and gains energy until a new collision. The average position change of a group of N electrons in time interval is called the drift velocity
1.87
The drift velocity of electrons has the opposite direction to that of the applied electric field E. By Newton's law, the average change in the momentum of a free electron is equal to the applied force
1.88
where the mean time between the successive collisions of electrons with atom lattice, called the relaxation time, is given by
1.89
in which is length of the mean free path of electrons between collisions. Equating the right-hand sides of (1.84) and (1.88), we obtain
1.90
yielding the average drift velocity of electrons
1.91
where the mobility of electrons in a conductor is
1.92
The volume charge density in a conductor is
1.93
where n is the concentration of free (conduction or valence) electrons in a conductor, which is equal to the number of conduction electrons per unit volume of a conductor. The resulting flow of electrons is known as the conduction (or drift) current. The conduction (drift) current density, corresponding to the motion of charge forced by electric field E, is given by
1.94
where the conductivity of a conductor is
1.95
and the resistivity of a conductor is
1.96
Hence, the point (or microscopic) form of Ohm's law (1827) for conducting materials is
1.97
The typical value of mobility of electrons in copper is /V·s. At , the average drift velocity of electrons in copper is . The thermal velocity of electrons between collisions is /s. Due to collisions of electrons with atomic lattice and the resulting loss of energy, the velocity of individual electrons in the direction opposite to the electric field E ismuch lower than the thermal velocity. The average drift velocity is much lower than the thermal velocity by two orders on magnitude. The average time interval between collisions of electrons is called the relaxation time and its typical value for copper is . The convection current and the displacement current do not obey Ohm's law, whereas the conduction current does it.
To illustrate Ohm's law, consider a straight round conductor of radius and resistivity carrying a DC current I. The current is evenly distributed in the conductor. Thus, the current density is
1.98
According to Ohm's law, the electric field intensity in the conductor is
1.99
1.5.6 Biot–Savart's Law
Hans Oersted discovered in 1819 that currents produce magnetic fields that form closed loops around conductors (e.g., wires). Moving charges are sources of the magnetic field. Jean Biot and Félix Savart arrived in 1820 at a mathematical relationship between the magnetic field H at any point P of space and the current I that generates H. Current I is a source of magnetic field intensity H. The Biot–Savart's law allows us to calculate the differential magnetic field intensity produced by a small current element . Figure 1.10 illustrates the Biot–Savart's law. The differential form of the Biot–Savart's law is given by
1.100
where is the current element equal to a differential length of a conductor carrying electric current I and points in the direction of the current I, and is the distance vector between and an observation point P with field H. The vector is perpendicular to both and to the unitvector directed from to P. The magnitude of is inversely proportional to , where R is the distance from to P. The magnitude of is proportional to sin , where is the angle between the vectors and . The Biot–Savart's law is analogous to Coulomb's law that relates the electric field E to an isolated point charge Q, which is a source of radial electric field .
Figure 1.10 Magnetic field produced by a small current element
The total magnetic field H induced by a current I is given by the integral form of the Biot–Savart's law
1.101
The integral must be taken over the entire current distribution.
1.5.7 Maxwell's Equations
Maxwell6 assembled the laws of Faraday, Ampère, and Gauss (for both electric and magnetic fields) into a set of four equations to produce a unified EM theory. Maxwell's equations (1865), together with the law of conservation of charge (the continuity equation), form a foundation of a unified and coherent theory of electricity and magnetism. They couple electric field E, magnetic field H, current density J, and charge density . These equations provide the qualitative and quantitative description of static and dynamic EM fields. They can be used to explain and predict electromagnetic phenomena. In particular, they govern the behavior of EM waves.
Maxwell's equations in differential (point or microscopic) forms in the time domain at any point in space and at any time are given by
1.102
1.103
1.104
and
1.105
where is the displacement current density. The conductive current density (corresponding to the motion of charge) and the displacement current density are sources of EM fields , , , and the volume charge density is a source of the electric fields and , where is permeability and is the permittivity of a material. Maxwell's equations include two Gauss's7 laws. Gauss's law states that charge is a source of electric field. In contrast, Gauss's magnetic law states that magnetic field is sourceless (divergenceless), that is, there are no magnetic sources or sinks. This law also indicates that magnetic flux lines close upon themselves. Two Maxwell's equations are partial differential equations because magnetic and electric fields, current, and charge may vary simultaneously with space and time.
Neglecting the generation and recombination of carrier charges like in semiconductors, the continuity equation or the law of local conservation of electric charge must be satisfied at all times
1.106
This law states that the time rate of change of electric charge is a source of electric current density field J. This means that the current density is continuous and charge can be neither created nor destroyed. It can only be transferred. The continuity equation is a point form of KCL known in circuit theory. The script letters are used to designate instantaneous field quantities, which are functions of position and time, for example, . Maxwell's equations are the cornerstone of electrodynamics. A time-varying magnetic field is always accompanied by an electric field, and a time-varying electric field is always accompanied by a magnetic field. For example, a radio antenna generates radiofrequency (RF) waves that consist of both the electric and magnetic fields. The divergence of B
